Joint Time Series and Cross-Section Limit Theory under Mixingale Assumptions

TL;DR

This paper extends joint time series and cross-section limit theory to include serial correlation, providing new limiting distributions under strict stationarity and independence assumptions, relevant for models with aggregate uncertainty.

Contribution

It introduces a framework allowing serial correlation in joint limit theory, expanding the applicability of previous results to more realistic data scenarios.

Findings

Derived limiting distributions with long run variances for serially correlated data

Established conditions for stationarity and independence in joint convergence

Applicable to estimators combining time series and cross-section data

Abstract

In this paper we complement joint time series and cross-section convergence results of Hahn, Kuersteiner and Mazzocco (2016) by allowing for serial correlation in the time series sample. The implications of our analysis are limiting distributions that have a well known form of long run variances for the time series limit. We obtain these results at the cost of imposing strict stationarity for the time series model and conditional independence between the time series and cross-section samples. Our results can be applied to estimators that combine time series and cross-section data in the presence of aggregate uncertainty in models with rationally forward looking agents.

Equations

$$y_{i,t}={\rm Y}\left({\nu }_t,u_i\right).$$

$$\mu \left({\nu }_t\right)$$

$${\sigma }^2\left({\nu }_t\right)$$

$${\tilde{y}}_{i,1}\left(d\right)={\pi }_{1}d+u_i$$

$$\sqrt{n}\left(\hat{\beta }-\beta \right)=\left(1+{\nu }_1^2\right)\sqrt{n}\left({\hat{\pi }}_1-{\pi }_1\right)-2{\pi }_1{\nu }_1\sqrt{\frac{n}{\tau }}\sqrt{\tau }\left(\hat{\varphi }-\varphi \right)+o_p\left(1\right).$$

$$\hat{\pi }\left(\nu \right)=\hat{\beta }/\left(1+{\nu }^2\right).$$

$$\sqrt{n}\left(\hat{\pi }\left(\nu \right)-\pi \left(\nu \right)\right)=\frac{1+{\nu }_1^2}{1+{\nu }^2}\sqrt{n}\left({\hat{\pi }}_1-{\pi }_1\right)-\frac{2{\pi }_1{\nu }_1}{1+{\nu }^2}\sqrt{\frac{n}{\tau }}\sqrt{\tau }\left(\hat{\varphi }-\varphi \right)+o_p\left(1\right)$$

$$E\left[\bar{\zeta }\exp \left(i{\ell }^{\prime }{U}_{n,\tau }\right)\right]\to E\left[\bar{\zeta }\exp \left(i{\ell }^{\prime }U\right)\right]$$

$$\underset{n,\tau }{\lim }E\left[\exp \left(i{\ell }^{\prime }{U}_{n,\tau }\right)|\mathcal{C}\right]=E\left[\exp \left(i{\ell }^{\prime }U\right)|\mathcal{C}\right]\text{ a.s.}$$

$$\left|E\left[\bar{\zeta }\left(\exp \left(i{\ell }^{\prime }{U}_{n,\tau }\right)-\exp \left(i{\ell }^{\prime }U\right)\right)\right]\right|$$

$$\le E\left[\left|\bar{\zeta }\right|\left|E\left[\exp \left(i{\ell }^{\prime }{U}_{n,\tau }\right)-\exp \left(i{\ell }^{\prime }U\right)|\mathcal{C}\right]\right|\right].$$

$$\frac{1}{\sqrt{n}}\sum _{i=1}^n{y}_{i,1}$$

$$\underset{n\to \infty }{\lim }E\left[e^{it{Y}_n}|\mathcal{A}\right]=e^{-\frac{1}{2}{t}^2{\sigma }^2\left({\nu }_1\right)}\text{ a.s.}$$

$$\underset{\tau \to \infty }{\lim }{\Vert E\left[\phi \left(Z_{\tau }\right)-\int \phi \left(x\sqrt{\eta }\right)g\left(x\right)dx|{\mathcal{M}}_k\right]\Vert }_1=0$$

$$\eta =E\left[{\nu }_0^2\right]+2\sum _{s=1}^{\infty }E\left[{\nu }_0{\nu }_s\right]$$

$$P\left(Z\le z|\mathcal{A}\right)={\int }_{-\infty }^{z/\sqrt{\eta }}g\left(x\right)dx\text{ a.s.}$$

$$P\left(Y\le y|\mathcal{A}\right)={\int }_{-\infty }^{y/\sqrt{{\sigma }^2\left({\nu }_1\right)}}g\left(x\right)dx\text{ a.s.}$$

$$y_{i,t}={{\rm Y}}_t\left({\nu }_1,\dots ,{\nu }_T,u_i\right).$$

$$\underset{n\to \infty }{\lim }E\left[e^{it{Y}_n}|{\mathcal{M}}_T\right]=e^{-\frac{1}{2}{t}^2{\sigma }^2\left({\nu }_1,...,{\nu }_T\right)},$$

$$\frac{1}{\sqrt{\tau }}\sum _{s=1}^{\tau }{\nu }_s=\frac{1}{\sqrt{\tau }}\sum _{s=1}^T{\nu }_s+\frac{\sqrt{\tau -T}}{\sqrt{\tau }}\frac{1}{\sqrt{\tau -T}}\sum _{s=T+1}^{\tau }{\nu }_s=\frac{1}{\sqrt{\tau -T}}\sum _{s=T+1}^{\tau }{\nu }_s+o_p\left(1\right)$$

$$\left(\sqrt{n}\left({\hat{\pi }}_1-{\pi }_1\right),\sqrt{\tau }\left(\hat{\varphi }-\varphi \right)\right){\to }_d\left(Y,Z\right)(\mathcal{A}\text{-stably})$$

$$\sqrt{n}\left({\hat{\pi }}_1-{\pi }_1\right)=\sqrt{n}\sum _{i=1}^n{w}_i{u}_i.$$

$$\sqrt{n}\left(\hat{\beta }-\beta \right){\to }_d\left(1+{\nu }_1^2\right)\left(2{\sigma }_u{\xi }_y\right)-2{\pi }_1{\nu }_1\sqrt{\kappa }\left(\sqrt{\eta }{\xi }_{\nu }\right)(\mathcal{A}\text{-stably}).$$

$$4\left({\left(1+{\nu }_1^2\right)}^2{\sigma }_u^2+\kappa \eta {\pi }_1^2{\nu }_1^2\right),$$

$${{\rm Y}}_n=\frac{\sqrt{n}\left(\hat{\beta }-\beta \right)}{2{\left({\left(1+{\nu }_1^2\right)}^2{\sigma }_u^2+\kappa \eta {\pi }_1^2{\nu }_1^2\right)}^{1/2}}$$

$${{\rm Y}}_n{\to }_d\xi \sim N\left(0,1\right).$$

$${\widehat{{\rm Y}}}_n=\frac{\sqrt{n}\left(\hat{\beta }-{\beta }_0\right)}{2{\left({\left(1+{\hat{\nu }}_1^2\right)}^2{\hat{\sigma }}_u^2+\kappa \hat{\eta }{\hat{\pi }}_1^2{\hat{\nu }}_1^2\right)}^{1/2}},$$

