2.9 Three General Test Procedures

Under the classical assumptions of the MLRM, in the section on the testing hypotheses we have derived appropriate finite sample test statistics in order to verify linear restrictions on the coefficients. Nevertheless, these exact tests are not always available, so in these cases it is very useful to consider the following three approaches, which allow us to derive large sample tests, which are asymptotically equivalent. Several situations which require making use of these tests will be presented in following chapters of this book. In this chapter we now focus on their general derivation, and on their illustration in the context of an MLRM under classical assumptions.

All three test procedures are developed within the framework of ML estimation and they use the information included in the log-likelihood in different but asymptotically equivalent ways.

The general framework to implement these principles is defined as follows:

\begin{displaymath}\begin{array}{c} H_{0}:h(\theta)=0 \\ H_{A}:h(\theta)\neq 0 \end{array}\end{displaymath} (2.180)

where $ h(\theta)$ includes $ q$ restrictions ($ q \leq k$) on the elements of the parameter vector $ \theta$ which has $ k$ dimension. Furthermore, suppose we have estimated by both unrestricted and restricted ML, so that we have the vectors $ \theta$ and $ \tilde{\theta}_{R}$.

Formal derivation of these tests can be found, for example, in Davidson and MacKinnon (1993).


2.9.1 Likelihood Ratio Test (LR)

This test is based on the distance between the log-likelihood function evaluated at the ML and the RML estimators. Thus, it is defined as:

$\displaystyle LR=2[\ell(\tilde{\theta})-\ell(\tilde{\theta}_{R})]$ (2.181)

which, under the null hypothesis (2.180) is asymptotically distributed as a $ \chi^{2}_{q}$. This result is obtained through a Taylor expansion of second order of the restricted log-likelihood around the ML estimator vector.

Taking (2.181) it can be thought that if the restriction $ h(\theta)=0$ is true when it is included, the log-likelihood should not reduce its value by a significant amount and thus, both $ \ell(\tilde{\theta})$ and $ \ell(\tilde{\theta}_{R})$ should be similar. Given that the inequality $ \ell(\tilde{\theta}) \geq \ell(\tilde{\theta}_{R})$ always holds (because a maximum subject to restrictions is never larger than an unrestricted maximum), significant discrepancies between both estimated log-likelihoods can be thought of as evidence against $ H_{0}$, since the RML estimator moves far away from the unrestricted ML.

Another way of understanding what underlies this test focuses on the asymptotic properties of the ML estimators under correct specification. Given several regularity conditions, the ML estimators are consistent, asymptotically efficient and their asymptotic distribution is normal. Moreover, it is shown that the RML estimators are consistent when the restrictions are true (correct a priori information). According to these results, we can say that, if $ H_{0}$ is true, then both ML and RML estimators are consistent, so it is expected that $ \ell(\tilde{\theta}) \cong
\ell(\tilde{\theta}_{R})$. Thus, small values of (2.181) provide evidence in favour of the null hypothesis.

As it was earlier described, the decision rule consists of, for a fixed significance level $ \epsilon$, comparing the value of the LR statistic for a given sample ($ LR^{*}$) with the corresponding critical point $ \chi^{2}_{\epsilon}$ (with $ q$ degrees of freedom), and concluding the rejection of $ H_{0}$ if $ LR^{*}>\chi^{2}_{\epsilon}$. Equivalently, we reject $ H_{0}$ if the p-value is less than $ \epsilon$.


2.9.2 The Wald Test (W)

This test is based on the distance between $ \tilde{\theta}_{R}$ and $ \tilde{\theta}$, so it tries to find out if the unrestricted estimators nearly satisfy the restrictions implied by the null hypothesis. The statistic is defined as :

$\displaystyle W=nh(\tilde{\theta})^{\top }[H_{\tilde{\theta}}^{\top }(I_{\infty}(\tilde{\theta}))^{-1}H_{\tilde{\theta}}]^{-1}h(\tilde{\theta})$ (2.182)

where $ H_{\theta}$ is the $ k \times q$ matrix of the derivatives $ \frac{\partial h(\theta)}{\partial\theta}$, in such a way that $ H_{\tilde{\theta}}$ means that it is evaluated at the unrestricted ML estimators.

According to the result:

$\displaystyle \sqrt{n}h(\theta)\rightarrow_{as}N[0,H_{\theta}^{\top }(I_{\infty}({\theta}))^{-1}H_{\theta}]$ (2.183)

The construction of a quadratic form from (2.183), under $ H_{0}$ and evaluated at the ML estimators, leads to the conclusion that (2.182) follows a $ \chi^{2}_{q}$.

