Estimation and Inference in the Presence of Fractional d=1/2 and Weakly Nonstationary Processes

TL;DR

This paper develops new limit theory for weakly nonstationary processes, including fractional processes with d=1/2, and introduces a robust specification test for regression models that remains valid regardless of the regressor's persistence.

Contribution

It provides the first comprehensive limit theory for WNPs and proposes a new chi-squared based specification test that is robust to the persistence level of regressors.

Findings

The new test maintains size control across various data generating processes.

Simulation results show the test outperforms existing methods like Wang and Phillips (2012).

The theory applies to fractional and autoregressive processes near nonstationarity.

Abstract

We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity -- what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional processes with d=1/2, and arrays of autoregressive processes with roots drifting slowly towards unity. We first apply the theory to study inference in parametric and nonparametric regression models involving WNPs as covariates. We then use these results to develop a new specification test for parametric regression models. By construction, our specification test statistic has a chi-squared limiting distribution regardless of the form and extent of persistence of the regressor, implying that a practitioner can validly perform the test using a fixed critical value, while remaining agnostic about the mechanism generating the regressor. Simulation exercises confirm that the test controls size across a wide range of data generating processes, and outperforms a comparable test due to Wang and Phillips (2012, Ann. Stat.) against many alternatives.

Tables

Table 1. Table 1: Size: maximum rejection frequency over ρ ∈ { − 0.5 , 0 , 0.5 } 𝜌 0.5 0 0.5 \rho\in\{-0.5,0,0.5\} ; α = 0.1 𝛼 0.1 \alpha=0.1

$𝒅$	$𝒏$		WP (2012)				$𝒑=\mathrm{𝟏𝟕}$				$𝒑=\mathrm{𝟐𝟓}$
$h=n^b$, $b=$			-0.2	-0.1	-0.05		-0.2	-0.1	-0.05		-0.2	-0.1	-0.05
$0.25$	100		0.04	0.02	0.01		0.07	0.05	0.04		0.09	0.06	0.04
	200		0.05	0.02	0.01		0.07	0.06	0.04		0.09	0.07	0.05
	500		0.06	0.02	0.01		0.07	0.06	0.04		0.09	0.07	0.05
$0.50$	100		0.05	0.02	0.01		0.07	0.06	0.04		0.09	0.07	0.05
	200		0.06	0.03	0.02		0.08	0.07	0.05		0.11	0.09	0.07
	500		0.07	0.04	0.03		0.07	0.07	0.05		0.10	0.09	0.07
$0.75$	100		0.06	0.04	0.03		0.08	0.07	0.06		0.11	0.09	0.07
	200		0.08	0.06	0.04		0.08	0.08	0.07		0.10	0.10	0.09
	500		0.08	0.07	0.05		0.07	0.08	0.07		0.09	0.09	0.09
$1.00$	100		0.08	0.06	0.05		0.09	0.09	0.08		0.13	0.11	0.11
	200		0.09	0.07	0.06		0.08	0.10	0.10		0.11	0.12	0.12
	500		0.09	0.08	0.08		0.09	0.09	0.09		0.10	0.11	0.11

Table 2. Table 2: Size-adjusted power when d ∈ { 0.5 , 1.0 } 𝑑 0.5 1.0 d\in\{0.5,1.0\} ; α = 0.1 𝛼 0.1 \alpha=0.1

	$𝒏$		WP (2012)				$𝒑=\mathrm{𝟏𝟕}$				$𝒑=\mathrm{𝟐𝟓}$
$h=n^b$, $b=$			-0.2	-0.1	-0.05		-0.2	-0.1	-0.05		-0.2	-0.1	-0.05
$𝒅=0.5$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.08	0.06	0.04		0.08	0.09	0.09		0.11	0.11	0.11
	200		0.14	0.13	0.11		0.11	0.14	0.14		0.15	0.17	0.17
	500		0.37	0.42	0.41		0.15	0.18	0.19		0.20	0.22	0.22
${\phi }_1\left(2x\right)$	100		0.07	0.04	0.03		0.07	0.08	0.06		0.10	0.09	0.08
	200		0.12	0.09	0.07		0.09	0.10	0.09		0.13	0.13	0.12
	500		0.29	0.27	0.23		0.13	0.15	0.14		0.19	0.19	0.18
${\left|x\right|}^{-2}\wedge 1$	100		0.06	0.03	0.02		0.11	0.12	0.12		0.15	0.15	0.14
$(\times 0.5)$	200		0.10	0.06	0.04		0.16	0.19	0.18		0.21	0.23	0.22
	500		0.19	0.16	0.12		0.12	0.14	0.14		0.16	0.16	0.16
${\left|x\right|}^{-1}\wedge 1$	100		0.06	0.03	0.02		0.07	0.09	0.08		0.10	0.11	0.10
$(\times 0.5)$	200		0.07	0.05	0.04		0.10	0.14	0.14		0.15	0.17	0.17
	500		0.14	0.13	0.11		0.14	0.18	0.19		0.19	0.21	0.22
${\left|x\right|}^{1.5}$	100		0.11	0.09	0.07		0.03	0.04	0.04		0.05	0.06	0.05
$(\times 0.02)$	200		0.22	0.23	0.22		0.04	0.05	0.06		0.06	0.07	0.07
	500		0.61	0.66	0.66		0.04	0.09	0.11		0.08	0.12	0.13
$x^2$	100		0.07	0.05	0.04		0.06	0.07	0.07		0.09	0.09	0.09
$(\times 0.02)$	200		0.13	0.12	0.11		0.08	0.12	0.13		0.12	0.15	0.16
	500		0.37	0.42	0.43		0.07	0.12	0.13		0.12	0.15	0.15
$𝒅=1.0$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.08	0.07	0.06		0.04	0.06	0.06		0.08	0.09	0.09
	200		0.10	0.09	0.08		0.02	0.05	0.07		0.06	0.09	0.10
	500		0.12	0.12	0.12		0.02	0.05	0.05		0.05	0.07	0.09
${\phi }_1\left(2x\right)$	100		0.08	0.06	0.05		0.02	0.04	0.05		0.07	0.08	0.07
	200		0.09	0.08	0.07		0.01	0.03	0.04		0.05	0.06	0.07
	500		0.10	0.09	0.09		0.02	0.02	0.03		0.03	0.04	0.05
${\left|x\right|}^{-2}\wedge 1$	100		0.08	0.06	0.05		0.05	0.08	0.08		0.10	0.12	0.12
$(\times 0.5)$	200		0.09	0.07	0.07		0.04	0.08	0.10		0.09	0.12	0.14
	500		0.09	0.09	0.08		0.02	0.06	0.08		0.05	0.09	0.11
${\left|x\right|}^{-1}\wedge 1$	100		0.08	0.06	0.05		0.04	0.07	0.07		0.09	0.10	0.10
$(\times 0.5)$	200		0.08	0.07	0.07		0.03	0.07	0.08		0.08	0.11	0.12
	500		0.11	0.11	0.11		0.02	0.06	0.08		0.06	0.09	0.12
${\left|x\right|}^{1.5}$	100		0.10	0.09	0.08		0.04	0.07	0.08		0.09	0.10	0.11
$(\times 0.02)$	200		0.13	0.13	0.13		-0.02	0.04	0.07		0.05	0.08	0.11
	500		0.19	0.21	0.22		-0.10	-0.01	0.00		-0.04	0.01	0.02
$x^2$	100		0.09	0.07	0.06		0.04	0.06	0.08		0.07	0.09	0.09
$(\times 0.02)$	200		0.11	0.11	0.11		0.00	0.01	0.01		0.01	0.01	0.01
	500		0.16	0.19	0.19		0.00	0.00	0.00		0.00	0.00	0.00

Table 3. Table 3: MI processes: x t = ( 1 − κ n − 1 ) ​ x t − 1 + ξ t subscript 𝑥 𝑡 1 superscript subscript 𝜅 𝑛 1 subscript 𝑥 𝑡 1 subscript 𝜉 𝑡 x_{t}=(1-\kappa_{n}^{-1})x_{t-1}+\xi_{t} ; κ n = n α κ subscript 𝜅 𝑛 superscript 𝑛 subscript 𝛼 𝜅 \kappa_{n}=n^{\alpha_{\kappa}} Size: maximum rejection frequency over ρ ∈ { − 0.5 , 0 , 0.5 } 𝜌 0.5 0 0.5 \rho\in\{-0.5,0,0.5\} ; α = 0.1 𝛼 0.1 \alpha=0.1

${𝜶}_{𝜿}$	$𝒏$		WP (2012)				$𝒑=\mathrm{𝟏𝟕}$				$𝒑=\mathrm{𝟐𝟓}$
$h=n^b$, $b=$			-0.2	-0.1	-0.05		-0.2	-0.1	-0.05		-0.2	-0.1	-0.05
$0.25$	100		0.02	0.01	0.00		0.06	0.03	0.02		0.07	0.03	0.02
	200		0.03	0.01	0.00		0.06	0.04	0.02		0.08	0.04	0.03
	500		0.04	0.01	0.00		0.07	0.04	0.03		0.08	0.05	0.03
$0.50$	100		0.04	0.02	0.01		0.07	0.05	0.04		0.09	0.06	0.04
	200		0.06	0.03	0.02		0.08	0.06	0.05		0.10	0.08	0.06
	500		0.08	0.05	0.03		0.08	0.07	0.07		0.10	0.09	0.08
$0.75$	100		0.06	0.03	0.02		0.08	0.06	0.05		0.10	0.08	0.06
	200		0.08	0.05	0.04		0.07	0.07	0.06		0.10	0.09	0.08
	500		0.09	0.07	0.06		0.08	0.08	0.08		0.09	0.10	0.10
$1.00$	100		0.06	0.04	0.03		0.08	0.07	0.06		0.11	0.09	0.08
	200		0.08	0.05	0.04		0.08	0.08	0.08		0.10	0.10	0.10
	500		0.09	0.07	0.07		0.08	0.08	0.09		0.09	0.10	0.11

Table 4. Table 4: Fractional type II processes Size-adjusted power when d ∈ { 0.25 , 0.75 } 𝑑 0.25 0.75 d\in\{0.25,0.75\} ; α = 0.1 𝛼 0.1 \alpha=0.1

	$𝒏$		WP (2012)				$𝒑=\mathrm{𝟏𝟕}$				$𝒑=\mathrm{𝟐𝟓}$
$h=n^b$, $b=$			-0.2	-0.1	-0.05		-0.2	-0.1	-0.05		-0.2	-0.1	-0.05
$𝒅=0.25$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.07	0.05	0.04		0.11	0.10	0.09		0.14	0.12	0.10
	200		0.16	0.14	0.12		0.15	0.17	0.16		0.19	0.20	0.19
	500		0.47	0.53	0.51		0.20	0.20	0.20		0.24	0.23	0.23
${\phi }_1\left(2x\right)$	100		0.08	0.05	0.03		0.10	0.08	0.07		0.13	0.10	0.08
	200		0.15	0.12	0.08		0.14	0.13	0.11		0.18	0.15	0.13
	500		0.47	0.45	0.37		0.18	0.17	0.16		0.22	0.21	0.20
${\left|x\right|}^{-2}\wedge 1$	100		0.06	0.03	0.02		0.14	0.13	0.13		0.18	0.16	0.15
$(\times 0.5)$	200		0.11	0.07	0.04		0.20	0.22	0.21		0.24	0.25	0.24
	500		0.35	0.27	0.20		0.14	0.12	0.13		0.16	0.14	0.14
${\left|x\right|}^{-1}\wedge 1$	100		0.05	0.02	0.01		0.09	0.09	0.08		0.12	0.11	0.09
$(\times 0.5)$	200		0.06	0.04	0.03		0.13	0.15	0.15		0.16	0.18	0.17
	500		0.14	0.13	0.11		0.19	0.22	0.22		0.23	0.25	0.25
${\left|x\right|}^{1.5}$	100		0.10	0.09	0.07		0.03	0.04	0.04		0.05	0.05	0.04
$(\times 0.02)$	200		0.25	0.25	0.23		0.03	0.04	0.04		0.06	0.06	0.05
	500		0.73	0.78	0.77		0.04	0.07	0.06		0.06	0.09	0.08
$x^2$	100		0.06	0.04	0.03		0.05	0.06	0.05		0.07	0.07	0.06
$(\times 0.02)$	200		0.12	0.11	0.10		0.06	0.08	0.07		0.09	0.10	0.09
	500		0.39	0.44	0.44		0.10	0.14	0.16		0.13	0.18	0.19
$𝒅=0.75$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.07	0.06	0.05		0.06	0.08	0.08		0.10	0.11	0.10
	200		0.12	0.11	0.10		0.07	0.10	0.11		0.11	0.14	0.14
	500		0.20	0.23	0.23		0.07	0.12	0.14		0.12	0.16	0.17
${\phi }_1\left(2x\right)$	100		0.07	0.05	0.04		0.05	0.06	0.06		0.08	0.08	0.08
	200		0.10	0.08	0.06		0.05	0.07	0.07		0.09	0.10	0.09
	500		0.14	0.13	0.12		0.04	0.07	0.08		0.08	0.11	0.11
${\left|x\right|}^{-2}\wedge 1$	100		0.07	0.04	0.03		0.09	0.11	0.11		0.12	0.14	0.14
$(\times 0.5)$	200		0.09	0.06	0.05		0.10	0.13	0.14		0.15	0.18	0.18
	500		0.11	0.10	0.08		0.07	0.12	0.13		0.12	0.15	0.16
${\left|x\right|}^{-1}\wedge 1$	100		0.07	0.04	0.03		0.07	0.08	0.08		0.09	0.11	0.11
$(\times 0.5)$	200		0.08	0.07	0.06		0.07	0.11	0.12		0.11	0.14	0.15
	500		0.11	0.11	0.11		0.08	0.13	0.15		0.13	0.16	0.18
${\left|x\right|}^{1.5}$	100		0.10	0.09	0.07		0.03	0.05	0.05		0.06	0.07	0.07
$(\times 0.02)$	200		0.18	0.18	0.18		0.04	0.06	0.08		0.07	0.09	0.10
	500		0.37	0.42	0.44		0.01	0.08	0.11		0.06	0.12	0.14
$x^2$	100		0.07	0.06	0.05		0.08	0.11	0.13		0.12	0.14	0.16
$(\times 0.02)$	200		0.13	0.13	0.12		0.04	0.09	0.10		0.08	0.11	0.12
	500		0.26	0.31	0.32		0.00	0.00	0.00		0.01	0.00	0.00

Table 5. Table 5: MI processes: x t = ( 1 − κ n − 1 ) ​ x t − 1 + ξ t subscript 𝑥 𝑡 1 superscript subscript 𝜅 𝑛 1 subscript 𝑥 𝑡 1 subscript 𝜉 𝑡 x_{t}=(1-\kappa_{n}^{-1})x_{t-1}+\xi_{t} ; κ n = n α κ subscript 𝜅 𝑛 superscript 𝑛 subscript 𝛼 𝜅 \kappa_{n}=n^{\alpha_{\kappa}} Size-adjusted power when α κ ∈ { 0.25 , 0.50 } subscript 𝛼 𝜅 0.25 0.50 \alpha_{\kappa}\in\{0.25,0.50\} ; α = 0.1 𝛼 0.1 \alpha=0.1

	$𝒏$		WP (2012)				$𝒑=\mathrm{𝟏𝟕}$				$𝒑=\mathrm{𝟐𝟓}$
$h=n^b$, $b=$			-0.2	-0.1	-0.05		-0.2	-0.1	-0.05		-0.2	-0.1	-0.05
${𝜶}_{𝜿}=0.25$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.06	0.03	0.02		0.11	0.09	0.07		0.13	0.11	0.08
	200		0.15	0.13	0.09		0.17	0.17	0.15		0.20	0.20	0.17
	500		0.50	0.54	0.50		0.21	0.22	0.23		0.24	0.24	0.25
${\phi }_1\left(2x\right)$	100		0.09	0.04	0.02		0.13	0.10	0.07		0.16	0.12	0.08
	200		0.22	0.16	0.10		0.19	0.17	0.13		0.23	0.20	0.16
	500		0.61	0.58	0.48		0.19	0.18	0.19		0.22	0.21	0.22
${\left|x\right|}^{-2}\wedge 1$	100		0.07	0.03	0.01		0.12	0.12	0.10		0.15	0.13	0.11
$(\times 0.5)$	200		0.17	0.10	0.05		0.20	0.22	0.21		0.23	0.24	0.24
	500		0.50	0.39	0.26		0.15	0.15	0.16		0.17	0.16	0.17
${\left|x\right|}^{-1}\wedge 1$	100		0.03	0.01	0.00		0.08	0.06	0.05		0.09	0.08	0.06
$(\times 0.5)$	200		0.06	0.03	0.02		0.12	0.12	0.11		0.15	0.15	0.12
	500		0.15	0.12	0.09		0.20	0.24	0.24		0.23	0.26	0.27
${\left|x\right|}^{1.5}$	100		0.07	0.05	0.03		0.04	0.02	0.01		0.05	0.03	0.02
$(\times 0.02)$	200		0.22	0.20	0.16		0.04	0.03	0.02		0.05	0.04	0.03
	500		0.72	0.76	0.74		0.04	0.04	0.03		0.06	0.05	0.04
$x^2$	100		0.03	0.02	0.01		0.04	0.03	0.02		0.05	0.03	0.02
$(\times 0.02)$	200		0.09	0.07	0.05		0.05	0.04	0.03		0.06	0.05	0.04
	500		0.33	0.35	0.33		0.06	0.07	0.07		0.09	0.09	0.08
${𝜶}_{𝜿}=0.50$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.08	0.06	0.04		0.11	0.12	0.11		0.15	0.14	0.13
	200		0.18	0.18	0.16		0.16	0.17	0.18		0.20	0.21	0.21
	500		0.47	0.54	0.54		0.17	0.19	0.19		0.22	0.22	0.22
${\phi }_1\left(2x\right)$	100		0.08	0.05	0.03		0.11	0.09	0.07		0.14	0.12	0.09
	200		0.15	0.12	0.09		0.14	0.13	0.12		0.18	0.17	0.15
	500		0.34	0.33	0.29		0.15	0.17	0.16		0.20	0.21	0.20
${\left|x\right|}^{-2}\wedge 1$	100		0.07	0.04	0.02		0.15	0.15	0.15		0.19	0.18	0.17
$(\times 0.5)$	200		0.12	0.08	0.05		0.19	0.22	0.22		0.24	0.25	0.25
	500		0.24	0.20	0.15		0.12	0.11	0.10		0.14	0.12	0.11
${\left|x\right|}^{-1}\wedge 1$	100		0.05	0.03	0.01		0.10	0.11	0.10		0.13	0.13	0.11
$(\times 0.5)$	200		0.08	0.06	0.05		0.15	0.17	0.17		0.19	0.21	0.21
	500		0.17	0.18	0.17		0.17	0.18	0.19		0.21	0.21	0.21
${\left|x\right|}^{1.5}$	100		0.12	0.10	0.09		0.03	0.03	0.03		0.05	0.04	0.04
$(\times 0.02)$	200		0.31	0.33	0.31		0.04	0.06	0.06		0.07	0.08	0.08
	500		0.76	0.83	0.85		0.06	0.11	0.14		0.10	0.15	0.17
$x^2$	100		0.07	0.05	0.04		0.05	0.06	0.05		0.07	0.07	0.07
$(\times 0.02)$	200		0.16	0.17	0.16		0.09	0.12	0.13		0.13	0.15	0.17
	500		0.50	0.60	0.61		0.04	0.07	0.07		0.08	0.09	0.09

Table 6. Table 6: MI processes: x t = ( 1 − κ n − 1 ) ​ x t − 1 + ξ t subscript 𝑥 𝑡 1 superscript subscript 𝜅 𝑛 1 subscript 𝑥 𝑡 1 subscript 𝜉 𝑡 x_{t}=(1-\kappa_{n}^{-1})x_{t-1}+\xi_{t} ; κ n = n α κ subscript 𝜅 𝑛 superscript 𝑛 subscript 𝛼 𝜅 \kappa_{n}=n^{\alpha_{\kappa}} Size-adjusted power when α κ ∈ { 0.75 , 1.00 } subscript 𝛼 𝜅 0.75 1.00 \alpha_{\kappa}\in\{0.75,1.00\} ; α = 0.1 𝛼 0.1 \alpha=0.1

	$𝒏$		WP (2012)				$𝒑=\mathrm{𝟏𝟕}$				$𝒑=\mathrm{𝟐𝟓}$
$h=n^b$, $b=$			-0.2	-0.1	-0.05		-0.2	-0.1	-0.05		-0.2	-0.1	-0.05
${𝜶}_{𝜿}=0.75$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.08	0.06	0.05		0.08	0.09	0.09		0.11	0.12	0.11
	200		0.13	0.13	0.12		0.09	0.12	0.13		0.14	0.16	0.16
	500		0.25	0.29	0.29		0.09	0.15	0.17		0.14	0.19	0.21
${\phi }_1\left(2x\right)$	100		0.07	0.05	0.03		0.06	0.07	0.06		0.10	0.09	0.08
	200		0.11	0.09	0.07		0.06	0.08	0.08		0.10	0.11	0.11
	500		0.17	0.17	0.15		0.06	0.09	0.09		0.10	0.13	0.14
${\left|x\right|}^{-2}\wedge 1$	100		0.07	0.04	0.03		0.11	0.13	0.13		0.15	0.16	0.15
$(\times 0.5)$	200		0.09	0.07	0.05		0.13	0.17	0.18		0.18	0.20	0.21
	500		0.13	0.11	0.09		0.09	0.14	0.15		0.15	0.18	0.18
${\left|x\right|}^{-1}\wedge 1$	100		0.06	0.03	0.02		0.08	0.09	0.10		0.11	0.12	0.11
$(\times 0.5)$	200		0.09	0.06	0.05		0.10	0.13	0.14		0.14	0.16	0.17
	500		0.13	0.14	0.14		0.10	0.16	0.17		0.15	0.20	0.21
${\left|x\right|}^{1.5}$	100		0.11	0.09	0.08		0.03	0.04	0.05		0.06	0.06	0.06
$(\times 0.02)$	200		0.21	0.23	0.22		0.04	0.07	0.08		0.06	0.10	0.11
	500		0.45	0.52	0.54		0.00	0.07	0.09		0.06	0.11	0.12
$x^2$	100		0.08	0.06	0.05		0.08	0.11	0.11		0.11	0.13	0.13
$(\times 0.02)$	200		0.14	0.15	0.14		0.06	0.11	0.12		0.10	0.14	0.14
	500		0.33	0.39	0.42		0.00	0.00	0.00		0.00	0.00	0.00
${𝜶}_{𝜿}=1.00$			Size adj. power				Size adjusted, relative to WP
${\phi }_1\left(x\right)$	100		0.08	0.06	0.05		0.06	0.08	0.07		0.09	0.10	0.09
	200		0.11	0.10	0.09		0.06	0.10	0.11		0.10	0.13	0.14
	500		0.17	0.19	0.19		0.04	0.09	0.10		0.08	0.13	0.14
${\phi }_1\left(2x\right)$	100		0.07	0.05	0.04		0.05	0.05	0.06		0.08	0.08	0.07
	200		0.10	0.08	0.06		0.04	0.06	0.06		0.07	0.09	0.09
	500		0.13	0.12	0.11		0.03	0.05	0.06		0.05	0.08	0.10
${\left|x\right|}^{-2}\wedge 1$	100		0.07	0.04	0.03		0.09	0.11	0.11		0.13	0.14	0.14
$(\times 0.5)$	200		0.09	0.06	0.05		0.10	0.13	0.14		0.14	0.17	0.17
	500		0.11	0.09	0.08		0.05	0.10	0.13		0.10	0.14	0.16
${\left|x\right|}^{-1}\wedge 1$	100		0.06	0.04	0.03		0.06	0.08	0.08		0.10	0.10	0.10
$(\times 0.5)$	200		0.09	0.07	0.06		0.07	0.11	0.12		0.11	0.14	0.15
	500		0.12	0.12	0.11		0.05	0.10	0.13		0.09	0.14	0.17
${\left|x\right|}^{1.5}$	100		0.09	0.08	0.07		0.03	0.04	0.05		0.05	0.06	0.06
$(\times 0.02)$	200		0.16	0.17	0.16		0.04	0.07	0.08		0.07	0.10	0.11
	500		0.29	0.33	0.34		-0.04	0.03	0.05		0.01	0.06	0.07
$x^2$	100		0.07	0.06	0.05		0.08	0.11	0.13		0.12	0.14	0.15
$(\times 0.02)$	200		0.11	0.12	0.11		0.04	0.06	0.07		0.07	0.08	0.08
	500		0.22	0.26	0.27		0.00	0.00	0.00		0.00	0.00	0.00