$$E_{{\omega }^{\prime }}\left[y_{i,1}^2|\mathcal{A}\right]\left(\omega \right)=E\left[y_{i,1}^2|\mathcal{A}\right]\left(\omega \right)Q_{{\omega }^{\prime }}\text{-a.s.}$$

$${\nu }_1\left(\omega \right)={\nu }_1\left({\omega }^{\prime }\right)Q_{{\omega }^{\prime }}\text{-a.s.;}$$

$$\frac{1}{n{S}^2\left({\nu }_1\left({\omega }^{\prime }\right)\right)}\sum _{i=1}^n{E}_{{\omega }^{\prime }}\left[y_{i,1}^{2}1\left\{\left|y_{i,1}\right|>\epsilon \sqrt{n}\right\}|\mathcal{A}\right]\le \frac{E_{{\omega }^{\prime }}\left[{\left|y_{1,1}\right|}^{2+\delta }|\mathcal{A}\right]\left(\omega \right)}{{\sigma }^2\left({\nu }_1\left({\omega }^{\prime }\right)\right){\left(\epsilon \sqrt{n}\right)}^{\delta }}\to 0,Q_{{\omega }^{\prime }}\text{ a.s.}$$

$$E_{{\omega }^{\prime }}\left[{\left|y_{1,1}\right|}^{2+\delta }|\mathcal{A}\right]\left(\omega \right)=E\left[{\left|y_{1,1}\right|}^{2+\delta }|\mathcal{A}\right]\left(\omega \right)Q_{{\omega }^{\prime }}\text{-a.s.}$$

Full text

Keywords: joint limiting distribution, data combination, stable convergence, external validity

1 Introduction

Aggregate shocks play an important role in decision making of rational agents. The presence of aggregate shocks complicates model specification and inference in cross-sectional settings. In Hahn, Kuersteiner and Mazzocco (2015,2019; henceforth HKM2019) we present three economic models that illustrate the challenges applied researchers face when trying to account for aggregate uncertainty. Because cross-sectional samples contain no information about the stochastic processes that generate aggregate uncertainty, we proposed to combine cross-sectional data with long historical time series in structural models that explicitly take into account how agents react to aggregate shocks.

Aggregate uncertainty is also considered by Rosenzweig and Udry (2019) in the context of treatment effect estimators. In this paper we use a stylized version of their model to illustrate how the methods proposed in HKM2019 can be used to evaluate causal parameters under hypothetical scenarios regarding the realization of an aggregate shock. Our method can be seen as a way to address problems of external validity in the presence of aggregate shocks.

A rigorous foundation of our proposed empirical strategy requires central limit theorems for the joint convergence of time series and cross-sectional averages. The analysis of joint limits is complicated by the fact that scaling factors resulting from linearized versions of our non-linear estimators often depend on aggregate shocks, leading to a lack of stochastic independence between the cross-section and time series samples. This lack of independence does not disappear in large samples due to cross-sectional non-ergodicity with regard to aggregate shocks, which we addressed by considering stable111The concept of stable convergence was introduced by Renyi (1963). It was further developed by Aldous and Eagleson (1978) and has found numerous applications in probability theory, statistics and econometrics. rather than weak convergence in distribution. Our previous asymptotic results were predicated on some martingale assumption of the time series data, which may not be satisfied in various applications. Therefore it is useful to develop further stable convergence results under different sets of assumptions.** **

This paper develops a joint time-series and cross-sectional central limit theorem (CLT) for strictly stationary mixingale time series and, conditional on an aggregate shock, independently distributed cross-sectional random variables. Mixingales introduced by Gordin (1973) and McLeish (1975) are a flexible tool to display temporal dependence in time series settings. We use the concept to capture situations where estimators are based on moment conditions or pseudo-likelihood functions that suffer from some form of misspecification. The form of misspecification we have in mind is not strong enough to affect consistency or convergence in distribution. However, because the criterion function is not exactly a conditional mean or the score of a correctly specified likelihood, the martingale difference properties associated with these criteria do not apply to the estimators we have in mind in this paper. ** **Rather, limiting variances display the type of long run variance expressions typically found in heteroskedasticity and autocorrelation consistent (HAC) variance estimators.

The results obtained in this paper differ both in terms of assumptions, scope and proof strategies from our related work in Hahn, Kuersteiner and Mazzocco (2016, HKM2016 hereafter). In that paper, we do not assume stationarity of the time series data or independence of the cross-sectional data conditional on aggregate shocks. On the other hand, we do impose martingale difference conditions both in the cross-section and time series dimension. The results in HKM2016 then are directed towards the case of correctly specified estimators.222In addition, our (2016) paper derives functional central limit theorems while in this paper we focus on the finite dimensional case.

The proof strategy for finite dimensional convergence in the two papers also differs markedly. Here, we exploit conditional independence between the time series and cross-section and base our proofs on marginal stable convergence of time series sample averages and almost sure conditional convergence in distribution of cross-sectional sample averages. In HKM2016, we do not assume conditional independence between cross-section and time series and directly establish a joint stable martingale difference CLT by nesting the time-series and cross-section samples in a spatial filtration similar to filtrations proposed in Kuersteiner and Prucha (2013). The proof of the martingale difference CLT then is based on establishing weak convergence in $L_{1}$ of the characteristic function.333See Aldous and Eagleson (1978) for a definition of weak convergence in $L_{1}$. Since it is not clear how to map cross-sectional data into a framework of strict stationarity, a direct proof of joint convergence for the time series and cross-sections as in HKM2016 does not seem to be obvious under the assumptions imposed in this paper.

In this paper, we get around the above-mentioned difficulty by establishing two separate limiting results, and combining them using a conditional independence assumption.** **First, we establish cross-sectional stable convergence by adapting an argument in Eagleson (1975) to show that the conditional (on aggregate shocks) characteristic function of the cross-sectional average converges almost surely, which implies stable convergence. Second, we use the stable central limit theorem for strictly stationary mixingale time series by Dedecker and Merlevède (2002) to establish stable convergence of our time series average.444To the best of our knowledge Dedecker and Merlevède (2002), and Ikeda (2017) are the only two stable CLTs for mixingales. Ikeda’s results, in particular his assumptions about conditional variances (Ikeda (2017), Theorem 1, Assumption (e)), are geared towards applications in finance. It is not clear that they could easily be adapted to the applications we have in mind. We thus rely on Dedecker and Merlevède (2002). Finally, we combine the two limiting results under the assumption that the time series and cross-section samples are independent conditional on the aggregate shock. This is done by exploiting insights from Barndorff-Nielsen, Hansen, Lunde and Shephard (2008) who analyzed joint stable convergence of a random sequence that converges stably and a random sequence that converges conditionally in law.

2 Model and Motivating Example

We consider asymptotics for a combined cross-section and time series data set where the cross-sectional units are iid and the time series units are strictly stationary. For concreteness we assume an explicit generating mechanism that describes the relationship between the time series variables $\nu_{t}$ and an iid individual specific random variable $u_{i}$ such that the cross-sectional variables $y_{i,t}$ are generated by the model****

[TABLE]

We consider two samples, $\left\{y_{i,t}:i=1,\ldots,n\right\}$ for some fixed $t$ and $\left\{\nu_{t}:t=1,\ldots,\tau\right\}$, and are interested in the joint distribution of normalized averages of the random variables in these two samples. Here, the function $\Upsilon$ is measurable with respect to an underlying probability space to be defined in more detail below. The idea behind the specification for $y_{i,t}$ is that the individual agent’s behavior can be fundamentally assumed to be a function of the two vectors $\nu_{t}$ and $u_{i}$. We also assume strict exogeneity in the sense that the collection $\left\{u_{i}\right\}_{i=1}^{\infty}$ is independent of the collection $\left\{\nu_{s}\right\}_{s=1}^{\infty}$.