Given that $ \tilde{\theta}$ is consistent, if $ H_{0}$ is true, we expect that $ h(\tilde{\theta})$ takes a value close to zero and consequently the value of W for a given sample ($ W^{*}$) adopts a small value. However, $ H_{0}$ is rejected if $ W^{*} >
\chi^{2}_{\epsilon}$. This amounts to saying that $ H_{0}$ is rejected if $ h(\tilde{\theta})$ is "very distant" from zero.

Finally, we must note that the asymptotic information matrix, which appears in (2.182), is usually non observable. In order to be able to implement the test, $ I_{\infty}(\theta)$ is substituted by $ \frac{I_{n}(\theta)}{n}$. Thus, the W statistic for a given sample of size $ n$ is written as:

$\displaystyle W_{n}=nh(\tilde{\theta})^{\top }\left[H_{\tilde{\theta}}^{\top }\...
...}}^{\top }(I_{n}(\tilde{\theta}))^{-1}H_{\tilde{\theta}}]^{-1}h(\tilde{\theta})$ (2.184)

which converges to the statistic $ W$ of (2.182) as n increases.


2.9.3 Lagrange Multiplier Test (LM)

This test is also known as the Rao efficient score test. It is defined as:

$\displaystyle LM=\frac{1}{n}\tilde{\lambda}^{\top }H_{\tilde{\theta}_{R}}^{\top }[I_{\infty}(\tilde{\theta_{R}})]^{-1}H_{\tilde{\theta}_{R}}\tilde{\lambda}$ (2.185)

which under the null hypothesis has an asymptotic distribution $ \chi^{2}_{q}$. Remember that $ \tilde{\lambda}$ is the estimated Lagrange multiplier vector which emerges if one maximizes the likelihood function subject to constraints by means of a Lagrange function.

This asymptotic distribution is obtained from the result:

$\displaystyle \frac{\tilde{\lambda}}{\sqrt{n}}\rightarrow_{as} N[0,(H_{\theta}^{\top }[I(\theta)]^{-1}H_{\theta})^{-1}]$ (2.186)

This result allows us to construct a quadratic form that, evaluated at $ \tilde{\theta}_{R}$ and under $ H_{0}$ leads to the $ \chi^{2}_{q}$ distribution of (2.185).

The idea which underlies this test can be thought of as follows. If the restrictions of $ H_{0}$ are true, the penalization for including them in the model is minimum. Thus, $ \tilde{\lambda}$ is close to zero, and $ LM$ also tends to zero. Consequently, large values of the statistic provide evidence against the null hypothesis.

Again, we must note that expression (2.185) contains the asymptotic information matrix, which is a problem for implementing the test. In a similar way to that described in the Wald test, we have:

$\displaystyle LM_{n}=\frac{1}{n}\tilde{\lambda}^{\top }H_{\tilde{\theta}_{R}}^{...
...}}^{\top }[I_{n}(\tilde{\theta_{R}})]^{-1}H_{\tilde{\theta}_{R}}\tilde{\lambda}$ (2.187)

which converges to (2.185) as $ n$ tends to $ \infty$.

If we remember the restricted maximization problem, which in our case is solved by means of the Lagrange function:

$\displaystyle \Im=\ell(\theta)-\lambda^{\top }h(\theta)
$

the set of first order conditions can be expressed as:

$\displaystyle \frac{\partial\Im}{\partial\theta}=g(\theta)-H\lambda=0
$

$\displaystyle \frac{\partial\Im}{\partial\lambda}=h(\theta)=0$ (2.188)

with $ g(\theta)$ being the gradient or score vector, that is to say, the first derivatives of the log-likelihood with respect to the parameters.

From the first set of first-order conditions in (2.188) one can deduce that $ g(\theta)=H\lambda$. Thus, $ H\tilde{\lambda}$ can be substituted by $ g(\tilde{\theta}_{R})$, leading to the known score form of the LM test (or simply the score test):

$\displaystyle LM=\frac{1}{n}g(\tilde{\theta}_{R})^{\top }[I_{\infty}(\tilde{\theta}_{R})]^{-1}g(\tilde{\theta}_{R})$ (2.189)

So, when the null hypothesis is true, the restricted ML estimator is close to the unrestricted ML estimator, so we expect that $ g(\tilde{\theta}_{R})\cong 0$, since we know that in the unrestricted optimization it is maintained that $ g(\tilde{\theta})=0$. It is evident that this test is based on the distance between $ g(\tilde{\theta}_{R})$ and $ g(\tilde{\theta})$. Small values of the statistic provide evidence in favour of the null hypothesis.

