Question

Perform hypothesis tests and confidence intervals for one coefficient

Answer 1

Hypothesis test for one coefficient

For the null hypothesis , assume we take $\beta_{i}$ as the true coefficient on ith regressor which takes value $\beta_{i,0}$. This would form the null hypothesis . Here for generality , we are assuming a two tailed test

Null hypothesis = H₀ :  $\beta_{i}$ =  $\beta_{i,0}$.

Alternative hypothesis = H_a: $\beta_{i}\neq \beta_{i,0}$

Steps to perform hypothesis test

1) Define null and alternative hypothesiis which is done above

2) Determine the level of significance to be used

3) Calculate t statistic by first finding the standard error of the coefficient

The formula for t statistic is

z= $(\bar{X}-\mu )/(\sigma /\sqrt{n})$

where n is the number of obervations

$\sigma /\sqrt{n}$ is the standard error of mean

4) Determine critical values.

5) Find whether to reject or not reject null hypothesis

6)State the conclusion

The confidence interval for one coefficient

For 95% confidence interval

95% CI=

P(μ-1.96 * ơNn < X < μ 1.96 * σ/Vn) = 0.95

Rearranging it for solving for we get

P(X-1.96 *o/Vn < μ < X + 1.96 *o/v/r) = 0.95

So the confidence interval would be

(X-1.96 * σ/v/r. X + 1.96 * σ/vn)