# 经济代写|计量经济学代写Econometrics代考|ECON3007

## 经济代写|计量经济学代写Econometrics代考|Efficient GMM Estimators

It is not completely straightforward to answer the question of whether GMM estimators are asymptotically efficient, since a number of separate issues are involved. The first issue was raised at the beginning of the last section, in connection with estimation by instrumental variables. We saw there that, for a given set of empirical moments $\boldsymbol{W}^{\top}(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})$, a whole family of estimators can be generated by different choices of the weighting matrix $\boldsymbol{A}(\boldsymbol{y})$ used to construct a quadratic form from the moments. Asymptotically, the most efficient of these estimators is obtained by choosing $\boldsymbol{A}(\boldsymbol{y})$ such that it tends to a nonrandom probability limit proportional to the inverse of the limiting covariance matrix of the empirical moments, suitably weighted by an appropriate power of the sample size $n$. This turns out to be true quite generally, as we now show.
Theorem 17.3. A Necessary Condition for Efficiency
A necessary condition for the estimator obtained by minimizing the quadratic form (17.13) to be asymptotically efficient is that it should be asymptotically equal to the estimator defined by minimizing (17.13) with $\boldsymbol{A}(\boldsymbol{y})$ independent of $\boldsymbol{y}$ and equal to the inverse of the asymptotic covariance matrix of the empirical moments $n^{-1 / 2} \boldsymbol{F}^{\top}(\boldsymbol{\theta}) \boldsymbol{\iota}$.
Note that, when the necessary condition holds, the form of the asymptotic covariance matrix of the GMM estimator $\hat{\theta}$ becomes much simpler. For arbitrary limiting weighting matrix $\boldsymbol{A}_0$, that matrix was given by (17.31). If the necessary condition is satisfied, then $\boldsymbol{A}_0$ in (17.31) may be replaced by the inverse of $\boldsymbol{\Phi}$, which, according to its definition (17.29), is the asymptotic covariance of the empirical moments. Substituting $\boldsymbol{A}_0=\boldsymbol{\Phi}^{-1}$ into (17.31) gives the simple result that
$$\boldsymbol{V}\left(n^{1 / 2}\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)\right)=\left(\boldsymbol{D}^{\top} \boldsymbol{\Phi}^{-1} \boldsymbol{D}\right)^{-1} .$$

## 经济代写|计量经济学代写Econometrics代考|Estimation with Conditional Moments

The moment conditions that we have used up to now have all been unconditional ones. In practice, however, it is the exception rather than the rule for an econometric model to be specified solely in terms of unconditional moments. In the literature on rational expectations models, for instance, economic theory requires that agents’ errors of prediction should be orthogonal to all variables in their information sets at the time the predictions are made. In the simple context of the linear regression model $\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}$, it is usual to assume not only that an error term $u_t$ is uncorrelated with the regressors $\boldsymbol{X}$ but also that its expectation conditional on the regressors is zero, which carries the additional implication that it is uncorrelated with any function of the regressors. In a time-series context, it is very common to suppose that the error $u_t$ has expectation zero conditional on all the past regressors as well as on the current ones.

Formally, it is easy to write down a set of parameter-defining equations in terms of conditional moments. Often there is only one such equation, which may be written as
$$E\left(f_t\left(y_t, \boldsymbol{\theta}\right) \mid \Omega_t\right)=0 \quad \text { for all } t=1, \ldots, n,$$
where $\Omega_t$ is the information set for observation $t$. We will make the simplifying assumption that $\Omega_t \subseteq \Omega_s$ for $t<s$. In (17.46) we are interpreting $f_t\left(y_t, \boldsymbol{\theta}\right)$ as some sort of error, such as a prediction error made by economic agents. The case of IV estimation of a linear regression model provides a simple example. In that case, (17.46) is interpreted as saying that the errors, just one per observation, are orthogonal to the information set defined by the set of instruments. It would be possible for there to be several parameter-defining equations like (17.46), as in the case of a multivariate regression model, but for simplicity we will in this section assume that there is just one.

In theory, there is no identification problem posed by the fact that there is only a single parameter-defining equation, because there is an infinite number of possible instruments in the sort of information set we consider. In practice, of course, one has to choose a finite number of these in order to set up a criterion function for GMM estimation. Most of this section will be taken up with establishing some results that affect this choice. First, we will demonstrate that the more instruments are used, the more precise is the GMM estimator. Next we show that, despite this, the asymptotic covariance matrices of the GMM estimators which can be constructed from instruments contained in the information sets $\Omega_t$ are bounded below. The lower bound, which is akin to the Cramér-Rao lower bound introduced in Chapter 8, is often called the GMM bound. In theory at least, there exists an optimal set of instruments which allows the GMM bound to be achieved, and the optimal instruments can in some cases be computed or estimated.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Efficient GMM Estimators

$$\boldsymbol{V}\left(n^{1 / 2}\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_0\right)\right)=\left(\boldsymbol{D}^{\top} \boldsymbol{\Phi}^{-1} \boldsymbol{D}\right)^{-1}$$

## 经济代写|计量经济学代写Econometrics代考|Estimation with Conditional Moments

$$E\left(f_t\left(y_t, \theta\right) \mid \Omega_t\right)=0 \quad \text { for all } t=1, \ldots, n,$$

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