translated by damien from Marco Avarucci

Basic Econometrics - Unit 4 Multiple Regression Analysis with Qualitative Information

 计量经济学基础第4单元定性信息多元回归分析

• Qualitative Information
   定性信息

  • Examples: gender, race, industry, region, rating grade, …
     例如：性别，种族，行业，地区，等级…

  • A person is either male or female, a worker belongs to a union or not…
     一个人不是男性就是女性，一个工人是否属于工会…

  • Qualitative variables may appear as the dependent or as independent variables
     定性变量可以作为因变量或自变量出现

A single dummy independent variable

 单个虚拟自变量

\[\begin{array}{c|lcr} \hline peson & wage & educ & exper & female & married & \\\\ \hline 1 & 3.10 & 11 & 2 & 1 & 0 \\\\ \hline 2 & 3.24 & 12 & 22 & 1 & 1 \\\\ \hline 3 & 3.00 & 11 & 2 & 0 & 0 \\\\ \hline 4 & 6.00 & 8 & 44 & 0 & 1 \\\\ \hline 5 & 5.30 & 12 & 7 & 0 & 1 \\\\ \hline \vdots & \vdots & \vdots & \vdots & \vdots \\\\ \hline 525 & 11.56 & 16 & 5 & 0 & 1 \\\\ \hline 526 & 3.50 & 14 & 5 & 1 & 0 \end{array}\]

• Graphical Illustration
   图示

• Dummy variable trap
   *虚拟变量陷阱**

• Estimated wage equation with intercept shift
   带截距偏移的估计工资方程

• Does that mean that women are discriminated against?
   这是否意味着妇女受到歧视？

  • Not necessarily. Being female may be correlated with other produc-tivity characteristics that have not been controlled for.
    不一定。女性可能与其他尚未控制的生产特征有关。

Using dummy explanatory variables in equations for log(y)

 在对数（y）方程中使用虚拟解释变量

Using dummy variables for multiple categories

 对多个类别使用虚拟变量

1. Define membership in each category by a dummy variable
    通过虚拟变量定义每个类别中的成员身份

2. Leave out one category (which becomes the base category)
    省略一个类别（成为基本类别）

• This marriage premium for men has long been noted by labour economists.
   长期以来，劳工经济学家一直注意到男性的这种“婚姻溢价”。

  • Does marriage make men more productive?
     婚姻能让男人更有效率吗？

  • Is being married a signal to employers (say, of stability and reliability)?
     结婚是雇主的信号吗（比如说，稳定和可靠）？

  • Is there a selection issue in that more productive men are likely to be married, on average?
     平均而言，生产力更高的男性很可能已婚，这是否存在选择问题？

  • The regression cannot tell us which explanation is correct.
     回归不能告诉我们哪种解释是正确的。

• A married woman, at given levels of the other variables, earns about 19.8% less than a single man.
   在给定的其他变量水平下，已婚女性的收入比单身男性低19.8%。

• A single woman earns about 11.0% less than a comparable single man. (p-value 0.048.)
   单身女性的收入比同等单身男性低11.0%。（p-value0.048。）

• What if we want to compare married women and single women?
   如果我们想比较已婚女性和单身女性呢？

  • slope for married women =. 321 −. 198
     已婚妇女的坡度 =. 321 −. 198

  • slope for single women =. 321 −. 110
     单身女性的坡度 =. 321 −. 110

  • difference = −. 198 − (−. 110) = −. 088
     差异 = −. 198 − (−. 110) = −. 088

• so married women earn about 8.8% less than single women (controlling for other factors).
   因此，已婚女性的收入比单身女性低8.8%（控制其他因素）。

• We cannot tell from the previous output whether this difference is statistically significant.
   我们无法从先前的结果判断这种差异是否具有统计学意义。

• Choose marrfem as the base group, re-estimate the model (including the other thee categories)
   选择marrfem作为基组，重新估计模型（包括其他类别）

