0
$\begingroup$

Basically from what i learned, I can use F-test to test for joint hypothesis. But if we have a regression model $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3$ , and the null hypothesis is $\beta_1 = \beta_2, \beta_3= 0$, what test statistics can we use for this simultaneous test?

$\endgroup$

2 Answers 2

4
$\begingroup$

One needs to formulate the hypotheses.

Specifically, \begin{align}\mathcal H_0&: \mathbf R\boldsymbol\beta-\mathbf q =\mathbf 0\\ \textrm{vs.}&\\\mathcal H_1 &: \mathbf R\boldsymbol\beta-\mathbf q \ne\mathbf 0\tag 1\label 1.\end{align}

Here $\boldsymbol\beta:=\begin{bmatrix}\beta_1\\\beta_2\\\beta_3\end{bmatrix};~\mathbf R$ is the matrix of coefficients of restrictions.

For $\beta_1 = \beta_2, \beta_3= 0,$ consider

$$\mathbf R=\begin{bmatrix}1&-1&0\\0&0&1\end{bmatrix};\tag 2\label 2$$

take $\mathbf q=\begin{bmatrix}0\\ 0\end{bmatrix}.$

The test then resorts to the usual F statistic derived here.


Reference:

$\rm [I]$ Econometric Analysis, William H. Greene, Pearson, $2018, $ sec. $5.3, $ pp. $119-120,~123.$

$\endgroup$
1
$\begingroup$

This is also an F-test, as we are also testing (two) joint hypotheses here.

This can be implemented with the linearHypothesis function in R, which specifies the correct F-statistic as a special case of the general F statistic as presented in, e.g., this discussion: Proof that F-statistic follows F-distribution

library(car)
n <- 200
k <- 3

X <- matrix(rnorm(n*k), ncol=k)
X1 <- X[,1]
X2 <- X[,2]
X3 <- X[,3]
y <- 2*X1 + 2*X2 + rnorm(n) # i.e. the DGP is such that your hypothesis is true

# y <- 2*X1 + 3*X2 + 0.5*X3 + rnorm(n) # i.e. the your hypothesis is false, should give large F statistics

reg <- lm(y~X1+X2+X3)
linearHypothesis(reg, c("X1 = X2", "X3=0"))
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.