The null and alternative hypotheses depend on what you want to know, not on the type of analysis you do to test them.
However the most common null hypothesis in linear regression is that the slope coefficient is zero. The most common alternative is that the slope coefficient is not zero. But any null hypothesis and alternative are possible.
Here are some examples:
- We regress income earned 10 years after graduating high school on high school GPA. Null hypothesis is the slope equals zero, meaning there is no systematic linear effect of GPA on income. The alternative is the slope is not equal zero as we are interested in any relation between GPA and income, positive or negative.
- We regress individual stock returns on the market excess return. Null hypothesis is that the constant term equals the risk-free rate of interest. Alternative is that it does not.
- We regress adult height on father’s height and mother’s height (this was the original regression, the one that gives the tool its name). Null hypothesis is that the slope coefficients add up to one and the constant term is zero, that is, children’s height are a weighted average of parents’ heights. Again the alternative is merely that the null hypothesis is not true, as we’re interested in any deviation.