Suppose I have a multiple linear regression of the form:
$$y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} +\beta_3 x_{i3}+\epsilon$$
Then suppose I am interested in testing this set of hypotheses:
$$H_0: \beta_1 *\beta_2 = 0$$ $$H_A: \beta_1 *\beta_2 \neq 0$$
I am interested in deriving a score test statistic for this hypothesis. I understand that I need to derive this under the null hypothesis. However, I am under the impression that score tests do not work for nonlinear hypotheses of this form, correct? Additionally, you see that there is a nuisance parameter to also take into consideration: $\beta_3$.
Any input would be much appreciated, thank you!
Update: this is quite similar to my question here. However, this setting has a nuisance parameter, whereas the other does not.