Finite Sample Behavior

The foregoing discussion has rested upon asymptotic theory. In finite samples, such theory can only provide an approximation. It is therefore important to assess the quality of this approximation in the types of model and sample sizes that are encountered in economics and finance. Intuition suggests that the qual­ity is going to vary from case to case depending on the form of the nonlinearity and the dynamic structure. A number of simulation studies have examined this question; see inter alia Tauchen (1986), Kocherlakota (1990) and the seven papers included in the July 1996 issue of Journal of Business and Economics Statistics. It is
beyond the scope of this article to provide a comprehensive review of these studies.35 However, it should be noted that in certain circumstances of interest the quality of the approximation is poor. In view of this evidence, it is desirable to develop methods which improve the quality of finite sample inferences. One such method is the bootstrap, and this has been explored in the context of GMM by Hall and Horowitz (1996).

Notes

* I am grateful to Atsushi Inoue, Fernanda Peixe, James Stock, and three anonymous reviewers for comments on an earlier draft of this paper.

1 Hall (1993) and Ogaki (1993) provide an overview of the areas in which GMM has been applied.

2 Also see Hall (1998) for a survey of hypothesis tests based specifically on GMM estimators.

3 This generic approach is known as the consumption based asset pricing model.

4 It is possible to generalize the arguments to allow for certain types of nonstationarity; see Gallant and White (1988), Potscher and Prucha (1997).

5 E. g. see Apostol (1974, p. 361).

6 E. g. see Dhrymes (1984, Proposition 92, p. 111).

7 For example see Quandt (1983) or Gallant (1987, ch. 2).

8 This property is not guaranteed by pointwise convergence of QT(0). See Apostol (1974, ch. 9) for a useful discussion of the difference between pointwise and uniform convergence.

9 E. g. see Apostol (1974, p. 355).

10 E. g. see Fuller (1976, p. 199). Hansen (1982) and Wooldridge (1994) provide formal proofs of the theorem.

11 For example see Hamilton (1994, pp. 279 -80).

12 See Hamilton (1994, pp. 261-2) for a discussion of the properties of autocovariance matrices.

13 For example see White (1994, Theorem 8.27, p. 193).

14 The requirement that ST be positive semi-definite (p. s.d.) is the matrix generalization of a nonnegative scalar variance. This property is not guaranteed for estimators of the generic form in (11.22). For example aiT = 1 does not yield a p. s.d. matrix; see Newey and West (1987).

15 See inter alia Newey and West (1987), Gallant (1987), Andrews (1991), Andrews and Monahan (1992).

16 Andrews (1991) and Newey and West (1994) propose data-based methods for band­width selection.

17 See Dhrymes (1984, p. 17).

18 See Rao (1973, ch. 8) for a discussion of the singular normal distribution.

19 If p = q then the asymptotic variance of PT is MSM = (G’0S-1G0)-1.

20 See Hall (2000b, ch. 3).

21 It is common to impose this assumption in both theoretical treatments and applica­tions of these long-run variance estimators in the context of GMM.

22 See Hall (2000b, ch. 5).

23 White (1982) refers to such an estimator as quasi-maximum likelihood.

24 See Hansen and Singleton (1982).

25 This difference facilitates the analysis but makes no difference to the ultimate result.

26 See Newey (1993) for a formal proof.

27 See Hall (1993) or Theil (1971, pp. 451-3).

28 Notice that if E„|^ = o2Is then the o2 factor cancels out as in our example.

29 Chamberlain’s (1987) analysis is based on a form of semiparametric maximum likeli­hood subject to (11.37). Also see Newey (1993, pp. 423-4).

30 One exception is the case in which ut(00) is a martingale difference case for which Hansen (1985) shows Theorem 7 extends directly with only a slight modification to make allowance for conditional heteroskedasticity.

31 In the linear model, global and local identification are equivalent because (11.6) is no longer an approximation but is an identity which holds over 0.

32 This terminology parallels the distinction between exact and near collinearity in the linear regression model.

33 Equation (11.43) implies the explanatory variable is a triangular array {xtT; t = 1, 2 . . . T; T = 1, 2 . . .} but we suppress the second subscript for notational brevity.

34 Notice that the data generation process for xt changes with T and it is for this reason that the expectations operator is indexed by T.

35 The interested reader is refered to Hall (1999b, ch. 6).

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