Resampling the likelihood ratio statistic: bootstrap methods

The bootstrap is a data based simulation method for statistical inference. The bootstrap approach involves approximating the distribution of a function of the observed data by the bootstrap distribution of the quantity. This is done by substituting the empirical distribution function for the unknown distribution and repeating this process many times to obtain a simulated distribution. Its recent development follows from the requirement of a significant amount of computa­tional power. Obviously there is no advantage to utilizing bootstrap procedures when the exact sampling distribution of the test statistic is known. However, it has been demonstrated that when the sampling distribution is not known, the substitution of computational intensive bootstrap resampling can offer an improvement over asymptotic theory. The use of non-pivotal bootstrap testing procedures does not require the mean adjustment facilitated by (13.43) and (13.44). However, pivotal (or bootstrap-t) procedures require both mean and variance adjustments in order to guarantee asymptotic pivotalness.

Utilizing a parametric bootstrap we present below a simple algorithm for resampling the likelihood ratio statistic which we then use to construct the em­pirical distribution function of the test statistic. For the purpose of exposition the algorithm is presented for the non-pivotal bootstrap.

1. Generate R samples of size T by sampling from the fitted null model ft(0T).

2.

image364

For each rth simulated sample, the pair (0T, yi (0T)) represent the parameter estimates obtained by maximizing the loglikelihoods

where yr(0T) denotes the rth bootstrap-sample conditional upon 0 = 0T. We then compute the simulated loglikelihood ratio statistic

Trf = Lf (0T) – Lg(Y * (0t)).

3. By constructing the empirical cdf of {Tf : 1 < r < R}, we can compare the ob­served test statistic, Tf = Lf(0T) – Lg(y*(0T)), with critical values obtained from the R independent (conditional) realizations of Tj. The p-value based upon the bootstrap procedure is given by24

X 1(Tfr > Tf)

Pr = , (13.46)

where 1(.) is the indicator function.

The bootstrap procedure outlined above simply resamples the likelihood ratio statistic without pivoting. There are a number of alternative test statistics which by using pivotal methods are conjectured to represent an improvement over classical first order methods (see for example, Beran (1988) and Hall (1988)). An evaluation of both the size and power properties of a number of simulation and bootstrap-based tests applied to linear versus loglinear regression models and a number of variants of threshold autoregressive models is provided in Kapetanios and Weeks (1999).

Notes

1 Therefore our focus is distinct from Chow (1981) who, in examining a similar prob­lem, assumes that the set of models under consideration contains a general model from which all other competing models may be obtained by the imposition of suitable parameter restrictions.

2 See, for example, Friedman and Meiselman (1963) on alternative consumption models, Barro (1977), Pesaran (1982b) and McAleer, Pesaran, and Bera (1990) on alternative explanations of the unemployment rate; Jorgenson and Siebert (1968), Dixit and Pindyck (1994) and Bernanke, Bohn, and Reiss (1988) on alternative models of investment behavior; and McAleer, Fisher, and Volker (1982) and Smith and Smyth (1991) on nonnested money demand functions.

3 An excellent survey article on nonnested hypothesis testing can be found in Gourieroux and Monfort (1994).

4 In cases where one or more elements of zt are discrete, as in probit or tobit specifica­tions cumulative probabality distribution functions can be used instead of probability density functions.

5 See Engle, Hendry, and Richard (1983).

6 A formalization of the concept of globally nonnested models can be found in Pesaran (1987). Also see Section 6.

7 Note that under Щ, E(yt) = E{exp(ln yt)} = exp(01 + 0.502).

8 See, Pesaran (1984, pp. 249-50).

9 There is substantial literature on nonnested tests of linear versus loglinear regres­sion models. Earlier studies include Aneuryn-Evans and Deaton (1980), Godfrey and Wickens (1981) and Davidson and MacKinnon (1985). In a more recent study Pesaran and Pesaran (1995) have examined the properties of a simulation-based variant of the Cox test.

10 A review of the model selection literature is beyond the scope of the present paper. See, for example, Leamer (1983) for an excellent review. A recent review focusing upon the selection of regressors problems is to be found in Lavergne (1998). Two excellent texts are Grasa (1989) and Linhart and Zucchini (1986). Maddala (1981) edited a special issue of the Journal of Econometrics which focuses on model selection.

11 For the case of the classical linear regression model examples of model selection criteria include Theil’s )2, with more general loss functions based upon information criteria including Akaike’s (1973) information criteria and Schwarz’s (1978) Bayesian information criterion.

12 The concepts of globally and partially nonnested models are defined in Pesaran (1987).

13 See also Dastoor (1981) for further discussion.

14 The cases where ~Lf Ф 0 (respectively Ф 0) but nevertheless £f a0 = 0 (respectively ZgP0 = 0) are discussed in Pesaran (1987, p. 74).

15 See Pesaran (1974) for details of the derivations.

16 For example, it is not possible to test whether X = 1/2, which could have been of interest in assessing the relative weights attached to the two rival models.

17 For a proof see McAleer and Pesaran (1986).

18 Chao and Swanson (1997) provide some asymptotic results for the /-test in the case of nonnested models with 1(1) regressors.

19 Recall that the encompassing condition (13.37) for the regression coefficients implies the condition (13.38) for error variance encompassing but not vice versa.

20 Notice that the normality assumption is not needed and can be relaxed.

21 Similarly, the variance encompassing statistic for testing Hf EHg is given by roT – [6T + T-1a’T X MZX«T ].

22 See also McAleer (1983).

23 See McAleer and Pesaran (1986) for additional details.

24 If T is discrete then repeat values of T can occur requiring that we make an adjust­ment to (13.46).

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