Then, conditional on $\nu_{t}$, the mean and variance of $y_{i,t}$ can be written as

[TABLE]

where $f_{u}\left(\cdot\right)$ denotes the marginal density of $u_{i}$. In our applications, the cross sectional moments are correctly specified which is equivalent to $\mu\left(\nu_{t}\right)=0$. To allow for serial correlation, we assume that $\left\{\nu_{s}\right\}_{s=1}^{\infty}$ is a mixingale. Mixingales are defined as processes whose mean conditional on past values eventually converges to zero as the conditioning set is moved to the more distant past. They therefore allow for serial correlation for any finite time horizon. A more precise technical definition is given below in Section 3.2. For simplicity, we assume that the cross-sectional sample is observed at time $t=1$ and that the time-series sample is collected starting at $t=1.$

Below we provide an example illustrating that cross-sectional parameters of interest may depend on parameters governing the time-series process. Consistent estimation of these cross-sectional parameters can be achieved by using plug-in estimates based on time-series estimates. Asymptotic approximations to the two-step cross-sectional estimator require joint limiting distributions of the cross-sectional and time-series estimates. Because the cross-section depends on the time series by (1), limit results cannot be obtained for the cross-sectional and time-series samples separately. Rather, convergence needs to be shown to hold jointly between the cross-sectional and time-series samples as well as an initial realization of the time-series process $\nu_{1}.$ This is accomplished by establishing joint stable convergence of $\left(\frac{1}{\sqrt{n}}\sum_{i=1}^{n}y_{i,1},\frac{1}{\sqrt{\tau}}\sum_{s=1}^{\tau}\nu_{s}\right)$. To prove joint stable convergence we use an argument that combines several results. The details are laid out in the subsequent sections.

To motivate the need for joint limiting distributions consider the following simple stylized example of our general framework. Assume that $\tilde{y}_{i,t}$ is an outcome of interest, observed for individual $i$ at time time $t.$ The individual is subject to a policy experiment $d_{i,t}.$ The outcome $\tilde{y}_{i,1}$ in period $t=1$ is determined by the following linear potential outcomes model

[TABLE]

where $d$ is a fixed constant and $\pi_{t}$ is the time varying causal effect of interest. For example, when $d_{i,t}$ is binary such that $d\in\left\{0,1\right\}$ the model describes the potential outcomes $\tilde{y}_{i,1}\left(0\right)$ and $\tilde{y}_{i,1}\left(1\right)$ with causal effect $\tilde{y}_{i,1}\left(1\right)-\tilde{y}_{i,1}\left(0\right)=\pi_{1}$. Typical treatment parameters such as the average treatment effect or the treatment effect on the treated are defined as cross-sectional averages $E\left[\tilde{y}_{i,1}\left(1\right)-\tilde{y}_{i,1}\left(0\right)\right]$ and $E\left[\tilde{y}_{i,1}\left(1\right)-\tilde{y}_{i,1}\left(0\right)|d_{i,1}=1\right]$ and coincide with $\pi_{1}.$ This is because we abstract from treatment heterogeneity in our example such that the parameter $\pi_{1}$ does not vary with $i$. Realized or observed values of $\tilde{y}_{i,1}$ satisfy the constraint that $\tilde{y}_{i,1}=\tilde{y}_{i,1}\left(d_{i,1}\right)$ for realized values of the policy variable $d_{i,1}$.

The parameter $\pi_{1}$ is identified from a random sample in the cross-section with random assignment of individuals to treatment. Assume that we observe a sample $\left(\tilde{y}_{i,1},d_{i,1}\right)$ of size $n.$** **Assume that (i) $\left(d_{i,1},u_{i}\right)$ is IID; (ii) $d_{i,1}$ is independent of $u_{i}$; and (iii) $\Pr\left(d_{i,1}=1\right)=\frac{1}{2}$, which is known to the econometrician.555Knowledge of the propensity score is used to simplify the calculation of standard errors in Section 4. A consistent estimator for $\pi_{1}$ is obtained from $\hat{\pi}_{1}=\sum_{i=1}^{n}w_{i}\tilde{y}_{i,1}$ where $w_{i}=d_{i,1}/\sum_{i=1}^{n}d_{i,1}-\left(1-d_{i,1}\right)/\sum_{i=1}^{n}\left(1-d_{i,1}\right)$.

The parameter $\pi_{t}$ possibly varies with time. For example, a job training program may be less effective in reducing unemployment during a recession than during an economic boom. If an investigator had cross-sectional data at a particular point in time and the goal were to make general predictions about the efficacy of such a job training program then the evidence obtained from a randomized study may not be informative about the promise of the program at a given future point in time. We adopt the approach of HKM2019, and address this problem by parametrizing the functional form of $\pi_{t}$. In HKM2019, we motivate such parametrizations with structural economic models that explain individual decision making by rational economic agents. Similarly, Rosenzweig and Udry (2019) consider models of investment behavior where the return to investment depends on the realization of an aggregate shock.

To keep the exposition simple we postulate a simple stylized parametric model for $\pi_{t}$ in this paper. Assume that there is an aggregate economic shock $\nu_{t}$ that affects the causal effect of the policy in a multiplicative way such that $\pi_{t}=\left.\beta\right/\left(1+\nu_{t}^{2}\right)$. Here, $\beta$ is interpreted as a deep structural parameter that can be used to impute the causal effect for values of the economic shock $\nu_{t}$ that may not have realized during the time period when the cross-sectional sample was observed. Further assume that we have time series data $z_{s}$ that satisfies the dynamic equation $z_{s}=\phi+\nu_{s}$ where $\phi$ is a fixed time invariant constant and $\nu_{s}$ is an unobserved stationary mixingale. If the parameter $\phi$ were known, the structural parameter $\beta$ could be identified from the relationship $\beta=\pi_{1}\left(1+\left(z_{1}-\phi\right)^{2}\right)$. The solution we propose in HKM2019 consists in using a time series sample $z_{s}$ for $s=1,...,\tau$ to estimate the parameter $\phi.$ In our simple setting this amounts to forming the time series average $\hat{\phi}=\tau^{-1}\sum_{s=1}^{\tau}z_{s}=\phi+\tau^{-1}\sum_{s=1}^{\tau}\nu_{s}.$

The specification of $\beta$ captures the idea that the causal effect of $d_{i,t}$ on $y_{i,t}$ is dampened in periods where $z_{t}$ deviates from its long term stationary mean $\phi.$ The parameter $\phi$ could measure an optimal growth path with deviations from it reducing the efficacy of policy interventions or investments. Since deviations may be persistent, it is plausible that $\nu_{s}$ exhibits serial correlation. For example, if $\nu_{s}=\theta\nu_{s-1}+\varepsilon_{s}$ with $\varepsilon_{s}\sim N\left(0,1\right)$ and $\left|\theta\right|<1$ it is easy to see that $E\left[\nu_{s}|\nu_{s-k}\right]=\theta^{k}\nu_{s-k}$ and thus that $\nu_{s}$ is a mixingale according to the definition given in Section 3.2. While the parameter $\phi$ can be consistently estimated as a simple sample average, the sample average is not the maximum likelihood estimator in this scenario of a Gaussian autoregressive process. As a result, the influence function of $\hat{\phi}$ is not a martingale difference sequence, as would be the case for the maximum likelihood estimator. Thus, the results in HKM2016, which require martingale difference sequences, cannot be applied to the case where the investigator is using an inefficient estimator for the parameter $\phi.$ In this paper, we develop the necessary asymptotic theory to handle cases where the influence function of $\phi$ is a more general mixingale process.

