# 经济代写|计量经济学代写Econometrics代考|EC5216

## 经济代写|计量经济学代写Econometrics代考|The Generalized Method of Moments

We saw in the last chapter that if a model is correctly specified, there will often be conditional moments which are zero. The essential idea of the generalized method of moments, or GMM, is that moment conditions can be used not only to test model specification but also to define model parameters, in the sense of providing a parameter-defining mapping for a model. The very simplest example of this is a model in which the only parameter of interest is the expectation of the dependent variable. This is a special case of what is called a location model. If each observation on a dependent variable $y$ is a drawing from a distribution with expectation $m$, then the moment $E(y-m)$ must be zero. This fact serves to define the parameter $m$, since if $m^{\prime} \neq m$, $E\left(y-m^{\prime}\right) \neq 0$. In other words, the moment condition is satisfied only by the true value of the parameter.

According to the (ordinary) method of moments, if one has a sample of independent drawings from some distribution, one can estimate any moment of the distribution by the corresponding sample moment. This procedure is justified very easily by invoking the law of large numbers in its simplest form. Thus, for the location model, if the sample is denoted by $y_t, t=1, \ldots, n$, the method of moments estimator of $m$ is just the sample mean
$$\hat{m}=\frac{1}{n} \sum_{t=1}^n y_t$$
When one speaks of the generalized method of moments, several generalizations are in fact implied. Some involve no more than relaxing regularity conditions, for instance, the assumption of i.i.d. observations. Since many different laws of large numbers can be proved (recall the list in Section 4.7), there is no reason to limit oneself to the i.i.d. case. But the essential generalizations follow from two facts. The first is that conditional moments may be used as well as unconditional ones, and the second is that moments may depend on unknown parameters.

## 经济代写|计量经济学代写Econometrics代考|Criterion Functions and M-Estimators

In Chapter 7 , the IV estimator for the linear regression model was defined by the minimization of the criterion function
$$(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})^{\top} \boldsymbol{P}_W(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta}) ;$$
see equation (7.15). Let $k$ denote the number of regressors and $l \geq k$ the number of instruments. In the just identified case, in which $l=k$, the minimized value of the criterion function is zero. This value is achieved at the value of $\boldsymbol{\beta}$ given by the simple IV estimator, defined by the $k$ conditions (17.05). When $l>k$, the minimized value is in general greater than zero, since it is not in general possible to solve what is now the set of $l$ conditions (17.05) for $k$ unknowns.

The overidentified case in the GMM context is similar. There are $l$ estimator-defining equations (17.07) but just $k$ unknown parameters. Instead of solving a set of equations, the left-hand sides of these equations are used to define a criterion function which is subsequently minimized to provide parameter estimates. Consider (17.08) again. If we write it as
$$(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})^{\top} \boldsymbol{W}\left(\boldsymbol{W}^{\top} \boldsymbol{W}\right)^{-1} \boldsymbol{W}^{\top}(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta}),$$
we see that the expression is a quadratic form made up from the empirical moments $\boldsymbol{W}^{\top}(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})$ and the inverse of the positive definite matrix $\boldsymbol{W}^{\top} \boldsymbol{W}$. This positive definite matrix is, under homoskedasticity and serial independence of the error terms, proportional to the covariance matrix of the vector of moments, the factor of proportionality being the variance of the error terms. Omitting this factor of proportionality does not matter, because the $\boldsymbol{\beta}$ which minimizes (17.09) is unchanged if (17.09) is multiplied by any positive scalar.
It is not necessary to use the covariance matrix of the empirical moments $\boldsymbol{W}^{\top}(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})$ if one merely wishes to obtain consistent, rather than efficient, estimates of $\boldsymbol{\beta}$ by the minimization of the criterion function.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|The Generalized Method of Moments

$$\hat{m}=\frac{1}{n} \sum_{t=1}^n y_t$$

## 经济代写|计量经济学代写Econometrics代考|Criterion Functions and M-Estimators

$$(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})^{\top} \boldsymbol{P}_W(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})$$

GMM 上下文中过度识别的情况类似。有l估计量定义方程 (17.07) 但只是 $k$ 末知参数。这些方程的左侧用 于定义一个准则函数，而不是求解一组方程，随后将其最小化以提供参数估计。再次考虑 (17.08)。如果 涐们把它写成
$$(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta})^{\top} \boldsymbol{W}\left(\boldsymbol{W}^{\top} \boldsymbol{W}\right)^{-1} \boldsymbol{W}^{\top}(\boldsymbol{y}-\boldsymbol{X} \boldsymbol{\beta}),$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: