Question

Given the following regression model: Y = B0 + B1X1 + B2X2 + B3X3 + u

If we want to test the null hypothesis H0: B0 + B1 = 5, which of the following is correct?

1) Estimate Y = gamma0 + gamma1(X1 - 1) + gamma2X2 + gamma3X3 + u and test the hypothesis gamma0 = 5

2) Estimate Y = gamma0 + gamma1(X1 + 1) + gamma2X2 + gamma3X3 + u and test the hypothesis gamma1 = 5

3) Estimate Y = gamma0 + gamma1(X1 + 5) + gamma2X2 + gamma3X3 + u and test the hypothesis gamma0 = 5

2) Estimate Y = gamma0 + gamma1(X1 - X2) + gamma2X2 + gamma3X3 + u and test the hypothesis gamma2 = 5

Answer 1

The main model: Y = B0 + B1X1 + B2X2 + B3X3 + u

We want to test null hypothesis H0: B0 + B1 = 5

Consider option 1) with the model  Y = gamma0 + gamma1(X1 - 1) + gamma2X2 + gamma3X3 + u

Compare it with the main model by comparing the coefficient of X1 and the constant term. When we do both of them, we get 2 equations:

gamma0 - gamma1 = B0 - (1) constant term comparison

gamma1 = B1 - (2) X1 coefficient comparison

substitute the second in the first equation we get:

=> gamma0 - B1 = B0

=> gamma0 = B1 + B0

So, testing   B1 + B0 = 5 is the same as testing gamma0 = 5, which means option 1) is correct.