Abstract
In the presence of heteroscedasticity and autocorrelation of unknown forms, the covariance matrix of the parameter estimator is often estimated using a nonparametric kernel method that involves a lag truncation parameter. Depending on whether this lag truncation parameter is specified to grow at a slower rate than or the same rate as the sample size, we obtain two types of asymptotic approximations: the small-b asymptotics and the fixed-b asymptotics. Using techniques for probability distribution approximation and high order expansions, this paper shows that the fixed-b asymptotics provides a higher order refinement to the first order small-b asymptotics. This result provides a theoretical justification on the use of the fixed-b asymptotics in empirical applications. On the basis of the fixed-b asymptotics and higher order small-b asymptotics, the paper introduces a new and easy-to-use F* test that employs a finite sample corrected Wald statistic and uses an F-distribution as the reference distribution. Finally, the paper develops a novel bandwidth selection rule that is testing-optimal in that the bandwidth minimizes the type II error of the F* test while controlling for its type I error. Monte Carlo simulations show that the F* test with the testing-optimal bandwidth works very well in finite samples.
Highlights
In linear and nonlinear models with moment restrictions, it is standard practice to employ the generalized method of moments (GMM) to estimate model parameters
We show that the di¤erence between the high-order corrected F critical value and the ...rst-order 2 critical value depends on the bandwidth parameter b, the number of joint hypotheses, and the kernel function used in the heteroskedasticity and autocorrelation robust (HAR) estimation
With more sophisticate and tedious arguments as in Sun and Phillips (2009), the term O(T 1=2 log T ) can be reduced to O(T 1=2): Here we are content with the weaker result as our main interest is to capture the e¤ect of b on the sampling distribution of FT : Given the similarity of the two expansions, the qualitative results for the location model in the previous subsection apply to the GMM setting
Summary
In linear and nonlinear models with moment restrictions, it is standard practice to employ the generalized method of moments (GMM) to estimate model parameters. In terms of distributional approximations, both the conventional small-b asymptotics and nonstandard ...xed-b asymptotics are considered in the literature In the former case, b is assumed to be small in that it goes to zero at certain rate with the sample size. The fourth objective is to operationalize the asymptotic F test by determining suitable values of the bandwidth parameter b: At present it is standard practice to use the bandwidth parameter that is optimal for the point estimation of the covariance matrix of the parameter estimator. On basis of the high order expansion, the section describes approximate measures of the type I and type II errors of the asymptotic F test It gives an explicit and closed-form expression for the testing-optimal bandwidth for the F test.
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