The $t$-test

Having estimated the coefficients in a regression model, we often want to go a step further. We may also ask whether the data provide convincing evidence for or against a particular claim about one of these coefficients.

Most commonly, we ask whether a coefficient estimate is significantly different from zero. For instance, suppose we estimate a positive relationship between education and wages. Could this simply have arisen through random sampling, even if the true coefficient were zero? Or is our estimate sufficiently far from zero to provide convincing evidence of a genuine relationship?

To answer questions like this, we use the $t$-test. We compare our estimated coefficient with the value claimed under a certain hypothesis. But we also need to take account of the uncertainty in our estimate, measured by its standard error.

Together, these ideas lead us to the test statistic. This is a random variable that has a known probability distribution if the null hypothesis we are testing is true. We can therefore ask how surprising our observed result would be under this hypothesis.

We begin by explaining the need for the $t$-test with an example, before developing the test statistic itself. We then work through an example testing whether a coefficient is zero. Finally, we extend exactly the same method to hypotheses about other possible coefficient values.

The regression $t$-test is one of the most frequently used tools in econometrics, and provides our first major application of the standard errors developed earlier.