2. 2010, ECON 77101 Hypothesis Testing 1: Single Coefficient Review of hypothesis testing Testing single coefficient Interval estimation Objectives

3. 2010, ECON 77102 s.e. (9.3421) (0.8837) R 2 = 0.7431, N = 20, SER = 8.5018 Explaining weight by height (Table 1.1) Can X really explain Y? When X=0, what is Y? If we suspect that the coefficient of X is 5, can we find support from the data?

4. 2010, ECON 77103 1.Hypothesis testing: Revision The principle of hypothesis testing The value of the parameter to be tested is assumed in H 0. The estimate of this parameter is compared with that assumed value. If the estimate is far from the assumed value, then H 0 is rejected. Otherwise, H 0 is not rejected.

5. 2010, ECON 77104 Procedures of Hypothesis Testing 1. Determine null and alternative hypotheses. 2. Specify the test statistic and its distribution as if the null hypothesis were true. 3. Select  and determine the rejection region. 4. Calculate the sample value of test statistic. 5. State your conclusions. 1. Revision

6. 2010, ECON 77105 2.Testing a Regression Coefficient Population Y i =  0 +  1 X 1i +  2 X 2i + … +  K X Ki +  i Sample: 3 types of tests (k = 0, 1, 2, , K): H o :  k = c; H A :  k  c H o :  k  c; H A :  k > c H o :  k  c; H A :  k < c c is any number meaningful in your study

7. 2010, ECON 77106 Probability Distribution of Least Squares Estimators 2. Testing

8. 2010, ECON 77107 Student's t - statistic t has a Student-t Distribution with N – K – 1 degrees of freedom. 2. Testing

9. 2010, ECON 77108 Two-Tail t-test 1.State the null & alternative hypotheses H 0 :  k = c H A :  k  c 2. Compute the estimated t-value c  ˆ     ˆ Se t k k 3. Choose a level of significance (  ) and degrees of freedom (N – K – 1). Then find a critical t-value from the t-table (t c = t N-K-1,  /2 ). 2. Testing

10. 2010, ECON 77109 Two-Tail t-test (cont.) 4. State the decision rule. Version I: If |t| > t c, then reject H 0. Version II: If t > t c or t < -t c, then reject H 0. 5. Conclusion Acceptance region 0 tctc -t c rejection region 2. Testing

11. 2010, ECON 771010 Example 1: In the following regression results, test whether the estimated coefficient of X 1 and X 2 are significantly different from zero. (  = 5%) Y = 14.32 + 0.798 X 1 – 0.101 X 2 se (6.1361) (0.2535) (0.08333) R 2 = 0.2718, N = 30. Hypotheses: H 0 :  1 = 0; H A :  1  0. First test: 2. Testing

12. 2010, ECON 771011 Computed t-value: Table t-value: For  = 0.05 and 30 – 2 – 1 = 27 degrees of freedom, a critical value is t 27,0.025 = 2.052. Decision rule: If |t| > 2.052, then reject H 0. Conclusion: Since |t| = 3.112 > 2.052, H o can be rejected. The estimated coefficient of X 1 is significantly different from zero. 2. Testing

13. 2010, ECON 771012 One-Tail t-test Step 1: State the null & alternative hypotheses Right-tail test: Test whether  k > c. H 0 :  k  c; H A :  k > c. Left-tail test: Test whether  k < c. H 0 :  k  c; H A :  k < c. 2. Compute the estimated t-value (same as before) 2. Testing

14. 2010, ECON 771013 3. Choose a level of significance (  ) and degrees of freedom (N – K – 1). Then find a critical t-value from the t-table (tc = t N-K-1,  ). 4. State the decision rule. Right-tail test: Reject H 0 if t > t c. Left-tail test: Reject H 0 if t < -t c. One-Tail t-test (cont.) 2. Testing

15. 2010, ECON 771014 0 tctc < t Right-tail 0 -t c t < left-tail 2. Testing One-Tail t-test (cont.) 5. Conclusion

16. 2010, ECON 771015 Example 3: Right-tail test: Test whether  1 is greater than 0.35 at 5% level of significance. 2. Testing Dependent Variable: Y Method: Least Squares Sample: 1 30 Included observations: 30 CoefficientStd. Errort-StatisticProb. C 16.427825.936234 2.7673810.0099 X 0.71770.246886 2.9070070.0071 R-squared 0.231839 Mean dependent var 32.6 Adjusted R-squared 0.204405 S.D. dependent var 12.71871 S.E. of regression 11.3446 Akaike info criterion7.759701 Sum squared resid 3603.597 Schwarz criterion7.853114 Log likelihood-114.3955 Hannan-Quinn criter.7.789585 F-statistic8.450689 Durbin-Watson stat1.338091 Prob(F-statistic)0.007061

17. 2010, ECON 771016 Example 4: Left-tail test: Test whether  1 in Example 3 is smaller than 1.2. (  = 0.05) 2. Testing

18. 2010, ECON 771017 A Special case H o :  k = 0 H A :  k  0 Statistic 2. Testing It is the lowest level of significance at which we could reject the H o that a parameter is zero. The p-values Reported by Regression Software

19. 2010, ECON 771018 The t-statistics and P-values Dependent Variable: Y Method: Least Squares Sample: 1 30 Included observations: 30 CoefficientStd. Errort-StatisticProb. C 16.427825.936234 2.7673810.0099 X 0.71770.246886 2.9070070.0071 R-squared 0.231839 Mean dependent var 32.6 Adjusted R-squared 0.204405 S.D. dependent var 12.71871 S.E. of regression 11.3446 Akaike info criterion7.759701 Sum squared resid 3603.597 Schwarz criterion7.853114 Log likelihood-114.3955 Hannan-Quinn criter.7.789585 F-statistic8.450689 Durbin-Watson stat1.338091 Prob(F-statistic)0.007061 2. Testing