The estimators $\hat{\pi}_{1}$ and $\hat{\phi}$ obtained from the cross-section and time series samples respectively can be combined to form the estimator $\hat{\beta}=\hat{\pi}_{1}\left(1+\left(z_{1}-\hat{\phi}\right)^{2}\right).$ In HKM2019 we are mostly interested in inference for the parameter $\beta$. Using a Taylor series expansion one obtains

[TABLE]

Under an asymptotic sequence where $\sqrt{\frac{n}{\tau}}\rightarrow\sqrt{\kappa}$ for $0<\kappa<\infty$ the limiting distribution of $\hat{\beta}$ is determined by the joint limiting distribution of $\hat{\pi}_{1}$ and $\hat{\phi}$ conditional on $\nu_{1}.$ Because both the cross-sectional and the time series data depend on $\nu_{1},$ establishing joint convergence requires special care. In HKM2016 we use the concept of stable convergence. In this paper, a somewhat different approach, based on almost sure conditional convergence is used. We also show that the latter implies the former.

In addition to the structural parameter $\beta$ we may be interested in causal effects outside of the observed cross-sectional sample. Using $\hat{\beta}$ we are in a position to impute the causal effects for arbitrary values of the aggregate shock $\nu_{s}$. For a fixed value of $\nu,$ and thus indirectly, $z=\phi+\nu,$ define the counterfactual causal effect $\pi\left(\nu\right)=\left.\beta\right/\left(1+\nu^{2}\right)$ which can be estimated by

[TABLE]

Inference for $\hat{\pi}\left(\nu\right)$ is based on a Taylor expansion and the result in (3)

[TABLE]

and relies on the same joint asymptotic limits as inference for $\hat{\beta}$.

3 Limit Theory

The analysis of estimators such as $\hat{\beta}$ or $\hat{\pi}\left(\nu\right)$ depends on the joint convergence of the tuple $\left(U_{n,\tau},\zeta\right)$ as $n,\tau\rightarrow\infty$, where $\zeta$ is any measurable function of $\nu_{1}$, and where $U_{n,\tau}$ is defined as $U_{n,\tau}\equiv\left(\frac{1}{\sqrt{n}}\sum_{i=n}^{n}y_{i,1},\frac{1}{\sqrt{\tau}}\sum_{s=1}^{\tau}\nu_{s}\right)$. For ease of exposition we assume that $y_{i,1}$ and $\nu_{s}$ take values in $\mathbb{R}$. Extensions to the multivariate case using the Cramer-Wold theorem are straight forward.

This form of convergence where the component $\zeta$ does not depend on $n$ or $\tau$ was first discussed by Renyi (1963). To formally define it, let $\left(\Omega,\mathcal{F},P\right)$ be a probability space with a sub-sigma field $\mathcal{C}$ and assume that $B$ is any $\mathcal{C}$-measurable event. Following Aldous and Eagleson (1978), the sequence $U_{n,\tau}$ is said to converge $\mathcal{C}$-stably to a random variable $U$ defined on $\left(\Omega,\mathcal{F},P\right)$ if $\lim_{n,\tau}P\left(U_{n,\tau}\leq x,B\right)$ exists for a countable dense set of points $x$ and every set $B$ that is measurable with respect to $\mathcal{C}$. As pointed out by Aldous and Eagleson (1978), stable convergence implies that for every $B$ with $P\left(B\right)>0$ the conditional (on $B$) distribution of $U_{n,\tau}$ converges to a well defined limit. Aldous and Eagleson (1978) also show that stable convergence is equivalent to weak $L_{1}$ convergence of the characteristic function of $U_{n,\tau}.$ Formally, let $\bar{\zeta}$ be any bounded $\mathcal{C}$-measurable random variable and $\ell$ any fixed vector conforming with $U_{n,\tau}.$ Then, $U_{n,\tau}$ converges weakly in $L_{1}$ to $U$ if

[TABLE]

for $i=\sqrt{-1}.$ The notion of weak $L_{1}$ convergence of the characteristic function plays a key role in the proof of stable central limit theorems in Hall and Heyde (1980) and Kuersteiner and Prucha (2013). The proof strategy in these papers can be traced back to McLeish (1974). Eagleson (1975) on the other hand proves a stable CLT for martingale differences by showing that the conditional characteristic function converges almost surely. Formally, this is stated as

[TABLE]

Using iterated expectations, which can be defined because $U_{n,\tau},$ $U$ and $\bar{\zeta}$ all live on the same probability space, it follows that

[TABLE]

By the fact that the characteristic function is uniformly bounded and using the Lesbesgue convergence theorem it follows that almost sure convergence of the conditional characteristic function implies stable convergence.

In HKM2016 we directly establish (5) for a martingale difference array by nesting $U_{n,\tau}$ into an increasing spatial filtration. The construction used in that paper does not require stationarity but exploits existing central limit theorems that are specific to martingale difference sequences. In this paper, we relax the martingale difference assumption at the cost of imposing stationarity. Another feature of our setting in this paper not required in HKM2016 is crucial for the argument: We assume that conditional on $\mathcal{C}$ the components $Y_{n}=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}y_{i,1}$ and $Z_{\tau}=\frac{1}{\sqrt{\tau}}\sum_{s=1}^{\tau}\nu_{s}$ are independent for all $n$ and $\tau.$ This allows to multiplicatively separate $E\left[\exp\left(i\ell^{\prime}U_{n,\tau}\right)|\mathcal{C}\right]$ into two components. We then show that the conditional characteristic function of $Y_{n}$ converges almost surely. This almost sure conditional convergence turns out to be the critical component in showing that (5) holds jointly. This argument is made rigorous in the subsequent discussion.

3.1 Conditional Cross-Sectional CLT

This section establishes a conditional central limit theorem for

[TABLE]

conditional on $\nu_{1}$. Recall that $\mu\left(\nu_{1}\right)$ and $\sigma^{2}\left(\nu_{1}\right)$ denote the conditional mean and variance of $y_{i,1}=\Upsilon\left(\nu_{1},u_{i}\right)$ given $\nu_{1}$, and that we assume $\mu\left(\nu_{1}\right)=0$ in our application.