Again $ I_{\infty}(\cdot)$ is substituted by $ \frac{I_{n}(\cdot)}{n}$, and the expression of the statistic for a sample of size $ n$ is given by:

$\displaystyle LM_{n}=g(\tilde{\theta}_{R})^{\top }[I_{n}(\tilde{\theta}_{R})]^{-1}g(\tilde{\theta}_{R})$ (2.190)

Very often, the $ LM$ test statistic is asymptotically equal to $ n$ times the non centered $ R^{2}$ of an artificial linear regression (for a more detailed description of this approach, see Davidson and Mackinnon (1984).


2.9.4 Relationships and Properties of the Three General Testing Procedures

The main property of these three statistics is that, under $ H_{0}$, all of them tend to the same random variable, as $ n\rightarrow\infty$. This random variable is distributed as a $ \chi^{2}_{q}$. In other words, in the asymptotic context we are dealing with the same statistic, although they have very different definitions. Thus, in large samples, it does not really matter which of the three tests we use. The choice of which of the three test statistics to use depends on the convenience in their computation. LR requires obtaining two estimators (under $ H_{0}$ and $ H_{A}$). If $ \tilde{\theta}$ is easy to compute but $ \tilde{\theta}_{R}$ is not, as may be the case of non linear restrictions in a linear model, then the Wald statistic becomes attractive. On the other hand, if $ \tilde{\theta}_{R}$ is easier to compute than $ \tilde{\theta}$, as is often the case of tests for autocorrelation and heteroskedasticity, then the $ LM$ test is more convenient. When the sample size is not large, choosing from the three statistics is complicated because of the fact that they may have very different finite-sample properties.

These tests satisfy the "consistency of size $ \varepsilon$" property, and moreover, they are "locally uniformly more powerful". The first property means that, when $ H_{0}$ is true, the probability of deciding erroneously (rejecting $ H_{0}$) is equal or less than the fixed significance level. The second property implies that these tests have maximum power (the probability of rejecting $ H_{0}$ when it is false) against alternative hypotheses such as:

$\displaystyle h(\theta)=\frac{b}{\sqrt{n}}
$

with $ b\neq 0$.

We shall now examine some questions related to the use of these tests. In a finite sample context, the asymptotic distribution used for the three tests is different from the exact distribution, which is unknown (except in some situations, as an MLRM under the classical assumptions), and furthermore, may not be equal in the three tests.

Moreover, once a significance level $ \epsilon$ is adopted, the same critical point $ \chi^{2}_{\epsilon}$ is used in the three cases because they have the same asymptotic distribution. But the values the three tests take, given the same sample data, are different, so this can lead to opposite conclusions. Specifically, it has been shown that for most models the following holds:

$\displaystyle W \geq LR \geq LM
$

that is to say, one test may indicate rejecting the null hypothesis whereas the other may indicate its acceptance.


2.9.5 The Three General Testing Procedures in the MLRM Context

In this subsection we try to present the implementation of the LR, W and LM tests in the framework of an MLRM under the classical assumptions, when we want to test a set of linear restrictions on $ \beta $, as was established earlier :

\begin{displaymath}\begin{array}{c} H_{0}:R\beta=r \\ H_{A}:R\beta \neq r \end{array}\end{displaymath} (2.191)

Our aim is mainly didactic, because in the framework of the MLRM there are finite sample tests which can be applied, and the asymptotic tests here derived will be unnecessary. Nevertheless, the following derivations allows us to illustrate all the concepts we have to know in order to construct these tests, and moreover, allows us to show some relationships between them.

