A COMPANION TO Theoretical Econometrics

Practical Problems

In Section 5 we noted that the motivation for the Cox test statistic was based upon the observation that unless two models, say f(-) and g( ) are nonnested then the expectation

T-% [Lf (0) - Lg(y)j, (13.40)

does not evaluate to zero and as a result standard likelihood ratio statistics are not appropriate. Cox (1961, 1962) proposed a procedure such that a centered (modified) loglikelihood ratio has a well-defined limiting distribution. In Section 5.1 we demonstrated that in the case of the linear regression we may obtain a closed form consistent estimate of (13.40). However, this is the exception rather than the rule and the use of the Cox test has been restricted to a relatively small number of applications due to problems in constructing a consistent estimate of the expected loglikelihood ratio statistic. There are two principal problems. First, in order to estimate (13.40) we require a consistent estimate of the pseudo-true value, y (0O). Second, in most cases even given such an estimate, the expectation (13.40) will still be intractable. An exception is the application of the Cox test to both binary and multinomial probit and logit models. Independent of the dimen­sion of the choice set, the expected difference between the two loglikelihoods under the null has a relatively simple, closed form expression (see Pesaran and Pesaran, 1993).

Following the work of Pesaran and Pesaran (1993, 1995) and Weeks (1996), a simulation-based application of the modified likelihood principle has been used to affect adjustments to the test statistic in order to improve the finite sample size and power properties. A drawback of this approach is that it is still reliant upon a reference distribution which is valid asymptotically. In addition, Orme (1994) attests to the existence of a large number of asymptotically equivalent (AE) vari­ants of the Cox test statistic which represents a formidable menu of choices for the applied econometrician. In the case of the numerator, various test statistics are based upon the use of alternative consistent estimators of the Kullback - Leibler measure of closeness. An additional set of variants of the Cox test statistic depends upon the existence of a number of AE ways of estimating the variance of the test statistic.

An alternative approach based upon the seminal work of Efron (1979), with contributions by Hall (1986), Beran (1988), Hinkely (1988), and Coulibaly and Brorsen (1998), applies bootstrap-based procedures to directly evaluate the em­pirical distribution function of the loglikelihood ratio statistic. In this context the focus is upon correcting the reference distribution rather than centering the loglikelihood ratio statistic and utilizing limiting distribution arguments. This type of adjustment may, in a number of cases, be theoretically justified through Edgeworth expansions and can under certain conditions result in improvements over classical asymptotic inference. The existence of a large menu of broadly equivalent test statistics is also relevant in the context of bootstrap-based infer­ence. Recent surveys by Vinod (1993), Jeong and Maddala (1993), and Li and

Maddala (1996), review a large number of variants including the double, recur­sive, and weighted bootstrap. Similarly, Hall (1988) notes that in many applica­tions the precise nature of the bootstrap design is not stated.

7.1 A simulation application of the modified likelihood principle

The essence of the Cox nonnested test is that the mean adjusted ratio of the maximized loglikelihoods of two nonnested models has a well-defined limiting

Подпись: distribution under the null hypothesis. Using the notation set out in Section 2 above we may write the numerator of the Cox test statistic as

Подпись: (13.41)S/g = T lLR/g у).

The last term on the right-hand side of (13.41), C/g(0T, у), represents a consistent estimator of C/g(0o, y*(0o)), the KLIC measure of closeness of g( ) to /(■). This may be written as C/g(0T, у) = Е/[T_1(L/(0T) - Lg(y))j, and is an estimator of the differ­ence between the expected value of the two maximized loglikelihoods under the distribution given by /(■); у is any consistent estimator for y*(0o). Weeks (1996), in testing probit and logit models of discrete choice, distinguished between three variants, у = {yT, yR(0T), уT}. уT is the MLE of у, у is due to Kent (1986) and is an estimator derived from maximizing the fitted loglikelihood, and y*R(0T) = rR XR=1Y * (0t) is a simulation-based estimator where y* (0T) is the solution to

Подпись: arg max jig (y)image19(13.42)

where yГ(0Т) is the rth draw of yt under H/ using 0T and R is the number of simulations. Note that for both R ^ ™ and T ^ ™ then y*R(0T) ^ y*(00).

C/g, R(0T, y*R(0T))

X [Lf (0T) - Lg (Y *R(0T ))j.

However (13.43) represents one approach to centering the loglikelihood ratio statistic, whereby both 0T and y*R(0T) are treated as fixed parameters. An alter-

native method of mean adjustment is given by the following estimator of KLIC

1 Л

s-i /А1 At? 1 /А ч 7?/ Aw x Xі гт» /А»ч т, / /А w п /л * *

C&r(0, ..., 0RT, Y*(0t), ..., yR(0t)) = — X[L/(0T) - Lg(Y*(0t))], (13.44)

T R r=1

image361 Подпись: TR Подпись: (13.43)

A simulation-based estimator of C/g(0o, y*(0o)) has been suggested by Pesaran and Pesaran (1993) and is given by

where the parameter arguments to both L/( ) and Lg( ) are allowed to vary across each rth replication. (See Coulibaly and Brorsen, 1998.)

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