If a Generated Regressor X is orthogonal to a non-generated regressor W, would the variance and tests on W be correct?

[Joro Kolev]

Good morning,

Professor Jeff Wooldridge or anybody else who knows the answer, or has encountered the topic before and can provide a reference:

I have a Generated Regressor X, and a set of standard variables W (which might be one, or might be more, let's say for the sake of concreteness that they are two W1 and W2). The generated regressor X is orthogonal to the set of standard regressors W. I am fitting the regression

Y = a + b*X + c*W1 + g*W2 + e

and I want to test Ho: c=g=0.

Would the test statistics, i.e., the variance matrix of W1 and W2 and the F/Chi statistic for joint significance, from the regression output be correct for this Ho?

In other words, is

reg Y X W1 W2

test W1 W2

a correct way to test Ho?

I think the answer is Yes, because there is such a result in the measurement error literature. If X is subject to measurement error, and W1 and W2 are both orthogonal to X, then if I am interested in testing hypothesis only on W1 and W2 I can proceed disregarding the fact that X is subject to measurement error.

But I was not able to find any result like this for generated regressors after a day of googling.