# Formulating Null and Alternative Hypotheses in a Regression Model

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?!

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.

• Thank you. You said that if the coefficient is not negative that we don't reject the null hypothesis. Do you mean we don't reject the alternative? What if the coefficient was significantly positive, what would be the interpretation regarding the null and the alternative? Rejecting the null but not accepting the alternative? This is what confuses me! – Amiro Mar 20 '20 at 7:20
• Hi: I was always told that you frame the conclusion in terms of the null. So, you either say that there was evidence to reject the null or not enough evidence to reject the null. So, if the estimate of $\beta$ is $> 0$, then there's no evidence to reject the null because the "complement" of the alternative is $\beta >= 0$. So, you won't be rejecting the null, since result is consistent with that region of the parameter space that coincides with the null. Note that the null is always specified as equality but it's really better to think of it as the complement of the alternative. – mlofton Mar 21 '20 at 8:29