Introduction to Econometrics Chapter 4

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1 Introduction to Econometrics Chapter 4 Ezequiel Uriel Jiménez University of Valencia Valencia, September 2013

2 4 ypothesis testing in the multiple regression 4.1 ypothesis testing: an overview 4.2 Testing hypotheses using the t test 4.3 Testing multiple linear restrictions using the F test 4.4 Testing without normality 4.5 Prediction Exercises

3 4 ypothesis testing in the multiple regression Motivation Testing hypothesis can answer the following questions: 1. Is the marginal propensity to consume smaller than the average propensity to consume? 2. as income a negative influence on infant mortality? 3. Does the rate of crime in an area plays a role in the prices of houses in that area? 4. Is the elasticity expenditure in fruit/income equal to 1? Is fruit a luxury good? 5. Is the Madrid stock exchange market efficient? 6. Is the rate of return of the Madrid Stock Exchange affected by the rate of return of the Tokyo Stock Exchange? 7. Are there constant returns to scale in the chemical industry? 8. Advertising or incentives? 9. Is the assumption of homogeneity admissible in the demand for fish? 10. ave tenure and age jointly a significant influence on wage? 11. Is the performance of a company crucial to set the salaries of CEOs? [3] All these questions are answered in this chapter

4 4.1 ypothesis testing: an overview 4 ypothesis testing in the multiple regression TABLE 4.1. Some distributions used in hypothesis testing. 1 restriction 1 or more restrictions Known 2 N Chi-square 2 Unknown Student s t Snedecor s F [4]

5 4.1 ypothesis testing: an overview 4 ypothesis testing in the multiple regression Non Rejection Region NRR FIGURE 4.1. ypothesis testing: classical approach. c Rejection Region RR W [5]

6 4.2 Testing hypotheses using the t test 4 ypothesis testing in the multiple regression [6] FIGURE Density functions: normal and t for different degrees of freedom.

7 4.2 Testing hypotheses using the t test 4 ypothesis testing in the multiple regression [7] FIGURE 4.3. Rejection region using t: right-tail alternative hypothesis. FIGURE 4.4. p-value using t: right-tail alternative hypothesis.

8 4 ypothesis testing in the multiple regression [8] 4.2 Testing hypotheses using the t test EXAMPLE 4.1 Is the marginal propensity to consume smaller than the average propensity to consume? cons inc u : 0 : 0 FIGURE 4.5. Example 4.1: Rejection region using t with a right-tail alternative hypothesis. cons i = inc (0.350) (0.062) ˆ 0 1 ˆ t se( ˆ ) se( ˆ ) FIGURE 4.6. Example 4.1: p-value using t with right-tail alternative hypothesis. i

9 4.2 Testing hypotheses using the t test 4 ypothesis testing in the multiple regression [9] Rejection Region RR Non Rejection Region NRR t 0 n k tn k FIGURE 4.7. Rejection region using t: left-tail alternative hypothesis Non rejected for α>p-value p-value Rejected for ɑ<p-value t ˆ 0 j tn k FIGURE 4.8. p-value using t: left-tail alternative hypothesis.

10 4.2 Testing hypotheses using the t test EXAMPLE 4.2 as income a negative influence on infant mortality? 4 ypothesis testing in the multiple regression [10] FIGURE 4.9. Example 4.2: Rejection region using t with a left-tail alternative hypothesis. deathun5 gnipc ilitrate u deathun 5 = gnipc ilitrate : 0 : 0 i i i (5.93) ( ) (0.183) t ˆ se( ˆ ) FIGURE Example 4.2: p-value using t with a left-tail alternative hypothesis.

11 4.2 Testing hypotheses using the t test 4 ypothesis testing in the multiple regression [11] FIGURE Rejection region using t: two-tail alternative hypothesis. FIGURE p-value using t: twotail alternative hypothesis.

12 4.2 Testing hypotheses using the t test EXAMPLE 4.3 as the rate of crime play a role in the price of houses in an area? price rooms lowstat crime u ypothesis testing in the multiple regression price = rooms lowstat -3854crime i i i i (8022) (1210) (80.7) (960) TABLE 4.2. Standard output in the regression explaining house price. n=55. Variable Coefficient Std. Error t-statistic Prob. C rooms lowstat crime : 0 : t ˆ se( ˆ ) [12]

13 [13] 4.2 Testing hypotheses using the t test EXAMPLE 4.3 as the rate of crime play a role in the price of houses in an area? (Continuation) 4 ypothesis testing in the multiple regression FIGURE Example 4.3: p-value using t with a two-tail alternative hypothesis.