Equations

$$y_t=m(x_{t-1})+u_t$$

$$\frac{1}{n}\sum _{t=1}^{n}f({\beta }_n^{-1}{x}_t)\text{and}\frac{{\beta }_n}{n{h}_n}\sum _{t=1}^{n}K\left(\frac{x_t-x}{h_n}\right)$$

$$\frac{1}{n}\sum _{t=1}^{n}f({\beta }_n^{-1}{x}_t)\stackrel{d}{\to }{\int }_{\mathbb{R}}f(x+X^{-}){\phi }_{{\sigma }_{+}^2}(x)dx$$

$$\frac{{\beta }_n}{n{h}_n}\sum _{t=1}^{n}K\left(\frac{x_t-x}{h_n}\right)\stackrel{d}{\to }{\phi }_{{\sigma }_{+}^2}(-X^{-}){\int }_{\mathbb{R}}K(u)du.$$

$$x_t(n)=\sum _{j=0}^{t-1}{\varphi }_j(n)v_{t-j},\text{where}{v}_t=\sum _{i=0}^{\infty }{c}_i{\xi }_{t-i},$$

$$(1-L{)}^{1/2}{x}_t=v_{t}1\left\{t>0\right\},\text{where }a(L)v_t=b(L){\xi }_t,$$

$$x_t(n)=(1-{\kappa }_n^{-1})x_{t-1}(n)+v_t,$$

$${\varphi }_j={\varphi }_j(n)=(1-{\kappa }_n^{-1}{)}^j.$$

$$\frac{1}{n}\sum _{t=1}^{n}f({\beta }_n^{-1}{x}_t)$$

$$\mathbf{E}\left[h\left({\beta }_n^{-1}{X}_{t_n}^{+}(n)\right)\mid {\mathcal{F}}_0\right]\stackrel{p}{\to }{\int }_{\mathbb{R}}h(x){\Phi }_{X^{+}}(x)dx\text{; and}$$

$$\underset{n\ge n_0,t_0\le t\le n}{\sup }\underset{x}{\sup }{\mathcal{D}}_{n,t}(x)<\infty$$

$$\frac{1}{n}\sum _{t=1}^{n}g\left({\beta }_n^{-1}{X}_t(n)\right)=\frac{1}{n}\sum _{t=1}^n\mathbf{E}\left[g\left({\beta }_n^{-1}{X}_t(n)\right)\mid {\mathcal{F}}_0\right]+o_p(1).$$

$${\beta }_n^{-1}(x_{\lfloor nr\rfloor }^{+},x_{\lfloor ns\rfloor }^{+})\stackrel{d}{\to }({\eta }_r,{\eta }_s)$$

$${\beta }_n^{-1}(x_{\lfloor nr\rfloor }^{-},x_{\lfloor ns\rfloor }^{-})\stackrel{d}{\to }({\eta }^{-},{\eta }^{-})$$

$$\frac{1}{n}\sum _{t=1}^{n}f\left({\beta }_n^{-1}{X}_t\left(n\right)\right)\stackrel{d}{\to }{\int }_{\mathbb{R}}f\left(x+X^{-}\right){\Phi }_{X^{+}}(x)dx.$$

$$\varrho (x):=\{\begin{array}{ll}{\phi }_{1/2}(x-X^{-})& \text{under FR1}\\ {\phi }_1(x)& \text{under FR2, MI,}\end{array}$$

$$\frac{1}{n}\sum _{t=1}^{n}f({\beta }_n^{-1}{x}_t(n))\stackrel{d}{\to }{\int }_{\mathbb{R}}f(x)\varrho (x)dx.$$

$${\int }_{\mathbb{R}}f(x){\phi }_1(x)dx={\int }_{\mathbb{R}}{x}^2{\phi }_1(x)dx=1.$$

$$\frac{{\beta }_n}{h_{n}n}\sum _{t=1}^{n}K\left(\frac{x_t(n)-x}{h_n}\right),$$

$${\Omega }_n\left(\eta \right):=\left\{\left\{s,t\right\}\in \mathbb{N}:\lfloor \eta n\rfloor \le s\le \lfloor (1-\eta )n\rfloor ,\lfloor \eta n\rfloor +s\le t\le n\right\},$$

$$\underset{n\ge n_0,0\le s<t\le n,t-s\ge t_0}{\sup }\underset{x}{\sup }{\mathcal{D}}_{t,s,n}(x)<\infty .$$

$$\underset{\eta \downarrow 0}{\lim }\underset{n\to \infty }{\mathrm{limsup}}\underset{(s,t)\in {\Omega }_n(\eta )}{\sup }\underset{|x|\le q_0{\eta }^{q_1}}{\sup }\left|{\mathcal{D}}_{t,s,n}(x)-{\mathcal{D}}_{t,s,n}(0)\right|=0$$

$$\frac{{\beta }_n}{h_{n}n}\sum _{t=t_0}^{n}K\left(\frac{X_t(n)-x}{h_n}\right)=\frac{1}{n}\sum _{t=t_0}^n{\phi }_{{\epsilon }^2}({\beta }_n^{-1}{X}_t(n)){\int }_{\mathbb{R}}K(u)du+o_p(1),$$

$$\frac{1}{n}\sum _{t=1}^n{\phi }_{{\epsilon }^2}({\beta }_n^{-1}{X}_t(n))$$

$$\stackrel{a.s.}{\to }{\Phi }_{X^{+}}(-X^{-}),\text{as }\epsilon \to 0.$$

$$\frac{{\beta }_n}{h_{n}n}\sum _{t=1}^{n}K\left(\frac{X_t(n)-x}{h_n}\right)\stackrel{d}{\to }{\Phi }_{X^{+}}(-X^{-}){\int }_{\mathbb{R}}K(u)du.$$

$$\frac{{\beta }_n}{h_{n}n}\sum _{t=1}^{n}K\left(\frac{x_t(n)-x}{h_n}\right)\stackrel{d}{\to }\varrho (0){\int }_{\mathbb{R}}K(u)du.$$

$$y_t=\mu +\gamma g(x_{t-1})+u_t$$

$$g(x)={\kappa }_g(\lambda )H_g(x/\lambda )+R_g(x,\lambda ),$$

$$R_{g,n}^j:=\frac{1}{{\kappa }_g^j({\beta }_n)n}\sum _{t=1}^n\mathbf{E}{\left|R_g(x_t(n),{\beta }_n)\right|}^j=o(1).$$

$$\underset{\lambda \to \infty }{\lim }{\kappa }_g(\lambda {)}^{-1}\underset{x}{\sup }|R_g(x,\lambda )|=0$$

Full text

,

62M10,

62F12,

62G08,

Fractional Process,

Half Unit Root,

Kernel Regression,

Specification Testing,

Mildly Integrated Process,

Weakly Nonstationary Process,

Ioannis Kasparislabel=e2]kasparis@ucy.ac.cy

1 Introduction

Inference in regression models when data is temporally dependent is a challenging problem, which has engendered a voluminous literature. Previous work has investigated the asymptotics of parametric and nonparametric regression estimators under a variety of assumptions on the form and extent of that dependence, including, for example: regressors generated by autoregressive fractionally integrated moving average (ARFIMA) models, general linear and nonlinear processes, and by partial sums and arrays formed from such processes. Both stationary and nonstationary processes have been considered, and quite distinct arguments – relying on stationary laws of large numbers (LLNs) and central limit theorems (CLTs) for the former, and the weak convergence of stochastic processes for the latter – have been utilised to handle these cases.

It might therefore seem as though little work remains to be done on this problem. However, in its treatment of nonstationary processes, previous work has typically employed assumptions that prevent these from being wholly contiguous with their stationary counterparts, leaving some significant gaps in the domain of the existing theory. Thus, for example, when regressors are generated by arrays of ARIMA processes, both the cases of fixed stationary autoregressive roots and of an autoregressive root drifting towards unity at rate $n^{-1}$ (a ‘nearly integrated’ process, henceforth ‘NI’; see Chan and Wei, 1987, 1988; Phillips, 1987a,b) have been closely studied, but the intermediate case of roots drifting towards unity at a strictly slower rate than $n^{-1}$ (a ‘mildly integrated’ process, henceforth ‘MI’; see Giraitis and Phillips, 2006; Magdalinos and Phillips, 2007) has received less attention until very recently. Similarly, the asymptotics of regression estimators when applied to fractionally integrated regressors of order $d$ (henceforth, $I(d)$) are well understood both when $d\in(-1/2,1/2)$ or $d\in(1/2,3/2)$, but not nearly so well when $d=1/2$. In view of the importance of ARFIMA models to the statistical modelling of time series, it is striking that the limit theory for these processes still remains to be fully characterised.111Although recent work by Shimotsu and Phillips (2005) and Hualde and Robinson (2011) allow for cases where $d=1/2$, the limit theory developed by these authors is specific to functionals in the frequency domain that arise in the context of memory estimation.

Existing work on the asymptotics of regression estimators may thus be divided into two literatures: that dealing with stationary processes, and that with strongly dependent nonstationary processes, with a certain space left in between them. Though these literatures are too vast to be cited exhaustively here, some particularly notable contributions include the following. For stationary long memory processes, the asymptotics of nonparametric regression estimators were developed by Wu and Mielniczuk (2002) and Wu, Huang and Huang (2010). For nonstationary processes, parametric regression estimators have been studied by Chan and Wei (1987, 1988), Phillips (1987a,b), Phillips (1995), Park and Phillips (1999, 2001) and Chan and Wang (2015). Robinson and Hualde (2003), Christensen and Nielsen (2006), Hualde and Robinson (2010) and Johansen and Nielsen (2012a) consider fractional systems. Several papers consider the problem of inference in regressions with a NI covariate, including Mikusheva (2007), Phillips and Magdalinos (2009) and Kostakis, Magdalinos and Stamatogiannis (2015). Wang and Phillips (2009a,b, 2011, 2012) consider nonparametric methods for estimation and inference in regressions with a NI or nonstationary fractional covariate; some closely related work on nonparametric estimation in the setting of null recurrent Markov chains is the subject of the papers by Karlsen and Tjøstheim (2001) and Karlsen, Myklebust and Tjøstheim (2007).

The present work aims to fill the gap between these two literatures, by developing the asymptotics of regression estimators for a class of processes intermediate between the stationary and more strongly dependent nonstationary processes previously considered. We term these weakly nonstationary processes (WNPs), with leading examples being MI and $I(1/2)$ processes. To appreciate the significance of our results, consider the regression model

[TABLE]

where $u_{t}$ is a martingale difference sequence; suppose for concreteness that $x_{t}$ is $I(d)$ for an unknown $d\in(-1/2,3/2)$. The object of interest is the regression function $m(\cdot)$; e.g. we would like to test such a null hypotheses as $\mathcal{H}_{0}:m(x)=m_{0}(x)$ for a given $x$ and $m_{0}$. Since $\mathcal{H}_{0}$ places no restriction on the process followed by $x_{t}$, establishing the asymptotic validity of any test of $\mathcal{H}_{0}$ requires that its asymptotics be developed under all possible values of the nuisance parameter $d$ (and indeed, under appropriate drifting sequences $\{d_{n}\}$: see e.g. Mikusheva, 2007; Andrews, Cheng and Guggenberger, 2020).222Formally, the asymptotic size of a test $\phi_{n}\in\{0,1\}$ would be defined as $\limsup_{n\rightarrow\infty}\sup_{d\in(-1/2,3/2),m\in\mathscr{M}}\mathbb{P}_{d,m}\{\phi_{n}=1\}$, for $\mathscr{M}$ a class of functions respecting $\mathcal{H}_{0}$. This requires results for the case where $d=1/2$, no less than for $d\in(-1/2,1/2)$ and $d\in(1/2,3/2)$. Empirically, values of $d$ in the vicinity of $1/2$ have been systematically found to provide a good description of the dynamics of inflation and realised volatility series (see e.g. Hassler & Wolter, 1995; Baillie, Chung and Tieslau, 1996; and Andersen, Bollerslev, Diebold and Labys, 2001). Indeed, recent work by Hassler and Pohle (2019) finds that when forecasting such series, an ARFIMA($p$,$d$,[math]) model with $d$ fixed at $1/2$ gives superior forecasts to those produced by the same model with an estimated value of $d$. As such, our results should be particularly relevant for inference in regressions involving such series as r.h.s. variables.

In the context of (1), we show that nonparametric kernel estimators of $m$ are asymptotically (mixed) Gaussian when $x_{t}$ is a WNP; consequently, the $t$ statistic for testing $\mathcal{H}_{0}$ is asymptotically standard normal. This accords with previous results for both stationary and strongly dependent nonstationary processes (e.g. NI processes or $I(d)$ process with $d>1/2$), which establish the asymptotic normality of the $t$ statistic in these cases** **(see Wu and Mielniczuk, 2002; and Wang and Phillips, 2009a,b, 2011). It follows that conventional tests of $\mathcal{H}_{0}$, involving the comparison of the $t$ statistic to normal critical values, are asymptotically valid even when the regressor has an unknown, but possibly high, degree of persistence. This is of particular importance for practitioners, since it implies that tests of $\mathcal{H}_{0}$ can be conducted without having to in any way adjust for the persistence of $x_{t}$.

We also consider the case where $m$ is parametrised as $m(x)=\mu+\gamma g(x)$ for a known function $g$. When $x_{t}$ is a WNP, least squares estimators of $(\mu,\gamma)$ are shown to exhibit the elevated rates of convergence familiar from when regressors are more strongly dependent, but with limiting distributions that are (mixed) Gaussian, similarly to the case of stationary regressors. This result is less directly useful to practitioners, since it breaks down when $x_{t}$ is more strongly dependent, in which case the limiting distributions of these estimators are well known to be nonstandard (see e.g. Phillips, 1995; Marinucci and Robinson, 1998; Robinson and Hualde, 2003).

We build on these results to develop a test for parametric specifications of $m$ the form $m(x)=\mu+\gamma g(x)$. The test is based on a comparison of the fit provided by parametric and nonparametric estimates of $m$, and is designed so as to inherit the asymptotic (mixed) normality of the nonparametric estimator. It therefore has the attractive property that the limiting distribution of the test statistic is invariant to the persistence of $x_{t}$, implying that a practitioner can perform the test using a fixed set of critical values, while remaining agnostic about the dependence properties of the regressor. Simulations confirm that the size of the test is successfully controlled in finite samples, as the process generating the regressor varies between weakly dependent and stationary, weakly nonstationary, and strongly dependent nonstationary. Relative to the specification test proposed by Wang and Phillips (2012), our test appears to have greater power against a broad range of alternatives, with these power improvements being especially pronounced for integrable and asymptotically vanishing alternatives.

Underpinning our limit theory for regression estimators and specification tests are a collection of new technical results concerning the asymptotics of additive functionals of WNPs, of the form

[TABLE]

where $\beta_{n}^{2}=Var(x_{n})$, $f$ is locally integrable, $K$ is integrable, and $h_{n}$ denotes a bandwidth sequence. These new results are needed because WNPs are characterised by being: (a) sufficiently nonstationary to resist the application of existing LLNs; and (b) so weakly dependent that the finite-dimensional distributions of the process $r\mapsto\beta_{n}^{-1}x_{\lfloor nr\rfloor}$ converge to those of a nonseparable Gaussian process.333By the nonseparability of a process $G$, we mean there does not exist a countable $T\subset[0,1]$ such that for every open interval $I\subset[0,1]$, $\inf_{r\in I\cap T}G(r)=\inf_{r\in I}G(r)$ and $\sup_{r\in I\cap T}G(r)=\sup_{r\in I}G(r)$ (see Loève, 1978, p. 171). Since this convergence cannot be strengthened to weak convergence with respect to the uniform or Skorokhod topologies, the asymptotics of (2) are not amenable to an application of the continuous mapping theorem, or even to more general results on the convergence of integral functionals (see Gikhman and Skorokhod, 1969, p. 485, Thm 1). Despite this, we shall prove that for WNPs

[TABLE]

where $X^{-}\sim N[0,\sigma_{-}^{2}]$ (with possibly $\sigma_{-}^{2}=0$) and $\varphi_{\sigma_{+}^{2}}$ denotes the $N[0,\sigma_{+}^{2}]$ density, and

[TABLE]

The remainder of this paper is organised as follows. To help further motivate this work, two leading examples of WNPs, $I(1/2)$ and MI processes, are discussed in detail in Section 2. Our general results on the asymptotics of the functionals in (2) are presented in Section 3. These provide the basis for the asymptotics of both parametric and nonparametric estimators in regression models involving WNPs – or non-linear transformations thereof – developed in Section 4. These results are in turn used, in Section 5, to propose a specification test whose limiting distribution is invariant to the persistence of the regressor. Its finite-sample performance is evaluated through the simulation exercises presented in Section 6. Proofs of all results are given in the Appendices (in the Supplementary Material).

Notation

$\overset{a.s.}{\rightarrow}$, $\overset{p}{\rightarrow}$ and $\overset{d}{\rightarrow}$ respectively denote convergence almost surely, in probability, and in distribution. For deterministic sequences $\{a_{n}\}$ and $\{b_{n}\}$, $a_{n}\sim b_{n}$ denotes $\lim_{n\rightarrow\infty}a_{n}/b_{n}=1$ and $a_{n}\asymp b_{n}$ denotes $\lim_{n\rightarrow\infty}|a_{n}/b_{n}|\in(0,\infty)$. For a random variable $X$, $X\sim F$ denotes that $X$ has distribution $F$. For a positive real number $x$, $\lfloor x\rfloor$ denotes its integer part. $\mathbf{1}\{A\}$ denotes the indicator function for the set $A$. $\overline{\mathbb{R}}$, $\mathbb{R}_{+}$ and $\mathbb{R}_{+}^{\ast}$ are the extended, the nonnegative, and (strictly) positive real numbers respectively. $f^{(j)}(x)$ denotes the $j$th derivative of the function $f(x)$. All limits are taken as $n\rightarrow\infty$ unless otherwise indicated.

2 Leading examples of WNPs: $I(1/2)$ and MI processes

In the next section, we provide general limit theorems for the additive functionals in (2) under high-level conditions, which may be regarded as defining the class of weakly nonstationary processes (WNPs); these results will then be specialised to $I(1/2)$ and MI processes. Before doing so, we give a precise definition of these two processes, which helps to motivate our high-level conditions. To bring these processes into a common framework, consider a linear process array of the form

[TABLE]

$t=1,...,n\in\mathbb{N}$, and the coefficients $\phi_{j}(n)$ and $c_{i}$ will be specified below. (Where there is no possibility of ambiguity, we shall generally denote $x_{t}(n)$ as simply $x_{t}$, for ease of notation.) $\{\xi_{t}\}_{t\in\mathbb{Z}}$ satisfies

Assumption \AlphAlph.

1. (i)

$\xi_{t}$* is i.i.d. with $\mathbf{E}\xi_{1}=0$ and $Var(\xi_{1})=\sigma_{\xi}^{2}<\infty$.* 2. (ii)

$\xi_{1}$* has an absolutely continuous distribution, and a characteristic function $\psi_{\xi}(\lambda)$ that satisfies $\int_{\mathbb{R}}|\psi_{\xi}(\lambda)|^{\theta}d\lambda<\infty$, for some $\theta\in\mathbb{N}$.*

2.1 $I(1/2)$ processes

The definition of a ‘fractionally integrated process’ used this paper closely follows that of Marinucci and Robinson (1999). These authors classify a non-stationary fractional process $x_{t}$ as type I or type II according to the ‘type’ of the fractional Brownian motion (fBM) to which the finite dimensional distributions of $\beta_{n}^{-1}x_{\lfloor nr\rfloor}$ converge, where $\beta_{n}^{2}:=Var(x_{n})$. Although these authors consider processes with a long memory component that is specified ‘parametrically’ – via the expansion of an autoregressive lag polynomial $(1-L)^{d}$ (see also Remark 2.1(c)** **below) – their classification extends straightforwardly to the case where this is instead formulated ‘semi-parametrically’, in terms of the decay rate of the coefficients $\{\phi_{j}\}$ in (5). Thus we shall say that for $d\in(1/2,1)$, $x_{t}$ is an $I(d)$ process of

• •

type I: if $\phi_{j}=1$, $c_{s}\sim\ell(s)s^{d-2}$, $\sum_{s=0}^{\infty}c_{s}=0$; and

• •

type II: if $\phi_{j}\sim\ell(j)j^{d-1}$, $\sum_{s=0}^{\infty}\lvert c_{s}\rvert<\infty$, and $\sum_{s=0}^{\infty}c_{s}\neq 0$;

where $\ell:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is locally integrable on $[1,\infty)$, positive-valued and slowly varying at infinity (henceforth, ‘SV’) in the sense of Bingham, Goldie and Teugels (1987, p. 6). Fractional processes of this kind have been widely studied when $d>1/2$: see e.g. Taqqu (1975), Kasahara and Maejima (1988) and Jeganathan (2004, 2008) for the type I case, and Robinson and Hualde (2003), Phillips and Shimotsu (2004), Shimotsu and Phillips (2005) and Hualde and Robinson (2011) for the type II case.

