ABSTRACT
In many cases, …rst-order asymptotic theory provides a poor approximation to the …nite sample distribution of the statistic of interest, and higher-order expansions also appear to be of limited help. For instance, the validity of the usual asymptotic theory often breaks down near the boundary of the parameter space when the limiting distribution changes discontinuously. Examples include near-integrated and near-cointegrated models, instrumental variable (IV) models with weak instruments, models with time-varying parameters whose variability is close to zero, etc. In these models, conventional asymptotic theory depends discontinuously on some underlying parameter (autoregressive, concentration or variance parameter) while the …nite sample distribution of the statistic of interest changes smoothly with the values of this parameter. For instance, if the largest autoregressive (AR) root in dynamic models is strictly less than one, its estimator is asymptotically distributed as a normal random variable but its asymptotic distribution is non-normal (Dickey–Fuller) if the AR root is exactly equal to one. As a result, neither of these asymptotic distributions can provide accurate approximations when the AR root is in the vicinity of (but not exactly at) unity. It has been shown that pretesting (if the parameter is equal to one, for instance) and conventional bootstrap methods also fail to deliver valid inference in this setup. This chapter discusses alternative asymptotic approximations that do not
treat the parameter, which gives rise to potential discontinuity, as …xed but reparameterize it as a drifting sequence that depends explicitly on the sample size. As the sample size increases, this sequence shifts towards the boundary of the parameter space. This arti…cial statistical device ensures a smooth transition of the asymptotic theory that mimics the behavior of the …nite sample distribution. It is now a popular analytical tool in models with highly persistent variables and weak instruments which are frequently encountered in the analysis of economic data. A potential drawback of this asymptotic framework is the introduction of additional nuisance parameters that are not consistently estimable, which complicates the inference procedure. The devel-
opment of uniformly valid and practically appealing inference methods based on these drifting parameter sequences is an active research topic. Another important situation where resorting to drifting parameter sequences
proves valuable is the analysis of models with many regressors/instruments as well as long-run forecasts and impulse responses. Conventional asymptotic theory assumes that the number of moment conditions or forecast horizons is …xed and the limit is taken only with respect to the sample size. However, when the number of moment conditions or forecast horizons is a nontrivial fraction of the sample size (for example, 20 or 30 with 100 observations), standard asymptotic theory fails to provide a reasonable approximation to the …nite sample distribution of interest. It is often bene…cial to parameterize this number as a function of the sample size by allowing it to grow at some rate as the sample size approaches in…nity. This type of drifting parameterization has become an essential tool in dealing with the many instruments problem as well as conditional forecasting and impulse response analysis at long horizons.1