14 4 ypothesis testing in the multiple regression [14] 4.2 Testing hypotheses using the t test EXAMPLE 4.4 Is the elasticity expenditure in fruit/income equal to 1? Is fruit a luxury good? ln( fruit) ln( inc) househsize punders u ln( fruit ) = ln( inc ) househsize punder5 i i i i (3.701) (0.512) (0.179) (0.013) TABLE 4.3. Standard output in a regression explaining expenditure in fruit. n=40. Variable Coefficient Std. Error t-statistic Prob. C ln(inc) househsize punder : 1 : : ˆ 0 2 ˆ t se( ˆ ) se( ˆ )

15 4.2 Testing hypotheses using the t test 4 ypothesis testing in the multiple regression [15] EXAMPLE 4.5 Is the Madrid stock exchange market efficient? D Pt + Dt + At Rate of total return: RAt P t-1 Rate of return due to increase in quotation Proportional change: DPt RA1 Change in logarithms: RA2t DlnPt t Pt -1 rmad92 rmad92t rmad92t- 1 t 1 2rmad92t 1 ut (0.0007) (0.0629) 2 R = n= : 2 1 ˆ : 2 1 t 2.02 se( ˆ ) EXAMPLE 4.6 Is the rate of return of the Madrid Stock Exchange affected by the rate of return of the Tokyo Stock Exchange? rmad92 rtok92 u t 1 2 t t : 1 : rmad rtok92 R 2 t 2 (0.0007) (0.0375) = n= 235 ˆ t 3.32 se( ˆ ) t

16 4.2 Testing hypotheses using the t test [16] 0,99 0,95 0, FIGURE Confidence intervals for marginal propensity to consume in example ypothesis testing in the multiple regression

17 4.2 Testing hypotheses using the t test EXAMPLE 4.7 Are there constant returns to scale in the chemical industry? ln( output) ln( labor) ln( capital) u ypothesis testing in the multiple regression [17] ln( output ) = ln( labor ) ln( capital ) i i i (0.327) (0.126) (0.085) TABLE 4.4. Standard output of the estimation of the production function: (4-20). Variable Coefficient Std. Error t-statistic Prob. constant ln(labor) ln(capital ) : 1 :

18 4.2 Testing hypotheses using the t test EXAMPLE 4.7 Are there constant returns to scale in the chemical industry? (Cont.) 4 ypothesis testing in the multiple regression TABLE 4.5. Covariance matrix in the production function. constant ln( labor) ln( capital ) constant ln(labor)) ln(capital ) a) Procedure: using covariance matrix of estimators. ˆ ˆ se( ) var( ˆ ˆ ) ˆ ˆ ˆ ˆ var( ) var( ) var( ) 2 covar( ˆ, ˆ ) se( ˆ ˆ ) [18] t ˆ ˆ 2 3 ˆ ˆ se( ˆ ˆ )

19 4.2 Testing hypotheses using the t test EXAMPLE 4.7 Are there constant returns to scale in the chemical industry? (Cont.) b) Procedure: reparameterizing the by introducing a new parameter. 4 ypothesis testing in the multiple regression [19] ln( output) ( 1)ln( labor) ln( capital) u ln( output / labor) ln( labor) ln( capital / labor) u 1 3 TABLE 4.6. Estimation output for the production function: reparameterized. Variable Coefficient Std. Error t-statistic Prob. constant ln(labor) ln(capital/labor) ˆ : 1 0 t : 0 se( ˆ )

20 4.2 Testing hypotheses using the t test EXAMPLE 4.8 Advertising or incentives? 4 ypothesis testing in the multiple regression sales advert incent u TABLE 4.7. Standard output of the regression for example 4.8. Variable Coefficient Std. Error t-statistic Prob. constant advert incent [20]

21 4.2 Testing hypotheses using the t test EXAMPLE 4.8 Advertising or incentives? (Continuation) 4 ypothesis testing in the multiple regression [21] TABLE 4.8. Covariance matrix for example 4.8. C advert incent constant advert incent : 0 : se( ˆ ˆ ) t ˆ ˆ 3 2 ˆ ˆ se( ˆ ˆ )

22 4.2 Testing hypotheses using the t test EXAMPLE 4.9 Testing the hypothesis of homogeneity in the demand for fish 4 ypothesis testing in the multiple regression [22] ln( fish ln( fishpr) ln( meatpr) ln( cons) u ln( fish ln( fishpr ) ln( meatpr ) ln( cons ) i i i i (2.30) (0.133) (0.112) (0.137) omogeneity resstriction: ln( fish ln( fishpr) ln( meatpr fishpr) ln( cons fishpr) u ln( fish ln( fishpr ) ln( meatpr ) ln( cons ) i i i i (2.30) (0.1334) (0.112) (0.137) ˆ t 3.44 se( ˆ )

23 4.3 Testing multiple linear restrictions using the F test 4 ypothesis testing in the multiple regression [23] Non Rejection Region NRR Fqn, k F qn, k Rejection Region RR FIGURE Rejection region and non rejection region using F distribution. Rejected for p-value Fqn, k F Non rejected for <p-value p-value FIGURE p-value using F distribution.