The proof follows the arguments in Eagleson (1975) and van der Vaart and Wellner (1996, Lemma 2.9.5) by specializing the regularity conditions in Eagleson (1975) to the conditionally independent case. Some additional notation is required to clearly define the probability space involved in the construction. Consider the product space $\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}$ with Borel fields666The sigma algebra $\mathcal{B}^{\infty}$ is the smallest sigma algebra that contains all the sets $\left\{x\in\mathbb{R}^{\infty}|x=\left(x_{1},x_{2}...,\right),x_{1}\in I_{1},x_{2}\in I_{2},...x_{m}\in I_{m}\right\}$ where $I_{j}$ is an interval $(a,b]$ on $\mathbb{R}$ - see Shiryaev (1995, p.146). $\mathcal{B}^{\infty}\times\mathcal{B}^{\infty}$ and product measure $P=P_{\nu}\times P_{u}$. The infinite dimensional vectors $\nu=\left(\nu_{1},\nu_{2},...\right)^{\prime}$ and $u=\left(u_{1},u_{2},...\right)^{\prime}$ take values in $\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}$. Assume that $\left(\nu,u\right)$ is $\mathcal{X}$-measurable where $\mathcal{X=B}^{\infty}\times\mathcal{B}^{\infty}$. A probability space of this form can be constructed using Kolmogorov’s existence theorem (see Billingsley 1995, p.486). Suppose that $\nu_{1}$ is a $\mathcal{A}$-measurable random variable, where $\mathcal{A}$ is a sub-sigma field of $\mathcal{X}$ such that $\mathcal{A}\subset\mathcal{X}$. Consider the probability space $\left(\mathbb{R}^{\infty}\times\mathbb{R}^{\infty},\mathcal{X},P\right)$. By Breiman (1992, Theorem 4.34 and A.46), as long as the sample space is a complete separable metric space, a regular conditional distribution on $\mathcal{X}$ given $\mathcal{A}\subset\mathcal{X}$ exists. As in Eagleson, let $\omega^{\prime}\in\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}$ and consider the regular conditional probability denoted by $Q_{\omega^{\prime}}\left(B,\mathcal{A}\right)=Q_{\omega^{\prime}}\left(B\right)$. It follows that for fixed $B\in\mathcal{X}$, $Q_{\omega^{\prime}}\left(B,\mathcal{A}\right)$ is a version of $P\left(\left.B\right|\mathcal{A}\right)$ and for fixed $\omega^{\prime}\in\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}$, $Q_{\omega^{\prime}}\left(\cdot\right)$ is a probability measure on $\mathcal{X}$.

Consider the measure space $\left(\mathbb{R}^{\infty}\times\mathbb{R}^{\infty},\mathcal{X},Q_{\omega^{\prime}}\right)$ with expectation $E_{\omega^{\prime}}$. By a Lemma in Eagleson (1975, p.558) the following holds: Let $\mathcal{G}$ be a sub-sigma field of $\mathcal{X}$ such that $\mathcal{A}\subseteq\mathcal{G}$.777It is easy to see that the proof of Lemma 1 in Eagleson (1975) goes through when $\mathcal{A=G}$. Then, for $P$ almost all $\omega^{\prime}\in\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}$ and a random variable $Y$ with $E\left|Y\right|<\infty$ it follows that $E_{\omega^{\prime}}\left[\left.Y\right|\mathcal{G}\right]\left(\omega\right)=E\left[\left.Y\right|\mathcal{G}\right]\left(\omega\right)$ $Q_{\omega^{\prime}}$-a.s.

Define $S_{n}\equiv\sum_{i=1}^{n}y_{i,1}$, and $Y_{n}\equiv\left.S_{n}\right/\sqrt{n}$. We now have the following result which is established by combining arguments in the proofs of Eagleson (1975, Theorem 2) and van der Vaart and Wellner (1996, Lemma 2.9.5).888Lemma 2.9.5 of van der Vaart and Wellner (1996) does not directly apply to our context, because of the conditional nature of our problem. However, our proof uses some of the arguments in their proof.

Theorem 1

Assume that for some $\delta>0$ it follows that $E\left[\left.\left|y_{i,1}\right|^{2+\delta}\right|\mathcal{A}\right]\leq K<\infty$ for some constant $K.$ Then, conditional on $\mathcal{A}$, $S_{n}$ converges in law as $n\rightarrow\infty$ to a normal distribution. In addition

[TABLE]

Proof. See Appendix.

3.2 Stable Time Series CLT

We follow Dedecker and Merlevède (2002) in defining a stationary sequence. Consider the probability space $\left(\mathbb{R}^{\infty},\mathcal{B}^{\infty},P_{\nu}\right)$ which is the second coordinate of $\left(\mathbb{R}^{\infty}\times\mathbb{R}^{\infty},\mathcal{X},P\right).$ Let $T:\mathbb{R}^{\infty}\rightarrow\mathbb{R}^{\infty}$ be a bijective bimeasurable transformation preserving $P_{\nu}.$ Let $\mathcal{M}_{0}$ be a sigma-algebra of $\mathcal{B}^{\infty}$** **and let $\nu_{0}$ be $\mathcal{M}_{0}$-measurable. Assume that $T$ satisfies $T^{-1}\mathcal{M}_{0}\supset\mathcal{M}_{0}$.999A mapping $T$ with this property is called compressible, see Halmos (1956, p.11). Define the non-decreasing filtration $\mathcal{M}_{i}=T^{-i}\left(\mathcal{M}_{0}\right)$. Define the sequence $\nu_{s}=\nu_{0}\left(T^{s}\omega\right)$ for $\omega\in\mathbb{R}^{\infty}$. Let $I$ be an invariant set $T^{-1}I=I$. Let $\mathcal{I}$ be the sigma-algebra of all invariant sets.

Now impose the following assumptions that correspond to Dedecker and Merlevède (2002, Theorem 1, s2.). Let $\left\|X\right\|_{p}\equiv\left(E\left[\left|X\right|^{p}\right]\right)^{1/p}$ for any random variable $X\in\mathbb{R}$ defined on the probability space $\left(\mathbb{R}^{\infty}\times\mathbb{R}^{\infty},\mathcal{X},P\right)$.

Condition 1

*Let $S_{\nu,\tau}\equiv\sum_{s=1}^{\tau}\nu_{s}$ and $Z_{\tau}\equiv\left.S_{\nu,\tau}\right/\sqrt{\tau}$.

i) The sequence $Z_{\tau}^{2}$ is uniformly integrable.

ii) The sequence $\left\|E\left[Z_{\tau}|\mathcal{M}_{0}\right]\right\|_{1}\rightarrow 0$ as $\tau\rightarrow\infty.$

iii) There exists a nonnegative $\mathcal{M}_{0}$ measurable variable $\eta$ with $\eta\left(\omega\right)=\eta\left(T\omega\right)$ for all $\omega\in\mathbb{R}^{\infty}$ such that $\left\|E\left[\left.Z_{\tau}^{2}-\eta\right|\mathcal{M}_{0}\right]\right\|_{1}\rightarrow 0$ as $\tau\rightarrow\infty.$*

Let $\varphi$ be a continuous function $\varphi:\mathbb{R\rightarrow}\mathbb{R}$ such that $\left|\left(1+x^{2}\right)^{-1}\varphi\left(x\right)\right|$ is bounded. Let $g\left(x\right)$ be the standard Gaussian density. If Condition 1 holds then it follows from Dedecker and Merlevède (2002, Theorem 1) that

[TABLE]

for every positive integer $k$. In particular, it holds for $\mathcal{M}_{1}$, which can be taken to be equivalent to $\mathcal{A}$ in the previous section. The next Lemma formally establishes a relationship between (6) and the $L_{1}$ based definition of stable convergence given in Aldous and Eagleson (1978).

