Autocorrelation Robust Testing in Regression Models

Testing hypotheses on regression coefficients in linear models with correlated disturbances is a topic of central interest in econometrics and statistics. Even in a Gaussian setting this is a non-trivial testing problem due to the presence of the (possibly infinite-dimensional) nuisance parameters that govern the dependence structure. Most tests available in the literature are F-tests that are corrected for the autocorrelation in the data (also known as "autocorrelation-consistent" or "autocorrelation robust" tests), and are justified on the basis of a standard asymptotic analysis. Recently, Preinerstorfer and Pötscher (2013) have shown analytically that in finite samples these procedures typically break down in that either the size of these autocorrelation-corrected F-type tests is equal to one, or the nuisance- minimal power is equal to zero (which of the two cases arises depends on an observable quantity being either above or below a certain threshold). Furthermore, they identified the cause for this effect, namely a concentration effect due to strong correlation. Exploiting this observation they suggested an adjustment procedure for autocorrelation-corrected F-type tests that can render such a test immune to the concentration effect. For the adjustment procedure to work, assumptions concerning the behavior of the correlation structure at its `singular boundary` and the number of its so-called concentration subspaces have to be satisfied (which is, e.g., the case for autoregressive models of order 1). The goal of the proposed project is to understand the testing problem for more complex correlation models that do not satisfy these assumptions, and to design appropriate adjustment procedures that perform well in terms of finite sample size and power properties of the resulting tests in these more difficult settings.

In applied econometrics and statistics one often encounters situations where hypotheses are to be tested on parameters in regression models. When the errors in such a model are potentially autocorrelated, e.g., because the observations are measurements over time, the autocorrelation is an additional nuisance parameter. Then the testing problem becomes non-trivial, even if one is willing to assume that the errors are normally distributed, and tests that are commonly used in practice overreject dramatically under the null hypothesis or have bad power properties as shown in Preinerstorfer and Pötscher (2016). Using such procedures in practice might therefore lead to wrong decisions with high probability. In case strongly correlated errors are well approximated by autoregressive models of first order (in a certain technical sense) Preinerstorfer and Pötscher (2016) have furthermore suggested an adjustment procedure to obtain substantially improved tests. The goal of the present project was to understand the testing problem for more complex correlation models that do not satisfy the approximation condition imposed in Preinerstorfer and Pötscher (2016), and to design appropriate testing procedures that do not overreject under the null hypothesis, and also have good power properties.In this project we have developed a set of conditions that can easily be checked, are met for most design matrices, and if met allow the researcher to obtain tests with correct rejection probabilities under the null hypothesis (in finite samples) by using non-standard critical values that can be obtained numerically. The results were obtained by first establishing theoretical results in a general framework, which were then applied to regression models with autocorrelated errors. We furthermore obtained results and methods that can be used to substantially improve the power of the tests so obtained.