Using Dummy Variables to Incorporate Ordinal Information

 使用虚拟变量合并顺序信息

• The data set BEAUTY.DTA includes a ranking of physical attractiveness of each man or woman, on a scale of 1 to 5, with 5 being “strikingly beautiful or handsome.”
   数据集BEAUTY.DTA包括每个男人或女人的外表吸引力排名，从1到5分，其中5分为“惊人的美丽或英俊”

• As we move up the scale from 1 to 5, why should a one-unit increase mean the same amount of “beauty”?
   当我们从1上升到5时，为什么一个单位的增加意味着同样多的“美”？

• The “looks” variable is what we call an ordinal variable: we know that the order of outcomes conveys information (5 is better than 4, and 2 is better than 1) but we do not know that the difference between 5 and 4 is the same as 2 and 1.
   “looks”变量就是我们所说的序数变量：我们知道结果的顺序传递信息（5比4好，2比1好），但我们不知道5和4之间的差别与2和1是一样的。

• Very few people are at the extreme values 1 and 5 (less than 1% each).
   很少有人处于极端值1和5（每个都小于1%）。

• It makes sense to combine into three categories: belavg, avg, abvavg.
   将其分为三类是有意义的：belavg、avg、abvavg。

• avg is the base group:
   avg是基础组：

Incorporating ordinal information using dummy variables

 使用虚拟变量合并顺序信息

• Example: City credit ratings and municipal bond interest rates
   例如：城市信用评级和市政债券利率

Interactions involving dummy variables

 涉及虚拟变量的交互作用

• Allowing for different slopes
   考虑到不同的坡度

• Interesting hypotheses
   有趣的假设

• Graphical Illustration
   图示

• Estimated wage equation with interaction term
   带交互项的估计工资方程

F-statistic for
 F统计是为了

\[H _0: \beta _{female} = \beta _{female \cdot educ} = 0\]

Is equal to \( 34.33 ( df =2,518), \ p-value=0.0000 \)

Testing for differences in regression functions across groups

 测试各组回归函数的差异

• Unrestricted model (contains full set of interactions)
   无限制模型（包含全套交互）

• Restricted model (same regression for both groups)
   限制模型（两组回归相同）

\[cumgpa = \beta _0 + \beta _1 sat + \beta _2 hsperc + \beta _3 tothrs + u\]

• Null hypothesis
   空假设

• Estimation of the unrestricted model
   非限制模型的预测

• Joint test with F-statistic
   F统计的联合测试

Many regressors: adding all the interaction effects might be cumbersome
 许多回归因素：添加所有交互效应可能会很麻烦

Alternative way to compute the F-statistic (Chow test)

 计算F统计量的另一种方法（Chow检验）

• Run separate regressions for the groups (e.g. men and for women); the unrestricted SSR is given by the sum of the SSR of these two regressions.
   对各组（例如男性和女性）分别进行回归分析；无限制SSR由这两个回归的SSR之和得出。

• We necessarily get the same estimated intercepts and slopes as if we include female dummy and a full set of interaction.
   我们必须得到相同的估计截距和斜率，如果我们包括女性假人和一整套互动。

• Run regression for the restricted (pooled) model and store SSR.
   对受限（合并）模型运行回归并存储SSR。

• Null: equality of regression functions across two groups.

   Null：两组回归函数相等。

• Important: Test assumes a constant error variance accross groups.
   重要：测试假设各组间的误差方差恒定。

\[F = \frac{[SSR - (SSR _1 + SSR _2)]}{SSR _1 + SSR _2} \cdot \frac{[n - 2(k+1)]}{k+1}\]

We can allow for an intercept difference between the groups, and then test for slope difference.
 我们可以考虑组之间的截距差，然后测试斜率差。

Replace \( SSR _p \) in Chow F-stat with the residuals from a regression with a dummy.
 将Chow F-stat中的\( SSR _p \)替换为虚拟回归的残差。