Lemma 1

Assume that Condition 1 holds. Then, $Z_{\tau}\rightarrow_{d}\sqrt{\eta}\xi_{\nu}$ ($\mathcal{A}$-stably) where $\eta$ is $\mathcal{A}$-measurable and $\xi_{\nu}$ is standard Gaussian independent of $\mathcal{A}$.

Proof. See Appendix.

Low level conditions for Condition 1 can be given using the results of Dedecker and Doukhan (2003). Let $\left\{\nu_{s}\right\}_{s\geq 0}$ be a sequence of real valued stationary random variables. Define the mixingale coefficient $\gamma_{s}\equiv\left\|E\left[\nu_{s}|\mathcal{M}_{0}\right]-E\left[\nu_{s}\right]\right\|_{1}$. The definition of $\gamma_{t}$ goes back to Gordin (1973) and for an analogous definition based on the $L_{2}$ norm to McLeish (1975). Modified to account for stationarity and represented in terms of $L_{1}$ norms, McLeish’s definition of a mixingale can be stated as follows: A sequence $\left(\nu_{s},\mathcal{M}_{s}\right)$ is a mixingale if for finite non-negative constants $c$ and $\gamma_{t}$ and all $s\geq 0,$ $k\geq 0$ it follows that $\left\|E\left[\nu_{k}|\mathcal{M}_{0}\right]-E\left[\nu_{k}\right]\right\|_{1}\leq c\gamma_{k}$ and $\left\|\nu_{s}-E\left[\nu_{s}|\mathcal{M}_{s+k}\right]\right\|_{1}\leq c\gamma_{k+1}$ and $\gamma_{k}\rightarrow 0$ as $k\rightarrow\infty$. Note in particular that the definition in McLeish takes a slightly more complicated form in terms of the bounding constant $c$ to account for possible non-stationarity of the process. In addition, because here $\nu_{s}$** **is adapted to the filtration $\mathcal{M}_{s}$ by construction, it follows that the second condition of McLeish (1975) is automatically satisfied since $\left\|\nu_{s}-E\left[\nu_{s}|\mathcal{M}_{s+k}\right]\right\|_{1}=0$.

As in Dedecker and Doukhan (2003, Lemma 2), we also assume the following:

Condition 2

*One of the three conditions below hold:

(i) $P\left(\left|\nu_{s}\right|>x\right)\leq\left(c/x\right)^{r}$ for some $r>2$ and $\sum_{k\geq 0}\left(\gamma_{k}/2\right)^{\left(r-2\right)/\left(r-1\right)}<\infty$

(ii) $\left\|\nu_{s}\right\|_{r}\leq\infty$ for some $r>2$ and $\sum_{k\geq 0}k^{1/\left(r-2\right)}\gamma_{k}<\infty$

(iii) $E\left[\left|\nu_{s}\right|^{2}\left(\ln\left(1+\left|\nu_{s}\right|\right)\right)\right]<\infty$ and $\gamma_{k}=O\left(a^{k}\right)$ for some $a<1$.*

The mixingale type assumptions made in Condition 2 represent the typical trade-off between the tail thickness of the marginal distribution of $\nu_{t}$ and the rate of decay of the mixingale coefficients $\gamma_{i}$. An application of Dedecker and Doukhan (2003, Lemma 2) shows that the conditions in Condition 2 imply that the condition $D\left(2,\gamma/2,\nu\right)$, defined in Dedecker and Doukhan (2003, p. 71), holds. It then follows by Dedecker and Doukhan (2003, Proposition 2) that $\operatorname*{Var}\left(\tau^{-1/2}S_{\nu,\tau}\right)$ converges. In addition, by Dedecker and Doukhan (2003, Corollary

1. it follows that $\nu_{0}E\left[S_{\nu,\tau}|\mathcal{M}_{0}\right]$ converges in $L_{1}$. By Dedecker and Rio (2000, Theorem 1) it now follows that $E\left[\nu_{0}^{2}|\mathcal{I}\right]+2E\left[\nu_{0}S_{\nu,\tau}|\mathcal{I}\right]$ converges in $L_{1}$ to an $\mathcal{I}$-measurable random variable $\eta$**, where $\mathcal{I}$ **denotes the sigma-algebra of all invariant sets. If $T$ is ergodic then $\eta$ is a constant equal to

[TABLE]

by Dedecker and Rio (2000, Remark 1). Note that (7) is the usual variance formula in HAC type standard errors. By Dedecker and Merlevède (2002, Proposition 3) it then follows that Condition 1 holds with $\eta$ defined in (7).

3.3 Joint Stable Convergence

This section adopts an argument from Barndorff-Nielsen, Hansen, Lunde, and Shephard (2008, BHLS) to our setting where we assume conditional independence between $Z_{\tau}$ and $Y_{n}$. We adapt the proof of Proposition 5 (p.1524) of BHLS to our context. To establish joint stable convergence of $Z_{\tau}$ and $Y_{n}$, we show that $\exp\left(isZ_{\tau}+itY_{n}\right)$ converges weakly in $L_{1}$ for all $t$ and $s$ (see Aldous and Eagleson, 1978).

Using the results in Sections 3.1 and 3.2, and a construction in Aldous and Eagleson (1978, p.327)101010To be more specific consider the construction of $Z.$ Aldous and Eagleson show that $Z$ can be constructed as follows: enlarge the probability space from $\left(\mathbb{R}^{\infty}\times\mathbb{R}^{\infty},\mathcal{X},P\right)$ to $\left(\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}\times I,\mathcal{X\times A},P\times\lambda\right)$ where in $\left(I,\mathcal{A},\lambda\right),$ $I$ is the unit interval with Lebesgue measure $\lambda$. For each $\omega$ define $Z\left(\omega,l\right)$ as the inverse of the distribution function $P\left(Z\leq z|\mathcal{M}\right)\left(\omega\right)$ and where $l$ is a uniformly distributed random variable on $\left[0,1\right].$ Since in our case, conditional on $\mathcal{M}$, $Z$ is $N\left(0,\eta\right)$ it follows that $Z$ can be represented as $Z=\sqrt{\eta}\xi_{\nu}$ where $\xi_{\nu}$ is $N\left(0,1\right).$, there are random variables $Z$ and $Y$ on a possibly enlarged probability space such that

[TABLE]

and

[TABLE]

We establish the following main result of our paper.

Theorem 2

$\left(Z_{\tau},Y_{n}\right)\rightarrow_{d}\left(Z,Y\right)$* ($\mathcal{A}$-stably) as $n,\tau\rightarrow\infty$ where $Z\sim\sqrt{\eta}\xi_{\nu}$, $Y\sim\sigma\left(\nu_{1}\right)\xi_{y}$, $\left(\xi_{\nu},\xi_{y}\right)$ is multivariate standard Gaussian with $\operatorname*{Cov}\left(\xi_{\nu},\xi_{y}\right)=0$ and $\sim$ stands for two random variables having the same distribution. In addition, $\left(\xi_{\nu},\xi_{y}\right)$ is independent of any $\mathcal{A}$-measurable random variable.*

Proof. See Appendix.

