Inference in Linear Time Series Models with some Unit Roots

TLDR

Linear time series models often involve variables that are integrated or cointegrated, and this study focuses on vector autoregressions where some companion matrix elements are unit roots, possibly including polynomial time regressors and drifts. The paper aims to develop estimation and hypothesis testing procedures for linear time series models when some or all variables have unit roots. Parameters on mean‑zero, non‑integrated regressors have jointly normal asymptotic distributions converging at rate T′/2, whereas coefficients on integrated or polynomial regressors have nonnormal limits; the results identify which t or F tests (e.g., Granger causality) remain asymptotically valid and which exhibit nonstandard distributions.

Abstract

This paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots. Our motivating example is a vector autoregression with some unit roots in the companion matrix, which might include polynomials in time as regressors. In the general formulation, the variable might be integrated or cointegrated of arbitrary orders, and might have drifts as well. We show that parameters that can be written as coefficients on mean zero, nonintegrated regressors have jointly normal asymptotic distributions, converging at the rate T'/2. In general, the other coefficients (including the coefficients on polynomials in time) will have nonnormal asymptotic distributions. The results provide a formal characterization of which t or F tests-such as Granger causality tests-will be asymptotically valid, and which will have nonstandard limiting distributions.