The preceding extend naturally to $d=1/2$, and give the definitions of $I(1/2)$ processes (of each type) used throughout this paper, even though $\beta_{n}^{-1}x_{\lfloor nr\rfloor}$ will not converge weakly to an fBM of either type in the case. We shall accordingly develop our limit theory for these processes under

Assumption \AlphAlph.

$x_{t}(n)$* is generated by (5). $\ell$ is SV such that $L(n):=\int_{1}^{n}[\ell^{2}(x)/x]dx\rightarrow\infty$ as $n\rightarrow\infty$, and either:*

1. FR1

$\phi_{j}=1\ \forall j\geq 0$, $c_{s}\sim\ell(s)s^{-3/2}$, and $\sum_{s=0}^{\infty}c_{s}=0$; or 2. FR2

$\phi_{j}\sim\ell(j)j^{-1/2}$, $\phi_{0}\neq 0$, $\sum_{s=0}^{\infty}|c_{s}|<\infty$, $\sum_{s=0}^{\infty}c_{s}\neq 0$.

Remark 2.1*.*

(a) FR1 and FR2 respectively imply that $x_{t}(n)$ is an $I(1/2)$ process of types I and II.

(b) $Var(x_{n})\asymp L(n)$ under FR, where $L(n)$ is itself SV. The divergence or convergence of $L(n)$ effectively demarcates the boundary between WNPs and stationary long memory processes. Either is possible, depending on $\ell$: e.g. $\ell(n)=1$ gives $L(n)\sim\ln n$ and $\ell(n)=(\ln n)^{-1/2}$ gives $L(n)\asymp\ln\ln n$, whereas $\ell(n)=(\ln n)^{-1}$ gives a bounded $L(n)$. When $L(n)$ diverges, $L(n)^{-1/2}x_{n}$ will obey a CLT. But when $L(n)$ is bounded, no CLT applies: and indeed in this case we have under FR2 that $\sum_{j=0}^{\infty}\phi_{j}^{2}<\infty$, so that $x_{t}$ is stationary. Such processes fall within the purview of existing results, and so have been excluded by our assumption that $L(n)\rightarrow\infty$.

(c) **FR **encompasses both types of parametric ARFIMA models with $d=1/2$. For example, consider the ARFIMA($1/2$) type II model

[TABLE]

where $a$ and $b$ denote finite-order polynomials in the lag operator $L$. In this case, it is possible to write $x_{t}=\sum_{j=0}^{t-1}\phi_{j}v_{t-j}$, where $\{\phi_{j}\}_{j\geq 0}$ are the coefficients in the power series expansion of $(1-L)^{-1/2}$; and so $\phi_{0}=1$ and $\phi_{j}\asymp j^{-1/2}$ (see e.g. p. 673 in Johansen and Nielsen, 2012b). Further, if all the roots of $a$ lie outside the unit circle, $v_{t}=a(L)^{-1}b(L)\xi_{t}$ is a linear process with geometrically decaying coefficients. Thus FR2 is satisfied. Since $\ell(x)=1$, we have trivially that $L(n)\rightarrow\infty$. Similar arguments show that FR1 is consistent with an ARFIMA($1/2$) type I model, under which $x_{t}$ is the partial sum of a ARFIMA($-1/2$) type II process.

2.2 MI processes

Mildly integrated (MI) processes are closely related to the nearly integrated (NI) processes studied by Chan and Wei (1987) and Phillips (1987), and more recently extended by Buchmann and Chan (2007). Both MI and NI processes may be defined in terms of an array as

[TABLE]

where $x_{0}(n)=0$ and $v_{t}$ is a stationary process and $\kappa_{n}>1$ with $\kappa_{n}\rightarrow\infty$, so that the autoregressive coefficient approaches unity as $n$ grows. They can thus be encompassed within the framework of (5) if we allow $\phi_{j}$ to depend on $n$ as per

[TABLE]

Both NI and MI processes thus describe highly persistent autoregressive processes, which have a root in the vicinity of unity. They have accordingly been used to investigate the behaviour of various inferential procedures under local departures from unit roots (e.g. Mikusheva, 2007, and Duffy, 2020), and in the construction of robust inferential procedures (e.g. Magdalinos and Phillips, 2011; Kostakis, Magdalinos and Stamatogiannis, 2015; Demetrescu et al 2019; Yang, Long, Peng and Cai, 2019). The crucial difference between NI and MI processes concerns the assumed growth rate of the sequence $\kappa_{n}$. NI processes are defined by $\kappa_{n}/n\rightarrow c\neq 0$, with the consequence that $n^{-1/2}x_{\lfloor nr\rfloor}$ converges weakly to an Ornstein-Uhlenbeck process. MI processes have $\kappa_{n}/n\rightarrow 0$, which tilts $x_{t}(n)$ closer to stationarity: and as a consequence, FCLTs cannot be used to derive the asymptotics of functionals of these processes. Their variance also grows at a slower rate, since $Var(x_{n}(n))\asymp\kappa_{n}=o(n)$.

Formally, we define an MI process as follows, specifying regularity conditions that are helpful in unifying notation and simplifying some derivations.444Previous work on these processes has assumed that $v_{t}$ is short memory in the sense that $\sum_{s=0}^{\infty}\lvert c_{s}\rvert<\infty$. In a previous working paper version of the present work (available as arXiv:1812.07944v1), we allowed $v_{t}$ to have long memory in the sense that $c_{s}\sim s^{-m}$ for $m\in(1/2,1)$, thereby extending this previous work much in the manner of Buchmann and Chan’s (2007) extension of earlier work on NI processes. To keep the length of the paper manageable, this generalisation is not reported here.

Assumption \AlphAlph.

$x_{t}\left(n\right)$* and $\phi_{j}(n)$ are as in (5) and (8). $\{c_{s}\}_{s\in\mathbb{Z}}$ is such that $\sum_{s=0}^{\infty}\left|c_{s}\right|<\infty$ and $\sum_{s=0}^{\infty}c_{s}\neq 0$. $\{\kappa_{n}\}_{n\in\mathbb{N}}$ has $\kappa_{n}>1$, $\kappa_{n}=n^{\alpha_{\kappa}}\ell_{\kappa}(n)$ for $\ell_{\kappa}$ SV and $\alpha_{\kappa}\in[0,1)$, $\kappa_{n}\rightarrow\infty$ and $\sup_{n\geq 1}\sup_{1\leq t\leq n}\kappa_{n}^{-1}\kappa_{t}<\infty$.*

3 Limit theory for functionals of WNPs

3.1 Additive functionals of standardised processes

Consider

[TABLE]

where $f$ is locally integrable, and $\beta_{n}^{2}:=Var(x_{n}(n))$. Here we provide high-level conditions (Assumption HL) under which the asymptotics of (9) may be derived. These conditions, particularly HL0–2 and HL4 below, may be taken as providing an abstract definition of a WNP – with HL3, HL5 and HL6 being merely regularity conditions that may be dispensed with, if $f$ satisfies certain assumptions. These high-level conditions are stated in terms of a general random array denoted $\{X_{t}(n)\}$, to distinguish it from the linear process array $\{x_{t}(n)\}$ introduced in (5) above.

Assumption \AlphAlph (high-level conditions).

1. HL0

Let $\left\{X_{t}\left(n\right)\right\}_{t=1}^{n}$, $n\in\mathbb{N}$ be a random array and $\left\{\mathcal{F}_{t}\right\}_{t=-\infty}^{\infty}$ a filtration such that $X_{t}\left(n\right)$ is $\mathcal{F}_{t}$-measurable for all $t$ and $n$. Let $\{\beta_{n}\}$ denote a positive sequence with $\beta_{n}\rightarrow\infty$. 2. HL1

$X_{t}(n)=X_{t}^{+}(n)+X_{t}^{-}(n)+R_{t}(n)$, where $X_{t}(n)^{-}$ is $\mathcal{F}_{0}$-measurable, and $\sup_{1\leq t\leq n}\mathbf{P}\{\beta_{n}^{-1}\lvert R_{t}(n)\rvert>\epsilon\}\rightarrow 0$ for every $\epsilon>0$. 3. HL2

There are random variables $X^{+}$ and $X^{-}$, where $X^{+}$ has bounded continuous density $\Phi_{X^{+}}$ such that: for every $\delta\in(0,1)$ and $\left\{t_{n}\right\}$ with $\lfloor n\delta\rfloor\leq t_{n}\leq n$

1. (a)

$\beta_{n}^{-1}X_{t_{n}}^{+}(n)\overset{d}{\rightarrow}X^{+}$, conditionally on $\mathcal{F}_{0}$ in the sense that for all bounded and continuous $h:\mathbb{R}\rightarrow\mathbb{R}$

[TABLE] 2. (b)

$\beta_{n}^{-1}X_{t_{n}}^{-}(n)\overset{d}{\rightarrow}X^{-}$, and $\beta_{n}^{-1}[X_{n}^{-}(n)-X_{t_{n}}^{-}(n)]\overset{p}{\rightarrow}0$. 4. HL3

$\beta_{t}^{-1}X_{t}(n)$* has density $\mathcal{D}_{n,t}(x)$ such that for some $n_{0}\geq t_{0}\geq 1$,*

[TABLE] 5. HL4

For every bounded and Lipschitz continuous $g:\mathbb{R}\rightarrow\mathbb{R}$

[TABLE] 6. HL5

For some $\lambda\in\left(0,\infty\right)$ and $n_{0}\geq 1$

1. (a)

$\sup_{n\geq n_{0},1\leq t\leq n}\mathbf{E}\left|\beta_{n}^{-1}X_{t}\left(n\right)\right|^{\lambda}<\infty$; or 2. (b)

$\sup_{n\geq n_{0},1\leq t\leq n}\mathbf{E}\exp\left(\lambda\left|\beta_{n}^{-1}X_{t}\left(n\right)\right|\right)<\infty$. 7. HL6

$\sup_{n\in\mathbb{N}}\frac{\beta_{n}}{n}\sum_{t=t_{0}}^{n}\beta_{t}^{-1}<\infty$, for $t_{0}$ as in HL3.

Remark 3.1*.*

(a) When $X_{t}(n)$ is a linear process array formed from an underlying i.i.d. sequence $\{\xi_{t}\}$ as in (5), HL1 is trivially satisfied by splitting it into terms depending on $\{\xi_{s}\}_{s\leq 0}$ and $\{\xi_{s}\}_{s=1}^{t}$.

(b) HL2 expresses one of the key properties of a WNP: that its finite dimensional distributions should converge (upon standardisation), albeit not to those of a separable process. Its requirements may be illustrated by an $I(1/2)$ type I process with $\ell(x)=1$, denoted $\{x_{t}\}$. Per the previous remark, write $x_{t}=x_{t}^{+}+x_{t}^{-}$, where $x_{t}^{+}$ and $x_{t}^{-}$ are respectively weighted sums of $\{\xi_{s}\}_{s=1}^{t}$ and $\{\xi_{s}\}_{s\leq 0}$. Then a CLT for weighted sums of linear processes (see Abadir, Distaso, Giraitis, and Koul, 2014) yields that for every $r,s\in(0,1]$

[TABLE]

where $\beta_{n}^{2}\asymp\ln n$, and $\eta_{r}$ and $\eta_{s}$ are independent $N[0,1/2]$ random variables. We thus have the marginal convergence of each coordinate of $\beta_{n}^{-1}x_{\lfloor nr\rfloor}^{+}$ to identical distributional limits, as per HL2(a) – and if $\xi_{t}$ is i.i.d., the required conditional convergence holds trivially. In that case, (10) holds jointly with (and independently of)

[TABLE]

where $\eta^{-}\sim N[0,1/2]$. Note the degeneracy in the joint distribution of the limit in (11), consistent with the second part of HL2(b).

(c) HL3 is useful for establishing $L_{1}$-approximations to functionals of WNPs, which permit convergence results proved under the requirement that $f$ in (9) be bounded and continuous to be extended to a much broader class of integrable functions. High-level conditions similar to HL3 have been employed for similar purposes in many previous works, e.g. Jeganathan (2004, 2008), Pötscher (2004), Gao, King, Lu and Tjøstheim (2009), Wang and Phillips (2009a,b; 2012) among others.

(d) HL4 expresses the requirement that a WNP should not be too strongly dependent. It would fail both for $I(d)$ processes with $d>1/2$, and for NI processes – and indeed for any process for which $\beta_{n}^{-1}X_{\lfloor nr\rfloor}(n)$ converges weakly to a process with continuous sample paths.

Theorem 3.1**.**

Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is locally integrable, and HL0–4 and HL6 hold. Further suppose $f$ is bounded or that:

1. (i)

There is a $\mathcal{Y\subseteq}\mathbb{R}$ such that $\int_{\mathbb{R}}\left|f\left(x+y\right)\right|\Phi_{X^{+}}(x)dx<\infty$ for all $y\in\mathcal{Y}$, and $\mathbf{P}\left(X^{-}\in\mathcal{Y}\right)=1$; 2. (ii)

$n^{-1}\sum_{t=1}^{t_{0}-1}f(\beta_{n}^{-1}X_{t}(n))=o_{p}\left(1\right)$, for $t_{0}$ as in HL3; 3. (iii)

For some $\lambda^{\prime}\in\left(0,\lambda\right)$, where $\lambda$ is as in HL5, either:

1. (a)

$\left|f\left(x\right)\right|=O(\left|x\right|^{\lambda^{\prime}})$, as $\left|x\right|\rightarrow\infty$ and HL5(a) holds; or 2. (b)

$\left|f\left(x\right)\right|=O\left(\exp\left(\lambda^{\prime}\left|x\right|\right)\right)$, as $\left|x\right|\rightarrow\infty$ and HL5(b) holds.

Then as $n\rightarrow\infty$,

[TABLE]

Remark 3.2*.*

(a) If $f$ is bounded, conditions (i)–(iii) hold trivially, and HL5 is unnecessary for (12). If $f$ is additionally Lipschitz, then HL3 and HL6 may also be dispensed with.

(b) Condition (ii) of Theorem 3.1 is a technical requirement that has also been employed in other studies that develop limit theory for functionals of nonstationary processes, e.g. Jeganathan (2004) and Pötscher (2004). This condition is redundant if HL3 holds with $t_{0}=1$.

**(c) **Since $X_{t}(n)$ is continuously distributed under Assumption INN, (12) continues to hold if $f$ is modified on a set of Lebesgue measure zero. Thus e.g. if $f$ has an integrable pole at some $x_{0}\in\mathbb{R}$, and otherwise satisfies the requirements of Theorem 3.1, then (12) holds regardless of how $f$ is defined at $x_{0}$.

For $I(1/2)$ and MI processes, it may be shown that Assumption INN and each of FR and MI are sufficient for Assumption HL. Theorem 3.1 therefore specialises as follows. Recall that $\varphi_{\sigma^{2}}(x)$ denotes the $N[0,\sigma^{2}]$ density, and take $\beta_{n}^{2}=Var(x_{n}(n))$. For $X^{-}\sim N[0,1/2]$, let

[TABLE]

noting that this is a density, albeit a random one under FR1.

Theorem 3.2**.**

*Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is locally Lebesgue integrable, and Assumption INN and either **FR *or MI hold. Further suppose:

Either: (a) for each t^{\prime}\in\mathbb{N}\ fixed,$\ n^{-1}\sum_{t=1}^{t^{\prime}}f\left(\beta_{n}^{-1}x_{t}\left(n\right)\right)=o_{p}\left(1\right)$; or (b) Assumption INN holds with $\theta=1$. 2. 2.

For some $\lambda^{\prime}\in\left(0,\infty\right)$, as $\left|x\right|\rightarrow\infty$ either

1. (a)

$\lvert f(x)\rvert=O(\lvert x\rvert^{\lambda^{\prime}})$, and $\mathbf{E}\left|\xi_{1}\right|^{2\vee\lambda}<\infty$ for some $\lambda>\lambda^{\prime}$; or 2. (b)

$\lvert f(x)\rvert=O\left(\exp\left(\lambda^{\prime}\left|x\right|\right)\right)$, and $\xi_{1}$ has a finite m.g.f. in a neighbourhood of zero.

Then

[TABLE]

Remark 3.3*.*

(a) Theorem 3.2 bridges existing asymptotic results for $I(d)$ processes of order $d\in(-1/2,1/2)$ and $d\in(1/2,3/2)$. There are similarities and differences between (14) and the limit theory that applies in these two cases. Firstly, there is some analogy with the LLN results that hold when $d\in(-1/2,1/2)$. As in that case, the limit in (14) is determined by an expectation, but with the difference that the expectation in (14) is with respect to a limiting distribution, rather than the invariant distribution of a strictly stationary process. Secondly, the limiting density ($\varphi_{1}$ or $\varphi_{1/2}$) is obtained by an application of a CLT, and in this respect the limit theory is analogous to that for $d\in(1/2,3/2)$, which involves FCLTs. In that case, the weak limits of additive functionals are stochastic, being a functional of a limiting fBM. This nondegeneracy of the limit carries over to the $I(1/2)$ type I process (FR1), as evinced by the dependence of $\varrho$ on $X^{-}\sim N[0,1/2]$ in this case. (The type II process can be regarded as a truncated type I process, and the additional variability of the latter appears to make its behaviour closer to that of an $I(d)$ processes with $d>1/2$).

**(b) **Theorem 3.2 generalises the limit theory of Giraitis and Phillips (2006) and Phillips and Magdalinos (2007) for MI processes to general nonlinear functionals. Those two papers consider quadratic functions (i.e. $f(x)=x^{2}$) of MI processes driven by short memory linear processes errors. Using a direct approach they show that $(n\beta_{n}^{2})^{-1}\sum_{t=1}^{n}x_{t}\left(n\right)^{2}$ $\overset{p}{\rightarrow}1$. This result can be understood as a special case of Theorem 3.2; indeed in this case the r.h.s. of (14) is

[TABLE]

Theorem 3.2 also generalises a result due to Tanaka (1999, p. 555), who shows that $\frac{1}{n\ln n}\sum_{t=1}^{n}x_{t}^{2}=O_{p}(1)$ for an ARFIMA($1/2$) processes, in the course of deriving the asymptotics of the maximum likelihood estimator for $d$ over a parameter space that includes $d=1/2$.

(c) Assumption INN entails that $\xi_{t}$ has finite second moment, and so $\beta_{n}^{-1}x_{\lfloor nr\rfloor}(n)$ satisfies a CLT; the limiting distributions that appear in (14) are therefore Gaussian.555Some preliminary work of the authors’ shows that Theorem 3.2 can be extended to the case where $\xi_{t}$ is in the domain of attraction of an $\alpha$-stable law with parameter $\alpha\in(0,2)$, in which case other stable distributions will appear in the limit. We leave extensions of this kind for future work. Note that the process $r\mapsto\beta_{n}^{-1}x_{\lfloor nr\rfloor}(n)$ does not converge weakly (with respect to the uniform or Skorokhod topologies): as the example of an $I(1/2)$ type I process given in Remark 3.1(b) illustrates, this is impossible because the finite-dimensional distributions of $\beta_{n}^{-1}x_{\lfloor nr\rfloor}(n)$ converge to those of a nonseparable process. (Similar calculations show that if $x_{t}$ is $I(1/2)$ type II or MI, then the fidis converge to those of a Gaussian ‘white noise’ process $G$ for which $G(r)\sim N[0,1]$ is independent of $G(s)$ for all $r,s\in[0,1]$.) This accords with the results of Johansen and Nielsen (2012b), who show that for an $I(d)$ process with $d\in(1/2,3/2)$ of either type, $\{\xi_{t}\}$ must have moments of order greater than $(d-1/2)^{-1}$ if $\beta_{n}^{-1}x_{\lfloor nr\rfloor}(n)$ is to converge weakly to a fBM, a requirement that becomes progressively more demanding as $d$ approaches $1/2$.

3.2 Kernel functionals

We next consider kernel functionals of the form

[TABLE]

where $x\in\mathbb{R}$, $h_{n}$ is a bandwidth sequence, and $K$ is an integrable kernel function satisfying

Assumption \AlphAlph (kernel).

$K:\mathbb{R}\rightarrow\overline{\mathbb{R}}$* is such that $K$ and $K^{2}$ are Lebesgue integrable.*

Whereas in (9) the nonlinear transformation $f$ is applied to the standardised process $\beta_{n}^{-1}x_{t}(n)$, in (15) $K$ is applied to the unstandardised process $x_{t}(n)$. This leads to a different limit theory, which is partly reflected in the different normalisations of the sums in (9) and (15). A notable difference between (15) and the usual expression for a kernel density estimator is the appearance of $\beta_{n}$. The more variable that $x_{t}$ is, the less frequent are its visits to the support of $K[(\cdot-x)/h_{n}]$ – and since the variance $\beta_{n}$ of a WNP grows with $n$, these visits accumulate only at rate $nh_{n}/\beta_{n}$, a fact reflected exactly in the normalisation in (15).

Nonetheless, the results of the preceding section turn out to be highly relevant for the asymptotics of (15). To explain why this is the case, we return to the setting of Assumption HL above, which we now augment by the following additional smoothness conditions on the density of the increments of $X_{t}(n)$. To state these, let

[TABLE]

for $\eta\in(0,1)$.

Assumption \AlphAlph (continued).

1. HL7

Let $X_{0}(n):=0$ and $t>s\geq 0$. Conditionally on $\mathcal{F}_{s}$, $\beta_{t-s}^{-1}(X_{t}(n)-X_{s}(n))$ has density \mathcal{D}$${}_{t,s,n}(x) such that for some $n_{0},t_{0}\geq 1$

[TABLE] 2. HL8

For all $q_{0},q_{1}>0$

[TABLE] 3. HL9

For $t_{0}$ as in HL7:

1. (a)

$\lim_{\eta\rightarrow 0}\lim_{n\rightarrow\infty}\frac{\beta_{n}}{n}\sum_{t=\lfloor(1-\eta)n\rfloor}^{n}\beta_{t}^{-1}=0$; 2. (b)

$\lim_{\eta\rightarrow 0}\lim_{n\rightarrow\infty}\frac{\beta_{n}}{n}\sum_{t=t_{0}}^{\lfloor\eta n\rfloor}\beta_{t}^{-1}=0$; 3. (c)

$\lim\sup_{n\rightarrow\infty}\frac{\beta_{n}}{n}\sum_{t=t_{0}}^{n}\beta_{t}^{-1}<\infty$; 4. (d)

there exist $l_{0},l_{1}>0$ such that $\liminf_{n\rightarrow\infty}\beta_{n}^{-1}\inf_{(s,t)\in\Omega_{n}(\eta)}\beta_{t-s}\geq\eta^{l_{1}}/l_{0}$ for all $\eta\in(0,1)$; 5. (e)

$\sup_{n\geq 1}\sup_{1\leq t\leq n}\beta_{n}^{-1}\beta_{t}<\infty$.

