F test for multiple restrictions

[Itzhak Rasooly]

Hi everyone,

I am regressing a (binary) dependent variable y on a bunch of (binary) independent variables x_1 x_2 x_3 x_4.

I want to test the joint hypothesis that x_4 - x_3 = x_3 - x_2 AND x_4 - x_3 = x_3 - x_2 AND x_4 - x_3 = x_3 - x_2

To do this, I typed

reg y x_1 x_2 x_3 x_4, robust
test (x_4 - x_3 = x_3 - x_2) (x_4 - x_3 = x_3 - x_2) (x_4 - x_3 = x_3 - x_2)

Can I check that I am doing this correctly? The results are a bit unexpected.

Thank you,

Itzhak

[Andrew Musau]

What is the difference between these three?

x_4 - x_3 = x_3 - x_2 AND x_4 - x_3 = x_3 - x_2 AND x_4 - x_3 = x_3 - x_2

Note that \(x_4-x_3 = x_3-x_2 = (x_4-x_3-x_3+x_2=0) = (x_4-2x_3+x_2=0)\) and you are just testing \(x_4-2x_3+x_2=0\). Only one restriction will be kept even if you specify it multiple times.

[Joro Kolev]

I am also curious what outcome OP expected by specifying the same restriction three times...

[Itzhak Rasooly]

Dear all,

My apologies -- I don't know what happened there! The restrictions I wanted to test are

x_4 - x_3 = x_3 - x_2 AND x_3 - x_2 = x_2 - x_1 AND x_2 - x_1 = x_1

leading to my attempted STATA code

reg y x_1 x_2 x_3 x_4, robust
test (x_4 - x_3 = x_3 - x_2) (x_3 - x_2 = x_2 - x_1) (x_2 - x_1 = x_1)

Does that look right?

One thing I found a bit unexpected is that the p-value from the F test is NOT the product of the p-values obtained from testing the three restrictions individually. But I guess it doesn't need to be... (dependence between tests?)

Thank you again,

Itzhak

[Andrew Musau]

Dear all,

My apologies -- I don't know what happened there! The restrictions I wanted to test are

x_4 - x_3 = x_3 - x_2 AND x_3 - x_2 = x_2 - x_1 AND x_2 - x_1 = x_1

leading to my attempted STATA code

reg y x_1 x_2 x_3 x_4, robust
test (x_4 - x_3 = x_3 - x_2) (x_3 - x_2 = x_2 - x_1) (x_2 - x_1 = x_1)

Does that look right?

Yes, that will jointly test the restrictions.

One thing I found a bit unexpected is that the p-value from the F test is NOT the product of the p-values obtained from testing the three restrictions individually. But I guess it doesn't need to be... (dependence between tests?)

Where did you get this idea from? The p-value of the joint test is not a linear function of the p-values of the tests done separately.