24 4 ypothesis testing in the multiple regression [24] 4.3 Testing multiple linear restrictions using the F test EXAMPLE 4.10 Wage, experience, tenure and age ln( wage) educ exper tenure age u ln( wage ) = educ exper tenure age RSS = i i i i i : 0 : is not true ln( wage) 1 2educ 3exper u ln( wagei ) = educi experi RSS = RSS RSS / q ( ) / 2 R UR F RSS /( n k) 5.954/48 UR

25 4 ypothesis testing in the multiple regression [25] 4.3 Testing multiple linear restrictions using the F test EXAMPLE 4.10 Wage, experience, tenure and age. (Continuation) 0 Non rejected Region NRR F FIGURE Example 4.10: Rejection region using F distribution (α values are from a F 2.40 ) Rejection Region RR

26 4.3 Testing multiple linear restrictions using the F test 4 ypothesis testing in the multiple regression [26] EXAMPLE 4.11 Salaries of CEOs ln ln ( ) ( ) : : is not true 1 0 salary sales roe ros u ln salary = ln sales roe ros i i i i R 2 = n= 209 F 3,205 p-value 0,0000 0,10 0,05 0,01 2,12 FIGURE Example 4.11: p-value using F distribution (α values are for a F 3,140 ) 2,67 3,92 26,93

27 4 ypothesis testing in the multiple regression [27] 4.3 Testing multiple linear restrictions using the F test EXAMPLE 4.11 Salaries of CEOs. (Continuation) TABLE 4.9. Complete output from E-views in the example Dependent Variable: LOG(SALARY) Method: Least Squares Date: 04/12/12 Time: 19:39 Sample: Included observations: 209 Variable Coefficient Std. Error t-statistic Prob. C LOG(SALES) ROE ROS R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

28 4 ypothesis testing in the multiple regression [28] 4.3 Testing multiple linear restrictions using the F test EXAMPLE 4.12 An additional restriction in the production function. (Continuation of example 4.7) ln( ) ln( ) ln( ) output 1 2 labor 3 capital u RSS UR : 1 0 : is not true 1 0 ln( output) (1 ) ln( labor) ln( capital) u ln( output / labor) ln( capital / labor) u RSS R RSS RSS / q ( ) / 2 R UR F RSS / ( n k) / (27 3) UR

29 4.3 Testing multiple linear restrictions using the F test 4 ypothesis testing in the multiple regression FIGURE Relationship between a F 1,n-k and t n-k. [29]

30 4 ypothesis testing in the multiple regression [30] 4.5 Prediction EXAMPLE What is the expected score in the final exam with 7 marks in the first short exam? Fitted : Fitted with the regressor shortex1º=7: The point prediction for shortex1º=7: =7.593 The lower and upper bounds of a 95% CI: The point prediction by an alternative way: Estimation of the se of 2 ˆ i i s (0.715) (0.123) finalmrk = shortex 1 = R = n = 16 [ ] finalmrk = + shortex - ˆ = R = n = i i 7 s (0.497) (0.123) ˆ0 q = qˆ - se( qˆ ) t = = / ˆ /2 se ˆ t14 q = q + ( q ) = = ê 2 finalmrk = = se( eˆ ) se( yˆ ) ˆ where is the S. E. of regression obtained from the E-views output directly. The lower and upper bounds of a 95% probability interval: y yˆ se( eˆ ) t y yˆ se( eˆ ) t

31 4.5 Prediction EXAMPLE 4.14 Predicting the salary of CEOs 4 ypothesis testing in the multiple regression [31] salary = assets tenure profits i i i i (104) (0.0013) (8.671) (0.0538) 2 sˆ = 1506 R = n= 447 TABLE Descriptive measures of variables of the on CEOs salary. assets tenure profits Mean Median Maximum Minimum Observations TABLE Predictions for selected values. Prediction ˆ0 Std. Error se ( ˆ 0 ) Mean values Median value Maximum values Minimum values

32 4.5 Prediction 4 ypothesis testing in the multiple regression EXAMPLE 4.15 Predicting the salary of CEOs with a log (continuation 4.14) ln( salary ) = ln( assets ) tenure profits i i i i (0.210) (0.0232) (0.0032) ( ) 2 sˆ = R = n= 447 Inconsistent prediction salary exp(ln( i = salaryi )) = exp( ln(10000) ) = 1207 Consistent prediction 2 salary = exp( / 2) 1207 = 1404 [32]

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