HL7 and HL9 may be regarded as strengthened versions of HL3 and HL6, and are closely related to Assumption 2.3 in Wang and Phillips (2009a). Under these conditions, an $L_{1}$-approximation argument developed by those authors and Jeganathan (2004) yields that, for $t_{0}$ as in HL7,

[TABLE]

as $n\rightarrow\infty$ and then $\varepsilon\rightarrow 0$. The leading order term on the r.h.s. clearly has the same form as the l.h.s. of (12) and is thus is amenable to a direct application of Theorem 3.1, which entails

[TABLE]

We thus have the following counterpart of Theorem 3.1 for kernel functionals.

Theorem 3.3**.**

Suppose that, in addition to Assumptions K, HL0–2, HL4 and HL6–9, the following hold:

1. (i)

$\{h_{n}\}$* is a positive sequence with $\beta_{n}^{-1}h_{n}+\beta_{n}(nh_{n})^{-1}\rightarrow 0$; and* 2. (ii)

for each $x\in\mathbb{R}$ and $t_{0}$ as in HL7, $\frac{\beta_{n}}{nh_{n}}\sum_{t=1}^{t_{0}-1}K\left(\frac{X_{t}(n)-x}{h_{n}}\right)=o_{p}\left(1\right)$.

Then

[TABLE]

For $I(1/2)$ and MI processes, the preceding specialises as follows.

Theorem 3.4**.**

Suppose that, in addition to Assumption K:

1. (i)

$x_{t}(n)$* satisfies Assumption INN,*** **and FR or MI; 2. (ii)

$\{h_{n}\}$* satisfies condition (i) of Theorem 3.3; and* 3. (iii)

for each $x\in\mathbb{R}$ and $t^{\prime}\in\mathbb{N}$, $\frac{\beta_{n}}{nh_{n}}\sum_{t=1}^{t^{\prime}}K\left(\frac{x_{t}(n)-x}{h_{n}}\right)=o_{p}\left(1\right)$.

Then

[TABLE]

Remark 3.4*.*

(a) Theorem 3.4 fills a gap in existing asymptotic theory for kernel functionals of linear processes. A general theory for stationary linear processes, including $I(d)$ processes with $\left|d\right|<1/2$, is given in Wu and Mielniczuk (2002). Supposing that $\int_{\mathbb{R}}K=1$, under their conditions kernel functionals converge in probability to the invariant density of the stationary process. Jeganathan (2004, 2008) provides limit theorems for kernel functionals of $I(d)$ processes with $1/2<d<3/2$. In that case, kernel functionals converge to the local time of a fractional Brownian motion (or fractional stable motion if innovations are in the domain of attraction of a stable law) – so their limit is an occupation density rather than the invariant density of some stationary process. The limiting behaviour of kernel functionals of $I(1/2)$ processes is intermediate between these two cases. These converge to the density of a random variable, rather than to an occupation density, but the density corresponds to a limiting random variate, rather than the invariant density of a stationary process.

(b) Theorem 3.4 nests a similar result provided by Duffy (2020) for bounded kernel functionals of MI processes, which unlike Assumption K requires $K$ to be bounded and Lipschitz continuous.

4 Estimation and inference in regressions with WNPs

The preceding results are fundamental to the asymptotics of parametric and nonparametric least squares estimators, in models involving WNPs as regressors. In this section, we show that these estimators have either Gaussian or mixed Gaussian limit distributions, and in consequence their associated $t$ statistics are asymptotically standard Gaussian. These results are in turn used, in Section 5, to derive the asymptotic distribution of a proposed regression specification test statistic, and in particular to show that it is asymptotically pivotal, being unaffected by the persistence of the regressor process.

4.1 Parametric regression

Consider the ordinary least squares (OLS) estimator of $(\mu,\gamma)$ in the model

[TABLE]

given by $(\hat{\mu},\hat{\gamma}):=\text{argmin}_{(a,b)}\sum_{t=1}^{n}[y_{t}-a-bg(x_{t-1})]^{2}$, where $g$ is a known nonlinear transformation. Since the regressor is predetermined (i.e. $\mathcal{F}_{t-1}$-measurable) relative to the error $u_{t}$, (17) is an instance of a so-called ‘predictive’ or ‘reduced form’ regression model. If $x_{t}$ is stationary, the OLS estimator will be asymptotically normal; whereas if $x_{t}$ is strongly dependent, the OLS estimator has a non-standard limiting distribution, unless either $g$ is itself integrable, or $x_{t}$ and $u_{t}$ satisfy a very restrictive ‘long-run orthogonality’ condition (see e.g. Park and Phillips, 1999, 2001).

When $x_{t}$ is a WNP, the OLS estimator is either asymptotically normal or mixed normal, depending on the type of process. In either case, the $t$ statistic is asymptotically $N[0,1]$, due to self-normalisation. In this respect, the asymptotics are similar to those when $x_{t}$ is stationary; but since the variance of a WNP grows without bound, the analysis requires arguments more appropriate to nonstationary processes. In particular, the following property of $g$, first introduced by Park and Phillips (1999, 2001), plays a key role.

Definition \AlphAlph (asymptotically homogeneous function).

Let $\{x_{t}(n)\}$ denote a random array and $\beta_{n}^{2}=Var(x_{n}(n))$. $g:\mathbb{R}\rightarrow\overline{\mathbb{R}}$ is asymptotically homogeneous for $\{x_{t}(n)\}$, if for each $\lambda>0$ it admits the decomposition

[TABLE]

where $\kappa_{g}:\mathbb{R}_{+}^{\ast}\rightarrow\mathbb{R}_{+}^{\ast}$, $H_{g}:\mathbb{R}\rightarrow\overline{\mathbb{R}}$, $R_{g}:\mathbb{R}\times\mathbb{R}_{+}^{\ast}\rightarrow\overline{\mathbb{R}}$ and for $j=\left\{1,2\right\}$,

[TABLE]

AHFs encompass a wide range of commonly used regression functions, such as polynomial functions, cumulative distribution functions (with $H_{g}(u)=\mathbf{1}\{u>0\}$), and logarithmic functions (with $H_{g}(u)=1$); see Park and Phillips (1999, 2001) for some further examples. Such a condition as

[TABLE]

is sufficient, but not necessary, for (18) to hold. The relevance of AHFs for the OLS estimator can be seen most easily in the case where $\mu=0$ is known and imposed, so that the OLS estimator for $\gamma$ satisfies

[TABLE]

Upon standardisation, the denominator on the r.h.s. is directly amenable to an application of Theorem 3.2. Since the numerator is a sum of martingale differences, it can be handled via an appropriate martingale CLT (either Hall and Heyde, 1980; or Wang, 2014); here Theorem 3.2 is used to verify the stability condition pertaining to its conditional variance.

Reasoning along these lines yields our main result on parametric OLS esitmators. To state it, let $MN[0,\varsigma^{2}]$ denote a mixed normal distribution with mixing variate $\varsigma^{2}$ (i.e. which has characteristic function $u\mapsto\mathbf{E}e^{-\varsigma^{2}u^{2}/2}$), and $\sigma(\{\xi_{s},u_{s}\}_{s\leq t})$ denote the $\sigma$-field generated by $\{\xi_{s},u_{s}\}_{s\leq t}$.

Theorem 4.1**.**

Let $\{y_{t}\}_{t=1}^{n}$ be generated by (17), $\mathcal{F}_{t}:=\sigma(\{\xi_{s},u_{s}\}_{s\leq t})$, and suppose that:

1. (i)

$x_{t}(n)$* satisfies (5), Assumption INN and either FR or MI;* 2. (ii)

$\{u_{t},\mathcal{F}_{t}\}_{t\geq 1}$* is a martingale difference sequence such that $\mathbf{E}[u_{t}^{2}\mid\mathcal{F}_{t-1}]=\sigma_{u}^{2}$ a.s. for some constant $\sigma_{u}^{2}<\infty$;* 3. (iii)

$\sup_{1\leq t\leq n}\mathbf{E}[u_{t}^{2}\mathbf{1}\{\lvert u_{t}\rvert\geq A_{n}\}\mid\mathcal{F}_{t-1}]=o_{p}(1)$* for non-random $A_{n}\rightarrow\infty$;* 4. (iv)

$g(x)$* is AHF for $\{x_{t}(n)\}$, with limit homogeneous component $H_{g}$ that is not a.e. constant, and is such that $H_{g}^{2}$ satisfies the conditions of Theorem 3.2.*

Then

[TABLE]

Remark 4.1*.*

(a) For $I(1/2)$ type II and MI processes $\hat{\gamma}_{n}$ is asymptotically normal; for $I(1/2)$ type I processes it is mixed normal, because then $\varrho(x)=\varphi(x-X^{-})$ is random. In either case, the $t$ statistics for testing hypotheses about $\mu$ or $\gamma$ will be asymptotically standard normal, so that inferences may be drawn in the usual manner. This contrasts with the case where regressors are $I(d)$ for $d>1/2$, e.g. see Phillips (1995), Park and Phillips (1999, 2001), Robinson and Hualde (2003).

(b) In a linear regression model, i.e. $g(x)=x$, we have $H_{g}(u)=u$ and $\kappa_{g}(\lambda)=\lambda$, and it follows that the OLS estimator for $\gamma$ has convergence rate $\beta_{n}n^{1/2}$, which is faster than the $n^{1/2}$-convergence rate that obtains when the regressor is stationary. For $I(1/2)$ processes the gain in convergence rate is given by the slowly varying factor $L^{1/2}(n)$ (see Remark 2.1(b) above).

**(c) **Suppose instead that $y_{t}=\mu+\gamma g(x_{t})+u_{t}$, so that the regressor is no longer predetermined. In this case, the asymptotics of the OLS estimator are different from (19): for example, if it is known that $\mu=0$ and $g(x)=x$, (so that $\kappa_{g}(\beta_{n})=\beta_{n}$) we have

[TABLE]

In this case there is a severe reduction in the convergence rate, by a factor of $n^{1/2}/\beta_{n}$, due to the endogeneity of $x_{t}$. This result is comparable to Theorem 5.2 of Marinucci and Robinson (1998) which gives the asymptotics of the OLS estimator when $x_{t}$ is $I(d)$ with $d\in(1/2,1)$. We expect that in this setting such methods as narrowband LS (see e.g. Marinucci and Robinson, 1998; Robinson and Hualde 2003; Christensen and Nielsen 2006) or those that use lagged regressors as instruments will be more efficient. We leave the exploration of alternative estimation procedures for future work.

4.2 Nonparametric regression

We next consider the nonparametric estimation of $m$ in the predictive regression

[TABLE]

In particular, we consider the kernel regression (Nadaraya–Watson; NW) estimator

[TABLE]

and the local linear (LL) estimator

[TABLE]

where $K_{th}(x):=K[(x_{t-1}-x)/h_{n}]$. The following theorem is a direct consequence of Theorem 3.4 and certain martingale central limit theorems, and is complementary to the recent work of Wang and Phillips (2009a,b; 2012) who develop estimation and testing procedures in the context of nonparametric regression with NI and $I(d)$ processes with $d\in(1/2,3/2)$. Let

[TABLE]

and $\nu_{K,i}:=\int x^{i}K(x)dx$ for $i\in\mathbb{N}$.

Theorem 4.2**.**

Let $\{y_{t}\}_{t=1}^{n}$ be generated by (20) and suppose that:

1. (i)

conditions (i)–(iii) of Theorem 4.1 hold, $\nu_{K,0}=1$ and $\nu_{K,2}\neq\nu_{K,1}^{2}$. 2. (ii)

$x^{j}[K(x)+K^{2}(x)]$* are bounded and integrable for $j\in[0,3]$;* 3. (iii)

$h_{n}+\beta_{n}/nh_{n}\rightarrow 0$;

If $m$ has a bounded first derivative and $nh_{n}^{3}/\beta_{n}\rightarrow 0$, then

[TABLE]

Alternatively, if $m$ has a bounded second derivative and $nh_{n}^{5}/\beta_{n}\rightarrow 0$, then* ***

[TABLE]

Remark 4.2*.*

(a) Since $\beta_{n}\rightarrow\infty$, the convergence rate of both $\hat{m}$ and $\tilde{m}$ when $x_{t}$ is a WNP is slower than when $x_{t}$ is stationary. For $I(1/2)$ processes this convergence rate is reduced by the slowly varying factor $L(n)^{1/2}$.

(b) Let $\tilde{\sigma}_{u}^{2}$ denote a consistent estimator of $\sigma_{u}^{2}$, and consider the nonparametric $t$ statistic for the hypothesis $\mathcal{H}_{0}:m(x)=m_{0}(x)$ based on the local linear estimator, as given by

[TABLE]

It follows directly from Theorem 4.2 that $\tilde{t}(x;m_{0})\overset{d}{\rightarrow}N[0,1]$, and similarly when $(\tilde{m},Q_{11})$ is replaced by $(\hat{m},\int K^{2})$.

Thus in conjunction with the existing literature, Theorem 4.2 implies that kernel nonparametric $t$ statistics are asymptotically standard Gaussian across a wide range of regressor processes, including: stationary fractional (with $-1/2<d<1/2$; Wu and Mielniczuk, 2002), weakly nonstationary (fractional with $d=1/2$ or mildly integrated), nonstationary fractional ($1/2<d<3/2$) and (nearly) integrated processes (Wang and Phillips, 2009a,b, 2011, 2012). This is in marked contrast to parametric $t$ statistics, which when regressors are nonstationary have limiting distributions that are typically nonstandard and dependent on nuisance parameters relating to the persistence of the regressor, which cannot be consistently estimated – a fact that greatly complicates parametric inference in these models (for an overview of this problem and the relevant literature, see Phillips and Lee, 2013, pp. 251–254).

(c) Suppose that $x_{t-1}$ on the r.h.s. of (20) is replaced by $x_{t}$, so that the regressor is no longer predetermined. If $x_{t}$ is stationary, then the correlation between it and $u_{t}$ prevents $m$ from being consistently estimated. However, for the case where $x_{t}$ is NI, Wang and Phillips (2009b) show that the nonparametric regression estimator is consistent for $m$ and asymptotically mixed Gaussian, even when $x_{t}$ is correlated with $u_{t}$, and $u_{t}$ is serially dependent. In other words, for NI covariates the asymptotics of the nonparametric regression estimator are unaffected by whether $x_{t}$ or $x_{t-1}$ appears in (20). We conjecture that a similar results also holds for WNPs, but leave an examination of this for future work.

**(d) **Our smoothness assumptions on $m$ could be relaxed along the lines of Wang and Phillips (2009a,b) and Wang and Phillips (2011), for the NW and LL estimators respectively; we have refrained from doing so here to permit Theorem 4.2 to be more concisely stated.

5 Specification testing when a regressor has an unknown degree of persistence

In this section, we exploit the asymptotic normality of the nonparametric $t$ statistic to develop a specification test statistic for parametric regression models that has the same asymptotic distribution regardless of the extent of the persistence of the regressor, and indeed regardless of whether that persistence is modelled in terms of long memory (i.e. as $I(d)$ for some $d\in(-1/2,3/2)$) or in terms of an autoregressive root localised to unity (as in an MI or NI process). The proposed test can thus be validly conducted, in a straightforward manner, without requiring practitioners either to make an assumption on the persistence of the regressor, or to somehow estimate this and take account of it when carrying out the test.

The hypothesis to be tested is that the true regression function $m$ in (20) belongs to a certain parametric family, as e.g. postulated in (17). Formally, the null is

[TABLE]

where $g$ is a known function;** **the alternative is that no such $\mu$ and $\gamma$ exist. Tests of $\mathcal{H}_{0}$, in a setting with (possibly) nonstationary regressors, have also been considered by Gao, King, Lu and Tjøstheim (2009), Wang and Phillips (2012; hereafter ‘WP’), and Dong, Gao, Tjøstheim and Yin (2017). WP test a parametric fit in the presence of a NI regressor, while Dong et al (2017) test for a parametric fit in regressions with a $d=0$ and a $d=1$ covariate. The test statistic of WP closely resembles that of Gao et al (2009), who propose a studentised U-statistic formed of kernel-weighted OLS regression residuals.666Gao et al (2009) apply their statistic to the problem of testing the null of a random walk (of the form $x_{t}=x_{t-1}+\xi_{t}$), against a (possibly nonlinear) stationary alternative. The underlying idea is to test for a neglected nonlinear component in an autoregression, whose presence would make the process stationary. The specification test proposed in this paper could also be potentially used for this purpose, but we leave explorations in this direction for future work.

We propose to test $\mathcal{H}_{0}$ by comparing parametric OLS and kernel nonparametric estimates of $m$. The model specified in (23) can be estimated parametrically by OLS regression, and also nonparametrically at each $x$ as

[TABLE]

which under $\mathcal{H}_{0}$ has no asymptotic bias, even if $h$ remains fixed as $n\rightarrow\infty$.** **If $g(x)=x$, so that the null of linearity is being tested, $\tilde{m}_{g}(x)$ specialises to the local linear regression estimator; but in general $\tilde{m}_{g}$ should be chosen consistent with the model under test, so that it has no bias under the null. Provided that $g^{(1)}(x)\neq 0$, a slight modification of the proof of Theorem 4.2 shows that $\tilde{m}_{g}(x)$ has the same limiting distribution as displayed in (22), under $\mathcal{H}_{0}$. Letting $(\hat{\mu},\hat{\gamma})$ denote the OLS estimates of $(\mu,\gamma)$, we can therefore compare the fit provided by the parametric and local nonparametric estimates of the model via an ensemble of $t$ statistics of the form

[TABLE]

where $\tilde{\sigma}_{u}^{2}(x):=\left[\sum_{t=2}^{n}K_{th}(x)\right]^{-1}\sum_{t=2}^{n}\left[y_{t}-\hat{\mu}-\hat{\gamma}g(x_{t-1})\right]K_{th}(x)$. Under $\mathcal{H}_{0}$, both the parametric and nonparametric estimators converge to identical limits and so for each $x\in\mathbb{R}$,

[TABLE]

where the equality is due to the relatively faster convergence rate of the parametric estimator, and the distributional limit follows as per Remark 4.2(b). Under the alternative, only the nonparametric estimator is consistent for $m$, and thus $|\tilde{t}|\stackrel{{\scriptstyle p}}{{\rightarrow}}\infty$.

Our proposed specification test statistic is based on these $t$ statistics evaluated at a set of $p$ points $\mathcal{X}\subset\mathbb{R}$, constructed as per

[TABLE]

$\tilde{F}$ is related to the ‘non-predictability sum test’ developed by Kasparis, Andreou and Phillips (2015), who were concerned with testing the null that $x_{t-1}$ cannot predict $y_{t}$, which in the present framework can be expressed as $m(x)=\mu$ for all $x\in\mathbb{R}$. Relative to other specification tests available in the literature, the principal advantage of a test based on $\tilde{F}$ is that the limiting distribution of this statistic is invariant to the extent of persistence in the regressor, making valid inference in the presence of data with an unknown degree of persistence straightforward.777The use of kernel methods in specification testing does not in and of itself lead to conventional inference: for example, a test recently proposed by Dong et al (2017) is also based on nonparametric methods, but the limiting distribution of their test statistic, and therefore the critical values for their test, depends on precise assumptions as to the form and extent of the persistence of the regressor.

5.1 Asymptotics for WNPs

Our final result gives the limiting distribution of $\tilde{F}$ under $\mathcal{H}_{0}$, and under a sequence of local alternatives of the form

[TABLE]

for some $g_{1}$, where $r_{n}\rightarrow 0$. This formulation of the alternative is similar to that of Horowitz and Spokoiny (2001) and WP. We only provide explicit results for the boundary case where $\{x_{t}\}$ is a WNP, which is the main focus of the present work. However, as discussed in Section 5.2 below, analogous results may be derived for the stationary fractional ($-1/2<d<1/2$), nonstationary fractional ($1/2<d<3/2$) and nearly integrated cases, on the basis of the limit theory presented in Wu and Mielniczuk (2002) and Wang and Phillips (2009a,b).

For the purposes of the next result, assume $g_{1}(x)$ is either integrable or an asymptotically homogeneous function (AHF) of asymptotic order $\kappa_{g_{1}}$; this helps to characterise the limiting behaviour of the test statistic under $\mathcal{H}_{1}$. Define $\mu_{\ast}:=-[\intop H_{g^{2}}\varrho\intop H_{g_{1}}\varrho-\intop H_{g}\varrho\intop H_{g}H_{{}_{g_{1}}}\varrho]/[\intop H_{g^{2}}\varrho-(\intop H_{g}\varrho)^{2}]$, which is the distributional limit of $-[\kappa_{g_{1}}(\beta_{n})r_{n}]^{-1}(\hat{\mu}-\mu)$ when $g_{1}$ is AHF.

Theorem 5.1**.**

Suppose that $\mathcal{X}$ has $p$ elements and

1. (i)

conditions (i)-(iii) of Theorem 4.2 hold, $x^{j}[K(x)+K^{2}(x)]$ are bounded and integrable for $j\in[0,4]$, $\sup_{t}\mathbf{E}[u_{t}^{4}\mid\mathcal{F}_{t-1}]<\infty$ a.s., and $nh_{n}^{5}/\beta_{n}\rightarrow 0$; and 2. (ii)

$g$* is AHF for the array $\{x_{t}(n)\}$, with limit homogeneous component $H_{g}$ such that $\kappa_{g}(\beta_{n})\rightarrow\infty$ and $H_{g}^{2}$ satisfies the requirements of Theorem 3.2. Further, $g$ has bounded second derivative, and $g^{(1)}(x)\neq 0$ for each $x\in\mathcal{X}$.*

Then under $\mathcal{H}_{0}$,

[TABLE]

Suppose that in addition $p\geq 2$ and:

1. (iii)

$r_{n}\rightarrow 0$* and either*

1. (a)

$g_{1},$* $g\cdot g_{1}$ are bounded and integrable, $r_{n}^{-1}n^{-1/2}\beta_{n}\rightarrow 0$; or* 2. (b)

$g_{1}$* is AHF for the array $\{x_{t}(n)\}$, with limit homogeneous function $H_{g_{1}}$ such that $H_{g_{1}}H_{g}$ satisfies the requirements of Theorem 3.2, and $r_{n}^{-1}n^{-1/2}\kappa_{g_{1}}^{-1}(\beta_{n})\rightarrow 0$. Further, each of the following limits exist (allowing ‘convergence’ to $\infty$):*

[TABLE] 2. (iv)

$l_{n}nh_{n}/\beta_{n}\rightarrow\infty$* where*

[TABLE] 3. (v)

$g_{1}$* has bounded second derivative.* 4. (vi)

Either: (iii.a) holds and $g_{1}(x)\neq 0$ for some $x\in\mathcal{X}$; (iii.b) holds with $\kappa_{\ast}\in(0,\infty]$ and $g_{1}(x)\neq g_{1}(x^{\prime})$ for some $x,x^{\prime}\in\mathcal{X}$; or (iii.b) holds with $\kappa_{\ast}=\infty$ and $\mu_{\ast}\neq 0$ $a.s$.

