1 Econometrics (NA1031) Lecture 3 Interval Estimation and Hypothesis Testing

2 Interval Estimation Point vs interval estimate Replacing σ 2 with creates a random variable t: When b k and se(b k ) are estimated values (numbers), based on a given sample of data, then b k ± t c se(b k ) is called a 100(1-α)% interval estimate of b k. Equivalently it is called a 100(1-α)% confidence interval. Usually α = 0.01 or α = 0.05, so that we obtain a 99% confidence interval or a 95% confidence interval.

3 The t-distribution

4 Hypothesis test (t-test) A null hypothesis: H 0 : β k = c Possible Alternative hypotheses are: H 1 : β k > c H 1 : β k < c H 1 : β k ≠ c Level of significance: 0.05 or 0.01 Probability of rejecting H 0 when it is true. Avoid saying that you ‘‘accept’’ the null hypothesis. Either reject or don’t reject the null hypothesis.

5 Figure 3.2 Rejection region for a one-tail test of H 0 :β k = c against H 1 :β k > c Figure 3.3 Rejection region for a one-tail test of H 0 :β k = c against H 1 :β k < c

6 Figure 3.4 Rejection region for a test of H 0 :β k = c against H 1 :β k ≠ c

7 p-value If t is the calculated value of the t-statistic, then: if H 1 : β K > c p = probability to the right of t if H 1 : β K < c p = probability to the left of t if H 1 : β K ≠ c p = sum of probabilities to the right of |t| and to the left of – |t|

8 Figure 3.5 The p-value for a right-tail test. Figure 3.6 The p-value for a left-tail test.

9 Figure 3.7 The p-value for a two-tail test of significance.

10 Linear combinations of parameters λ = c 1 β 1 + c 2 β 2, where c 1 and c 2 are constants that we specify using LSE we can compute a variance: which can be used for testing.

11 Stata Start Stata mkdir C:\PE cd C:\PE copy chap03_15.do doedit chap03_15.do

12 Assignment Exercise 3.5 page 120 in the textbook.