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# 经济代写|计量经济学代写ECONOMETRICS代考|BEA472 Omitted Variable Bias: The Simple Case

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## 经济代写|计量经济学代写ECONOMETRICS代考|Omitted Variable Bias: The Simple Case

Now suppose that, rather than including an irrelevant variable, we omit a variable that actually belongs in the true (or population) model. This is often called the problem of excluding a relevant variable or underspecifying the model. We claimed in Chapter 2 and earlier in this chapter that this problem generally causes the OLS estimators to be biased. It is time to show this explicitly and, just as importantly, to derive the direction and size of the bias.

Deriving the bias caused by omitting an important variable is an example of misspecification analysis. We begin with the case where the true population model has two explanatory variables and an error term:
$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+u,$$
and we assume that this model satisfies Assumptions MLR.1 through MLR.4.
Suppose that our primary interest is in $\beta_1$, the partial effect of $x_1$ on $y$. For example, $y$ is hourly wage (or log of hourly wage), $x_1$ is education, and $x_2$ is a measure of innate ability. In order to get an unbiased estimator of $\beta_1$, we should run a regression of $y$ on $x_1$ and $x_2$ (which gives unbiased estimators of $\beta_0, \beta_1$, and $\beta_2$ ). However, due to our ignorance or data inavailability, we estimate the model by excluding $x_2$. In other words, we perform a simple regression of $y$ on $x_1$ only, obtaining the equation
$$\tilde{y}=\tilde{\beta}_0+\tilde{\beta}_1 x_1 .$$
We use the symbol ” $\sim$ ” rather than “N” to emphasize that $\tilde{\beta}_1$ comes from an underspecified model.

When first learning about the omitted variables problem, it can be difficult for the student to distinguish between the underlying true model, (3.40) in this case, and the model that we actually estimate, which is captured by the regression in (3.41). It may seem silly to omit the variable $x_2$ if it belongs in the model, but often we have no choice. For example, suppose that wage is determined by
$$\text { wage }=\beta_0+\beta_1 e d u c+\beta_2 a b i l+u .$$
Since ability is not observed, we instead estimate the model
$$\text { wage }=\beta_0+\beta_1 e d u c+v,$$
where $v=\beta_2 a b i l+u$. The estimator of $\beta_1$ from the simple regression of wage on educ is what we are calling $\tilde{\beta}_1$.

## 经济代写|计量经济学代写ECONOMETRICS代考|Omitted Variable Bias: More General Cases

Deriving the sign of omitted variable bias when there are multiple regressors in the estimated model is more difficult. We must remember that correlation between a single explanatory variable and the error generally results in all OLS estimators being biased. For example, suppose the population model
$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+u,$$
satisfies Assumptions MLR.1 through MLR.4. But we omit $x_3$ and estimate the model as

$$\tilde{y}=\tilde{\beta}_0+\tilde{\beta}_1 x_1+\tilde{\beta}_2 x_2 .$$
Now, suppose that $x_2$ and $x_3$ are uncorrelated, but that $x_1$ is correlated with $x_3$. In other words, $x_1$ is correlated with the omitted variable, but $x_2$ is not. It is tempting to think that, while $\tilde{\beta}_1$ is probably biased based on the derivation in the previous subsection, $\tilde{\beta}_2$ is unbiased because $x_2$ is uncorrelated with $x_3$. Unfortunately, this is not generally the case: both $\tilde{\beta}_1$ and $\tilde{\beta}_2$ will normally be biased. The only exception to this is when $x_1$ and $x_2$ are also uncorrelated.

Even in the fairly simple model above, it is difficult to obtain the direction of the bias in $\tilde{\beta}1$ and $\tilde{\beta}_2$. This is because $x_1, x_2$, and $x_3$ can all be pairwise correlated. Nevertheless, an approximation is often practically useful. If we assume that $x_1$ and $x_2$ are uncorrelated, then we can study the bias in $\tilde{\beta}_1$ as if $x_2$ were absent from both the population and the estimated models. In fact, when $x_1$ and $x_2$ are uncorrelated, it can be shown that $$\mathrm{E}\left(\tilde{\beta}_1\right)=\beta_1+\beta_3 \frac{\sum{i=1}^n\left(x_{i 1}-\bar{x}1\right) x{i 3}}{\sum_{i=1}^n\left(x_{i 1}-\bar{x}_1\right)^2} .$$

## 经济代写|计量经济学代写ECONOMETRICS代考|省略变量偏差:简单情况

$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+u,$$
，我们假设该模型满足假设MLR.1到MLR.4。

$$\tilde{y}=\tilde{\beta}_0+\tilde{\beta}_1 x_1 .$$

$$\text { wage }=\beta_0+\beta_1 e d u c+\beta_2 a b i l+u .$$

$$\text { wage }=\beta_0+\beta_1 e d u c+v,$$
，其中$v=\beta_2 a b i l+u$。通过对educ上工资的简单回归估计出$\beta_1$，我们称之为$\tilde{\beta}_1$。

## 经济代写|计量经济学代写ECONOMETRICS代考|省略的变量偏差:更多的一般情况

.

$$y=\beta_0+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+u,$$

$$\tilde{y}=\tilde{\beta}_0+\tilde{\beta}_1 x_1+\tilde{\beta}_2 x_2 .$$现在，假设 $x_2$ 和 $x_3$ 是不相关的，但是呢 $x_1$ 与 $x_3$。换句话说， $x_1$ 与省略的变量相关，但是 $x_2$ 不是。人们很容易认为，尽管 $\tilde{\beta}_1$ 可能是基于前一小节的推导， $\tilde{\beta}_2$ 是没有偏见的 $x_2$ 与 $x_3$。不幸的是，通常情况并非如此:两者皆是 $\tilde{\beta}_1$ 和 $\tilde{\beta}_2$ 通常会有偏见。唯一的例外是当 $x_1$ 和 $x_2$ 也是不相关的。

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