In order to obtain the form which the LR, W and LM adopt, when the $ q$ linear restrictions (2.191) are tested, it is convenient to remember some results that were obtained in the previous sections referring to the ML and the RML estimation. First, the set of parameters of the MLRM are denoted:

$\displaystyle \theta^{\top }=(\beta^{\top },\sigma^{2})
$

The log-likelihood function and the ML estimators ( $ \tilde{\theta}$) are rewritten as follows:

$\displaystyle \ell(\theta)=-\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\sigma^{2}-\frac{(y-X\beta)^{\top }(y-X\beta)}{2\sigma^{2}}$ (2.192)

$\displaystyle \tilde{\beta}=(X^{\top }X)^{-1}X^{\top }y$ (2.193)

$\displaystyle \tilde{\sigma}^{2}=\frac{\tilde{u}^{\top }\tilde{u}}{n}$ (2.194)

Analogously, the gradient vector and the information matrix are given by:

$\displaystyle g(\theta)= \begin{pmatrix}\frac{\partial\ell(\theta)}{\partial\be...
...}X\beta) \\ -\frac{n}{2\sigma^{2}}+\frac{u^{\top }u}{2\sigma^{4}} \end{pmatrix}$ (2.195)

$\displaystyle I_{n}(\theta)= \begin{pmatrix}\frac{X^{\top }X}{\sigma^{2}} & 0 \\ 0 & \frac{n}{2\sigma^{4}} \end{pmatrix}$ (2.196)

$\displaystyle [I_{n}(\theta)]^{-1}= \begin{pmatrix}\sigma^{2}(X^{\top }X)^{-1} & 0 \\ 0 &\frac{2 \sigma^{4}}{n}\ \end{pmatrix}$ (2.197)

In order to obtain the restricted maximum likelihood estimators (RML), the Lagrange function is expressed as:

$\displaystyle \Im=\ell(\theta)+2\lambda^{\top }(R\beta-r)$ (2.198)

The first-order condition of optimization

\begin{displaymath}
\begin{array}{ccc}
\frac{\partial\Im}{\partial\beta}=0; & \...
...ma^{2}}=0; &
\frac{\partial\Im}{\partial\lambda}=0
\end{array}\end{displaymath}

leads to the obtainment of the RML estimators included in $ \tilde{\theta}_{R}$:

$\displaystyle \tilde{\beta}_{R}= \tilde{\beta}+(X^{\top }X)^{-1}R^{\top }[R(X^{\top }X)^{-1}R^{\top }]^{-1}(r-R\tilde{\beta})$ (2.199)

$\displaystyle \tilde{\sigma}^{2}_{R}=\frac{\tilde{u}_{R}^{\top }\tilde{u}_{R}}{n}$ (2.200)

together with the estimated Lagrange multiplier vector:

$\displaystyle \tilde{\lambda}=\frac{[R(X^{\top }X)^{-1}R^{\top }]^{-1}(r-R\tilde{\beta})}{\tilde{\sigma}^{2}_{R}}$ (2.201)

To obtain the form of the LR test in the MLRM, given its general expression

$\displaystyle LR=2(\ell(\tilde{\theta})-\ell(\tilde{\theta}_{R}))
$

we substitute the parameters of expression (2.192) for the ML and RML estimators, obtaining $ \ell(\tilde{\theta})$ and $ \ell(\tilde{\theta}_{R})$, respectively:

$\displaystyle \ell(\tilde{\theta})=-\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\tilde{\si...
...^{2}-\frac{(y-X\tilde{\beta})^{\top }(y-X\tilde{\beta})}{2\tilde{\sigma}^{2}}=
$

$\displaystyle -\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\tilde{\sigma}^{2}-\frac{\tilde...
...{\sigma}^{2}}= -\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\tilde{\sigma}^{2}-\frac{n}{2}$ (2.202)

$\displaystyle \ell(\tilde{\theta}_{R})=-\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\tilde...
...y-X\tilde{\beta}_{R})^{\top }(y-X\tilde{\beta}_{R})}{2\tilde{\sigma}^{2}_{R}}=
$

$\displaystyle -\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\tilde{\sigma}^{2}_{R}-\frac{\t...
...^{2}_{R}}= -\frac{n}{2}\ln2\pi-\frac{n}{2}\ln\tilde{\sigma}^{2}_{R}-\frac{n}{2}$ (2.203)

Note that, in order to obtain the last terms of (2.202) and (2.203), the sums of squared residuals $ \tilde{u}^{\top }\tilde{u}$ and $ \tilde{u}_{R}^{\top }\tilde{u}_{R}$ are written as a function of $ \tilde{\sigma}^{2}$ and $ \tilde{\sigma}^{2}_{R}$ respectively, by taking their corresponding expressions (2.194) and (2.200).

