用R构建lm示例模型验证The Frisch-Waugh Theorem

The Frisch-Waugh Theorem is a statistical theorem that states that the coefficients of a multiple regression model can be estimated by running a series of separate regressions. In other words, instead of estimating all the coefficients in one regression, we can estimate them individually by controlling for the other variables.

To demonstrate the Frisch-Waugh Theorem using R, we can create a linear regression model with multiple independent variables and then estimate the coefficients using separate regressions.

Here's an example:

# Generate some random data
set.seed(123)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
x3 <- rnorm(n)
y <- 2*x1 + 3*x2 + 4*x3 + rnorm(n)

# Create the multiple regression model
model <- lm(y ~ x1 + x2 + x3)

# Estimate the coefficients using separate regressions
coef_x1 <- lm(model$residuals ~ x1)$coef[2]
coef_x2 <- lm(model$residuals ~ x2)$coef[2]
coef_x3 <- lm(model$residuals ~ x3)$coef[2]

# Print the estimated coefficients
cat("Coefficient for x1:", coef_x1, "\n")
cat("Coefficient for x2:", coef_x2, "\n")
cat("Coefficient for x3:", coef_x3, "\n")

In this example, we first generate some random data with three independent variables (x1, x2, and x3) and a dependent variable (y). We then create a multiple regression model using the lm() function.

To estimate the coefficients using separate regressions, we run three additional regressions, each time regressing the residuals of the multiple regression model on one of the independent variables. The estimated coefficients for each variable are stored in coef_x1, coef_x2, and coef_x3.

Finally, we print the estimated coefficients using the cat() function.

Note that the estimated coefficients using separate regressions should be the same as the coefficients obtained from the multiple regression model. This demonstrates the Frisch-Waugh Theorem, which states that we can estimate the coefficients individually by controlling for the other variables