Then under $\mathcal{H}_{1}$,

[TABLE]

Remark 5.1*.*

**(a) **The requirement $\kappa_{g}(\beta_{n})\rightarrow\infty$ is a technical condition that is satisfied in most specifications employed in empirical work, e.g. linear models. It can be relaxed at the cost of a more involved exposition.

(b) If (iii.b) holds with $\kappa_{\ast}=\infty$, then $\mu_{\ast}\neq 0$ a.s. is sufficient for the test to be consistent, in the sense that (27) holds. Consistency may still obtain when $\mu_{\ast}=0$, but that requires a more detailed analysis than we are able to provide here.

(c) The sequence $l_{n}nh_{n}/\beta_{n}$ gives the divergence rate of the test statistic under $\mathcal{H}_{1}$, which is closely related the power of the test. The maximal divergence rate (i.e. $nh_{n}/\beta_{n}$) is attained when $g_{1}$ is AHF of diverging asymptotic order ($\kappa_{g_{1}}(\beta_{n})\rightarrow\infty$) and $\kappa_{g_{1}}(\beta_{n})r_{n}\rightarrow\infty$. In such cases, the divergence rate is otherwise unaffected by $r_{n}$. When $\kappa_{g_{1}}(\beta_{n})\rightarrow\infty$ but $\kappa_{g_{1}}(\beta_{n})r_{n}\rightarrow 0$, the divergence rate reduces to $\kappa_{g_{1}}^{2}(\beta_{n})r_{n}^{2}nh_{n}/\beta_{n}$. The divergence rate is smallest (i.e. $r_{n}^{2}nh_{n}/\beta_{n}$) in cases where $g_{1}$ is integrable or AHF of vanishing asymptotic order (i.e. $\kappa_{g_{1}}(\beta_{n})\rightarrow 0$).

(d) We have assumed that the set $\mathcal{X}$ comprises a fixed number ($p$) of points. We can get some idea of the large-sample distribution of $\tilde{F}$ if $p$ is allowed to grow with the sample size from the fact that $(2p)^{-1/2}(\tilde{F}-p)\stackrel{{\scriptstyle d}}{{\rightarrow}}(2p)^{-1/2}(\chi_{p}^{2}-p)\stackrel{{\scriptstyle d}}{{\rightarrow}}N(0,1)$ as $n\rightarrow\infty$ and then $p\rightarrow\infty$.

5.2 Asymptotics in other cases

$\tilde{F}$ will have the same limiting distribution as given by (26) of Theorem 5.1, even when $x_{t}$ is a stationary, nonstationary fractional, or near-integrated process. This can be established using results available in the existing literature, by arguments outlined in the remainder of this section, and is confirmed by the simulation exercises presented in Section 6 below. (These also permit our results on the consistency of the test against local alternatives to be extended beyond weakly nonstationary processes.)

From the proof of Theorem 5.1, it is clear that (26) holds if

1. (i)

the OLS estimator for $(\mu,\gamma)$ converges faster than the nonparametric estimator for $m$ (e.g. NW/LL); 2. (ii)

the nonparametric estimator is asymptotically (mixed) Gaussian; and 3. (iii)

for every $\mathit{x},x^{\prime}\in\mathcal{X}$, such that $x\neq x^{\prime}$,

[TABLE]

for all $j,j^{\prime}\in\{0,1,2\}$, where $K_{j}(x):=x^{j}K(x)$.

This last requirement ensures that the component $t$ statistics in (24) are asymptotically independent of each other.

When $x_{t}$ is nonstationary fractional ($1/2<d<3/2$) or nearly integrated, the arguments of Park and Phillips (2001) (see also Christopeit, 2009 and the references therein) show that the convergence rates of $(\hat{\mu},\hat{\gamma})$ are as in (19); that the convergence rates of the kernel nonparametric regression estimators are slower follows from Wang and Phillips (2009a,b). Conditions (ii) and (iii) also follow from Wang and Phillips (2009a, and particularly pp. 1910–11 of 2009b for (28)).

When $x_{t}$ is stationary, it is well known that the OLS estimator is $n^{-1/2}$-consistent while the $(nh_{n})^{-1/2}$-consistency and asymptotic normality of the nonparametric estimators follows e.g. from the results of Wu and Mielniczuk (2002). Thus conditions (i)–(ii) hold. Finally, if $x_{t}$ has a bounded density $\mathcal{D}_{x}(u)$ (see Wu and Mielniczuk, 2002, Lemma 1) and under standard regularity conditions on $K$, we have

[TABLE]

so that condition (iii) holds, since $\beta_{n}\asymp 1$ in this case.

6 Simulations

We conducted simulations to evaluate the finite-sample performance of the proposed specification test, in terms of size and power against a range of alternatives. For this exercise, a natural comparison is with the specification test of WP, which is known to have a standard Gaussian limiting distribution when $x_{t}$ is NI. (In our simulation exercises, we assume that this also holds when $x_{t}$ is fractionally integrated and/or stationary, and so compare their statistic to normal critical values in these cases.)

For all simulation exercises, the null hypothesis is $\mathcal{H}_{0}:m(x)=\mu+\beta x$, so that the proposed specification test can be implemented by comparing the fit of a local linear regression with an OLS regression. The data generating process is

[TABLE]

with $x_{0}=0$, where $(u_{t},\xi_{t})$ are i.i.d. bivariate Gaussian with unit variances and correlation $\rho$. For each value of $d\in\{0.25,0.50,0.75,1.00\}$, we evaluate the size of the test by computing the maximum rejection frequency under the null (i.e. when $g_{1}(x)=0$) for $\rho\in\{-0.5,0.0,+0.5\}$ (with 5000 replications). We consider sample sizes $n\in\{100,200,500\}$, bandwidths of the form $h=n^{b}$ for $b\in\{-0.2,-0.1,-0.05\}$, and use the Gaussian kernel in all cases. For our test, which requires the choice of points at which to compare the nonparametric and parametric estimates of the regression function, we consider two choices: $p=17$ or $25$ points, evaluated at the quantiles of $\{x_{t}\}_{t=1}^{n}$ equally spaced between the $0.1$ and $0.9$ quantiles.

The results are displayed in Table 1, for a test having 10 per cent nominal significance level. They clearly illustrate that our test has good size control across the range of bandwidths considered, both when the data is in the stationary and nonstationary regions, and on the boundary between these, suggesting that the asymptotics developed in Sections 3–5 provide a good approximation to the finite-sample distribution of the statistic.

We also computed the size-adjusted power of our test, and of WP’s, against alternatives that are either integrable** **($g_{1}(x)=\varphi_{1}(x)$, $\varphi_{1}(2x)$, or $x^{-2}\wedge 1$), non-integrable but vanishing at infinity ($g_{1}(x)=x^{-1}\wedge 1$), or polynomials ($g_{1}(x)=|x|^{v}$ with $v\in\{1.5,2.0\}$).888By size-adjusted power, we mean that if the test is found to reject at rate $\hat{\alpha}>0.1$ under the null (as reported in the top panel), then the power of the test is adjusted downwards by subtracting $\hat{\alpha}-0.1$ from the rejection rate under each alternative (so if $\hat{\alpha}\leq 0.1$, no adjustment is made). The simulation designs are the same as for the size calculations, except that we here only report results for $\rho=0$ and $d\in\{0.5,1.0\}$ (results for $d\in\{0.25,0.75\}$ and MI processes are broadly similar, and are provided in Appendix F in the Supplementary Material). The alternatives are scaled by the factors indicated in Table 2 so as to ensure non-trivial power for these designs. To facilitate the comparison between the power of our test and that of WP, we report the size-adjusted power of their procedure in the first three columns of Table 2, and the relative size-adjusted power of our test alongside, i.e. the difference between the power of our test and of theirs.

It is noticeable that our test generally outperforms WP’s: indeed, the relevant entries of the table are almost uniformly positive, with the exception of a few cases where $g_{1}(x)=|x|^{1.5}$ and $d=1$. The most pronounced power improvements are for those cases where $d$ is smaller, and the alternatives are either integrable or asymptotically vanishing (i.e. tranformations that exhibit weaker signal and are therefore harder to detect); whereas the performance of the two tests is less easily distinguishable for polynomially growing alternatives, when $d=1$.

Appendix A Proofs under high-level conditions

This appendix provides proofs of our main results under high level conditions (Assumption** HL**), i.e. Theorems 3.1 and 3.3. We begin by stating some auxiliary technical lemmas, whose proofs appear in Appendix A.3.

A.1 Technical lemmas

Lemma A.1**.**

Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is locally integrable, and let $\varepsilon,\eta>0$. Then there is a Lipschitz continuous $f_{\varepsilon,\eta}$ such that $\int_{\left|x\right|\leq\eta}\left|f(x)-f_{\varepsilon,\eta}(x)\right|dx<\varepsilon$ and $f_{\varepsilon,\eta}(x)=0$ for $\left|x\right|>\eta$.

Lemma A.2**.**

Let $\{X_{n}\}$ and $\{Y_{n}\}$ be real valued random sequences on some probability space $(\Omega,\mathcal{A},\mathbf{P})$ and $\mathcal{F\subset A}$ a $\sigma$-field, for which

1. (i)

$X_{n}\overset{d}{\rightarrow}X\sim F_{X}$, conditionally on $\mathcal{F}$, in the sense that $\mathbf{E}\left(h(X_{n})\mid\mathcal{F}\right)\overset{p}{\rightarrow}\int_{\mathbb{R}}h(x)dF_{X}(x)$ for all $h:\mathbb{R}\rightarrow\mathbb{R}$ bounded and continuous; and 2. (ii)

$Y_{n}\overset{d}{\rightarrow}Y$, where $Y_{n}$ is $\mathcal{F}$-measurable for each $n$.

Then for all $g:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ bounded and Lipschitz continuous,

[TABLE]

Lemma A.3**.**

Suppose that:

1. (i)

HL0* and*** HL7–9*** hold;* 2. (ii)

$K$* satisfies Assumption*** K***; and* 3. (iii)

$\left\{h_{n}\right\}$* is such that $\beta_{n}^{-1}h_{n}+\beta_{n}(nh_{n})^{-1}\rightarrow 0$.*

Then for all $x\in\mathbb{R}$, and $t_{0}$ as in HL7,

[TABLE]

A.2 Proofs of Theorems 3.1 and 3.3

Proof of Theorem 3.1.

By Lemma A.1, for each $\varepsilon>0$ there is a Lipschitz continuous $f_{\varepsilon}\left(x\right)$ such that $\int_{\left|x\right|\leq\varepsilon^{-1}}\left|f\left(x\right)-f_{\varepsilon}\left(x\right)\right|dx<\varepsilon$ and $f_{\varepsilon}\left(x\right)=0$ for $\left|x\right|>\varepsilon^{-1}$. We shall prove that:

[TABLE]

as $n\rightarrow\infty$ then $\varepsilon\rightarrow 0$; that for each $\varepsilon>0$

[TABLE]

as $n\rightarrow\infty$; and that

[TABLE]

as $\varepsilon\rightarrow 0$, where the integral on the r.h.s. exists a.s. by condition (i) of Theorem 3.1. In view of (S.1)-(S.3), (12) then follows from Theorem 4.2 in Billingsley (1968).

Proof of (S.1). We prove (S.1)** **under condition (iii.a) of Theorem 3.1. The proof under (iii.b) is identical and therefore omitted. By condition (iii.a) we may choose $\varepsilon>0$ sufficiently small that $\left|f\left(x\right)\right|\leq C\left|x\right|^{\lambda^{\prime}}$ for all $\left|x\right|\geq\varepsilon^{-1}$. Decompose

[TABLE]

where $\int_{\left|x\right|\leq\varepsilon^{-1}}\left|f_{1,\varepsilon}\left(x\right)-f_{\varepsilon}\left(x\right)\right|dx<\varepsilon$, and $\left|f_{2,\varepsilon}\left(x\right)\right|\leq C\left|x\right|^{\lambda^{\prime}}1\{\lvert x\rvert>\mathbb{\varepsilon}^{-1}\}$. Letting $\tilde{X}_{t}\left(n\right):=\beta_{n}^{-1}X_{t}\left(n\right)$, in view of condition (ii) of Theorem 3.1, (S.1) will follow once we have shown that

[TABLE]

as $n\rightarrow\infty$ and then $\varepsilon\rightarrow 0$, for $t_{0}$ as in condition (ii).

To that end, note that by HL5(a) and condition (iii.a) of Theorem 3.1, there is an $n_{0}\geq 1$ and a $\lambda>\lambda^{\prime}$ such that $\sup_{n\geq n_{0},1\leq t\leq n}\mathbf{E}\lvert\tilde{X}_{t}(n)\rvert^{\lambda}<\infty$. Hence $\lvert\tilde{X}_{t}(n)\rvert^{\lambda^{\prime}}$ is uniformly integrable, and so for $n\geq n_{0}$

[TABLE]

as $\varepsilon\rightarrow 0$. This gives the required negligibility of the second l.h.s. term in (S.4). For the first l.h.s. term, we note HL3 implies that for $n$ sufficiently large

[TABLE]

as $\mathbb{\varepsilon}\rightarrow 0$, where the second inequality holds by HL6.

Proof of (S.2). Let $\mathbf{E}_{0}(\cdot):=\mathbf{E}\left(\cdot\mid\mathcal{F}_{0}\right)$, and $\delta\in(0,1)$. By HL4 and the boundedness of $f_{\varepsilon}$,

[TABLE]

as $n\rightarrow\infty$ and then $\delta\rightarrow 0$. Now let $\tilde{X}_{t}^{+}(n):=\beta_{n}^{-1}X_{t}^{+}(n)$ and $\tilde{X}_{t}^{-}(n):=\beta_{n}^{-1}X_{t}^{-}(n)$. Since $f_{\mathbb{\varepsilon}}$ is bounded and Lipschitz, it follows from HL1 that

[TABLE]

as $n\rightarrow\infty$ for each $\delta\in(0,1)$; and from HL2(b) that

[TABLE]

as $n\rightarrow\infty$ for each $\delta\in(0,1)$, where $l_{n}\in\{\lfloor n\delta\rfloor+1,\ldots,n\}$ may always be chosen such that the final equality holds. Finally, by HL2(a), Theorem 2.1 in Billingsley (1968) and Lemma A.2 we have as

[TABLE]

as $n\rightarrow\infty$, for each $\delta\in(0,1)$. Hence (S.2) follows from (S.5)-(S.8) and Theorem 4.2 in Billingsley (1968).

Proof of (S.3). Let $y\in\mathcal{Y}$ for $\mathcal{Y}$ as in condition (i) of Theorem 3.1. Noting $f_{\mathbb{\varepsilon}}(x)=0$ for $\lvert x\rvert>\mathbb{\varepsilon}$, and that $\Phi_{X^{+}}$ is bounded under **HL2, **we have

[TABLE]

as $\mathbb{\varepsilon}\rightarrow 0$, where the negligibility of the second r.h.s. term follows by condition (i) of Theorem 3.1 and the dominated convergence theorem. Noting that $\mathbf{P}\{X^{-}\in\mathcal{Y}\}=1$ completes the proof. ∎

Proof of Theorem 3.3.

By Lemma A.3 and condition (ii) of the theorem,

[TABLE]

as $n\rightarrow\infty$ and then $\varepsilon\rightarrow 0$. By Theorem 3.1 we have

[TABLE]

as $n\rightarrow\infty$. Finally, as noted in (16),

[TABLE]

as $\varepsilon\rightarrow 0$ by the continuity of $\Phi_{X^{+}}$ under HL2. The result then follows by Theorem 4.2 in Billingsley (1968). ∎

A.3 Proofs of Lemmas A.1–A.3

Proof of Lemma A.1.

See Theorem 2.26 in Folland (1999). ∎

Proof of Lemma A.2.

By Theorems 6.3 and 6.4 in Kallenberg (2001), there is a probability kernel $\nu_{n}$ from $\left(\Omega,\mathcal{F}\right)$ to $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$ such that for each $\mathcal{F}$-measurable random variable $\eta$,

[TABLE]

For each $y\in\mathbb{R}$, define $h_{n}(y):=\int_{\mathbb{R}}g(x,y)d\nu_{n}(x)$: so by the preceding and condition (i) of the lemma, we have

[TABLE]

Moreover, by the Lipschitz continuity of $g$

[TABLE]

Hence, $\left\{h_{n}\left(y\right)\right\}$ is stochastically equicontinuous on $\mathbb{R}$, whence $h_{n}\left(y\right)\overset{p}{\rightarrow}h\left(y\right)$ uniformly on every compact subset of $\mathbb{R}$ (see for example Theorem 1 and Lemma 1 in Andrews, 1992). Finally, fix $\varepsilon>0$ and choose $M_{\varepsilon}$ such that $\lim\sup_{n\rightarrow\infty}\mathbf{P}\left(\left|Y_{n}\right|>M_{\varepsilon}\right)<\varepsilon$, which is possible since $Y_{n}\overset{d}{\rightarrow}Y$. Then

[TABLE]

as $n\rightarrow\infty$, by the uniform convergence in probability of $h_{n}$ on compacta. In view of the preceding,

[TABLE]

where $\overset{d}{\rightarrow}_{(1)}$ is due to condition (ii) and the continuity of $h(y)$. ∎

Proof of Lemma A.3.

The result follows from Lemma 7 in Jeganathan (2004) and arguments similar to those used in Wang and Phillips (2009, pp. 725-728). Note, in particular, that if we define $x_{k,n}:=\beta_{n}^{-1}X_{k}(n)$ and $d_{l,k,n}:=\beta_{n}^{-1}\beta_{l-k}$, then HL7–9 ensure that their Assumption 2.3 is satisfied, with the exception that the density of $x_{k,n}$ is bounded only for $k\geq t_{0}$. If $t_{0}=1$, the result follows immediately. Otherwise, it follows via very minor modifications of the arguments leading to (5.2) in Wang and Phillips (2009), making use in particular of the additional condition imposed on $\{\beta_{n}\}$ by HL9(e). ∎

Appendix B Proofs under low-level conditions

This appendix provides proofs for the remaining results of Section 3, i.e. Theorems 3.2 and 3.4. Technical lemmas stated in this appendix are proved in Appendix C.

B.1 Sufficient conditions for Assumption HL

Preliminary to the proof of Theorem 3.2, we first present a set of ‘intermediate-level conditions’ for general linear processes (Assumption** LP** below); their sufficiency for Assumptions** HL0–HL6** is established by Proposition B.1 below. These conditions are of interest in their own right, insofar as they allow the conclusions of Theorem 3.1 to be extended to a broad class of linear processes. They also allow the proof of Theorem 3.2 to be reduced essentially to verifying that $I(1/2)$ and MI processes satisfy Assumption LP (see Appendix B.3).

Our conditions on linear processes shall be stated in terms of arrays of the form

[TABLE]

where $\{\xi_{t}\}$ is the i.i.d. sequence appearing in Assumption INN. We showed in Section 2 that it was possible to write both $I(1/2)$ and MI processes in the form

[TABLE]

for appropriate coefficients $\{c_{i}\}$ and $\{\phi_{j}(n)\}$. Suppressing the dependence of these quantities on $n$ for the sake a readability, as we shall do freely below, (S.10) implies that

[TABLE]

Note, in particular, that $a_{k,t}$ does not depend on $t$ for $1\leq k\leq t-1$, and we accordingly denote these coefficients by simply $a_{k}$. For $k\geq t$, the notation $a_{k,t}^{-}$ reminds us that these coefficients refer to innovations dated $t\leq 0$.

The following conditions on linear processes do not require these to have been generated according to a specific time series model (e.g.** Assumption FR/MI**), or indeed as in (S.10). We shall, however, impose one restriction consistent with that model: that the coefficients $a_{k,t}$ should not depend on $t$ for $1\leq k\leq t-1$. Our conditions thus envisage an array of the form

[TABLE]

associated to which, define

[TABLE]

and set $\beta_{n}:=\beta_{n,n}$, for $n\in\mathbb{N}$ and $t\in\{1,\ldots,n\}$. The following will always be applied in conjunction with Assumption** INN**, and are stated in terms of the $\theta$ and the i.i.d. sequence $\{\xi_{t}\}$ appearing in that assumption.

Assumption \AlphAlph (linear process).

1. LP1

$x_{t}(n)$* is as in (S.12), with $\beta_{n,t}\in(0,\infty)$ for all $n,t\in\mathbb{N}$.* 2. LP2

Either:

1. (a)

$\mathbf{E}\lvert\xi_{1}\rvert^{\lambda}<\infty$* for some $\lambda\in[2,\infty)$; or* 2. (b)

$\xi_{1}$* has a finite moment generating function (m.g.f.) in a neighbourhood of zero.* 3. LP3

There exists $t_{0}\in\mathbb{N}$ such that

1. (a)

$\liminf_{n\rightarrow\infty}\inf_{t_{0}\leq t\leq n}\beta_{t}^{-1}\beta_{n,t}^{+}>0$; 2. (b)

$\limsup_{t\rightarrow\infty}\sup_{n\geq t}(\beta_{n,t}^{+})^{-1}\max_{0\leq k\leq t-1}\lvert a_{k}(n)\rvert=0$** 4. LP4

There exist $\delta>0$ and $n_{0}\in\mathbb{N}$ such that: for each $1\leq t\leq t_{0}$ (as in LP3) and $n\geq n_{0}$, there exist $\{k_{1},\ldots k_{\theta}\}\subset\mathbb{N}$ with

[TABLE] 5. LP5

$\lim_{n\rightarrow\infty}n^{-1/2}\beta_{n}^{-1}\sum_{k=0}^{n-1}\lvert a_{k}(n)\rvert=0$. 6. LP6

For any $\delta\in(0,1)$ and $\{t_{n}\}$ such that $\lfloor n\delta\rfloor\leq t_{n}\leq n$:

1. (a)

$\beta_{n}^{-1}[\beta_{n,t_{n}}^{+},\beta_{n,t_{n}}^{-}]\rightarrow[\sigma_{+},\sigma_{-}]$* with $\sigma_{+}>0$ and $\sigma_{-}\geq 0$;* 2. (b)

$\beta_{n}^{-1}(\max_{0\leq k\leq n-1}\lvert a_{k}(n)\rvert+\sup_{1\leq t\leq n,k\geq t}\lvert a_{k,t}^{-}(n)\rvert)\rightarrow 0$; 3. (c)

$\beta_{n}^{-2}\sum_{l=-\infty}^{0}[a_{t_{n}-l,t_{n}}^{-}(n)-a_{n-l,n}^{-}(n)]^{2}\rightarrow 0$. 7. LP7

$\sup_{n\geq n_{0},1\leq t\leq n}\beta_{n}^{-1}\beta_{n,t}<\infty$* for some $n_{0}\in\mathbb{N}$.*

Remark B.1*.*

The principal relationships between the preceding conditions, and the high-level conditions (Assumption** HL**) as they would be applied to $X_{t}(n)=x_{t}(n)$ may be summarised as follows; these are formally established by Proposition B.1 below.