We now substitute the two last expressions in the general form of the LR test, to obtain:

$\displaystyle LR=2\left[-\frac{n}{2}\ln\tilde{\sigma}^{2}+\frac{n}{2}\ln\tilde{\sigma}^{2}_{R}\right]=n\ln\frac{\tilde{\sigma}^{2}_{R}}{\tilde{\sigma}^{2}}$ (2.204)

Thus, (2.204) is the expression of the LR statistic which is used to test linear hypothesis in an MLRM under classical assumptions.

In order to derive the form of the Wald test in this context, we remember the general expression of this test which is presented in (2.184):

$\displaystyle W_{n}=(h(\tilde{\theta}))^{\top }[H_{\tilde{\theta}}^{\top }(I_{n}(\tilde{\theta}))^{-1}H_{\tilde{\theta}}]^{-1}h(\tilde{\theta})
$

The elements of the last expression in the context of the MLRM are:

$\displaystyle h(\theta)=R\beta-r$ (2.205)

$\displaystyle H_{\theta}= \begin{pmatrix}\frac{\partial h(\theta)}{\partial\bet...
...\theta)}{\partial\sigma^{2}} \end{pmatrix} = \begin{pmatrix}R\\ 0 \end{pmatrix}$ (2.206)

with $ H_{\theta}$ being a $ (k+1)\times q$ matrix.

Then, from (2.197) and (2.206) we get:

$\displaystyle H_{\theta}^{\top }[I_{n}(\theta)]^{-1}H_{\theta}=\sigma^{2}R(X^{\top }X)^{-1}R^{\top }
$

Making the inverse of this matrix, and evaluating it at the ML estimators, we have:

$\displaystyle [H_{\tilde{\theta}}^{\top }[I_{n}(\tilde{\theta})]^{-1}H_{\tilde{\theta}}]^{-1}=\frac{1}{\tilde{\sigma}^{2}}[R(X^{\top }X)^{-1}R^{\top }]^{-1}
$

Thus, in the context we have considered, the Wald statistic can be expressed as:

$\displaystyle W_{n}=\frac{1}{\tilde{\sigma}^{2}}(R\tilde{\beta}-r)^{\top }[R(X^{\top }X)^{-1}R^{\top }]^{-1}(R\tilde{\beta}-r)$ (2.207)

or equivalently, in agreement with the equality given in (2.167), it can be written:

$\displaystyle W_{n}=\frac{\tilde{u}^{\top }_{R}\tilde{u}_{R}-\tilde{u}^{\top }\...
...tilde{u}}=n\frac{\tilde{\sigma}^{2}_{R}-\tilde{\sigma}^{2}}{\tilde{\sigma}^{2}}$ (2.208)

(Note that the vector $ (r-R\tilde{\beta})$ in (2.167) and $ (R\tilde{\beta}-r)$ in (2.207) leads to the same quadratic form).

With respect to the LM test, it must be remembered that it had two alternative forms, and both will be considered in this illustration.

If we focus on the first one, which was written as:

$\displaystyle LM_{n}=\tilde{\lambda}^{\top }H_{\tilde{\theta}_{R}}^{\top }[I_{n}(\tilde{\theta}_{R})]^{-1}H_{\tilde{\theta}_{R}}\tilde{\lambda}
$

their elements were previously defined, so we can write:

$\displaystyle H_{\tilde{\theta}_{R}}^{\top }[I_{n}(\tilde{\theta}_{R})]^{-1}H_{\tilde{\theta}_{R}}=\tilde{\sigma}^{2}_{R}R(X^{\top }X)^{-1}R^{\top }$ (2.209)

Thus, the LM statistic in an MLRM under classical assumptions gives:

$\displaystyle LM_{n}=\tilde{\lambda}^{\top }H_{\tilde{\theta}_{R}}^{\top }[I_{n}(\tilde{\theta}_{R})]^{-1}H_{\tilde{\theta}_{R}}\tilde{\lambda}=
$

$\displaystyle \frac{(r-R\tilde{\beta})^{\top }[R(X^{\top }X)^{-1}R^{\top }]^{-1...
...[R(X^{\top }X)^{-1}R^{\top }]^{-1}(r-R\tilde{\beta})}{\tilde{\sigma}^{2}_{R}}=
$

$\displaystyle \frac{1}{\tilde{\sigma}^{2}_{R}}(r-R\tilde{\beta})^{\top }[R(X^{\top }X)^{-1}R^{\top }]^{-1}(r-R\tilde{\beta})$ (2.210)

Again, we consider the equality (2.167), so we obtain:

