Formulating Null and Alternative Hypotheses in a Regression Model

Question

I have a simple question regarding formulating research null and alternative hypotheses in a simple econometric regression model.

In my research Hypothesis, I expect X to negatively affect Y (from intuition as well as empirical literature).

So, in order to state the null and alternative hypotheses, which form is accurate:

Case 1:

H0: X has no impact on Y.

H1: X has an impact on Y.

Case 2:

H0: X has no Impact on Y

H1: X has a negative impact on Y.

And let's say X was found significantly affecting Y, Can I say the null is rejected, but the alternative in case (in case 2), or my research hypothesis is not accepted?!

THANK YOU LOADS!! :D

Answer 1

Hi: When you state a hypothesis, it should be about the coefficients.

So, suppose your regression model was $ Y = \beta_{0} + \beta_{1} X$.

Then, BEFORE you run the regression, your hypothesis would be:

$H_0: ~~ \beta_1 = 0 $

$H_1: ~~ \beta_1 < 0$.

Then, when you consider the output of your regression, you can't use it directly because the default in the regression output is that the alternative hypothesis is that the coefficient does not equal zero. So, don't use the output.

For your hypothesis, you need to carry out the hypothesis test by hand. Take the coefficient and first, check if it's negative.

A) If it is negative, then you still need to check if it's significant by carrying out the standard t-test at whatever significance level you are interested in testing for. Any standard regression-data analysis will explain the t-test in detail.

B) If it is not negative, then you don't even need to do the test because it's already obvious that you don't reject the null hypothesis.

I hope this helps.