Test statistics for simultaneous test with two conditions ($\beta_1 = \beta_2, \beta_3= 0$) for multiple regression

Question

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?

Answer 1

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.

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Reference:

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

Answer 2

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"))