Canonical correlation analysis of stochastic trends via functional approximation

TL;DR

This paper introduces a new semiparametric method combining functional approximation and canonical correlation analysis to infer the number of common trends and their loadings in time series data, with proven asymptotic properties.

Contribution

It develops a novel approach for inference on common trends in I(1)/I(0) systems using functional approximation and canonical correlations, with comprehensive testing and estimation procedures.

Findings

Tests on the number of trends are asymptotically pivotal.

Estimators of loadings are consistent, efficient, and asymptotically normal.

Simulation results show good performance for sample sizes where T ≥ 10p.

Abstract

This paper proposes a novel approach for semiparametric inference on the number $s$ of common trends and their loading matrix $\psi$ in $I(1)/I(0)$ systems. It combines functional approximation of limits of random walks and canonical correlations analysis, performed between the $p$ observed time series of length $T$ and the first $K$ discretized elements of an $L^2$ basis. Tests and selection criteria on $s$, and estimators and tests on $\psi$ are proposed; their properties are discussed as $T$ and $K$ diverge sequentially for fixed $p$ and $s$. It is found that tests on $s$ are asymptotically pivotal, selection criteria of $s$ are consistent, estimators of $\psi$ are $T$-consistent, mixed-Gaussian and efficient, so that Wald tests on $\psi$ are asymptotically Normal or $\chi^2$. The paper also discusses asymptotically pivotal misspecification tests for checking model assumptions. The approach can be coherently applied to subsets or aggregations of variables in a given panel. Monte Carlo simulations show that these tools have reasonable performance for $T\geq 10 p$ and $p\leq 300$. An empirical analysis of 20 exchange rates illustrates the methods.