(a) LP2 and LP7 imply that $\beta_{n}^{-1}x_{t}(n)$ has uniformly bounded moments of a sufficient order (as per HL5). For $\beta_{n}^{-1}x_{t}(n)$ to have finite $\lambda$-moments, it is sufficient that $\xi_{t}$ also have finite $\lambda$-moments; $\beta_{n}^{-1}x_{t}(n)$ will have finite exponential moments if $\xi_{t}$ has a finite m.g.f. in a neighbourhood of zero.

(b) LP3 and LP4 ensure that $\beta_{t}^{-1}x_{t}(n)$ has a uniformly bounded density, as required by HL3. Under INN, a weighted sum involving at least $\theta$ of the innovations $\xi_{t}$ will have an integrable characteristic function (c.f.). $\beta_{t}^{-1}x_{t}^{+}(n)=\sum_{k=0}^{t-1}\beta_{t}^{-1}a_{k}(n)\xi_{t-k}$ will thus have a density bounded uniformly over $n$ and $t$, provided that the $L_{1}$ norm of its c.f. can be uniformly bounded. This in turn requires that: (i) the variance of $\beta_{t}^{-1}x_{t}^{+}(n)$ can be bounded away from zero; and (ii) it is never dominated by less than $\theta$ of the innovations that contribute to it. Both are ensured by LP3, at least for $n$ and $t$ sufficiently large. LP4 entails that for every $t$, a sufficient number of coefficients $\{a_{k,t}(n)\}$ are bounded away from zero; together with LP3 it is sufficient for HL3 to hold with $t_{0}=1$.

(c) LP5 can be understood as a kind of weak dependence condition, which is used solely to verify HL4.

(d) **LP6 **permits a central limit theorem for weighted sums to be applied to each of $\beta_{n}^{-1}x_{t}^{+}(n)$ and $\beta_{n}^{-1}x_{t}^{-}(n)$, as required by HL2. LP6(a) determines the limiting variance of each of these two terms, while LP6(b) is a negligibility requirement on the linear process coefficients, akin to a Lindeberg condition. Finally, LP6(c) implies the second part of HL2(b).

Proposition B.1** (LP $\Rightarrow$ HL).**

Suppose Assumptions* LP1** and INN hold. Then HL0–1 hold for $\mathcal{F}_{t}:=\sigma(\{\xi_{r}\}_{r\leq t})$, $X_{t}^{+}(n):=x_{t}^{+}(n)$, $X_{t}^{-}(n):=x_{t}^{-}(n)$, and $\beta_{n}^{2}=Var[x_{n}(n)]$. Moreover:*

1. (i)

LP6* $\Rightarrow$* HL2*** with $(X^{+},X^{-})\sim N[0,\mathrm{diag}\{\sigma_{+}^{2},\sigma_{-}^{2}\}]$.* 2. (ii)

1. (a)

LP3 $\Rightarrow$* HL3;* 2. (b)

**LP3 **and LP4 $\Rightarrow$ HL3 with $t_{0}=1$. 3. (iii)

LP5* $\Rightarrow$* HL4***.* 4. (iv)

1. (a)

**LP2(a) ***and LP7 $\Rightarrow$ ***HL5(a) **with $\lambda\geq 2$; 2. (b)

LP2(b),** LP6(b)*** and LP7 $\Rightarrow$ HL5(b).*

The proof of Proposition B.1 (and subsequently, Theorem 3.4) requires the following technical lemma, whose proof is deferred to Appendix C.2. Let $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}.$

Lemma B.1**.**

Let $\{\xi_{t}\}$ be as in Assumption INN with $\sigma_{\xi}^{2}=1$, $\{\vartheta_{k}\}_{k\in\mathbb{N}_{0}}$ be a real sequence, and $\delta>0$.

1. (i)

Suppose $\sigma_{\vartheta}^{2}:=\sum_{k=0}^{\infty}\vartheta_{k}^{2}>0$ and $\max_{k\in\mathbb{N}_{0}}\vartheta_{k}^{2}\leq\sigma_{\vartheta}^{2}/2\theta$. Then there exists a function $G(A;\sigma^{2},\psi_{\xi})$, not otherwise depending on $\{\vartheta_{k}\}$, such that $\sigma^{2}\mapsto G(A;\sigma^{2},\psi_{\xi})$ is weakly decreasing in $\sigma^{2}$,

[TABLE]

where $C<\infty$ depends only on $\psi_{\xi}$, and $\lim_{A\rightarrow\infty}G(A;\sigma_{\vartheta}^{2},\psi_{\xi})=0$. 2. (ii)

Suppose there exist $\{k_{1},\ldots,k_{\theta}\}\subset\mathbb{N}_{0}$ such that $\lvert\vartheta_{k_{i}}\rvert>\delta$ for all $i\in\{1,\ldots,\theta\}$. Then

[TABLE]

where $C<\infty$ depends only on $\psi_{\xi}$.

Proof of Proposition B.1.

That HL0–1 are satisfied (with $R_{t}(n)=0$) is immediate from (S.12), and the fact that $\beta_{n}=\beta_{n,n}\in(0,\infty)$ by LP1.

(i). Since $\{\xi_{t}\}$ is i.i.d., in view of (S.12) we have

[TABLE]

where $\left\{\xi_{k}^{\ast}\right\}\overset{d}{=}\left\{\xi_{k}\right\}$ and $\left\{\xi_{k}^{\ast}\right\}\perp\left\{\xi_{k}\right\}$. Using LP6(a)–(b), the weak convergence of these quantities to a $N[0,\mathrm{diag}\{\sigma_{+}^{2},\sigma_{-}^{2}\}]$ distribution, as required by HL2, follows immediately from Lemma 2.1 of Abadir, Distaso, Giraitis and Koul (2014). Finally, note that due to LP6(c)

[TABLE]

which establishes the final part of HL2(b).

**(ii). **By (S.12) and the Fourier inversion theorem (e.g. Feller, 1971, Theorem XV.3), it suffices for HL3 to show that the c.f. of $\beta_{t}^{-1}x_{t}^{+}(n)$ has an $L^{1}$ norm that is bounded uniformly for all $n$ and $t\leq n$ sufficiently large. Since $\beta_{t}^{-1}x_{t}^{+}(n)$ is a linear process with coefficients $\vartheta_{k,n,t}:=\beta_{t}^{-1}a_{k}(n)\boldsymbol{1}\{0\leq k\leq t-1\}$ and variance $\sigma_{\vartheta,n,t}^{2}:=\beta_{t}^{-2}(\beta_{n,t}^{+})^{2}$, we can do this with the aid of Lemma B.1. LP3(b) implies that there exists $n_{0},t_{0}\in\mathbb{N}$ such that

[TABLE]

for all $k\in\mathbb{N}_{0}$, $t_{0}\leq t\leq n$ and $n\geq n_{0}$, while LP3(a) implies that $n_{0}$ and $t_{0}$ may be additionally chosen such that

[TABLE]

for some $\epsilon>0$, for all $t_{0}\leq t\leq n$ and $n\geq n_{0}$. By Lemma B.1(i), the $L^{1}$ norm of $\mathbf{E}[\mathrm{i}\lambda\beta_{t}^{-1}x_{t}^{+}(n)]$ is bounded by $C\epsilon^{-1}<\infty$ for such $n$ and $t$, and thus HL3 holds for some $t_{0}\in\mathbb{N}$.

HL3 will hold with $t_{0}=1$ if we can additionally bound the $L^{1}$ norm of $\mathbf{E}[\mathrm{i}\lambda\beta_{t}^{-1}x_{t}(n)]$ (note the deliberate omission of the ‘$+$’ superscript) for $1\leq t\leq t_{0}$ and all $n\geq n_{0}$. To this end, note that $\beta_{t}^{-1}x_{t}(n)$ is a linear process with coefficients $\vartheta_{k,n,t}:=\beta_{t}^{-1}a_{k,t}(n)\boldsymbol{1}\{k\geq 0\}$, and that for $\delta>0$ as in LP4, for each $1\leq t\leq t_{0}$ and $n\in\mathbb{N}$ there exist $\{k_{1},\ldots,k_{\theta}\}\in\mathbb{N}_{0}$ such that

[TABLE]

where the r.h.s. is nonzero and independent of $n$ and $t$. Applying Lemma B.1(ii) completes the proof.

(iii). The argument is similar to that of Wu and Mielniczuk (2002, p. 1452). Let $\mathbf{E}_{t}[\cdot]:=\mathbf{E}[\cdot\mid\mathcal{F}_{t}]$, and decompose

[TABLE]

so that

[TABLE]

where each $M_{n,s}$ is a sum of martingale differences.

By (S.9) we can write

[TABLE]

where $\xi^{\ast}\overset{d}{=}\xi_{0}$ is independent of $\{\xi_{t}\}$. Since $g$ is Lipschitz,

[TABLE]

whence, by the orthogonality of martingale differences

[TABLE]

where the final inequality follows since $a_{s,t}=a_{s}$ for $0\leq s\leq t-1$. Deduce from (S.14) and the preceding that

[TABLE]

Finally, note that if each $x_{t}$ were divided by a positive constant, the preceding would hold with the r.h.s. divided by the same constant. Hence by LP5

[TABLE]

which yields HL4.

(iv). Suppose LP2(a) holds for some $\lambda\geq 2$. By Theorem 2 of Whittle (1960), there exists a $C$ (depending only $\lambda$ and $\mathbf{E}\lvert\xi_{1}\rvert^{\lambda}$) such that

[TABLE]

and thus HL5(a) follows by LP7.

Next, suppose LP2(b) holds. For any $\lambda\in(0,\infty)$, we have

[TABLE]

We shall show that the first r.h.s. term is finite for sufficiently small $\lambda$; the same argument delivers the bound for the second term and thence HL5(b). Since $\mathbf{E}\exp(\mu\xi_{1})<\infty$ for sufficiently small $\mu$, there exist $\varsigma_{\xi},b_{\xi}\in(0,\infty)$ such that $\mathbf{E}\exp(\mu\xi_{t})\leq\exp(\mu^{2}\varsigma_{\xi}^{2})$ for all $\lvert\mu\rvert<b_{\xi}$ (see e.g. Theorem 2.13 in Wainwright, 2019). In view of LP6(b), we may choose $\lambda\in(0,\infty)$ such that

[TABLE]

for all $n$ sufficiently large. Hence by Fatou’s lemma, for such $n$,

[TABLE]

where the final term is bounded uniformly over $1\leq t\leq n$ and $n\geq n_{0}$ by LP7. ∎

B.2 Lemmas for $I(1/2)$ and MI processes

The proof of Theorem 3.2 is now mostly a matter of verifying that each of FR and MI (in conjunction with INN) imply LP, whereupon the result will follow by appeals to Proposition B.1 and Theorem 3.1. The following lemmas establish some key properties of $I(1/2)$ and MI processes under these assumptions, which will be used in the proofs of each of Theorems 3.2 and 3.4. Their proofs appear in Appendix C.1.

Lemma COEF.

*Suppose that FR or **MI *holds. Then there exist $D_{1},D_{2}\in(0,\infty)$ and $n_{0},k_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$

1. (i)

$\sup_{1\leq t\leq n,k\in\mathbb{N}_{0}}\lvert a_{k,t}(n)\rvert\leq D_{2}$;

and under MI,

1. (ii)

$\min_{k_{0}\leq k\leq\kappa_{n}\wedge(n-1)}\lvert a_{k}(n)|\geq D_{1}.$**

For the purposes of the next two lemmas, define

[TABLE]

By Proposition 1.5.9a in BGT, $L(n)=\int_{1}^{n}x^{-1}\ell^{2}(x)dx$ is a slowly varying function, a fact that shall be used freely throughout the following. Recall also that $L(n)\rightarrow\infty$ under FR. MI entails that $\kappa_{n}$ is regularly varying with index $\alpha\in[0,1)$, and thus in all cases, $\gamma_{n}$ is regularly varying with index $\alpha\in[0,1/2)$.

Lemma LVAR.

Suppose INN and either FR or MI holds. Let $t_{n}\in[\lfloor rn\rfloor,n]\cap\mathbb{N}$, where $0<r\leq 1$. Then

1. (i)

Under FR1:

[TABLE]

and for any $s\in(0,r]$,

[TABLE] 2. (ii)

Under FR2 or MI:

[TABLE]

Lemma CLT.

Under the assumptions of Lemma Lemma,

[TABLE]

where $\Sigma=\mathrm{diag}\{\mathcal{V}_{\infty}/2,\mathcal{V}_{\infty}/2\}$ under FR1, and $\Sigma=\mathrm{diag}\{\mathcal{V}_{\infty},0\}$ under FR2 or MI.

B.3 Proofs of Theorems 3.2 and 3.4

We first state a preliminary lemma on regularly varying sequences, whose proof is given in Appendix C.2 below.

Lemma B.2**.**

Let $\{\varphi_{j}\}_{j\in\mathbb{N}}$ be a real sequence, and $\varsigma$ a positive-valued, slowly varying function, that is locally integrable on $[1,\infty)$.

1. (i)

Suppose $\varphi_{j}\sim j^{l}\varsigma(j)$ for $l>-1$. Then for all $0\leq s<r<\infty$ as $n\rightarrow\infty$

[TABLE] 2. (ii)

Suppose $\varphi_{j}\sim j^{l}\varsigma(j)$ for $l<-1$. Then as $n\rightarrow\infty$

[TABLE] 3. (iii)

Suppose $S(x):=\int_{1}^{x}u^{-1}\varsigma(u)du$ is such that $S(x)\rightarrow\infty$ as $x\rightarrow\infty$. Then for any $n_{0}\geq 1$, as $n\rightarrow\infty$

[TABLE]

Proof of Theorem 3.2.

With the aid of Proposition B.1, we show that the conditions of** **Theorem 3.1 are satisfied for $X_{t}(n)=x_{t}(n)$. We first verify Assumptions HL0–6, before turning to the other conditions of the theorem.

HL0–1. Under either of **FR **or MI, $x_{t}(n)$ can be written in the form (5); that $\beta_{n,t}<\infty$ for each $n,t\in\mathbb{N}$ follows from $\mathbf{E}v_{1}^{2}=\sigma_{\xi}^{2}\sum_{k=0}^{\infty}c_{k}^{2}<\infty$. Recall also from (S.9) and (S.11) that we may write $x_{t}(n)=\sum_{k=0}^{\infty}a_{k,t}(n)\xi_{t-k}$, where $a_{k,t}(n)=\sum_{j=0}^{(t-1)\wedge k}\phi_{j}(n)c_{k-j}$. Under FR, $\phi_{0}(n)=\phi_{0}\neq 0$, while under MI, $\phi_{0}(n)=1-\kappa_{n}^{-1}\rightarrow 1$ and has $\phi_{0}(n)>0$ for all $n$, since $\kappa_{n}>1$. Now let $k^{\ast}$ denote the smallest $k\in\mathbb{N}$ such that $c_{k}\neq 0$. Then

[TABLE]

for all $n,t\in\mathbb{R}$. Thus LP1 holds, whence HL0–1 follows by Proposition B.1.

HL2. Once we have verified LP6, by Proposition B.1(i) this will hold with $(X^{+},X^{-})\sim N[0,\mathrm{diag}\{\sigma_{+}^{2},\sigma_{-}^{2}\}]$. Lemma Lemma implies that LP6(a) holds with $(\sigma_{+}^{2},\sigma_{-}^{2})=(1/2,1/2)$ under FR1 and $(\sigma_{+}^{2},\sigma_{-}^{2})=(1,0)$ under FR2/MI. It further implies that $\beta_{n}\asymp\gamma_{n}\rightarrow\infty$, and thus LP6(b) follows by Lemma Lemma. With respect to LP6(c), it follows from (S.13) above that

[TABLE]

for $t_{n}\in\{\lfloor nr\rfloor,\ldots,n\}$ for some $r\in(0,1]$. By Lemma Lemma, the r.h.s. is immediately $o(1)$ under FR1. Under FR2 or MI, that same result implies $\gamma_{n}^{-2}Var[x_{t_{n}}^{-}(n)]=o(1)$, and thus

[TABLE]

HL3 (for some $\boldsymbol{t_{0}\in\mathbb{N}}$). By Proposition B.1(ii), it suffices to verify** LP3**. Consider FR first: in this case, $x_{t}(n)$ does not depend on $n$. Thus $\beta_{n,t}^{+}=\beta_{t}^{+}\asymp\gamma_{t}$ and $\beta_{t}\asymp\gamma_{t}$ as $t\rightarrow\infty$ by Lemma Lemma, whence LP3(a) holds. By** **Lemma Lemma, $\max_{0\leq k\leq t-1}\lvert a_{k}(n)\rvert\leq D_{2}$, which together with $\gamma_{t}=L^{1/2}(t)\rightarrow\infty$ delivers LP3(b).

Next suppose MI holds. By Lemma Lemma, we have

[TABLE]

Since $C_{1}:=\sup_{n\geq 1,1\leq t\leq n}\kappa_{t}/\kappa_{n}<\infty$ by Assumption MI, where trivially $C_{1}\geq 1$, we have

[TABLE]

since $\kappa_{t}=o(t)$ as $t\rightarrow\infty$. Thus there exists a $t_{0}\in\mathbb{N}$ and $C\in(0,\infty)$ such that $\beta_{n,t}^{+}\geq C\kappa_{t}^{1/2}$ for all $t_{0}\leq t\leq n$. Since $\beta_{t}\asymp\gamma_{t}=\kappa_{t}^{1/2}\rightarrow\infty$ by Lemma Lemma, and $\max_{0\leq k\leq t}\lvert a_{k}(n)\rvert\leq D_{2}$ by Lemma Lemma, both parts of LP3 follow immediately.

HL4. By Proposition B.1(iii), we need to only to verify LP5. First suppose FR1 holds. Then recalling (S.11) above,

[TABLE]

as $k\rightarrow\infty$, where $=_{(1)}$ follows by $\sum_{k=0}^{\infty}c_{k}=0$, and $\asymp_{(2)}$ by Lemma B.2(ii), since $c_{j}\sim k^{-3/2}\ell(k)$. Hence, by Lemma B.2(i)

[TABLE]

under FR1. Alternatively, if FR2 holds, then we also have

[TABLE]

by $\sum_{k=0}^{\infty}\lvert c_{k}\rvert<\infty$ and Lemma B.2(i). Thus under FR, Lemma Lemma implies that

[TABLE]

where the final convergence follows by Proposition 1.5.9a in BGT.

Next suppose MI holds. In this case $a_{k}(n)=\sum_{j=0}^{k}\rho_{n}^{j}c_{k-j}$ for $\rho_{n}:=1-\kappa_{n}^{-1}$, and so for all $n$ sufficiently large that $\rho_{n}>0$,

[TABLE]

where $\leq_{(1)}$ holds since $\{c_{i}\}$ is absolutely summable. Hence by Lemma Lemma

[TABLE]

HL5. In view of Proposition B.1(iv), having already verified LP6(b), we need to verify LP7. Suppose first that FR holds. Then since $x_{t}(n)$ does not depend on $n$, we have by Lemma Lemma that, as $t\rightarrow\infty$

[TABLE]

which is clearly monotone increasing. Thus there exist $C_{1},C_{2}\in(0,\infty)$ such that

[TABLE]

for all $t\leq n$, whence LP7 holds trivially.

Suppose next that MI holds. In this case, $x_{t}(n)=\sum_{k=0}^{\infty}a_{k,t}(n)\xi_{t-k}$ with $a_{k,t}=\sum_{j=0}^{(t-1)\wedge k}\phi_{j}(n)c_{k-j}$, where $\sum_{i=0}^{\infty}\lvert c_{i}\rvert<\infty$ and $\phi_{j}(n)=(1-\kappa_{n}^{-1})^{j}>0$ for all $n$ sufficiently large. Now define a new process $\{\overline{x}_{t}(n)\}$ by $\overline{x}_{t}(n)=\sum_{k=0}^{\infty}\overline{a}_{k,t}(n)\xi_{t-k}$, where $\overline{a}_{k,t}=\sum_{j=0}^{(t-1)\wedge k}\phi_{j}(n)\lvert c_{k-j}\rvert$. Then $\{\overline{x}_{t}(n)\}$ also satisfies MI, and it is easily verified that

[TABLE]

for all $1\leq t\leq n$, whence

[TABLE]

It follows from** **Lemma Lemma that $\overline{\beta}_{n}^{2}\asymp\kappa_{n}\asymp\beta_{n}^{2}$, whence

[TABLE]

Thus by Proposition B.1(iv), either HL5(a) or (b) holds depending on the assumptions made on $\{\xi_{t}\}$ – i.e. depending on whether condition 2(a) or 2(b) of Theorem 3.2 is maintained.

HL6. By Lemma Lemma, $\beta_{n}\asymp\gamma_{n}$, which is regularly varying with index $\alpha\in[0,1)$. Thus HL6 follows by Lemma B.2(i).

We have thus verified that HL0–6 hold under our assumptions, with $(X^{+},X^{-})\sim N[0,\mathrm{diag}\{\sigma_{+}^{2},\sigma_{-}^{2}\}]$. The conclusion of Theorem 3.1 accordingly holds for bounded $f$, with

[TABLE]

where $\varrho$ is as defined in (13).

For more general $f$, we need to verify conditions (i)–(iii) of Theorem 3.1. With respect to condition (i), since is assumed that $f$ is locally integrable and satisfies $\lvert f(x)\lvert=O(e^{\lambda^{\prime}\lvert x\lvert})$ for some $\lambda^{\prime}<\infty$ as $\lvert x\lvert\rightarrow\infty$, we thus have

[TABLE]

for every $y\in\mathbb{R}$, as required.

Condition (iii) of Theorem 3.1 is implied by condition 2 of Theorem 3.2, and the verification of HL5 given above.

