# 统计代写|线性回归分析代写linear regression analysis代考|Could the same bias occur when adjusting for quantitative

## 统计代写|线性回归分析代写linear regression analysis代考|Could the same bias occur when adjusting for quantitative

Yes, this is possible. However, I am not aware of any formula that could calculate this bias. And, for what it is worth, my intuition tells me this would be less pronounced than it is for controlling for a categorization.
As a brief informal example, from the model in Section 2.9,
$$(\widehat{\text { income }})_i=-34,027+5395 \times(\text { educ })_i+367 \times(a f q t)_i$$
I found that adjusting for the AFQT score decreased the operative variation in the years-of-schooling variable (measured as the variance) more for those in the top third of AFQT scores relative to the others.
Furthermore, in these models:
$$\begin{gathered} \text { income }=\beta_0+\beta_1 \times(\text { educ })+\beta_2 \times(\text { black })+\varepsilon \ \text { income }=\beta_0+\beta_1 \times(\text { educ })+\beta_2 \times(\text { black })+\beta_3 \times(\text { afq })+\varepsilon \end{gathered}$$
when I include the AFQT score (again, a quantitative variable), the variance of years-of-schoolingadjusted-for-AFQT decreases more for Blacks than non-Blacks, causing the effective weight for Blacks to decrease by about 2.1 percentage points. Whereas this is not a large amount, it shows that adding a quantitative variable could impact the effective weights across groups in the model. This is something that I think should be investigated by people smarter than I am.

## 统计代写|线性回归分析代写linear regression analysis代考|How to choose the best set of control variables

In this section, I discuss another highly-misunderstood topic: choosing the optimal set of control variables. This is often called model selection. Recall that control variables are those that are included in a model to help identify the true causal effects of the key-explanatory variables (or the key empirical relationships for other regression objectives). Opinions vary widely on the criteria for choosing the right set of control variables. Poor choices in choosing control variables contribute to the plethora of incorrect research results.

The purpose of using control variables is to address any potential PITFALLS, particularly omitted-factors bias and improper reference groups. That is, for the objective of estimating causal effects, control variables help toward ruling out alternative explanations for why the empirical relationship is what it is. At the same time, including the wrong control variables might introduce PITFALLS (using mediating factors or outcomes as controls, and improper reference groups), causing bias in the coefficient estimate.

The generally easy part of choosing which control variables to use is avoiding mediating factors and outcomes of the key- $\mathrm{X}$ variable. The more difficult part is determining what control variables to include to address the three types of omitted-factors bias. The third type, whether there is a replacement action for receiving low values of the treatment, is also part of PITFALL #6 of improper reference groups. The other issues with improper reference groups (having the correct counterfactual and whether the control group has a lower-intensity amount of the treatment) are more about characterizing the treatment correctly than choosing the correct set of control variables.

Recall the steps for assessing whether there is omitted-factors bias from spurious correlation:

• Step 1: Determine the factors of the key-X variable. Think about what causes the key-X variable to be high for some observations and low for others. Or, for a dummy variable, what causes some observations to get the treatment (have a value of 1 ) vs. not get the treatment $(0)$ ?
• Step 2: Determine if any of those factors could affect the outcome independently (i.e., beyond through its effects on the key- $\mathrm{X}$ variable) and are not fully held constant in the model.

So we want to control for any factor that could affect the key- $X$ variable and the outcome. That is, we want to convert such bad variation from operative to held-constant variation. Thus, we need to determine the main sources of variation in the key- $X$ variable.

# 线性回归代考

## 统计代写|线性回归分析代写linear regression analysis代考|Could the same bias occur when adjusting for quantitative

$$(\text { income })_i=-34,027+5395 \times(\text { educ })_i+367 \times(a f q t)_i$$

$$\text { income }=\beta_0+\beta_1 \times(\text { educ })+\beta_2 \times(\text { black })+\varepsilon \text { income }=\beta_0+\beta_1 \times(\text { educ })+\beta_2$$

## 统计代写|线性回归分析代写linear regression analysis代考|How to choose the best set of control variables

• 第 1 步：确定 key-X 变量的因子。想一想是什么导致 key-X 变量对于某些观察值高而对于其他观察值低。或者，对于虚拟变量，是什么导致某些观察结果得到处理（值为 1 ）与未得到处理(0) ?
• 第 2 步：确定这些因素中的任何一个是否会独立影响结果（即，通过其对关键因素的影响之外）X变量）并且在模型中没有完全保持不变。

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