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Article Excerpt
1. INTRODUCTION
The bootstrap is a general method for estimating the sampling distribution of a statistic. Under suitable conditions, the bootstrap distribution is asymptotically first-order equivalent to the asymptotic distribution of the statistic of interest. The consistency of the bootstrap distribution, however, does not guarantee the consistency of the variance of the bootstrap distribution (the "bootstrap variance") as an estimator of the asymptotic variance, because it is well known that convergence in distribution of a random sequence does not imply convergence of moments (see, e.g., Billingsley 1995, thm. 25.12). For the sample median and smooth functions of sample means, examples of the inconsistency of bootstrap variance estimators in the iid context have been given by Ghosh, Parr, Singh, and Babu (1984) and Shao (1992).
For time series observations, the moving blocks bootstrap (MBB) introduced by Kunsch (1989) and Liu and Singh (1992a) has been shown to consistently estimate the variance of the sample mean under weak dependence and heterogeneity assumptions (see Goncalves and White 2002). For more general statistics, conditions for the consistency of the bootstrap variance estimator do not appear to be available.
The main purpose of this article is to provide sufficient conditions for the consistency of MBB variance estimators when the statistic of interest is the least squares (LS) estimator in possibly misspecified linear regression models with dependent data. Our framework includes linear regression with iid observations as a special case. In related work, Liu and Singh (1992b) showed the consistency of the iid bootstrap variance estimator for regressions with fixed regressors and iid errors. Our results allow for stochastic regressors and autocorrelated errors. Although the consistency of the MBB distribution of the LS estimator is well established in the literature (see, e.g., Fitzenberger 1997; Politis, Romano, and Wolf 1997), the consistency of the bootstrap variance of the LS estimator has not received much attention. As we remarked earlier, the former does not necessarily imply the latter, so that currently available results do not justify bootstrapping the standard errors of the LS estimates using the MBB.
Our result is important in that many applied studies have used bootstrap standard error estimates as a measure of the precision of their parameter estimates (see, e.g., Efron 1979; Freedman and Peters 1984; Efron and Tibshirani 1986; Li and Maddala 1999). We also emphasize that this result plays an important role in justifying bootstrap applications based on Studentized statistics, for which asymptotic refinements of the bootstrap can be expected. The construction of Studentized statistics involves normalization by the standard error of the estimator. Our results formally justify using the bootstrap in computing such standard errors. This feature is especially convenient in cases when asymptotic closed-form solutions are not available or are too cumbersome to be calculated. In addition, we present simulation evidence that suggests that inference based on bootstrap estimates of standard errors may be considerably more accurate in small samples than inference based on asymptotic closed-form standard error estimates. For a multiple linear regression model with autocorrelated (and heteroscedastic) errors, we find that confidence intervals that rely on bootstrap standard errors tend to perform better than confidence intervals that rely on asymptotic closed-form variances. In particular, the coverage errors of symmetric MBB percentile-t confidence intervals based on bootstrap standard error estimates are substantially smaller than the coverage errors typically found for other (asymptotic theory-based and bootstrap-based) confidence intervals in this setting, especially under strong autocorrelation.
The remainder of the article is organized as follows. Section 2 presents the theoretical results. Section 3 compares the accuracy of the bootstrap estimator with that of closed-form estimators of the variance. Section 4 provides concluding remarks, and an Appendix gives all of the proofs.
2. LINEAR REGRESSION
In this section we prove the asymptotic validity of the MBB for variance estimation in the context of linear regressions when the data-generating process (DGP) is near--epoch-dependent (NED) on a mixing process (Billingsley 1968; McLeish 1975; Gallant and White 1988). NED processes allow for considerable dependence and heterogeneity. They include as a special case the more conventional mixing processes, which can be overly restrictive for applications in economics [see, e.g., Andrews 1984 for an example of a simple AR(1) process that fails to be strong mixing]. NED processes cover a variety of nonlinear time series models, including the bilinear, generalized autoregressive conditional heteroscedastic, and threshold autoregressive models (see Davidson 2002).
We define {[Z.sub.t]} to be [L.sub.q]-NED on a mixing process {[V.sub.t]} provided that E([Z.sub.t.sup.q]) < [infinity] and [v.sub.k] [equivalent to] [sup.sub.t] ||[Z.sub.t] - [E.sub.t-k.sup.t+k]([Z.sub.t])||[.sub.q] tends to as k [right arrow] [infinity] at an appropriate rate, where q [greater than or equal to] 2. In particular, if [v.sub.k] = O([k.sup.-a-[delta]]) for some [delta] > 0, then we say that {[Z.sub.t]} is [L.sub.q]-NED (on{[V.sub.t]}) of size -a. Here and in what follows, ||[Z.sub.t]||[.sub.q] [equivalent to] (E|[Z.sub.t]|[.sup.q])[.sup.1/q] denotes the [L.sub.q] norm of the random vector [Z.sub.t], with |[Z.sub.t]| its Euclidean norm, and [E.sub.t-k.sup.t+k](*) [equivalent to] E(* |[F.sub.t-k.sup.t+k]), where [F.sub.t-k.sup.t+k] [equivalent to] [sigma]([V.sub.t-k],..., [V.sub.t+k]) is the [sigma]-field generated by [V.sub.t-k],..., [V.sub.t+k]. The sequence {[V.sub.t]} is assumed to be strong mixing, that is, [[alpha].sub.k] [equivalent to] [sup.sub.m][sup.sub.{A[member of][F.sub.-[infinity].sup.m],B[member of][F.sub.m+k.sup.[infinity]]}]|P(A [intersection] B) - P (A) P (B)| [right arrow] as k [right arrow] [infinity] at an appropriate rate.
Gallant and White (1988) studied the asymptotic properties of quasi--maximum likelihood estimators (QMLEs) for heterogeneous NED data and nonlinear dynamic models. Recently, Goncalves and White (2004) established the first-order asymptotic validity of the MBB for the framework of Gallant and White (1988). In particular, Goncalves and White (2004) showed that the MBB consistently estimates the asymptotic distribution of the QMLE. But as Goncalves and White (2004) remarked, their results do not justify using the variance of the bootstrap distribution to consistently estimate the asymptotic variance of the QMLE. Here we fill this gap for the special case of the LS estimator for linear dynamic models. In particular, we give explicit conditions that justify bootstrapping the variance of the LS estimator in possibly misspecified linear dynamic models when the DGP is NED on a mixing process.
Assumption 1 is a version of the Gallant and White (1988) and Goncalves and White (2004) assumptions specialized to the case of linear dynamic models.
Assumption 1. a. Let ([OMEGA], F, P) be a complete probability space. The observed data are a realization of a strictly stationary stochastic process {[Z.sub.t] = ([Y.sub.t], [X'.sub.t])':[OMEGA] [right arrow] [R.sup.p+1], t = 1, 2,...},...
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