# Econometric Theory/Statistical Inference/Hypothesis Testing

## Basic Concepts

To conduct a successful hypothesis test, the following are required:

• Testable Hypothesis

We need to have a null (${\displaystyle H_{0}}$) and alternate (${\displaystyle H_{1}}$) hypothesis.

• Feasible test statistic

A test statistic is a random variable whose value for given sample data determines whether the null is rejected or retained. It is feasible when:

• Its probability distribution is known when the null hypothesis (${\displaystyle H_{0}}$) is true.
• Its value can be calculated from the given sample data
• Decision rule

A decision rule clearly delineates the:

• Rejection region - the set of values of the test statistic for which ${\displaystyle H_{0}}$ is to be rejected.
• Non-rejection region - the set of values of the test statistic for which the ${\displaystyle H_{0}}$ is to be retained.

## Procedure for testing a hypothesis

1. Formulate ${\displaystyle H_{0}}$ and ${\displaystyle H_{1}}$.
2. Specify the test statistic and its distribution.
3. Calculate the sample value of the test statistic under ${\displaystyle H_{0}}$ for the given sample data.
4. Select a significance level (α) and determine the corresponding critical values (for the particular distribution).
5. Apply the decision rule and state the conclusion (or inference) implied by the sample value of the test statistic.

## Notes

• The null hypothesis will always be the position where there is an equality (either strong or weak), and the alternate hypothesis will have the inequality.