Asymptotic Tests and Related Topics

4.5.1 Likelihood Ratio and Related Tests

Let Ux, 0) be the joint density of a Г-vector of random variables x = (Xi, x2,. . . , xTY characterized by a ЛГ-vector of parameters 6. We assume all the conditions used to prove the asymptotic normality (4.2.23) of the maximum likelihood estimator 6. In this section we shall discuss the asymp­totic tests of the hypothesis

h(0) = O, (4.5.1)

where h is a ^-vector valued differentiable function with q<K. We assume that (4.5.1) can be equivalently written as

Подпись:В = r(a),

where a is a p-vector of parameters such that p = K— q. We denote the constrained maximum likelihood estimator subject to (4.5.1) or (4.5.2) as в = Ha).

image328 image329

Three asymptotic tests of (4.5.1) are well known; they are the likelihood ratio test (LRT), Wald’s test (Wald, 1943), and Rao’s score test (Rao, 1947). The definitions of their respective test statistics are

image330 image331

Maximization of log L subject to the constraint (4.5.1) is accomplished by setting the derivative of log L — A’h(0) with respect to в and A toj), where A is the vector of Lagrange multipliers. Let the solutions be в and A. Then they satisfy

Подпись: В image333

Inserting this equation into the right-hand side of (4.5.5) yields Rao = — A’BA where

Silvey (1959) showed that В is the asymptotic variance-covariance matrix of A and hence called Rao’s test the Lagrange multiplier test. For a more thorough discussion of the three tests, see Engle (1984).

All three test statistics can be shown to have the same limit distribution, X2(q), under the null hypothesis. In Wald and Rao, a2 log L/двдв’ can be replaced with T plim T~1d1 log L/двдв’ without affecting the limit distribu­tion. In each test the hypothesis (4.5.1) is to be rejected when the value of the test statistic is large.

We shall prove LRT —» x2(q). By a Taylor expansion we have

log L(0O) = log Цв) + ~^qT~

 

000-0)

 

(4.5.6)

 

image334

image335

image336

image337

Treating Дг(а)] = Да) as a function of a, we similarly obtain

 

log Да) – log Дао) = і Да – Оо)’За(а – а„),

 

(4.5.9)

 

(4.5.10)

(4.5.11)

(4.5.12)

(4.5.13)

(4.5.14)

(4.5.15)

(4.5.16)

(4.5.17)

 

where

 

image338

and

 

М,-

 

Since

 

aiogL

 

= R/_LdjogL

 

image339

we have from (4.5.11)—(4.5.15)

 

image340

we obtain

LRT = €'(l ~ jybSjR’jy*)*. (4.5.18)

But, because

Подпись: (4.5.19)3a = R’JeR,

I — 3y2R3~lR’3 У2 can be easily shown to be an idempotent matrix of rank q. Therefore, by Theorem 2 of Appendix 2, LRT —* хЧя)-

The proof of Wald —*хЧя) and Rao —► хЧя) are omitted; the former is very easy and the latter is as involved as the preceding proof.

Next we shall find explicit formulae for the three tests (4.5.3), (4.5.4), and

(4.5.5) for the nonlinear regression modeU4.3.1) when the error u is normal. Let P be the NLLS estimator of Д, and let P be the constrained NLLS, that is, the value of P that minimizes (4.3.5) subject to the constraint h(p) = 0. Also, define 6 = (df/dp’)j and G = (dt/dP’)j. Then the three test statistics are de­fined as

Подпись: (4.5.20) (4.5.21) LRT = T[log T’lST(P) – log T-‘SAP)],

image343

Wald

and

T[y-{(p)]’G{G’GrlG'[y-m]

Rao ———————— =—————– .

ST(fi)

Because (4.5.20), (4.5.21), and (4.5.22) are special cases of(4.5.3), (4.5.4), and

(4.5.5) , all three statistics are asymptotically distributed asx2(q) under the null hypothesis if u is normal.10 Furthermore, we can show that statistics (4.5.20),

(4.5.21) , and (4.5.22) are asymptotically distributed as хя) under the null even if u is not normal. Thus these statistics can be used to test a nonlinear hypothesis under a nonnormal situation.

In the linear model with linear hypothesis Q’P = 0, statistics (4.5.20)-

(4.5.22) are further reduced to

LRT = Г log [ST{P)/ST(P)],

(4.5.23)

Wald = T[ST(p) – ST(p)]/ST(p),

(4.5.24)

Rao = T[ST(P) – ST(P)]/ST(p).

(4.5.25)

Thus we can easily show Wald S LRT ё Rao. The inequalities hold also in the multiequation linear model, as shown by Bemdt and Savin (1977). Al­though the inequalities do not always hold for the nonlinear model, Mizon

(1977) found Wald S LRT most of the time in his samples.

Gallant and Holly (1980) obtained the asymptotic distribution of the three statistics under local alternative hypotheses in a nonlinear simultaneous equations model. Translated into the nonlinear regression model, their results can be stated as follows: If there exists a sequence of true values {fil) such that lim PI = fi0 and 6 = lim TuPl~ plim /?) is finite, statistics (4.5.20),

(4.5.21) , and (4.5.22) converge to chi-square with q degrees of freedom and noncentrality parameter A, where

image344(4.5.26)

Note that if is distributed as a ^-vector 2V(0, V), then ({ 4- /r)’V_1(£ + pi) is distributed as chi-square with q degrees of freedom and noncentrality parameter In other words, the asymptotic local power of the tests

based on the three statistics is the same.

There appear to be only a few studies of the small sample properties of the three tests, some of which are quoted in Breusch and Pagan (1980). No clear-cut ranking of the tests emerged from these studies.

A generalization of the Wald statistic can be used to test the hypothesis

(4.5.1) , even in a situation where the likelihood function is unspecified, as long as an asymptotically normal estimator fl of fi is available. Suppose fi is asymptotically distributed as N(fi, V) under the null hypothesis, with V estimated consistently by V. Then the generalized Wald statistic is defined by

image345(4.5.27)

and is asymptotically distributed as* 2(^) under the null hypothesis. Note that

(4.5.21) is a special case of (4.5.27).

Another related asymptotic test is the specification test of Hausman (1978). It can be used to test a more general hypothesis than (4.5.1). The only requirement of the test is that we have an estimator, usually a maximum likelihood estimator, that is asymptotically efficient under the null hypothesis but loses consistency under an alternative hypothesis and another estimator that is asymptotically less efficient than the first under the null hypothesis but remains consistent under an alternative hypothesis. If we denote the first

a <s<

estimator by 0 and the second by 0, the Hausman test statistic is defined by {6 — Q)’~6 — в), where V is a consistent estimator of the asymptotic variance-covariance matrix of (0—0). Under the null hypothesis it is asymptotically distributed as chi-square with degrees of freedom equal to the dimension of the vector 0.

If we denote the^ asymptotic variance-covariance matrix by V, it is well known that V(0 —0) = V(0) — V(0). This equality follows from V(0) = V12, where Vj2 is the asymptotic covariance between 0 and 0. To verify this equality, note that if it did not hold, we could define a new estimator 0 + [V(0) — V12][V(0— 0)]“‘£0 — 0), the asymptotic variance-covariance matrix of which is V(0) – [V(0) – V12][V(0 – 0)]"‘[V(0) – V12]’, which is smaller (in the matrix sense) than V( 0). But this is a contradiction because 0 is asymptotically efficient by assumption.

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