20. 2010, ECON 771019 The p-value of  1 -hat for a two-sided test t 0 f(t) -2.91 2.91 p/2 = 0.00355 red area = p-value = 0.0071 p/2 = 0.00355 2.048 -2.048 critical values 2. Testing

21. 2010, ECON 771020 3. Confidence Intervals for Regression Coefficients Y i =  0 +  1 X i + u i (i = 1,  n) The OLS estimators for  0 and  1 are point estimators.  The OLS estimates are likely to be different from the theoretical values  We have no idea of how close the OLS estimates to the theoretical values

22. 2010, ECON 771021 Interval estimation: We know the chance of including the population parameter (  k )in the intervals constructed from repeated samples. 3. Confidence Interval

23. 2010, ECON 771022 Confidence coefficient : 1 -  Level of significance :  Interval estimate : (  k * - ,  k * +  ) Population parameter:  k Estimator of  k : Estimate of  k :  k * Confidence limits 3. Confidence Interval

24. 2010, ECON 771023 Constructing Confidence Interval for  k Actual estimated  k could be fallen into these regions   ˆ f k  ˆ k    ˆ E k k 3. Confidence Interval

25. 2010, ECON 771024   ˆ f k  ˆ k    ˆ E k k a b () )( interval ainterval b Constructing Confidence Interval for  k 3. Confidence Interval

26. 2010, ECON 771025 () tata )( tbtb interval t a interval t b f(t)       ˆ Se ˆ t k k k 0 Constructing Confidence Interval for  k 3. Confidence Interval

27. 2010, ECON 771026 Probability statements P(-t c < t < t c ) = 1   P( t t c ) =  3. Confidence Interval

28. 2010, ECON 771027 A 95% confidence interval means that, using the interval estimator and drawing samples from the population, 95% of the interval estimates would include the population value . The probability that a particular interval estimate contains this population value is either 0 or 1. 3. Confidence Interval The (1-  )  100% CI for  k is

29. 2010, ECON 771028 Example 5 : Regressing WEIGHT on HEIGHT, 2005 3. Confidence Interval

30. 2010, ECON 771029 95% confidence limits for  1 : 95% confidence interval for  0 : 3. Confidence Interval The 95% confidence interval for  1 is (0.3765, 0.7811).

31. 2010, ECON 771030 4. Applied Examples Example 6 : Restaurant location (Section 3.2) Suppose you have been hired to determine a location for the next Woody’s. Woody’s is a moderately priced, 24-hour, family restaurant chain. Two choices are: Location A: NN = 4.4, PP = 104, II = 20.6 Location B: NN = 2, PP = 50, II = 20

32. 2010, ECON 771031 YY i =  0 +  N N i +  P PP i +  I II i +  i. -ve +ve ? YY: Number of customers served in thousand N: Number of direct market competitors PP: Population in thousand within a 3-mile radius II: Average household income in thousand Example 6 : Restaurant location (Cont’d) 4. Examples

33. 2010, ECON 771032 Woody’s: Null and Alternative Hypotheses 1. H o :  N  0; H A :  N < 0 2. H o :  P  0; H A :  P > 0 3. H o :  I = 0; H A :  I  0 YY = 102.19 *** – 9.07 *** N + 0.35 *** PP + 1.29 ** II se (2.0527) (0.07268) (0.5433) R 2 = 0.618, R 2 = 0.579, N = 33. ^ _ 4. Examples

34. 2010, ECON 771033 If  i is normally distributed, then hat is normally distributed with mean  k and variance var( ). Z = has standard normal distribution. Var( ) is unobservable. is used instead and is denoted by se. t = has t distribution with N – K – 1 degrees of freedom. 4. Examples

35. 2010, ECON 771034 Woody’s two-sided test Hypotheses: H o :  I = 0; H A :  I  0 Statistics: Decision rule: Let  = 0.05. From the table the critical values are t c =  t 29,0.025 =  2.045. Reject H o if |t| > 2.045. Computed t-value: Decision: Since t = 2.37 > 2.045, reject H o. Thus  I hat is significantly different from zero.

36. 2010, ECON 771035 Woody’s one-sided tests Hypotheses: H o :  N  0; H A :  N < 0 Decision rule: Let  = 0.05. From the table the critical value is t c = -t 29,0.05 = - 1.699. Reject H o if t < - 1.699. Computed t-value: Decision: Since t = - 4.42 < -1.699, reject H o. Thus  N hat is significantly smaller than zero. 4. Examples

37. 2010, ECON 771036 Example 7: Sales of Hamburger TR i =  0 +  p P i +  A A i +  i. ? + Data: Weekly observations for a hypothetical hamburger chain TR : Weekly revenue in $1,000 P : Price in $ A : Advertising expenditure in $1,000 4. Examples

38. 2010, ECON 771037 TR = 113.83*** – 10.26***P + 2.68***A se (1.6007) (0.1189) R 2 = 0.8739, N = 78. ^ Regression results: a.Is the demand significantly elastic or inelastic in price? b.Is the increase in total revenue stimulated by more advertisements significantly greater than the corresponding increased cost of advertising? Let  = 0.01. Answer the following two questions statistically.