3.4 Extension to Short Panels

Suppose that $T\geq 2$ with $T$ fixed and finite. Let’s assume the same “sufficient statistic” structure, i.e.,

[TABLE]

We then have $y_{i}\equiv\left(y_{i,1},\ldots,y_{i,T}\right)^{\prime}$ iid given $\mathcal{M}_{T}$, a sigma-algebra relative to which $\left\{\nu_{t},t\leq T\right\}$ is measurable. The proof of Theorem 1 goes through without modification once $\mathcal{A}$ is replaced with $\mathcal{M}_{T}$. Let $Y_{n}\equiv\frac{1}{\sqrt{n}}\sum_{i=1}^{n}q\left(y_{i}\right)$ for some real-valued function $q$.** **It follows that

[TABLE]

where $\sigma^{2}\left(\nu_{1},...,\nu_{T}\right)=E\left[q\left(y_{1}\right)^{2}|\mathcal{M}_{T}\right]$. As for $Z_{\tau}=\frac{1}{\sqrt{\tau}}\sum_{s=1}^{\tau}\nu_{s}$, note that (6) holds when we replace $\mathcal{A}$ with $\mathcal{M}_{T}.$ It follows that $Z_{\tau}\rightarrow_{d}Z$ ($\mathcal{M}_{T}$-stably) where $Z\sim\sqrt{\eta}\xi_{\nu}$ and $\eta$ is defined as before. We can write

[TABLE]

because $\frac{1}{\sqrt{\tau}}\sum_{s=1}^{T}\nu_{s}=o_{p}\left(1\right)$ and $\frac{\sqrt{\tau-T}}{\sqrt{\tau}}\rightarrow 1$ as $\tau\rightarrow\infty$. Then, Dedecker and Doukhan (2003, Corollary 1), Dedecker and Rio (2000, Theorem 1) and Proposition 3 of Dedecker and Merlevède (2002) can be applied to $\frac{1}{\sqrt{\tau-T}}\sum_{s=T+1}^{\tau}\nu_{s}$. Joint convergence then follows by the same argument as in Section 3.3 once $\mathcal{A}$ is understood to be $\mathcal{M}_{T}$. The joint limiting distribution is again given by $\left(Z_{\tau},Y_{n}\right)\rightarrow_{d}\left(Z,Y\right)$ ($\mathcal{A}$-stably).

4 Standard Errors

In this section we return to the example discussed in Section 2 as an illustration of how our joint limiting distributions can be used to obtain limiting results for the structural and treatment parameters of interest. The estimators for the structural parameter $\beta$ and the counterfactual causal effect $\pi\left(v\right)$ are based on estimators for the in-period treatment effect $\pi_{1}$ and the parameter governing the aggregate shock distribution $\phi.$ We recall that $\hat{\pi}_{1}=\sum_{i=1}^{n}w_{i}\tilde{y}_{i,1}$** **and $\hat{\phi}=\phi+\tau^{-1}\sum_{s=1}^{\tau}\nu_{s}$. Using the result in Theorem 2 we find that the joint distribution of $\left(\hat{\pi}_{1},\hat{\phi}\right)$ is

[TABLE]

where $\mathcal{M}$ is the sigma-field generated by $\nu_{1}$ and $\left(Y,Z\right)\sim\left(\sigma\left(\nu_{1}\right)\xi_{y},\sqrt{\eta}\xi_{\nu}\right).$ First note that

[TABLE]

It is straightforward to show that $\sqrt{n}\sum_{i=1}^{n}w_{i}u_{i}=n^{-1/2}\sum_{i=1}^{n}y_{i,1}+o_{p}\left(1\right)$ with $y_{i,1}=2\left(2d_{i,1}-1\right)u_{i}$.111111Arguments similar to the proof of Theorem 4 in Kuersteiner and Prucha (2013) justify the use of $o_{p}\left(1\right)$ notation in the context of stable convergence. Essentially, convergence in probability implies joint weak convergence and thus stable convergence.** **By Theorem 1 it follows that $\sigma^{2}\left(\nu_{1}\right)=4\operatorname*{Var}\left(u_{i}\right)$ and $\xi_{y}\sim N\left(0,1\right)$. Similarly, assume that $v_{s}$ is strictly stationary and ergodic and satisfies Condition 2. Then it follows from Lemma 1 that $\eta=E\left[\nu_{0}^{2}\right]+2\sum_{s=1}^{\infty}E\left[\nu_{0}\nu_{s}\right]$ and $\xi_{\nu}$ is standard Gaussian and independent of $\xi_{y}$.

We are now in a position to derive the limiting distribution of $\hat{\beta}$ and $\hat{\pi}\left(\nu\right).$ Using the expansion in (3) and under the additional assumption $\sqrt{\frac{n}{\tau}}\rightarrow\sqrt{\kappa}$ for $0<\kappa<\infty$ the limiting distribution for $\hat{\beta}$ is given by

[TABLE]

The form of the limiting distribution is mixed Gaussian with random variance

[TABLE]

where the randomness comes from the fact that $\nu_{1}$ is a random draw from the stationary distribution of the process $\nu_{s}$ and that $\pi_{1}$ depends on $\nu_{1}$. By the continuous mapping theorem it follows that the standardized statistic

[TABLE]

is asymptotically normal, i.e.,

[TABLE]

Similarly, the limiting distribution of $\hat{\pi}\left(\nu\right)$ can be obtained from the expansion in (4). Since in our example $\hat{\pi}\left(\nu\right)$ is a rescaled version of $\hat{\beta}$ the resulting limiting distribution is just a scaled version of (9) where the scale factor is $\left(1+\nu^{2}\right)$ and where $\nu,$ as opposed to $\nu_{1}$, is a fixed constant.

Feasible inference can be based on (10). Consider testing the null hypothesis that $\beta=\beta_{0}$ for some fixed value $\beta_{0}.$ A Wald test for this hypothesis is given by the t-ratio

[TABLE]

where $\hat{\nu}_{1}=z_{1}-\hat{\phi}$ and $\hat{\sigma}_{u}^{2}=n^{-1}\sum_{i=1}^{n}\hat{u}_{i}^{2}$ with $\hat{u}_{i}=\tilde{y}_{i,1}-\hat{\pi}_{1}d_{i,1}$. Since $\hat{\pi}_{1}-\pi_{1}=O_{p}\left(n^{-1/2}\right)$ and $\hat{\phi}-\phi=O_{p}\left(\tau^{-1/2}\right)$ by (8) it follows from standard arguments that $\hat{\nu}_{1}-\nu_{1}=o_{p}\left(1\right)$ and $\hat{\sigma}_{u}^{2}-\sigma_{u}^{2}=o_{p}\left(1\right).$ Consistent estimators for $\eta$ can be obtained from procedures proposed by Newey and West (1987, 1994), Andrews (1991) or Phillips (2005) under somewhat different conditions than imposed in Condition 2. For example, Newey and West (1987) or Andrews (1991) impose strong mixing but do not require stationarity. Since $\alpha$-mixing processes are also mixingales by McLeish (1975), a stationary strong mixing process can be defined in a way that it satisfies both the conditions of Theorem 2 and conditions in Newey and West (1987) or Andrews (1991). Assume that $\hat{\eta}$ is a consistent estimator for $\eta.$ It follows from the continuous mapping theorem that $\hat{\Upsilon}_{n}-\Upsilon_{n}=o_{p}\left(1\right)$ and that $\hat{\Upsilon}_{n}$ has a limiting $N\left(0,1\right)$ distribution under the null. As a consequence, the null that $\beta=\beta_{0}$ is rejected against a two-sided alternative at level $\alpha$ if $\left|\hat{\Upsilon}_{n}\right|>c_{1-\alpha/2}$ where $c_{1-\alpha/2}$ is the $1-\alpha/2$ quantile of the standard Gaussian distribution. Similarly, a $1-\alpha$ confidence interval for $\hat{\beta}$ can be constructed as $\hat{\beta}\pm c_{1-\alpha/2}\left.\left(2\left(\left(1+\hat{\nu}_{1}^{2}\right)^{2}\hat{\sigma}_{u}^{2}+\kappa\hat{\eta}\hat{\pi}_{1}^{2}\hat{\nu}_{1}^{2}\right)^{1/2}\right)\right/\sqrt{n}$.

