regression - How does Stata treat multiple factor variables in regression?

Question

I have a city-year level dataset, and run the follow regression with city fixed effects:

reg y x i.city 

I think this is equivalent to generating a dummy variable for each of 300 cities in the data, and run (city 1 as base level):

reg y x city2 ... city300

However, I need to include year dummies as well. I get the estimates using:

reg y x i.city i.year

Does anyone know what is going behind this regression in matrix form? Is that the same as generating one dummy for each year and run the following?

reg y x city2 ... city300 year2 ... year20

The reason I want to do this is try to code the command from scratch using matrix operations (X'X)^{-1}(X'y), where X includes the city dummies and year dummies.

标签： regressionstatacategorical-data

Answer 1

您正在使用的称为虚拟（0,1）变量的角点编码，其中 k-1 二进制（0,1）变量级别用于每个因子（分类变量）。如果您指定不应使用常数项：

reg y x i.city i.year, nocon

然后零和约束编码将用于二进制变量构造，其中将有一个用于 X 矩阵中的 city1 和 year1 的二进制变量。

如您所见（下图），当饮食中的视黄醇浓度 (retdiet) 回归male虚拟变量时，常数 (y-intcp) 的系数项是女性 (815) 中的平均值 y，系数为maledelta在男性和女性之间的 y 值。然而，当使用两个虚拟指标时 -fem和male， 和, nocon被指定（在逗号之后），回归系数的值fem和male是每组中 y （retdiet）的平均值。