Why do we use mostly two-tailed Student's t-statistics to find if an explanatory variable is significant in a regression?

Question

Let's assume a standard regression model: $$y=\beta x+u$$ We'd like to test if an variable $x_j$ is relevant in the respect of the model.

t-statistics: $$t=\frac{\hat{\beta_j}-b}{SE(\hat{\beta_j})}$$

Most of the time, we assume the following two-tailed hypothesis: $$H_0: \beta_j=b, \ H_1: \beta_j \neq b$$

My question is, why don't we use more often the one-tailed hypothesis to test the significance of a variable $x_j$? When I read papers and write assignments I hardly ever come across one-tailed t-tests in the respect to regression models. Why?

Answer 1

We'd like to test if an variable $x_j$ is relevant in the respect of the model

means that we want to test its "statistical significance", so the null hypothesis is $$\text{H}_0 : \beta = 0$$

(by the way, historically, that's why it is called the "null" hypothesis: a hypothesis of "null"-zero- effect).

The $t$-statistic for this test is

$$t=\frac{\hat{\beta_j}}{SE(\hat{\beta_j})}$$

Using a two-tailed test does not constrain us as regards the sign of the coefficient (the direction of the effect, if it exists). It may be positive or negative. If it is positive, the null if it is rejected, will be because the $t$-statistic takes a large positive value. But if the effect is negative, then the $t$-statistic will take a high negative value. So we want to test against either case, and this is why we use a "two-tailed" test.