A Primer on Bootstrap Testing of Hypotheses in Time Series Models : With an Application to Double Autoregressive Models

In this paper we discuss the general application of the bootstrap as a tool for statistical inference in econometric time series models. We do this by considering the implementation of bootstrap inference in the class of double-autoregressive [DAR] models discussed in Ling (2004). DAR models are particularly interesting to illustrate implementation of the bootstrap to time series: first, standard asymptotic inference is usually difficult to implement due to the presence of nuisance parameters under the null hypothesis; second, inference involves testing whether one or more parameters are on the boundary of the parameter space; third, under the alternative hypothesis, fourth or even second order moments may not exist. In most of these cases, the bootstrap is not considered an appropriate tool for inference. Conversely, and taking testing (non-)stationarity to illustrate, we show that although a standard bootstrap based on unrestricted parameter estimation is invalid, a correct implementation of a bootstrap based on restricted parameter estimation (restricted bootstrap) is first-order valid; that is, it is able to replicate, under the null hypothesis, the correct limiting null distribution. Importantly, we also show that the behaviour of this bootstrap under the alternative hypothesis may be different because of possible lack of finite second-order moments of the bootstrap innovations. This features makes - for some parameter configurations - the restricted bootstrap unable to replicate the null asymptotic distribution when the null is false. We show that this drawback can be fixed by using a new 'hybrid' bootstrap, where the parameter estimates used to construct the bootstrap data are obtained with the null imposed, while the bootstrap innovations are sampled with replacement from the unrestricted residuals. We show that this bootstrap, novel in this framework, mimics the correct asymptotic null distribution, irrespetively of the null to be true or false. Throughout the paper, we use a number of examples from the bootstrap time series literature to illustrate the importance of properly defining and analyzing the bootstrap generating process and associated bootstrap statistics