5 Conclusions

This paper complements results in Hahn, Kuersteiner and Mazzocco (2016) to allow for mis-specification in the time series and cross-section models. This is achieved by covering strictly stationary mixingales. A novel conditional CLT for the cross-sectional data is combined with a stable CLT for stationary mixingales to establish joint stable convergence of the combined time series and cross-section data sets. The asymptotic variance covariance matrix for the time series component has the familiar HAC structure. In our setting the variance of the cross-sectional component may be a function of underlying random shocks and thus be itself a random variable. Thus, the joint limit is in general mixed Gaussian rather than standard normal.

Our proofs rely on strict stationary and conditional independence between cross-sectional and time series data. Extending these results to allow for more general heterogenous mixingales is a topic of future research.

Appendix A Proofs

Proof of Theorem 1. The proof closely follows the argument in Eagleson (1975). For $P$-almost all $\omega^{\prime}\in\mathbb{R}^{\infty}\times\mathbb{R}^{\infty}$ it follows from Lemma 1 in Eagleson (1975) that

[TABLE]

Then, there is a $N_{1}\in\mathcal{X}$ such that $P\left(N_{1}\right)=0$ and for fixed $\omega^{\prime}\notin N_{1}$ it follows that $E_{\omega^{\prime}}\left[S_{n}|\mathcal{A}\right]\left(\omega\right)=0$ and $E_{\omega^{\prime}}\left[S_{n}^{2}|\mathcal{A}\right]\left(\omega\right)=nS^{2}\left(\nu_{1}\left(\omega\right)\right)$. Using the same argument as below (6) in Eagleson (1975) we make the following observation: since $\nu_{1}\left(\omega\right)$ is $\mathcal{A}$-measurable it follows that for fixed $\omega^{\prime}$,

[TABLE]

in other words for a set of $\omega$ with $Q_{\omega^{\prime}}$-measure one, the above equality holds, where the right hand side $\nu_{1}\left(\omega^{\prime}\right)$ is a constant. Then, for $N_{2}\in\mathcal{X}$ such that $P\left(N_{2}\right)=0$ and for fixed $\omega^{\prime}\notin N_{2}$ it follows that

[TABLE]

In (11) we have used the fact that, again by Lemma 1 in Eagleson (1975),

[TABLE]

Since by assumption, $E\left[\left.\left|y_{1,1}\right|^{2+\delta}\right|\mathcal{A}\right]\left(\omega\right)\leq K<\infty$ the convergence in (11) is established $Q_{\omega^{\prime}}$ almost surely. Then the conditions for the Lindeberg CLT hold at least for $\omega^{\prime}$ outside a set of $P$-measure zero. Thus, by the Lindeberg CLT, for $P$-almost all $\omega^{\prime}$ it follows that

[TABLE]

Proof of Lemma 1. To see that Dedecker and Merlevède (2002, Theorem 1), hereafter DM, implies $\mathcal{A}$-stable convergence, let $\vartheta$ be any bounded**, $\mathcal{A}$-measurable random variable such that $\left|\vartheta\right|<K$ for some bounded non-random constant $K$. As in DM, let $\mathcal{H}$ be the space of continuous functions $\varphi:\mathbb{R\rightarrow R}$ such that $\left|\left(1+x^{2}\right)^{-1}\varphi\left(x\right)\right|$ is bounded. Note that $\mathcal{H}$ contains the space of bounded continuous functions. **Then, $\tau^{-1/2}S_{\nu,\tau}$ converges $\mathcal{A}$ stably if for any $\varphi\in\mathcal{H}$

[TABLE]

From $\mathcal{A}\subseteq\mathcal{M}_{k}$ and the law of iterated expectations one obtains

[TABLE]

Taking the $\lim\sup$ on both sides of the inequality then establishes that Dedecker and Merlevède (2002, Theorem 1) implies stable convergence.

Proof of Theorem 2. For any bounded $\mathcal{A}$-measurable random variable $\vartheta$ consider

[TABLE]

where the second equality follows from the fact that conditional on $\mathcal{A},$ $Y_{n}$ and $Z_{\tau}$ are independent. Then, using that $\left|E\left[\exp\left(isZ_{\tau}\right)|\mathcal{A}\right]\vartheta\right|\leq K<\infty$ a.s. it follows that

[TABLE]

by the result in Section 3.1, noting that $\exp\left(itY_{n}\right)$ and $\exp\left(itY\right)$ are bounded. Next note that $E\left[\exp\left(itY\right)|\mathcal{A}\right]=e^{-\frac{1}{2}t^{2}\sigma^{2}\left(\nu_{1}\right)}$ a.s. is bounded. Let $\tilde{\vartheta}=\vartheta e^{-\frac{1}{2}t^{2}\sigma^{2}\left(\nu_{1}\right)}$ such that $\tilde{\vartheta}$ is $\mathcal{A}$-measurable and bounded. By stable convergence of $Z_{\tau}$, established in Section 3.2, it then follows that

[TABLE]

Together (12) and (13) imply that for $n,\tau\rightarrow\infty,$

[TABLE]

By the Cramer-Wold theorem this implies that $\left(Z_{\tau},Y_{n}\right)\rightarrow\left(Z,Y\right)$ $\mathcal{A}$-stably as $n,\tau\rightarrow\infty$. To identify the limits note that $E\left[\exp\left(itY\right)|\mathcal{A}\right]=E\left[\exp\left(it\sigma\left(\nu_{1}\right)\xi_{y}\right)|\mathcal{A}\right]=e^{-\frac{1}{2}t^{2}\sigma^{2}\left(\nu_{1}\right)}$ where $\xi_{y}$ is $N\left(0,1\right)$ and therefore independent of any $\mathcal{A}$-measurable random variable. Similarly, $E\left[\left.\exp\left(isZ_{\tau}\right)\right|\mathcal{A}\right]=E\left[\left.\exp\left(is\sqrt{\eta}\xi_{\nu}\right)\right|\mathcal{A}\right]$ where $\xi_{\nu}$ is $N\left(0,1\right)$ and independent of any $\mathcal{A}$-measurable random variable. Note that the last statement follows from (6) after setting $\varphi\left(.\right)=\exp\left(is.\right).$ The fact that $\xi_{\nu}$ and $\xi_{y}$ are independent of each other follows from the assumption, that conditional on $\mathcal{A}$, $Z_{\tau}$ and $Y_{n}$ are independent for all $n$ and $\tau.$