### 经济代写|劳动经济学代写Labor Economics代考|ECON656

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|劳动经济学代写Labor Economics代考|Multiple Regression Analysis

The preceding discussion has assumed that the only variable influencing quit rates, other than random (unexplained) factors, is a firm’s wage rate. The discussion of positive economics in this chapter stresses, however, that the prediction of a negative relationship between wages and quit rates is made holding all other factors constant. As we will discuss in chapter 10, economic theory suggests that there are many factors besides wages that systematically influence quit rates. These include characteristics both of firms (e.g., employee benefits offered, working conditions, and firm size) and of their workers (e.g., age and level of training). If any of these other variables that we have omitted from our analysis tend to vary across firms systematically with the wage rates that the firms offer, the resulting estimated relationship between wage rates and quit rates will be incorrect. In such cases, we must take these other variables into account by using a model with more than one independent variable. We rely on economic theory to indicate which variables should be included in our statistical analysis and to suggest the direction of causation.

To illustrate this procedure, suppose for simplicity that the only variable affecting a firm’s quit rate besides its wage rate is the average age of its workforce. With other factors kept constant, older workers are less likely to quit their jobs for a number of reasons (as workers grow older, ties to friends, neighbors, and coworkers become stronger, and the psychological costs involved in changing jobs-which often requires a geographic move-grow larger). To capture the effects of both wage rates and age, we assume that a firm’s quit rate is given by
$$Q_1=\alpha^{\prime} 0+\alpha_1^{\prime} W_i^{\prime}+\alpha_2^{\prime} A_i+\epsilon_i(1 A .4)$$
$A_i$ is a variable representing the age of firm is workers. Although $A_i$ could be measured as the average age of the workforce, or as the percentage of the firm’s workers older than some age level, for expositional convenience we have defined it as a dichotomous variable. $A_i$ is equal to 1 if the average age of firm $i$ ‘s workforce is greater than 40 , and it is equal to zero otherwise. Clearly, theory suggests that $\alpha_2^{\prime}$ is negative, which means that whatever values of $\alpha_0^{\prime}, \alpha_1^{\prime}$, and $W_i$ pertain (that is, keeping all else constant), firms with workforces having an average age above 40 years should have lower quit rates than firms with workforces having an average age equal to or below age $40 .$

The parameters of equation (1A.4)-that is, the values of $\alpha_0^{\prime}, \alpha_1^{\prime}$, and $^{\prime}{ }_2-$ can be estimated using multiple regression analysis, a method that is analogous to the one described earlier. This method finds the values of the parameters that define the best straight-line relationship between the dependent variable and the set of independent variables. Each parameter tells us the effect on the dependent variable of a one-unit change in the corresponding independent variable, holding the other independent variables constant. Thus, the estimate of $\alpha_1^{\prime}$ tells us the estimated effect on the quit rate $(Q)$ of a one-unit change in the wage rate $(W)$, holding the age of a firm’s workforce $(A)$ constant.

## 经济代写|劳动经济学代写Labor Economics代考|The Problem of Omitted Variables

If we use a univariate regression model in a situation calling for a multiple regression model-that is, if we leave out an important independent variable-our results may suffer from omitted variables bias. We illustrate this bias because it is an important pitfall in hypothesis testing, and because it illustrates the need to use economic theory to guide empirical testing.

To simplify our example, we assume that we know the true values of $\alpha_0^{\prime}, \alpha_1^{\prime}$, and $\alpha_2^{\prime}$ in equation (1A.4) and that there is no random error term in this model (each $\varepsilon_i$ is zero). Specifically, we assume that
$$Q_i=50-2.5 W_i-10 A_i(1 A .5)$$
Thus, at any level of wages, a firm’s quit rate will be 10 percentage points lower if the average age of its workforce exceeds 40 than it will be if the average age is less than or equal to $40 .$