It thus remains to verify condition (ii) of Theorem 3.1. Under condition 1(a) of Theorem 3.2, there is nothing to left prove. Suppose instead that condition 1(b) of of Theorem 3.2 is assumed, i.e. that INN holds with $\theta=1$. In that case, we need to verify that HL3 holds with $t_{0}=1$. By Proposition B.1(ii)(b), it suffices to verify LP4, since we have already verified LP3. Recall (S.9) and (S.11) above, and let $k^{\ast}$ denote the smallest $k\in\mathbb{N}$ such that $c_{k^{\ast}}\neq 0$. Then

[TABLE]

where the final inequality follows from arguments given in the course of verifying HL0–1 above. Thus LP4 holds. ∎

Proof of Theorem 3.4.

We verify the assumptions of** **Theorem 3.3 are satisfied for $X_{t}(n)=x_{t}(n)$.

HL0–2, HL4, HL6. These follow by the arguments given in the proof of Theorem 3.2.

HL7. We first note that the proof of Proposition B.1(ii), together with the arguments used to verify HL3 in the proof of Theorem 3.2, show that $\{\beta_{t}^{-1}x_{t}^{+}(n)\}$ has a density $\mathcal{D}_{t,n}^{+}$ that is uniformly bounded over all $n$ and $t_{0}\leq t\leq n$ for some $t_{0}\in\mathbb{N}$. Now let $0\leq s<t\leq n$ and note from (S.12) that we may write

[TABLE]

where $x_{s,t}^{\prime}$ is $\mathcal{F}_{s}$-measurable, and $x_{s+1,t}$ is independent of $\mathcal{F}_{s}$. Since

[TABLE]

we must have that the density $\mathcal{D}_{t,s,n}$ of $\{\beta_{t-s}^{-1}[x_{t}(n)-x_{s}(n)]\}$, conditional on $\mathcal{F}_{s}$, satisfies

[TABLE]

where $\chi_{s,t}(n):=\beta_{t-s}^{-1}[x_{s+1,t}(n)-x_{s}(n)]$ is $\mathcal{F}_{s}$-measurable (with the convention that $x_{0}(n):=0$). It follows that

[TABLE]

by the uniform boundedness of $\mathcal{D}_{r,n}^{+}$ noted above.

HL8. Let $\mathcal{D}_{r,n}^{\ast}$ denote the density of $\gamma_{n}^{-1}x_{r}^{+}(n)=(\gamma_{n}^{-1}\beta_{r})\cdot\beta_{r}^{-1}x_{r}^{+}(n)$, where $\{\gamma_{n}\}$ is as defined in (S.15), and $\delta_{\eta}:=q_{0}\eta^{q_{1}}$ for $\eta,q_{0},q_{1}>0$. Let $s,t\in\Omega_{n}(\eta)$, and $r:=t-s$. In view of (S.16),

[TABLE]

By the definition of $\Omega_{n}(\eta)$, HL8 will hold if we can show that when $r=r_{n}$ for any sequence $\{r_{n}\}$ with $r_{n}\in[\lfloor n\eta\rfloor,n]$, the r.h.s. converges to zero as $n\rightarrow\infty$ and then $\eta\rightarrow 0$.

By Lemma Lemma, $\beta_{r_{n}}^{-1}\gamma_{n}\asymp\gamma_{r_{n}}^{-1}\gamma_{n}\asymp 1$, since $\{\gamma_{n}\}$ is regularly varying. In particular, $\{\beta_{r_{n}}^{-1}\gamma_{n}\}$ is bounded above, and so (by redefinition of $q_{0}$) the result will follow if we can prove that

[TABLE]

as $n\rightarrow\infty$ and then $\eta\rightarrow 0$. By Lemma Lemma, $\gamma_{n}^{-1}x_{r_{n}}^{+}(n)\overset{d}{\rightarrow}N[0,\sigma_{+}^{2}]$ for some $\sigma_{+}^{2}>0$. By the argument given in the proof of Corollary 2.2 in Wang and Phillips (2009), it therefore suffices for (S.17) to show that the characteristic functions of $\{\gamma_{n}^{-1}x_{r_{n}}^{+}(n)\}$ are uniformly integrable: since in this case, $\mathcal{D}_{r_{n},n}^{\ast}$ converges uniformly to the $N[0,\sigma_{+}^{2}]$ density. To that end, note that $\gamma_{n}^{-1}x_{r_{n}}^{+}(n)$ is a linear process with variance converging to $\sigma_{+}^{2}>0$ (as $n\rightarrow\infty$) by Lemma Lemma, and coefficients $\{\gamma_{n}^{-1}a_{k}(n)\}_{k=0}^{r_{n}-1}$ with $\max_{0\leq k\leq r_{n}-1}\gamma_{n}^{-1}\lvert a_{k}(n)|\rightarrow 0$ by Lemma Lemma. Thus the required uniform integrability follows by Lemma B.1(i).

HL9. By Lemma Lemma, $\beta_{n}\asymp\gamma_{n}$, which is regularly varying with index $\alpha\in[0,1/2)$ and so can be written as $\gamma_{n}=n^{\alpha}\varsigma(n)$, for $\varsigma$ a positive-valued, slowly varying function. Thus HL9(a)–(c) follow immediately from Lemma B.2(i). For HL9(d), we note that

[TABLE]

for some $C>0$. Since $(t,s)\in\Omega_{n}(\eta)$ implies $t-s>\lfloor\eta n\rfloor$, the r.h.s. can be bounded below (up to a positive constant) by

[TABLE]

where the equality follows by Theorem 1.2.1 in BGT. Thus HL9(d) holds. Finally, since $\beta_{n}\asymp\gamma_{n}$, and both $\beta_{n}>0$ and $\gamma_{n}>0$ for all $n$, there exist $\underline{C},\overline{C}\in(0,\infty)$ such that $\beta_{n}\in[\underline{C}\gamma_{n},\overline{C}\gamma_{n}]$ for all $n$. Thus

[TABLE]

From (S.15), under FR $\gamma_{n}=L(n)$ is increasing, so $\sup_{n\geq 1}\sup_{1\leq t\leq n}\gamma_{n}^{-1}\gamma_{t}\leq 1$ trivially. Under MI, $\gamma_{n}=\kappa_{t}$, for which $\sup_{n\geq 1}\sup_{1\leq t\leq n}\kappa_{n}^{-1}\kappa_{t}<\infty$ is a maintained assumption. Thus HL9(e) holds. ∎

Appendix C Proofs auxiliary to Appendix B

This appendix provides proofs of the lemmas stated in Appendix B.

C.1 Proofs of Lemmas COEF, LVAR and CLT

Proof of Lemma Lemma.

(i). In all cases, $\lvert\phi_{j}(n)\rvert<C$ uniformly over $j,n\in\mathbb{N}$, and $\{c_{s}\}$ is absolutely summable. Hence from (S.11),

[TABLE]

(ii). Under MI, we have $\phi_{j}(n)=\rho_{n}^{j}$ for $\rho_{n}:=1-\kappa_{n}^{-1}$, and thus

[TABLE]

Since $\rho_{n}\in(0,1)$, we have for all $0\leq k\leq\lfloor\kappa_{n}\rfloor$ that

[TABLE]

from which follows that $\rho_{n}^{k}\in(e^{-1}/2,1]$ for all $k\in\{0,\ldots,\lfloor\kappa_{n}\rfloor\}$, for all $n\in\mathbb{N}$.

Thus to bound $a_{k}(n)$ away from zero, we need only to bound the bracketed term in (S.18) away from zero. To that end, note that for all $0\leq k\leq\lfloor\kappa_{n}\rfloor$

[TABLE]

as $n\rightarrow\infty$ by the dominated convergence theorem, since $\lvert\rho_{n}^{-j}-1\rvert\lvert c_{j}\rvert 1\{j\leq\kappa_{n}\}\rightarrow 0$ for each $j\in\mathbb{R}$, and by an analogous argument to (S.19) is (for sufficiently large $n$) bounded above by $\lvert c_{j}\rvert(1+2e)$, which is summable. Since $\sum_{j=0}^{\infty}c_{j}\neq 0$, deduce that there exist $n_{0},k_{0}\in\mathbb{N}$ such that

[TABLE]

for all $n\geq n_{0}$ and $k\in\{k_{0},\ldots,\lfloor\kappa_{n}\rfloor\}$. ∎

Proof of Lemma Lemma.

For convenience set $\sigma_{\xi}^{2}=1$. Under FR it is sufficient to prove the result with $t_{n}=n$; the result in the general case follows straightforwardly from the fact that $L(t_{n})/L(n)\rightarrow 1$ as $n\rightarrow\infty$, since $L(n)$ is slowly varying.

FR1. Suppose we show that

[TABLE]

and that for any sequences $\{\delta_{1,n}\}$, $\{\delta_{2,n}\}$ with $\delta_{1,n}\asymp n\asymp\delta_{2,n}$,

[TABLE]

Since $\gamma_{n}^{2}=L(n)$ and $a_{k,t}=\sum_{j=0}^{(t-1)\wedge k}c_{k-j}$, it follows from (S.20) that

[TABLE]

Further, note that we may write

[TABLE]

The second r.h.s. term is of the same form as (S.21) with $\delta_{1,n}=\delta_{2,n}=n$, and so is $o[L(n)]$. The first r.h.s. term expands as

[TABLE]

where we have used the fact that $\sum_{j=0}^{\infty}c_{j}=0$, and so by (S.20) and the Cauchy-Schwarz inequality,

[TABLE]

Thus it follows from (S.22) and (S.23) that $\gamma_{n}^{-2}Var(x_{n}^{-})\rightarrow 4$, as required. Finally, noting that

[TABLE]

we have for $0<s<r\leq 1$ that

[TABLE]

which is of the form (S.21) with $\delta_{1,n}=\lfloor ns\rfloor$ and $\delta_{2,n}=\lfloor nr\rfloor-\lfloor ns\rfloor$, and so is $o[L(n)]$ as required.

It remains to prove (S.20) and (S.21). Recall that $c_{j}\sim j^{-3/2}\ell(j)=:g(j)$, which is regularly varying with index $-3/2$. Since $\ell$ is defined only up to an asymptotic equivalence, it is without loss of generality to take $\ell$ to be such that $g$ is monotone decreasing (see Theorem 1.5.3 in BGT). For the first part of (S.20), using that $\sum_{j=0}^{\infty}c_{j}=0$ we have

[TABLE]

By Lemma B.2(ii),

[TABLE]

as $k\rightarrow\infty$, while by Lemma B.2(iii),

[TABLE]

whence by the Toeplitz lemma (Hall and Heyde, 1980, p. 31)

[TABLE]

as required. For the second part of (S.20), we have

[TABLE]

By the first part of (S.20), the r.h.s. is asymptotically equivalent to

[TABLE]

by the slow variation of $L(n)$.

Finally, we turn to (S.21). Since $c_{j}\sim g(j)$, $c_{j}$ is eventually strictly positive, and there exists a $C<\infty$ such that $0<c_{j}\leq Cg(j)$ for all $j$ sufficiently large. Hence, for all $n$ sufficiently large

[TABLE]

where the second inequality holds by the monotonicity of $g(j)$. By Lemma B.2(ii),

[TABLE]

Since $\delta_{1,n}\asymp n\asymp\delta_{2,n}$, it follows that

[TABLE]

by the slow variation of $\ell^{2}$, and Theorem 1.5.9.a in BGT.

FR2. Consider $x_{n}^{+}$ first. We have

[TABLE]

Fix $i,j\in\mathbb{N}$, taking $i\geq j$ without loss of generality. Then

[TABLE]

Since $\phi_{k}\sim k^{-1/2}\ell(k)$, which is regularly varying, $\phi_{k+(i-j)}/\phi_{k}\rightarrow 1$ as $k\rightarrow\infty$, while by Lemma B.2(iii),

[TABLE]

Hence by the Toeplitz Lemma (Hall and Heyde, 1980, p. 31),

[TABLE]

as $n\rightarrow\infty$, for each $i,j\in\mathbb{N}$ with $i\geq j$. Moreover, by the Cauchy-Schwarz inequality,

[TABLE]

for some $C<\infty$ not depending on $i$ or $j$. Since $\sum_{k=0}^{\infty}\lvert c_{k}\rvert<\infty$, it follows from (S.24)–(S.26) and the dominated convergence theorem that

[TABLE]

With respect to $x_{n}^{-}$, we have:

[TABLE]

where $\leq_{(1)}$ holds since $\phi_{k}\sim k^{-1/2}\ell(k)$, which can be taken to be monotone without loss of generality (see Theorem 1.5.3 in BGT), while $\leq_{(2)}$ follows from $\sum_{i=j}^{n-1}\lvert c_{n+k-i}\rvert\leq\sum_{i=0}^{\infty}\lvert c_{i}\rvert<\infty$. Letting $\overline{c}_{m}:=\sum_{k=m}^{\infty}\lvert c_{k}\rvert$, we thus have that $Var(x_{n}^{-})$ is bounded, up to multiplicative constant, by

[TABLE]

We claim that the r.h.s. is of smaller order than $\gamma_{n}^{2}=L(n)$. For the first term, this follows from $\overline{c}_{\lfloor n/2\rfloor}\rightarrow 0$ and $\sum_{j=0}^{\lfloor n/2\rfloor}\phi_{j}^{2}\sim L(\lfloor n/2\rfloor)\sim L(n)$ by Lemma B.2(iii) and the slow variation of $L(n)$. By the same argument,

[TABLE]

MI. By the same argument as which led to (S.24), and noting that $\phi_{j}(n)=\rho_{n}^{j}$ for $\rho_{n}=1-\kappa_{n}^{-1}$ in this case, we have

[TABLE]

Fix $i,j\in\mathbb{N}$, taking $i\geq j$ without loss of generality. Then

[TABLE]

Since $1-\rho_{n}^{2}=(1+\rho_{n})(1-\rho_{n})\sim 2\kappa_{n}^{-1}$, it follows that

[TABLE]

as $n\rightarrow\infty$ for each $i,j\in\mathbb{N}$ with $i\geq j$, since $\rho_{n}^{i-j}\rightarrow 1$ as $n\rightarrow\infty$ and

[TABLE]

since $\kappa_{n}=o(t_{n})$. Moreover, for all $i,j\in\{0,\ldots,t_{n}\}$ with $i\geq j$, and all $n\in\mathbb{N}$,

[TABLE]

Hence it follows by the absolute summability of $\{c_{s}\}$ and the dominated convergence theorem that

[TABLE]

Next, since the coefficients $\{a_{k,t}^{-}(n)\}$ are uniformly bounded by Lemma Lemma, we have

[TABLE]

Letting $\overline{c}_{m}:=\sum_{k=m}^{\infty}\lvert c_{k}\rvert$, we can bound the r.h.s., up to a multiplicative constant, by

[TABLE]

where $\leq_{(1)}$ holds for all $n$ sufficiently large that $\rho_{n}\in(0,1)$, and $=_{(2)}$ from $\overline{c}_{\lfloor t_{n}/2\rfloor}+\rho_{n}^{\lfloor t_{n}/2\rfloor}\rightarrow 0$ (as per (S.27) above), and

[TABLE]

Proof of Lemma Lemma.

By Lemma Lemma, we have

[TABLE]

for appropriate $\sigma_{+}$ and $\sigma_{-}$. Each of $\{\gamma_{n}^{-1}x_{t_{n}}^{+}(n)\}$ and $\{\gamma_{n}^{-1}x_{t_{n}}^{-}(n)\}$ are independent linear processes arrays with respective coefficients $\{\gamma_{n}^{-1}a_{k}(n)\}_{k=0}^{t_{n}-1}$ and $\{\gamma_{n}^{-1}a_{k,t_{n}}^{-}(n)\}_{k=t_{n}}^{\infty}$, which by Lemma Lemma have the property that

[TABLE]

The postulated weak convergence thus follows by Lemma 2.1 in Abadir, Distaso, Giraitis and Koul (2014). ∎

C.2 Proofs of Lemmas B.1 and B.2

Proof of Lemma B.1.

Without loss of generality, we will prove the result as stated, but with $\mathbb{N}$ in place of $\mathbb{N}_{0}$, so that $\{\vartheta_{k}\}$ is indexed by $k\in\{1,2,\ldots\}$. Let $\eta:=\sum_{k=1}^{\infty}\vartheta_{k}\xi_{k}$, and define

[TABLE]

where the second equality holds by independence. Since $\mathbf{E}\xi_{1}^{2}<\infty$, by a Taylor expansion of the log characteristic function, there exists a $\gamma\in(0,\infty)$ such that

[TABLE]

**(i). **Without loss of generality, suppose $\{\vartheta_{k}\}$ is ordered as so that $\lvert\vartheta_{1}\rvert\geq\lvert\vartheta_{2}\lvert\geq\cdots$. Define

[TABLE]

We claim that $\mathcal{K}$ is nonempty. To see this, observe that by definition of $\mathcal{K}$ and the stated condition on $\max_{k\in\mathbb{N}}\vartheta_{k}^{2}$,

[TABLE]

where we have used that $\sum_{k=1}^{\infty}k^{-2}=\pi/6$. Hence

[TABLE]

and so $\mathcal{K}$ is nonempty. Let $k_{0}$ denote its first element, noting that $k_{0}\geq\theta+1$ by construction.

We want to bound the integral of $\lvert\psi_{\eta}(\lambda)\rvert$ over $[A,\infty)$. To that end, decompose

[TABLE]

and consider the integral on each of these two pieces separately. We have

[TABLE]

Since $\lvert\lambda|\leq\vartheta_{k_{0}}^{-1}$ on the domain of integration, we have $\lvert\lambda\vartheta_{k}\rvert\leq\lvert\vartheta_{k_{0}}^{-1}\vartheta_{k}\rvert\leq 1$ for all $k\in\mathcal{K}$, and so by (S.28) and then (S.29),

[TABLE]

where $G_{1}$ depends on $\psi_{\xi}$ through $\gamma$. By a change of variables $G_{1}(A;\sigma_{\vartheta}^{2})\leq C_{\gamma}\sigma_{\vartheta}^{-1}$, for some $C_{\gamma}$ depending only on $\gamma$. Moreover, $\lim_{A\rightarrow\infty}G_{1}(A;\sigma_{\vartheta}^{2})=0$ and $\sigma^{2}\mapsto G_{1}(A;\sigma^{2})$ is decreasing in $\sigma^{2}$, as required,

Next, noting that the leading $k_{0}$ elements of $\{\vartheta_{k}\}$ (as ordered) must be nonzero, we have

[TABLE]

where the second equality follows from (S.28) and the fact that $\lvert\lambda\vartheta_{k}\rvert\geq\lvert\vartheta_{k_{0}}^{-1}\vartheta_{k}\rvert\geq 1$ on the domain of integration, for all $k\geq k_{0}$. By Hölder’s inequality,

[TABLE]

Now $\vartheta_{k}\geq\vartheta_{k_{0}}\geq c_{0}\sigma_{\vartheta}k_{0}^{-1}$ for $k\leq k_{0}$, where $c_{0}:=(3/2\pi)^{1/2}$, whence

[TABLE]

Thus,

[TABLE]

Since the final integral is bounded by $\int\lvert\psi_{\xi}(\lambda)\rvert^{\theta}d\lambda<\infty$, and $e^{-\gamma k}k\rightarrow 0$ as $k\rightarrow\infty$, it is evident that

[TABLE]

for some $C_{\psi_{\xi}}<\infty$ depending only on $\psi_{\xi}$.

Since each $G_{2,k}(A;\sigma^{2})$ is weakly decreasing in $\sigma^{2}$, so too must be $G_{2}(A;\sigma^{2})$. It remains therefore to show that $G_{2}(A;\sigma_{\vartheta}^{2})\rightarrow 0$ as $A\rightarrow\infty$. To that end, let $\epsilon>0$ and note that since $e^{-\gamma k}k\rightarrow 0$ as $k\rightarrow\infty$, we may choose $k^{\ast}$ such that $G_{2,k}(A;\sigma_{\vartheta}^{2})\leq\epsilon$ for all $k\geq k^{\ast}$. Hence

[TABLE]

as $A\rightarrow\infty$, since $G_{2,k}(A;\sigma_{\vartheta}^{2})\rightarrow 0$ as $A\rightarrow\infty$ for each $k$ fixed.

(ii). Without loss of generality, we may take $k_{i}=i$ for each $i\in\{1,\ldots,\theta\}$. Then by the same argument as which led to (S.30) above,

[TABLE]

Proof of Lemma B.2.

(i). The result follows by similar arguments to those given in Giraitis, Koul and Surgailis (2012, p. 20).

(ii). Since $g(j)=j^{l}\varsigma(j)$ with $l<-1$, by Theorem 1.5.3 in BGT, we may without loss of generality take $\varsigma$ to be such that $g$ is monotone decreasing. Since $\sum_{j=n}^{\infty}\varphi_{j}\sim\sum_{j=n}^{\infty}g(j)$ as $n\rightarrow\infty$, we have

[TABLE]

By monotonicity of $g$,

[TABLE]

By Theorem 1.5.11 in BGT,

[TABLE]

while by the preceding and Theorem 1.5.2 in BGT,

[TABLE]

(iii). Set $s(x):=x^{-1}\varsigma(x)$, and note that

[TABLE]

Let $\epsilon>0$. By Theorem 1.5.3 of BGT, $\underline{s}(x)\sim s(x)\sim\overline{s}(x)$ as $x\rightarrow\infty$ , and so we may choose $x_{0}\in\mathbb{N}$ with $x_{0}\geq 2$ such that

[TABLE]

for all $x\geq x_{0}$. Since $\underline{s}$ and $\overline{s}$ are monotone decreasing, we also have

[TABLE]

By (S.31) and (S.32),

[TABLE]

as $n\rightarrow\infty$, noting that since $S(n)\rightarrow\infty$ by hypothesis, $\int_{x}^{n}s(u)du\sim S(n)$ for each $x\geq 1$. Similarly,

[TABLE]

Deduce

[TABLE]

as $n\rightarrow\infty$. ∎

Appendix D Proofs of Theorems 4.1 and 4.2

Proof of Theorem 4.1.

Let $\kappa_{gn}:=\kappa_{g}(\beta_{n})$. By standard arguments,

[TABLE]

Consider $M_{n}$ first. We claim that

[TABLE]

for $j\in\{1,2\}$, where $H_{n,t}:=H_{g}(x_{n,t-1})$. When $j=1$, this follows immediately from the definition of an AHF. When $j=2$, the l.h.s. is bounded (in absolute value) by

[TABLE]

The second term is $o_{p}(1)$ by (18). Since $H_{g}^{2}$ satisfies the conditions of Theorem 3.2, we have

[TABLE]

whence the first term in (S.38) is also $o_{p}(1)$, by the Cauchy-Schwarz inequality. Thus by (S.37) and Theorem 3.2,

[TABLE]

which is a.s. nonsingular, since $H_{g}$ is not a.e. equal from a constant function by assumption.

