| Summary: | We investigate the finite sample behaviour of the ordinary least squares (OLS) estimator in vector autoregressive (VAR) models. The data generating process is assumed to be a purely nonstationary first-order VAR. Using Monte Carlo simulation and numerical optimization we derive response surfaces for OLS bias and variance in terms of VAR dimensions both under correct model specification and under several types of over-parameterization: we include a constant, a constant and trend, and introduce excess autoregressive lags. Correction factors are introduced that minimise the mean squared error (MSE) of the OLS estimator. Our analysis improves and extends one of the main finite-sample multivariate analytical bias results of Abadir, Hadri and Tzavalis (1999), generalises the univariate variance and MSE results of Abadir (1995) to a multivariate setting, and complements various asymptotic studies. The distribution of unit root test statistics generally contains nuisance parameters that correspond to the correlation structure of the innovation errors. The presence of such nuisance parameters can lead to serious size distortions. To address this issue, we adopt an approach based on the characterization of the class of asymptotically similar critical regions for the unit root hypothesis and the application of two new optimality criteria for the choice of a test within this class. The correlation structure of the innovation sequence takes the form of a moving average process, the order of which is determined by an appropriate information criterion. Limit distribution theory for the resulting test statistics is developed and simulation evidence suggests that our statistics have substantially reduced size while retaining good power properties. Stock return predictability is a fundamental issue in asset pricing. The conclusions of empirical analyses on the existence of stock return predictability vary according to the time series properties of the economic variables considered as potential predictors. Given the uncertainty about the degree of persistence of these variables, it is important to operate in the most general possible modelling framework. This possibility is provided by the IVX methodology developed by Phillips and Magdalinos (2009) in the context of cointegrated systems with no deterministic components. This method is modified in order to apply to multivariate systems of predictive regressions with an intercept in the model. The resulting modified IVX approach yields chi-squared inference for general linear restrictions on the regression coefficients that is robust to the degree of persistence of the predictor variables. In addition to extending the class of generating mechanisms for predictive regression, the approach extends the range of testable hypotheses, assessing the combined effects of different explanatory variables to stock returns rather than the individual effect of each explanatory variable.
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