Figure 1 A.2 graphically illustrates this assumed relationship between quit rates, wage rates, and workforce average age. For all firms that employ workers whose average age is less than or equal to $40, A_i$ equals zero and thus their quit rates are given by the line $Z_0$ $Z_0$. For all firms that employ workers whose average age is greater than $40, A_i$ equals 1 and thus their quit rates are given by the line $Z_1 Z_1$. The quit-rate schedule for the latter set of firms is 10 percentage points below the one for the former set. Both schedules indicate, however, that a $\$ 1$increase in a firm’s average hourly wage will reduce its annual quit rate by$2.5$percentage points (that is, both lines have the same slope). Now, suppose a researcher were to estimate the relationship between quit rates and wage rates, but ignored the fact that the average age of a firm’s workers also affects the quit rate. That is, suppose one were to omit a measure of age and estimate the following equation: $$Q_i=\alpha_0+\alpha_1 W_i+\epsilon_i(1 A .6)$$ Of crucial importance to us is how the estimated value of$\alpha_1$will correspond to the true slope of the quit/wage schedule, which we have assumed to be$-2.5$. ## 劳动经济学代考 ## 经济代写|劳动经济学代写Labor Economics代考|Multiple Regression Analysis 前面的讨论假设，除了随机 (无法解释的) 因素之外，影响离职率的唯一变量是公司的工资率。然而，本章对积 极经济学的讨论强调，工资与离职率之间负相关的预测是在所有其他因素不变的情况下做出的。正如我们将在第 10 章讨论的那样，经济理论表明，除了工资之外，还有许多因素会系统地影响戒烟率。这些包括公司的特征（例 如，提供的员工福利、工作条件和公司规模) 及其员工的特征 (例如，年龄和培训水平) 。如果我们在分析中忽 略的这些其他变量中的任何一个往往会随着公司提供的工资率在公司之间系统地变化，由此产生的工资率和离职 率之间的估计关系将是不正确的。在这种情况下，我们必须通过使用具有多个自变量的模型来考虑这些其他变 量。我们依靠经济理论来指出哪些变量应该包括在我们的统计分析中，并提出因果关系的方向。 为了说明这个过程，为了简单起见，除了工资率之外，影响公司离职率的唯一变量是其劳动力的平均年龄。在其 他因素保持不变的情况下，由于多种原因 (随着年龄的增长，与朋友、邻居和同事的联系变得更紧密，以及换工 作所涉及的心理成本一一这通常需要地理移动变大）。为了捕捉工资率和年龄的影响，我们假设公司的离职率由 下式给出 $$Q_1=\alpha^{\prime} 0+\alpha_1^{\prime} W_i^{\prime}+\alpha_2^{\prime} A_i+\epsilon_i(1 A .4)$$$A_i$是代表公司工人年齡的变量。虽然$A_i$可以用劳动力的平均年龄来衡量，或者用公司工人年龄超过某个年龄水平 的百分比来衡量，为了说明方便，我们将其定义为二分变量。$A_i$如果公司的平均年龄等于 1 的劳动力大于 40 ， 否则为零。显然，理论表明$\alpha_2^{\prime}$是负数，这意味着无论$\alpha_0^{\prime}, \alpha_1^{\prime}$，和$W_i$相关（即保持其他所有因素不变)，员工平 均年龄超过 40 岁的公司的离职率应该低于员工平均年龄等于或低于年齡的公司$40 .$等式 (1A.4) 的参数，即$\alpha_0^{\prime}, \alpha_1^{\prime}$，和${ }_2^{\prime}$一可以使用多元回归分析来估计，这是一种类似于前面描述的方法。此方法 查找定义因变量和自变量集之间的最佳直线关系的参数值。每个参数都告诉我们相应自变量变化一个单位对因变 量的影响，同时保持其他自变量不变。因此，估计$\alpha_1^{\prime}$告诉我们对戒烟率的估计影响$(Q)$工资率变化一个单位$(W)$， 持有公司劳动力的年龄$(A)$持续的。 ## 经济代写|劳动经济学代写Labor Economics代考|The Problem of Omitted Variables 如果我们在需要多元回归模型的情况下使用单变量回归模型一一也就是说，如果我们遗漏了一个重要的自变量 一一我们的结果可能会受到遗漏变量偏差的影响。我们说明了这种偏差，因为它是假设检验中的一个重要缺陷， 并且因为它说明了使用经济理论来指导实证检验的必要性。 为了简化我们的示例，我们假设我们知道$\alpha_0^{\prime}, \alpha_1^{\prime}$，和$\alpha_2^{\prime}$在等式 (1A.4) 中，并且该模型中没有随机误差项（每 个$\varepsilon_i$为零) 。具体来说，我们假设 $$Q_i=50-2.5 W_i-10 A_i(1 A .5)$$ 因此，在任何工资水平上，如果员工平均年龄超过 40 岁，公司的离职率将比平均年龄小于或等于 40 岁时低 10 个百分点。 40 . 图 1 A.2 以图形方式说明了离职率、工资率和劳动力平均年龄之间的这种假设关系。对于所有雇用平均年龄小于或 等于$40, A_i$等于零，因此他们的戒烟率由线给出$Z_0 Z_0$. 对于所有雇用平均年龄大于$40, A_i$等于 1 ，因此他们的戒 烟率由线给出$Z_1 Z_1$. 后一组公司的退出率比前一组低 10 个百分点。然而，这两个时间表都表明，$\$1$ 公司平均小 时工资的增加将使其年离职率降低2.5个百分点（即两条线的斜率相同）。

$$Q_i=\alpha_0+\alpha_1 W_i+\epsilon_i(1 A .6)$$

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