$\displaystyle LM_{n}=\frac{\tilde{u}^{\top }_{R}\tilde{u}_{R}-\tilde{u}^{\top }...
...{R}}= n\frac{\tilde{\sigma}^{2}_{R}-\tilde{\sigma}^{2}}{\tilde{\sigma}^{2}_{R}}$ (2.211)

Now, we shall consider the second form of the LM test, which is given by:

$\displaystyle LM=g(\tilde{\theta}_{R})^{\top }[I_{n}(\tilde{\theta}_{R})]^{-1}g(\tilde{\theta}_{R})
$

According to the expressions (2.195) and (2.197), evaluated at the RML estimator vector, we have:

$\displaystyle g(\tilde{\theta}_{R})=
\begin{pmatrix}
\frac{1}{\tilde{\sigma}^{2...
...trix}
\frac{1}{\tilde{\sigma}^{2}_{R}}X^{\top }\tilde{u}_{R}\\
0
\end{pmatrix}$

and consequently,

$\displaystyle LM_{n}=
\begin{pmatrix}
\frac{1}{\tilde{\sigma}^{2}_{R}}\tilde{u}...
...rix}
\frac{1}{\tilde{\sigma}^{2}_{R}}X^{\top }\tilde{u}_{R} \\
0
\end{pmatrix}$

$\displaystyle =\frac{1}{\tilde{\sigma}^{2}_{R}}\tilde{u}_{R}^{\top }X(X^{\top }X)^{-1}X^{\top }\tilde{u}_{R}$ (2.212)

It the restricted residual vector given in (2.162) is substituted in (2.212), one leads to the same expression (2.211) after some calculations.

Having presented the corresponding expressions of the three statistics for testing linear restrictions, it is convenient to derive a last result which consists in obtaining the each statistic as a function of the general F test given in (2.139) or (2.140). If we take this last expression, we have that:

$\displaystyle F=\frac{\frac{\tilde{u}_{R}^{\top }\tilde{u}_{R}-\tilde{u}^{\top ...
...}{n-k}}= \frac{n}{q\hat{\sigma}^{2}}(\tilde{\sigma}^{2}_{R}-\tilde{\sigma}^{2})$ (2.213)

According to the relationship between $ \tilde{\sigma}^{2}$ and $ \hat{\sigma}^{2}$, given by:

$\displaystyle \hat{\sigma}^{2}=\frac{n\tilde{\sigma}^{2}}{n-k}
$

it is straightforward that

$\displaystyle F=\frac{n-k}{q}\frac{(\tilde{\sigma}^{2}_{R}-\tilde{\sigma}^{2})}{\tilde{\sigma}^{2}}$ (2.214)

From this last result, we have:

$\displaystyle \frac{(\tilde{\sigma}^{2}_{R}-\tilde{\sigma}^{2})}{\tilde{\sigma}...
...}\Rightarrow \frac{\tilde{\sigma}^{2}_{R}}{\tilde{\sigma}^{2}}=1+\frac{Fq}{n-k}$ (2.215)

Expression (2.215) is substituted in (2.204), (2.208) and (2.211), in order to obtain $ LR$, $ W$ and $ LM$ tests as functions of F. The corresponding expressions are:

$\displaystyle LR=n\ln(1+\frac{Fq}{n-k})$ (2.216)

$\displaystyle W=\frac{nqF}{n-k}$ (2.217)

$\displaystyle LM=\frac{nq}{(n-k)F^{-1}+q}$ (2.218)

The set of results (2.216) to (2.218) allows us to understand some arguments mentioned in the previous subsection. Specifically, in the context of an MLRM under the classical assumptions, it is possible to obtain the exact distribution of each statistic, by means of their relationship with the F statistic. Moreover, we can see that such exact distributions are not the same as their corresponding asymptotic distributions, and furthermore the three statistics are different in the finite sample framework. This fact can lead to obtain opposite conclusions from their decision rules, because, as Berndt, and Savin (1977) show, it is satisfied $ W \geq LR \geq LM$.


2.9.6 Example

With the aim of testing the same restriction as in previous sections, in the quantlet XEGmlrm07.xpl we calculate the three asymptotic tests LR, W and LM. Note that the RML and ML estimation of $ \beta $ and $ \sigma ^{2}$, were obtained in the previous quantlet, 12554 XEGmlrm06 .

12558 XEGmlrm07.xpl

The p-values associated with the LR, W and LM statistics show that the null hypothesis is rejected in all the cases.