We turn next to $S_{n}$. We first note that

[TABLE]

by (18). Thus

[TABLE]

where the leading r.h.s. term is a vector martingale with conditional variance matrix

[TABLE]

by Theorem 3.2. Thus, if we can verify the requirements of Wang’s (2014) martingale CLT (his Theorem 2.1), the convergence

[TABLE]

will hold jointly with (S.39), where $\zeta\sim N[0,I_{2}]$ is independent of $X^{-}$ (and of the possibly random density $\varrho$ that depends on $X^{-}$) whence the result follows.

Regarding Wang’s CLT, the only condition that is not trivially satisfied in our setting is his Assumption 2. Since the mixing variate $X^{-}$ is the weak limit of $\beta_{n}^{-1}x_{n}^{-}(n)$, which is $\mathcal{F}_{0}$-measurable, it can be seen from the proof of Wang’s CLT that his condition (2.3) is unnecessary in our case. Thus we need only to verify that $n^{-1/2}\max_{2\leq t\leq n}\lvert H_{n,t-1}\rvert=o_{p}(1)$, which is equivalent (see e.g. Hall and Heyde, 1980, p. 53) to the Lindeberg condition that for each $\eta>0$,

[TABLE]

as $n\rightarrow\infty$. Recall that $H_{g}^{2}$ is assumed to satisfy the conditions of Theorem 3.2: therefore so too does $F_{A}(x):=H_{g}^{2}(x)\boldsymbol{1}\{\lvert H_{g}(x)\rvert>A\}$ for each $A\in\mathbb{R}$. Now

[TABLE]

by Theorem 3.2. By dominated convergence, $\Gamma(A)\overset{a.s.}{\rightarrow}0$ as $A\rightarrow\infty$, since $H_{g}^{2}\varrho$ is integrable. Therefore for a given $\epsilon>0$, we may choose $A_{\epsilon}$ such that $\mathbf{P}\{\Gamma(A_{\epsilon})\geq\epsilon\}\leq\epsilon$, whence

[TABLE]

where $\leq_{(1)}$ holds since $A_{\epsilon}<\eta n^{1/2}$ for all $n$ sufficiently large, and $\leq_{(2)}$ follows by the portmanteau theorem. Deduce $\Gamma_{n}(\eta n^{1/2})\overset{p}{\rightarrow}0$, i.e. (S.41) holds. ∎

Proof of Theorem 4.2.

(i). We first prove (21). By Theorem 3.4,

[TABLE]

jointly for $j\in\{1,2\}$. Since $m$ has bounded derivative, $\lvert m(x_{t-1})-m(x)\lvert\leq C\lvert x_{t-1}-x|$, and thus

[TABLE]

also by Theorem 3.4. Hence

[TABLE]

where the second r.h.s. term is $o_{p}(1)$ since $nh_{n}^{3}/\beta_{n}\rightarrow 0$ by assumption. The first r.h.s. term is a martingale with conditional variance

[TABLE]

by (S.42). Thus, similarly to the argument given in the proof of Theorem 4.1, if we can show the following Lindeberg condition holds, that

[TABLE]

for each $\eta>0$, then by Wang’s (2014) martingale CLT,

[TABLE]

jointly with (S.42), for $\zeta\sim N[0,1]$ independent of $\varrho(0)$. (21) then follows from (S.42), (S.43) and (S.45).

Finally, to verify (S.44), we note that for $n$ sufficiently large, the l.h.s of (S.44) is bounded by

[TABLE]

where $\overset{d}{\rightarrow}_{(1)}$ holds by Theorem 3.4 as $n\rightarrow\infty$, and $\overset{a.s.}{\rightarrow}_{(2)}$ by dominated convergence as $A\rightarrow\infty$. (S.44) thus follows by similar arguments as were used to prove (S.41) above.

(ii). We next prove (22), using arguments similar to those given in part (i). Let $X_{t}^{\prime}:=(1,x_{t-1}-x)$, $K_{th}:=K[h_{n}^{-1}(x_{t-1}-x)]$, $H_{n}:=\sum_{t=2}^{n}K_{th}X_{t}X_{t}^{\prime}$ and $\Lambda_{n}:=(nh_{n}/\beta_{n})^{1/2}\mathrm{diag}\{1,h_{n}\}$. Then the LL estimator can be written as

[TABLE]

and by standard arguments decomposes as

[TABLE]

where $\mathbf{m}(x):=[m(x),m^{(1)}(x)]^{\prime}$.

We consider each of $H_{n}$, $R_{n}$ and $S_{n}$ in turn. For $H_{n}$, we have

[TABLE]

by Theorem 3.4. For $R_{n}$,

[TABLE]

by Taylor’s theorem, for some $\overline{x}_{t-1}$ lying between $x_{t-1}$ and $x$. Thus,

[TABLE]

which since $m^{(2)}$ is bounded, is bounded (in norm) by a multiple of

[TABLE]

where $=_{(1)}$ is by Theorem 3.4, and $=_{(2)}$ by the assumption that $nh_{n}^{5}/\beta_{n}\rightarrow 0$. Finally, since $\Lambda_{n}^{-1}S_{n}$ is a martingale with conditional variance matrix

[TABLE]

by Theorem 3.4, it follows by Wang’s (2014) CLT and similar arguments as were used in part (i) that

[TABLE]

jointly with (S.48), where $\zeta\sim N[0,I_{2}]$ is independent of $\varrho(0)$. Since

[TABLE]

the result now follows from (S.48)–(S.51), noting that the r.h.s. of (S.48) is invertible under condition (i). ∎

Appendix E Proof of Theorem 5.1

The proof is divided into four parts. Part I derives the asymptotic distribution of $\tilde{F}$ under $\mathcal{H}_{0}$. Part II derives the asymptotics of $(\hat{\mu},\hat{\gamma})$, and part III those of $\tilde{\sigma}_{u}^{2}(x)$, under $\mathcal{H}_{1}$. Finally, part IV draws on the results of parts II and III to derive the asymptotics of $\tilde{F}$ under $\mathcal{H}_{1}$.

I. Asymptotics of $\boldsymbol{\tilde{F}}$ under $\boldsymbol{\mathcal{H}_{0}}$. Recall from (24) that $\tilde{F}$ is constructed from an ensemble of $t$ statistics of the form

[TABLE]

for $x$ taking values in some finite set $\mathcal{X}\subset\mathbb{R}$. Let $x\in\mathcal{X}$ be fixed. We shall show that:

[TABLE]

jointly over $x\in\mathcal{X}$, where $\zeta_{x}\sim N[0,1]$ is independent of $\zeta_{x^{\prime}}\sim N[0,1]$ for each $x,x^{\prime}\in\mathcal{X}$; and that

[TABLE]

for each $x\in\mathcal{X}$. Since $(\beta_{n}/nh_{n})\sum_{t=2}^{n}K[(x_{t-1}-x)/h_{n}]\overset{d}{\rightarrow}\varrho(0)$ by Theorem 3.4, and $\hat{\mu}$ and $\hat{\gamma}$ are consistent at rates $n^{-1/2}$ and $n^{-1/2}\kappa_{g}(\beta_{n})=O_{p}(n^{-1/2})$ respectively by Theorem 4.1, it will then follow that

[TABLE]

whence

[TABLE]

as required.

It thus remains to prove (S.52) and (S.53). Consider (S.52) first: the proof uses arguments similar to those used in part (ii) of the proof of Theorem 4.2. Define $X_{t}^{\prime}:=[1,g(x_{t-1})-g(x)]$, $K_{th}:=K[h_{n}^{-1}(x_{t-1}-x)]$, $\Lambda_{n}:=(nh_{n}/\beta_{n})^{1/2}\mathrm{diag}\{1,h_{n}\}$ and $H_{n}:=\sum_{t=2}^{n}K_{th}X_{t}X_{t}^{\prime}$, noting how the definition of $X_{t}$ has been modified. Recall that $\tilde{m}_{g}$ is obtained by a $K_{th}$-weighted regression of $y_{t}$ on a constant and $g(x_{t-1})-g(x)$. Therefore under $\mathcal{H}_{0}$, this estimator suffers from no approximation bias, and standard arguments give

[TABLE]

where $e_{1}^{\prime}:=(1,0)$.

We shall consider each of $H_{n}$ and $S_{n}$ in turn. We claim that

[TABLE]

and recall that $g^{(1)}(x)\neq 0$ by condition (ii). Consider e.g. the $(2,2)$ element of the l.h.s. matrix: this is equal to

[TABLE]

for some $\overline{x}_{t-1}$ lying between $x_{t-1}$ and $x$, by a Taylor expansion. By Theorem 4.2,

[TABLE]

whereas, since $K(u)u^{4}$ is integrable by condition (i)

[TABLE]

whence by the Cauchy–Schwarz inequality,

[TABLE]

This gives the claimed convergence for the $(2,2)$ element in (S.55); the result for the other elements follows by analogous arguments.

We next turn to $S_{n}$. For each $x\in\mathcal{X}$, $\Lambda_{n}^{-1}S_{n}(x)$ is a vector martingale with conditional variance

[TABLE]

where $\stackrel{{\scriptstyle d}}{{\rightarrow}}_{(1)}$ follows by arguments identical to those used to prove (S.55), with $K^{2}$ in place of $K$. Moreover, the conditional covariation of $\Lambda_{n}^{-1}S_{n}(x)$ with $\Lambda_{n}^{-1}S_{n}(x^{\prime})$ for $x^{\prime}\neq x$ is a $2\times 2$ matrix with $(i,j)$ element

[TABLE]

where $K_{th}(x):=K[h_{n}^{-1}(x_{t-1}-x)]$. We claim this is $o_{p}(1)$, in which case it will follow by an application of Wang’s (2014) CLT that $\Lambda_{n}^{-1}S_{n}(x)$ and $\Lambda_{n}^{-1}S_{n}(x^{\prime})$ are asymptotically independent, with

[TABLE]

jointly with (S.55) over all $x\in\mathcal{X}$, where $\zeta_{x}\sim N[0,I_{2}]$ is independent of $\zeta_{x^{\prime}}\sim N[0,I_{2}]$ for $x^{\prime}\neq x$.

To verify that (S.58) is indeed $o_{p}(1)$, note that by similar arguments to those given in (S.56) above, we can bound (S.58) by linear combinations of functionals of the form

[TABLE]

where $L_{a}(u)=\lvert K(u)u^{a}\rvert$ for $a\in\{0,1,2\}$, which is bounded and integrable by condition (i). Finally, note that by the proof of Theorem 3.2, $\{x_{t}(n)\}$ satisfies HL3 (for some $t_{0}\in\mathbb{N}$) and HL6. Hence for $t\geq t_{0}$ and $n\geq n_{0}$,

[TABLE]

Since $L_{4}$ is bounded by condition (i), it must be the case that $L_{b}(u)\rightarrow 0$ as $\lvert u\rvert\rightarrow\infty$ for $b\in\{0,1,2\}$, whence the r.h.s. integral converges to zero by the dominated convergence theorem as $h_{n}\rightarrow 0$. Thus

[TABLE]

as was required, where the final equality follows since HL6 holds, as was noted above.

(S.52) now follows from (S.54), (S.55), (S.59) and the fact that

[TABLE]

as may be verified by direct calculation.

We turn therefore to (S.53). We have

[TABLE]

Recognising

[TABLE]

for some $\overline{x}_{t-1}$ lying between $x_{t-1}$ and $x$, and that $g^{(2)}$ is bounded, it follows from Theorem 4.2 (arguing similarly as in (S.57) above) that

[TABLE]

Since $\left\{u_{t}^{2}-\sigma_{u}^{2},\mathcal{F}_{t}\right\}$ is a martingale difference sequence and $K_{th}$ is $\mathcal{F}_{t-1}$-measurable, $(\beta_{n}/nh_{n})\sum_{t=2}^{n}K_{th}(u_{t}^{2}-\sigma_{u}^{2})$ is a martingale with conditional variance

[TABLE]

where $\leq_{(1)}$ holds by $Z:=\sup_{t}\mathbf{E}[u_{t}^{4}\mid\mathcal{F}_{t-1}]<\infty$ a.s., and $=_{(2)}$ by Theorem 3.2. Thus $(\beta_{n}/nh_{n})\sum_{t=2}^{n}K_{th}(u_{t}^{2}-\sigma_{u}^{2})=o_{p}(1)$ by Corollary 3.1 in Hall and Heyde (1980), whence

[TABLE]

Since $(\beta_{n}/nh_{n})\sum_{t=2}^{n}K_{th}\overset{d}{\rightarrow}\varrho(0)$ by Theorem 3.2, (S.62), (S.63), the consistency of $(\hat{\mu},\hat{\gamma})$ (from Theorem 4.1) and applications of the Cauchy–Schwarz inequality to the cross-product terms on the r.h.s. of (S.61) yield

[TABLE]

i.e. (S.53) holds.

II. Asymptotics of $\boldsymbol{(\hat{\mu},\hat{\gamma})}$ under $\boldsymbol{\mathcal{H}_{1}}$. Recall that

[TABLE]

under $\mathcal{H}_{1}$. Let $\kappa_{gn}:=\kappa_{g}(\beta_{n})$ and $D_{n}:=n^{1/2}\mathrm{diag}\{1,\kappa_{gn}\}$.

(a). Suppose condition (iii.a) holds. Let $R_{n}^{\prime}:=r_{n}^{-1}n^{-1/2}\beta_{n}$, which is $o(1)$ by assumption. By (S.36) and subsequent arguments given in the proof of Theorem 4.1,

[TABLE]

where $M_{n}\overset{d}{\rightarrow}M_{H_{g}}$ and $S_{n}=O_{p}(1)$ by (S.39) and (S.40) respectively, and

[TABLE]

by Theorem 3.4, since $g_{1}$ and $g\cdot g_{1}$ are integrable by assumption, and $\kappa_{gn}\rightarrow\infty$. Since $R_{n}^{\prime}\rightarrow 0$, it follows that

[TABLE]

(b). Suppose condition (iii.b) holds. Let $\kappa_{g_{1}n}:=\kappa_{g_{1}}(\beta_{n})$ and $R_{n}^{\prime\prime}:=r_{n}^{-1}n^{-1/2}\kappa_{g_{1}n}^{-1}$; the latter is $o(1)$ by assumption. By analogous arguments to those given in part (a),

[TABLE]

where $=_{(1)}$ follows by straightforward calculations, since $g$ and $g_{1}$ are AHF, and $\overset{d}{\rightarrow}_{(2)}$ follows by Theorem 3.2.

III. Asymptotics of $\boldsymbol{\tilde{\sigma}_{u}^{2}(x)}$ under $\boldsymbol{\mathcal{H}_{1}}$. We have

[TABLE]

(a). Suppose condition (iii.a) holds. The argument here is similar to the proof of (S.53) given in part I. Recall (S.62), noting that this also holds with $g_{1}$ in place of $g$, since $g_{1}$ is also assumed to have bounded second derivative. Thus

[TABLE]

for $f\in\{g,g_{1}\}$. By the analysis of part II(a) of the proof, and noting that both diagonal elements of

[TABLE]

are divergent, it follows that $(\hat{\mu},\hat{\gamma})\overset{p}{\rightarrow}(\mu,\gamma)$. In view of $r_{n}=o(1)$ and (S.63), the preceding facts and applications of the Cauchy–Schwarz inequality to the cross-product terms on the r.h.s. of (S.67) yield

[TABLE]

(b). Suppose condition (iii.b) holds. Recall the analysis given in part II(b) of the proof, and note that

[TABLE]

The behaviour of these sequences, and thus that of $\tilde{\sigma}_{u}^{2}(x)$, will depend on $\kappa_{\ast\ast}:=\lim_{n\rightarrow\infty}r_{n}\kappa_{g_{1}n}$; we need to separately consider the cases where: $\kappa_{\ast\ast}=0$; $\kappa_{\ast\ast}\in(0,\infty)$; or $\kappa_{\ast\ast}=\infty$. By a suitable rescaling of $r_{n}$, it is without loss of generality to normalise $\kappa_{\ast\ast}=1$ in the second of these cases.

Suppose $\kappa_{\ast\ast}=0$. Then both diagonal element of $R_{n}^{\prime\prime}D_{n}$ are divergent, and $(\hat{\mu},\hat{\gamma})\overset{p}{\rightarrow}(\mu,\gamma)$. The same arguments given in part III(a) thus imply that $\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle p}}{{\rightarrow}}\sigma_{u}^{2}$.

Suppose $\kappa_{\ast\ast}=1$. Only the second element diagonal element of $R_{n}^{\prime\prime}D_{n}$ diverges, so $\hat{\gamma}\overset{p}{\rightarrow}\gamma$; and since the first diagonal element converges to $\kappa_{\ast\ast}^{-1}=1$, $\hat{\mu}-\mu\stackrel{{\scriptstyle d}}{{\rightarrow}}-\mu_{\ast}$ by (S.66). Thus in this case, we have

[TABLE]

where $\stackrel{{\scriptstyle d}}{{\rightarrow}}_{(1)}$ follows by $\sum_{t=2}^{n}K_{th}u_{t}=O_{p}(\beta_{n}/nh_{n})^{1/2}$, as follows e.g. from (S.45) in the proof of Theorem 4.2.

Suppose $\kappa_{\ast\ast}=\infty$. In this case, (S.69) implies that $\hat{\mu}-\mu$ is divergent at rate $r_{n}\kappa_{g_{1}n}$, and dominates $\hat{\gamma}-\gamma$. Thus by (S.66)

[TABLE]

In summary, in each case $\varsigma_{n}^{-1}\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle d}}{{\rightarrow}}\sigma_{\ast}^{2}$, where

[TABLE]

Note that in the case where $\kappa_{\ast\ast}=1$, we have $\lim_{n\rightarrow\infty}r_{n}\kappa_{g_{1}n}=\kappa_{\ast\ast}=1$, and so we may equivalently take either $\varsigma_{n}=1$ or $\varsigma_{n}=r_{n}^{2}\kappa_{g_{1}n}^{2}$; we present (S.70) in terms of the latter to facilitate the next part of the proof.

IV. Asymptotics of $\boldsymbol{\tilde{F}}$ under $\boldsymbol{\mathcal{H}_{1}}$. Since $g_{1}$ has bounded second derivative (by condition (v)), arguments similar to those given in part I of this proof and in the proof of Theorem 4.2 yield that, so long as $\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle p}}{{\rightarrow}}\sigma_{u}^{2}$ (and $nh_{n}^{5}/\beta_{n}\rightarrow 0$ as assumed),

[TABLE]

For future reference, we note here that

[TABLE]

and that if $\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle p}}{{\rightarrow}}\sigma_{u}^{2}$, then by Theorem 3.4

[TABLE]

which is a.s. nonzero.

We show below that in each of the cases contemplated by condition (iii), $\delta_{n}\tilde{t}(x;\hat{\mu},\hat{\gamma})\overset{d}{\rightarrow}\omega(x)$, where $\delta_{n}\rightarrow 0$, and $\omega(x)$ is nonzero for at least one $x\in\mathcal{X}$. That $\tilde{F}\overset{p}{\rightarrow}\infty$ then follows immediately from (24).

(a). Suppose condition (iii.a) holds. By the results of parts II(a) and III(a), and (S.68) in particular,

[TABLE]

Thus by (S.72) and recalling from part III(a) that $\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle p}}{{\rightarrow}}\sigma_{u}^{2}$

[TABLE]

whence, noting that $\beta_{n}/r_{n}^{2}nh_{n}\rightarrow 0$ by condition (iv),

[TABLE]

as required, by (S.73). By condition (vi), $g_{1}(x)\neq 0$ for at least one $x\in\mathcal{X}$.

(b). Suppose condition (iii.b) holds. Recall $\kappa_{\ast}:=\lim_{n\rightarrow\infty}\kappa_{g_{1}n}$ and $\kappa_{*\ast}:=\lim_{n\rightarrow\infty}\kappa_{g_{1}n}r_{n}$. We consider each of the cases contemplated by condition (iv), in turn.

Suppose $\kappa_{*}\in[0,\infty)$, which implies $\kappa_{\ast\ast}=0$. By the results of part II(b) and (S.69),

[TABLE]

By part III(b), $\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle p}}{{\rightarrow}}\sigma_{u}^{2}$ , and thus, similarly to the argument given in part IV(a), noting that $\beta_{n}/r_{n}^{2}nh_{n}\rightarrow 0$ also in the present case,

[TABLE]

and note that by condition (vi), $\kappa_{*}\mu_{*}+g_{1}(x)\neq 0$ for at least one $x\in\mathcal{X}$.

Suppose $\kappa_{*}=\infty$ and $\kappa_{\ast\ast}=0$. In this case, it remains true that $\tilde{\sigma}_{u}^{2}(x)\stackrel{{\scriptstyle p}}{{\rightarrow}}\sigma_{u}^{2}$ , by part III(b) of the proof. Moreover, by the results of part II(b) and (S.69),

[TABLE]

Thus under the assumption that $\beta_{n}/r_{n}^{2}\kappa_{g_{1}n}^{2}nh_{n}\rightarrow 0$, similar arguments as were used in the previous case yield

[TABLE]

which is nonzero a.s. by condition (vi).

Suppose $\kappa_{**}\in(0,\infty]$, which implies $\kappa_{*}=\infty$; as in part III(b), normalise $\kappa_{\ast\ast}=1$ if $\kappa_{\ast\ast}\in(0,\infty)$. We have

[TABLE]

for $\sigma_{\ast}^{2}$ as in (S.70). Thus $\tilde{\sigma}_{u}^{2}(x)$ is bounded away from zero, and possibly divergent, whence $t^{*}(x;\mu,\gamma)=O_{p}(1).$ In view of (S.74)–(S.75) and the fact that $\beta_{n}/nh_{n}\rightarrow 0$, it follows that

[TABLE]

which is nonzero a.s. by condition (vi). ∎

Appendix F Additional simulations for Section 6

This section provides the results of some additional simulations not reported in the main text. Table 4 is the counterpart of Table 2 in the main text, reporting size-adjusted power of the procedure for a type II fractional process, relative to Wang and Phillips (2012), for the cases where $d\in\{0.25,0.75\}$.

We also repeated the simulation exercise described in Section 6, but now allowing $x_{t}$ to be mildly (or nearly) integrated, that is of the form

[TABLE]

with $x_{0}(n)=0$. All other aspects of the simulation design are exactly as in Section 6. We generated data for $\alpha_{\kappa}\in\{0.25,0.50,0.75,1.00\}$; this last corresponds to a nearly integrated process. Size results are given in Table 3, while size-adjusted power relative to Wang and Phillips (2012) is reported in Tables 5 and 6.