Predictive Inference Under Model Misspecification with an Application to Assessing the Marginal Predictive Content of Money for Output

In this chapter we discuss model selection and predictive accuracy tests in the context of parameter and model uncertainty under recursive and rolling estimation schemes. We begin by summarizing some recent theoretical findings, with particular emphasis on the construction of valid bootstrap procedures for calculating the impact of parameter estimation error on the class of test statistics with limiting distributions that are functionals of Gaussian processes with covariance kernels that are dependent upon parameter and model uncertainty. We then provide an example of a particular test which falls in this class. Namely, we outline the so-called Corradi and Swanson (CS: 2002) test of (non)linear out-of-sample Granger causality. Thereafter, we carry out a series of Monte Carlo experiments examining the properties of the CS and a variety of other related predictive accuracy and model selection type tests. Finally, we present the results of an empirical investigation of the marginal predictive content of money for income, in the spirit of Stock andWatson (1989), Swanson (1998), Amato and Swanson (2001), and the references cited therein. We find that there is evidence of predictive causation when in-sample estimation periods are ended any time during the 1980s, but less evidence during the 1970s. Furthermore, recursive estimation windows yield better prediction models when prediction periods begin in the 1980s, while rolling estimation windows yield better models when prediction periods begin during the 1970s and 1990s. Interestingly, these two results can be combined into a coherent picture of what is driving our empirical results. Namely, when recursive estimation windows yield lower overall predictive MSEs, then bigger prediction models that include money are preferred, while smaller models without money are preferred when rolling models yield the lowest MSE predictors.