## Hypothesis Testing About Regression Coefficients

In this short tutorial, we would demonstrate Hypothesis Testing About Regression Coefficients using Stata. The demonstration is based on the Stata dataset we can bring into analysis through the code -sysuse auto- and commonly most of the Stata examples on linear regression follow the same data. In this tutorial, we will only highlight the different possible ways of Hypothesis Testing About Regression Coefficients and would not discuss further technicalities of regression and predictive modeling.

The Hypothesis Testing About Coefficients of a Regression Model can be discussed in two themes. First, we will demonstrate how to test a single coefficient while the second case will explain hypothesis testing about multiple regression coefficients. Lets begin the first case, then.

The single coefficient in regression model (either simple regression model with one independent variable (IV) or multiple regression with multiple IVs) of the given dependent variable (DV). The null hypothesis is usually given as: Ho: β=0 against the alternative that H1:β≠0. The hypothesis test is based on the t Statistic or Z statistic commonly reported by statistical softwares like Stata in a column against each coefficient. The Stata output for a multiple regression model can be seen here where we can see that each row of the IVs has a t Statistic and p-value. We can use these two columns separately to test the null hypothesis of Ho: β=0.

We can see in the above output that the regression coefficients for all IVs are given with t and p-values. The hypothesis testing following like:

Define the null and alternative hypothesis about any coefficient. Lets assume we wish to test hypothesis that turn has no effect on price against the alternative that turn has an effect on price. The null can be specified as: Ho: β(turn)=0 against the alternative hypothesis that H1: β(turn)≠0.

Determine a level of significance. We will follow the convention of α=0.05

Then we determine the critical region with benchmark values of t statistic. Given the normal routine, we will see that commonly the t-statistic at α=0.05 will be absolute of 1.96. It is named as tabulated t Statistic.

Then we compute the calculated t statistic for given coefficient (here for turn) and report it. Its given in the Stata output and is equal to -3.01.

Then finally, decisions to accept or reject the null hypothesis is given. If the tabulated t statistic is greater than calculated t statistic, we reject the null hypothesis and accept the alternative. As we can see, the t statistic for turn in above output is greater than 1.96 so we reject the null hypothesis that turn has no effect on price and accept the alternative.