# Inference about the slope: t-Test

- t-test for a population slope
- Is there a linear relationship between x and y?

- Null and alternative hypotheses
- H0: β
_{1}= 0 (no linear relationship) - H1: β
_{1}≠ 0 (linear relationship does exist)

- H0: β
- Test statistic
- Where:
- b1 = Sample regression slope coefficient
- β1 = Hypothesized slope
- sb1 = Estimator of the standard error of the slope

Example

- A risk analyst performs a simple linear regression on return data comprising three variables evolving in time and obtains, amongst others, the following statistics:

- Based on these data at a 95% confidence level, the analyst should conclude that:

- The intercept and “X Variable 2” are statistically significant
- “X Variable 1” and “X Variable 3” are statistically significant
- “X Variable 1”, “X Variable 2” and “X Variable 3” are all statistically not significant
- More information is required, such as the corresponding p-values, before any meaningful deductions may be made

Solution

**A.**

A is correct. (Relatively) small standard errors and high t-stats are one indication of indicate statistical significance

B is incorrect. – (Relatively) large standard errors and low t-stats are one indication of indicate statistical significance

C is incorrect. Negative t-stats are not an indication of statistical insignificance

D is incorrect. The p-values are redundant information if the t-stat is provided. That is, the p-values tell one nothing more than the t-stats do.