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

2010, ECON s.e. (9.3421) (0.8837) R 2 = , N = 20, SER = 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?

2010, ECON 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.

2010, ECON 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

2010, ECON 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

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

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

2010, ECON 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

2010, ECON 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 Conclusion Acceptance region 0 tctc -t c rejection region 2. Testing

2010, ECON 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 = X 1 – X 2 se (6.1361) (0.2535) ( ) R 2 = , N = 30. Hypotheses: H 0 :  1 = 0; H A :  1  0. First test: 2. Testing

2010, ECON 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 = Decision rule: If |t| > 2.052, then reject H 0. Conclusion: Since |t| = > 2.052, H o can be rejected. The estimated coefficient of X 1 is significantly different from zero. 2. Testing

2010, ECON 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

2010, ECON 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

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

2010, ECON 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 X R-squared Mean dependent var 32.6 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic)

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

2010, ECON 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

2010, ECON The t-statistics and P-values Dependent Variable: Y Method: Least Squares Sample: 1 30 Included observations: 30 CoefficientStd. Errort-StatisticProb. C X R-squared Mean dependent var 32.6 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Testing

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

2010, ECON 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

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

2010, ECON 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

2010, ECON 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

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

2010, ECON () 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

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

2010, ECON 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 Confidence Interval The (1-  )  100% CI for  k is

2010, ECON Example 5 : Regressing WEIGHT on HEIGHT, Confidence Interval

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

2010, ECON 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

2010, ECON 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

2010, ECON 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 = *** – 9.07 *** N *** PP ** II se (2.0527) ( ) (0.5433) R 2 = 0.618, R 2 = 0.579, N = 33. ^ _ 4. Examples

2010, ECON 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

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

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

2010, ECON 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, Examples

2010, ECON TR = *** – 10.26***P ***A se (1.6007) (0.1189) R 2 = , 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  = Answer the following two questions statistically.