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I am trying to perform a Seemingly Unrelated Regression (SUR) on Stata, with 4 dependent variables (that define the behaviour of a company) and 5 independent variables plus several control variables. Indeed, the regressors are the same for each dependent variable.

I am now trying to reduce the number of control variables. To do so, I am testing the coefficients of each of these variables (Null Hypothesis= In each equation, the coefficients of the considered control variable are equal to zero):

test [$y1list]$x_control1 =[$y2list]$x_control1 = [$y3list]$x_control1 = [$y4list]$x_control1

Since I had one control variable that was categorical (with 3 categories), I hot-coded it creating 2 dummy variables. Specifically, one dummy is called “Europe” and the other is “US”, as these two map the location of the company.

Is it correct to remove both the variables by testing separately the coefficients of the two dummies with this line of code? Is there another way to do so?

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By testing and rejecting the null hypothesis

$H_0 : \beta_{k} = \beta_{k+1} = \beta_{eu} = \beta_{us}$

it says

$H_1 : \text{At least one of them } \left\lbrace \beta_{k}, \beta_{k+1}, \beta_{eu}, \beta_{us} \right\rbrace \text{ is non-zero}$

But we don't know which one of (or which combination of) them is non-zero. So you might want to test these 4 control variables separately.

Also, say that you tested $\beta_{eu} = 0$ and $\beta_{us} = 0$ separately. And you found the coefficient for the EU dummy $\beta_{eu}$ is zero, while the coefficient for the US dummy $\beta_{us}$ is non-zero. In this case, you might want to drop the EU dummy while keeping the US dummy - meaning that there are in fact two categories (US, non-US), instead of three (US, EU, others).

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