用R构建lm模型验证The Frisch-Waugh Theorem

The Frisch-Waugh theorem is a statistical result that states that the coefficients in a multiple regression model can be estimated by running separate regressions on the dependent variable and the independent variables, after controlling for other variables.

To demonstrate the Frisch-Waugh theorem using R, we can start by loading a dataset and fitting a multiple regression model. Let's assume we have a dataset with three variables: "Y" as the dependent variable, "X1" and "X2" as the independent variables, and "Z" as the control variable.

# Load the necessary packages
library(lmtest)

# Load the dataset
data <- read.csv("dataset.csv")

# Fit the multiple regression model
model <- lm(Y ~ X1 + X2 + Z, data = data)

# Extract the residuals from the model
residuals <- residuals(model)

# Run the first stage regression on Y, controlling for X2 and Z
stage1_model <- lm(Y ~ X2 + Z, data = data)

# Extract the residuals from the first stage regression
stage1_residuals <- residuals(stage1_model)

# Run the second stage regression on X1, controlling for X2 and Z
stage2_model <- lm(X1 ~ X2 + Z, data = data)

# Extract the coefficients from the second stage regression
stage2_coefficients <- coef(stage2_model)

# Estimate the effect of X1 on Y, controlling for X2 and Z using the Frisch-Waugh theorem
FW_estimate <- lm(stage1_residuals ~ stage2_residuals)

# Print the coefficient estimate
print(coef(FW_estimate))

In this code, we first load the necessary packages, then load the dataset using the read.csv() function. We fit the multiple regression model using lm() function, with "Y" as the dependent variable and "X1", "X2", and "Z" as the independent variables.

Next, we extract the residuals from the model using residuals() function. We run the first stage regression on "Y", controlling for "X2" and "Z", and extract the residuals from this regression.

Then, we run the second stage regression on "X1", controlling for "X2" and "Z", and extract the coefficients from this regression.

Finally, we estimate the effect of "X1" on "Y", controlling for "X2" and "Z" using the Frisch-Waugh theorem by running a simple linear regression of the first stage residuals on the second stage residuals.

The estimated coefficient from the Frisch-Waugh regression represents the effect of "X1" on "Y" while controlling for "X2" and "Z"