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1. Introduction [1]
Modern time series analysis leads the researcher to consider a wide variety of data characteristics in determining whether a relation exists between variables. The researcher needs to be concerned about whether the variables are stationary or not, whether they each have a trend and how many lags to include in examining the data. The researcher needs to also be concerned with whether a relation between the variables is apparent between the levels of the variables, perhaps through cointegration, or whether the relation is only apparent in first (or higher-order) differences. The existence of autocorrelation or heteroscedasticity and possibly correcting for these problems are also issues with which the researcher often needs to deal. All of this analysis with time series data renders the researcher to having to take into account a wide variety of potential models, and many of these models are not nested within others.
Researchers using time series data in economics and finance have often proceeded in their analysis by estimating regression models and testing various hypotheses in a frequentist tradition, e.g. testing for a unit root, testing for cointegration, testing for autocorrelation and so forth. Hypothesis testing has had a dual role in finance, economics and other scientific disciplines. First, it is used to give us a minimum degree of confidence about rejecting a null hypothesis on some parameter restriction(s) by controlling for type I error at a particular level which is arbitrarily chosen, such as 5% [2]. Second, it is used for model selection – if the null hypothesis is not rejected we often consider the model with the null hypothesis as being acceptable, whereas if the null hypothesis is rejected we find the model without the parameter restrictions in the null hypothesis as acceptable. The first role is formally the most appropriate use of hypothesis testing, but the model selection usage is, in our opinion, the usage that predominates in finance and economics [3]. This is clear since economists are prone to repeated testing of various models to arrive at an acceptable one that fits the data without patterns in the residuals and that is hopefully robust. That process of repeated testing and discarding of models based upon it clearly affects the statistical size of the associated...