Automatic positive semidefinate HAC covariance matrix and GMM estimation

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Smith, Richard J. (2005)

This paper proposes a new class of heteroskedastic and autocorrelation consistent (HAC) covariance matrix estimators. The standard HAC estimation method reweights estimators of the autocovariances. Here we initially smooth the data observations themselves using kernel function–based weights. The resultant HAC covariance matrix estimator is the normalized outer product of the smoothed random vectors and is therefore automatically positive semidefinite. A corresponding efficient GMM criterion may also be defined as a quadratic form in the smoothed moment indicators whose normalized minimand provides a test statistic for the overidentifying moment conditions.
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    Proof of Theorem 3.1. As V is p+d+ from Assumption 3+1~d!, VZ T ~ bD ! is p+d+ and invertible w+p+a+1+ Let g~ b! [ E @ g~zt , b!# + Under Assumption 2+1, $ g~zt , b0 !%t` 1 is a stationary and a-mixing sequence ~White, 1984, Theorem 3+49, p+ 47! and, thus, ergodic ~White, 1984, Proposition 3+44, p+ 46!+ By a uniform weak law of numbers ~Smith, 2001, Lemma A+1!, if Assumptions 2+1-2+3 and 3+1 hold, supb B 7ST 1 g[ T ~ b! k1 g~ b!7 op~1! and g~ b! is continuous by the strictly stationary and ergodic version of Lemma 2+4 in Newey and McFadden ~1994, p+ 2129!+ Let Q~ b! g~ b!'V 1g~ b!+ Then, by Assumption 3+1~a!, Q ~ b ! is uniquely minimized at b0 and is continuous in b B+ Therefore, as lmin@VZ T ~ bD !# 0 w+p+a+1 where lmin@VZ T ~ bD !# is the smallest eigenvalue of VZ T ~ bD !, uniformly b p+ 2121!+ B+ The result follows by Theorem 2+1 in Newey and McFadden ~1994,

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