# Degrees of freedom in hypothesis testing with multiple constrained parameters in one constraint

Suppose you are estimating a (multivariate) model with a parameter vector $$\theta=(\theta_1,\dots,\theta_p)'$$. You have two constraints in the model, and would like to test them with either the likelihood ratio (LR) test, the Wald (W) test or Lagrange multiplier (LM) test. You have the following two constraints: \begin{align} H_0:\begin{cases} \theta_1 = 0 \\ \theta_2 = \theta_3 \end{cases} \end{align}

I am trying to determine the degrees of freedom for the relevant test, but seem not to be able to find it. Is it equal to the number of constraints (2 in this case) or the number of constrained parameters (3 in this case)?