经济代写|计量经济学代写Econometrics代考|Alternative Covariance Matrix Estimators

This expression cannot even be expressed as a function of $\hat{\tau}$ alone. To obtain an expansion of the test statistic that makes use of it, we must make use of the property of the normal distribution which tells us that $E\left(y_t^4\right)=3 \sigma^4$, or, in terms of $\tau, 3 e^{4 \tau} \cdot{ }^4$ Using this property, we can invoke a law of large numbers and conclude that the OPG information matrix estimator is indeed equal to $2+o(1)$ at $\tau=0$. Thus the third variant of the LM test statistic is
$$L M_3=\frac{n^2\left(e^{2 \hat{\tau}}-1\right)^2}{\sum_{t=1}^n\left(y_t^2-1\right)^2}=2 n \hat{\tau}^2+o(1) .$$
Once again, the leading term is $2 n \hat{\tau}^2$, but the form of $L M_3$ is otherwise quite different from that of $L M_1$ or $L M_2$.

Just as there are various forms of the LM test, so are there various forms of the Wald test. Any one of these may be formed by combining the unrestricted estimate $\hat{\tau}$ with some estimate of the information matrix, which in this case is actually a scalar. The simplest choice is just the true information matrix, that is, 2 . With this we obtain
$$W_1=2 n \hat{\tau}^2 .$$
It is easy to see that $W_2$, which uses the empirical Hessian, is identical to $W_1$, because (13.55) evaluated at $\tau=\hat{\tau}$ is just $-2 n$. On the other hand, use of the OPG estimator yields
$$W_3=\hat{\tau}^2 \sum_{t=1}^n\left(y_t^2 e^{-2 \bar{\tau}}-1\right)^2,$$
which is quite different from $W_1$ and $W_2$.
All of the above test statistics were based on $\tau$ as the single parameter of the model, but we could just as well use $\sigma$ or $\sigma^2$ as the model parameter. Ideally, we would like test statistics to be invariant to such reparametrizations. The LR statistic is always invariant, since $\hat{\ell}$ and $\tilde{\ell}$ do not change when the model is reparametrized. But all forms of the Wald statistic, and some forms of the LM statistic, are in general not invariant, as we now illustrate.

经济代写|计量经济学代写Econometrics代考|Classical Test Statistics and Reparametrization

The idea of a reparametrization of a parametrized model was discussed at length in Section 8.3. We saw there that one of the properties of maximum likelihood estimation is its invariance under reparametrizations. Since the classical tests are undertaken in the context of maximum likelihood estimation, it might be expected, or at least hoped, that the classical test statistics would likewise be parametrization invariant. That is true for the LR statistic, since, as was shown in Chapter 8, the value of a maximized loglikelihood function is invariant to reparametrization. But the results of the last section have shown that it cannot be true in general for the other two classical tests. In this section, we discuss the effects of reparametrization on the classical test statistics in more detail. In particular, we endeavor to determine what ingredients of the LM and Wald tests, and what ingredients of various information matrix estimators, are or are not responsible for the parametrization dependence of so many of the possible forms of the classical tests. We believe that these are important topics. However, the discussion is necessarily quite detailed, and some readers may wish to skip this section on a first reading.
First of all, we must make it clear that when we speak of invariance we mean different things when we are discussing different quantities. For example, if a model is reparametrized by a mapping $\boldsymbol{\eta}: \Theta \rightarrow \Phi$, where $\boldsymbol{\theta}$ and $\phi$ denote the parameter vectors under the two parametrizations, then by the invariance of the MLE under reparametrization it is certainly not meant that $\hat{\boldsymbol{\theta}}=\hat{\boldsymbol{\phi}}$, but rather that
$$\hat{\boldsymbol{\phi}}=\boldsymbol{\eta}(\hat{\boldsymbol{\theta}}) .$$
The notation here was used previously in Chapter 8, around equation (8.23), and will be used again below. We must distinguish between quantities expressed in terms of the $k$-vector of parameters $\boldsymbol{\theta}$ and quantities expressed in terms of the $k$-vector of parameters $\phi$. As in Chapter 8 , we will use primes to denote quantities expressed in terms of $\phi$.

For the maximized loglikelihood function, invariance means simply that
$$\ell(\hat{\boldsymbol{\theta}})=\ell^{\prime}(\hat{\boldsymbol{\phi}}) .$$
Thus, when we speak of parameter estimates being invariant under reparametrization, we mean that (13.59) holds, whereas when we speak of maximized loglikelihood functions, or test statistics, we mean that the actual numerical value is unchanged when calculated using different parametrizations.

计量经济学代考

经济代写|计量经济学代写Econometrics代考|Alternative Covariance Matrix Estimators

$$L M_3=\frac{n^2\left(e^{2 \hat{\tau}}-1\right)^2}{\sum_{t=1}^n\left(y_t^2-1\right)^2}=2 n \hat{\tau}^2+o(1) .$$

$$W_1=2 n \hat{\tau}^2 .$$

$$W_3=\hat{\tau}^2 \sum_{t=1}^n\left(y_t^2 e^{-2 \bar{\tau}}-1\right)^2,$$

经济代写|计量经济学代写Econometrics代考|Classical Test Statistics and Reparametrization

$$\hat{\boldsymbol{\phi}}=\boldsymbol{\eta}(\hat{\boldsymbol{\theta}}) .$$

$$\ell(\hat{\boldsymbol{\theta}})=\ell^{\prime}(\hat{\boldsymbol{\phi}}) .$$

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