# 经济代写|计量经济学代写Econometrics代考|Best27

## 经济代写|计量经济学代写Econometrics代考|Instrumental Variables

Up to this point, the only estimation technique we have considered is least squares, both ordinary and nonlinear. While least squares has many merits, it also has some drawbacks. One major drawback is that least squares yields consistent estimates only if the error terms are asymptotically orthogonal to the regressors or, in the nonlinear case, to the derivatives of the regression function. Consider, for simplicity, the linear regression model
$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma^2 \mathbf{I}\right),$$
where $\boldsymbol{X}$ is an $n \times k$ matrix of explanatory variables. The issues are the same whether the regression function is linear or nonlinear, and so we will deal with the linear case for simplicity. When the data are generated by the DGP
$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}0+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma_0^2 \mathbf{I}\right),$$ we have seen that the OLS estimate is $$\hat{\boldsymbol{\beta}} \equiv\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{y}=\boldsymbol{\beta}_0+\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{u}$$ It is obvious that if $\hat{\boldsymbol{\beta}}$ is to be consistent for $\boldsymbol{\beta}_0$, the condition $$\operatorname{plim}{n \rightarrow \infty}\left(n^{-1} \boldsymbol{X}^{\top} \boldsymbol{u}\right)=\mathbf{0}$$
must hold. If $\hat{\boldsymbol{\beta}}$ is to be unbiased, the stronger condition that $E\left(\boldsymbol{X}^{\top} \boldsymbol{u}\right)=\mathbf{0}$ must hold. These necessary conditions are not directly verifiable, since the orthogonality property of least squares ensures that regardless of whether $\boldsymbol{u}$ is correlated with $\boldsymbol{X}$ or not, the residuals $\hat{\boldsymbol{u}}$ are orthogonal to $\boldsymbol{X}$. This means that, no matter how biased and inconsistent least squares estimates may be, the least squares residuals will provide no evidence that there is a problem.
Suppose that plim $\left(n^{-1} \boldsymbol{X}^{\top} \boldsymbol{u}\right)=\boldsymbol{w}$, a nonzero vector. Then from (7.03) it is clear that $\operatorname{plim}(\hat{\boldsymbol{\beta}}) \neq \boldsymbol{\beta}0$. Moreover, the probability limit of $n^{-1}$ times the sum of squared residuals will be $$\operatorname{plim}{n \rightarrow \infty}\left(n^{-1} \boldsymbol{u}^{\top} \boldsymbol{M}X \boldsymbol{u}\right)=\sigma_0^2-\boldsymbol{w}^{\top} \operatorname{plim}{n \rightarrow \infty}\left(n^{-1} \boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{w}$$

## 经济代写|计量经济学代写Econometrics代考|Errors in Variables

Almost all economic variables are measured with error. This is true to a greater or lesser extent of all macroeconomic time series and is especially true of survey data and many other cross-section data sets. Unfortunately, the statistical consequences of errors in explanatory variables are severe, since explanatory variables that are measured with error are necessarily correlated with the error terms. When this occurs, the problem is said to be one of errors in variables. We will illustrate the problem of errors in variables with a simple example.

Suppose, for simplicity, that the DGP is
$$\boldsymbol{y}=\alpha_0+\beta_0 \boldsymbol{x}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma_0^2 \mathbf{I}\right),$$
where $\boldsymbol{x}$ is a vector that is observed with error. We actually observe $\boldsymbol{x}^$, which is related to $\boldsymbol{x}$ by $$\boldsymbol{x}^=\boldsymbol{x}+\boldsymbol{v}, \quad \boldsymbol{v} \sim \operatorname{IID}\left(\mathbf{0}, \omega^2 \mathbf{I}\right) .$$
The vector $v$ is a vector of measurement errors, which are assumed (possibly unrealistically) to have the i.i.d. property and to be independent of $\boldsymbol{x}$ and $\boldsymbol{u}$. Substituting $\boldsymbol{x}^-\boldsymbol{v}$ for $\boldsymbol{x}$ in (7.04), the DGP becomes $$\boldsymbol{y}=\alpha_0+\beta_0 \boldsymbol{x}^-\beta_0 \boldsymbol{v}+\boldsymbol{u} .$$
Thus the equation we can actually estimate is
$$\boldsymbol{y}=\alpha+\beta \boldsymbol{x}^+\boldsymbol{u}^,$$
where $\boldsymbol{u}^* \equiv \boldsymbol{u}-\beta_0 \boldsymbol{v}$. It is clear that $\boldsymbol{u}^$ is not independent of $\boldsymbol{x}^$. In fact
$$E\left(\boldsymbol{x}^{* \top} \boldsymbol{u}^\right)=E\left((\boldsymbol{x}+\boldsymbol{v})^{\top}\left(\boldsymbol{u}-\beta_0 \boldsymbol{v}\right)\right)=-\beta_0 E\left(\boldsymbol{v}^{\top} \boldsymbol{v}\right)=-n \beta_0 \omega^2,$$ where, as usual, $n$ is the sample size. If we assume for concreteness that $\beta_0>0$, the error term $\boldsymbol{u}^$ is negatively correlated with the regressor $\boldsymbol{x}^$. This negative correlation means that least squares estimates of $\beta$ will be biased and inconsistent, as will least squares estimates of $\alpha$ unless $\boldsymbol{x}^$ happens to have mean zero. Note that the inconsistency of $\hat{\beta}$ is a problem only if we care about the parameter $\beta$. If, on the contrary, we were simply interested in finding the mean of $\boldsymbol{y}$ conditional on $\boldsymbol{x}^*$, estimating equation (7.05) by least squares is precisely what we would want to do.

## 经济代写|计量经济学代写Econometrics代考|Instrumental Variables

$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma^2 \mathbf{I}\right),$$

$$\boldsymbol{y}=\boldsymbol{X} \boldsymbol{\beta} 0+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma_0^2 \mathbf{I}\right),$$

$$\hat{\boldsymbol{\beta}} \equiv\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{y}=\boldsymbol{\beta}_0+\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{u}$$

$$\operatorname{plim} n \rightarrow \infty\left(n^{-1} \boldsymbol{X}^{\top} \boldsymbol{u}\right)=\mathbf{0}$$

$$\operatorname{plim} n \rightarrow \infty\left(n^{-1} \boldsymbol{u}^{\top} \boldsymbol{M} X \boldsymbol{u}\right)=\sigma_0^2-\boldsymbol{w}^{\top} \operatorname{plim} n \rightarrow \infty\left(n^{-1} \boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{w}$$

## 经济代写|计量经济学代写Econometrics代考|Errors in Variables

$$\boldsymbol{y}=\alpha_0+\beta_0 \boldsymbol{x}+\boldsymbol{u}, \quad \boldsymbol{u} \sim \operatorname{IID}\left(\mathbf{0}, \sigma_0^2 \mathbf{I}\right),$$

$$x^{=} x+v, \quad v \sim \operatorname{IID}\left(0, \omega^2 \mathbf{I}\right) .$$

$$\boldsymbol{y}=\alpha_0+\beta_0 \boldsymbol{x}^{-} \beta_0 \boldsymbol{v}+\boldsymbol{u} .$$

$$\boldsymbol{y}=\alpha+\beta \boldsymbol{x}^{+} \boldsymbol{u}$$

\boldsymbol{x}^. 这种负相关意味着最小二乘估计 $\beta$ 将是有偏见和不一致的，最小二乘估计也是如此 $\alpha$ 除非

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