Wald test for one-sided hypothesis - strict inequality

[Ann Jun]

I am trying to test a one-sided hypothesis after running a regression to see if the sum of two coefficients (X1 and x2) is greater than 0. My regression looks like this:

reghdfe y x1 x2 x3, noabsorb vce(robust)


I have tried this command:
test _b[x1] + _b[x2]=0
local sign_d = sign(_b[x1] + _b[x2])
display "Ho: coef _b[x1] + _b[x2] >=0 t= " invttail(r(df_r), ttail(r(df_r),`sign_d'*sqrt(r(F)))) " p-value = " 1-ttail(r(df_r),`sign_d'*sqrt(r(F)))

This displays the correct p-value for the hypothesis, but I am looking to test the STRICT inequality (H0: _b[x1] + _b[x2] >0) instead of _b[x1] + _b[x2] >=0. Is there any way to do this?

[Leonardo Guizzetti]

It's been a bit since I've have to consider this nuance of a one-sided test, but I believe you would have the null hypothesis be of the "at least as extreme or more" flavor, so your alternative, directional, hypothesis is already precisely defined as either greater than or lesser than. So you are already calculating what you seek.

H0: mu ≥ mu_0
Ha: mu < mu_0
(Or vice versa for the other direction)

[FernandoRios]

I would say no
because the probability of a continuous variable to be exactly equal to a constant is Zero

[Joro Kolev]

You can test Ho: _b[x1] + _b[x2]=0 vs
Ha: _b[x1] + _b[x2]>0
or
Ha': _b[x1] + _b[x2]<0.

Standard hypotheses testing is ill equipped with dealing with set nulls of the type you are speaking about.

See

Decision Making and Underperformance in Competitive
Environments: Evidence from the National Hockey League
Gueorgui I. Kolev, Gonçalo Pina and Federico Todeschini, 2015, Kyklos.

I speak of these matters there.