# 经济代写|计量经济学代写Econometrics代考|ECON2271

## 经济代写|计量经济学代写Econometrics代考|Conditional Moment Tests

One important approach to model specification testing that we have not yet discussed is to base tests directly on certain conditions that the error terms of a model should satisfy. Such tests are sometimes called moment specification tests but are more frequently referred to as conditional moment, or CM, tests. They were first suggested by Newey (1985a) and Tauchen (1985) and have been further developed by White (1987), Pagan and Vella (1989), Wooldridge (1991a, 1991b), and others. The basic idea is that if a model is correctly specified, many random quantities which are functions of the error terms should have expectations of zero. The specification of a model sometimes allows a stronger conclusion, according to which such functions of the error terms have zero expectations conditional on some information set – whence the terminology of conditional moment tests.

Since an expectation is often referred to as a moment, the condition that a random quantity has zero expectation is generally referred to as a moment condition. Even if a population moment is zero, its empirical counterpart, which we will call an empirical moment, will (almost) never be so exactly, but it should not be significantly different from zero. Conditional moment tests are based directly on this fact.

Conditional moment tests can be used to test many different aspects of the specification of econometric models. Suppose that the economic or statistical theory behind a given parametrized model says that for each observation $t$ there is some function of the dependent variable $y_t$ and of the model parameters $\boldsymbol{\theta}$, say $m_t\left(y_t, \boldsymbol{\theta}\right)$, of which the expectation is zero when the DGP used to compute the expectation is characterized by $\theta$. Thus, for all $t$ and for all $\theta$,
$$E_{\boldsymbol{\theta}}\left(m_t\left(y_t, \boldsymbol{\theta}\right)\right)=0 .$$
We may think of (16.48) as expressing a moment condition. In general, the functions $m_t$ may also depend on exogenous or predetermined variables.
Even though there is a different function for each observation, it seems reasonable, by analogy with empirical moments, to take the following expression as the empirical counterpart of the moment in condition (16.48):
$$m(\boldsymbol{y}, \hat{\boldsymbol{\theta}}) \equiv \frac{1}{n} \sum_{t=1}^n m_t\left(y_t, \hat{\boldsymbol{\theta}}\right),$$
where $\hat{\boldsymbol{\theta}}$ denotes a vector of estimates of $\boldsymbol{\theta}$. Expression (16.49) is thus a form of empirical moment. A one-degree-of-freedom CM test would be computed by dividing it by an estimate of its standard deviation and would be asymptotically distributed as $N(0,1)$ under suitable regularity conditions. There might well be more than one moment condition, of course, in which case the test statistic could be calculated as a quadratic form in the empirical moments and an estimate of their covariance matrix and would have the chi-squared distribution asymptotically.

## 经济代写|计量经济学代写Econometrics代考|Information Matrix Tests

One important type of conditional moment test is the class of tests called information matrix, or IM, tests. These were originally suggested by White (1982), although the conditional moment interpretation is more recent; see Newey (1985a) and White (1987). The basic idea is very simple. If a model that is estimated by maximum likelihood is correctly specified, the information matrix must be asymptotically equal to minus the Hessian. If it is not correctly specified, that equality will in general not hold, because the proof of the information matrix equality depends crucially on the fact that the joint density of the data is the likelihood function; see Section 8.6.

Consider a statistical model characterized by a loglikelihood function of the form
$$\ell(\boldsymbol{y}, \boldsymbol{\theta})=\sum_{t=1}^n \ell_t\left(y_t, \boldsymbol{\theta}\right),$$
where $\boldsymbol{y}$ denotes an $n$-vector of observations $y_t, t=1, \ldots, n$, on a dependent variable, and $\boldsymbol{\theta}$ denotes a $k$-vector of parameters. As the subscript $t$ indicates, the contribution $\ell_t$ made by observation $t$ to the loglikelihood function may depend on exogenous or predetermined variables that vary across the $n$ observations. The null hypothesis for the IM test is that
$$\operatorname{plim}{n \rightarrow \infty}\left(\frac{1}{n} \sum{t=1}^n\left(\frac{\partial^2 \ell_t(\boldsymbol{\theta})}{\partial \theta_i \partial \theta_j}+\frac{\partial \ell_t(\boldsymbol{\theta})}{\partial \theta_i} \frac{\partial \ell_t(\boldsymbol{\theta})}{\partial \theta_j}\right)\right)=0,$$
for $i=1, \ldots, k$ and $j=1, \ldots, i$. Expression (16.65) is a typical element of the information matrix equality. The first term is an element of the Hessian, and the second is the corresponding element of the outer product of the gradient. Since the number of such terms is $\frac{1}{2} k(k+1)$, the number of degrees of freedom for an IM test is potentially very large.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Conditional Moment Tests

$$E_{\boldsymbol{\theta}}\left(m_t\left(y_t, \boldsymbol{\theta}\right)\right)=0 .$$

$$m(\boldsymbol{y}, \hat{\boldsymbol{\theta}}) \equiv \frac{1}{n} \sum_{t=1}^n m_t\left(y_t, \hat{\boldsymbol{\theta}}\right),$$

## 经济代写|计量经济学代写Econometrics代考|Information Matrix Tests

$$\ell(\boldsymbol{y}, \boldsymbol{\theta})=\sum_{t=1}^n \ell_t\left(y_t, \boldsymbol{\theta}\right)$$

$$\operatorname{plim} n \rightarrow \infty\left(\frac{1}{n} \sum t=1^n\left(\frac{\partial^2 \ell_t(\boldsymbol{\theta})}{\partial \theta_i \partial \theta_j}+\frac{\partial \ell_t(\boldsymbol{\theta})}{\partial \theta_i} \frac{\partial \ell_t(\boldsymbol{\theta})}{\partial \theta_j}\right)\right)=0$$

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