1. MEASURES OF GOODNESS OF FIT The sum of the squares of the actual values of Y (TSS: total sum of squares) could be decomposed into the sum of the squares of the fitted values (ESS: explained sum of squares) and the sum of the squares of the residuals. 1 Model Y =  1 +  2 X + u

2. 2 R 2, the usual measure of goodness of fit, was then defined to be the ratio of the explained sum of squares to the total sum of squares. MEASURES OF GOODNESS OF FIT Model Y =  1 +  2 X + u

3. 3 The indetermination coefficient. MEASURES OF GOODNESS OF FIT Model Y =  1 +  2 X + u

4. 4 MEASURES OF GOODNESS OF FIT Model 1: OLS, using observations 1-540 Dependent variable: EARNINGS CoefficientStd. Errort-ratiop-value const -13.9335 3.21985 -4.32740.00002*** S 2.45532 0.231851 10.5901<0.00001*** Mean dependent var 19.63622S.D. dependent var 14.41566 Sum squared resid 92688.67S.E. of regression 13.12569 R-squared 0.172498Adjusted R-squared 0.170960 F(1, 538) 112.1496p-value(F) 6.22e-24 Log-likelihood -2155.494Akaik criterion 4314.987 Schwarz criterion 4323.570Hannan-Quinn 4318.344 What would happen if we add more variables?

5. 4 MEASURES OF GOODNESS OF FIT Model 2: OLS, using observations 1-540 Dependent variable: EARNINGS coefficient std. error t-ratio p-value --------------------------------------------------------- const -14.6539 10.1243 -1.447 0.1484 S 2.42647 0.224538 10.81 9.24e-025 *** FEMALE -6.73090 1.09465 -6.149 1.53e-09 *** AGE 0.109493 0.235791 0.4644 0.6426 Mean dependent var 19.63622 S.D. dependent var 14.41566 Sum squared resid 86571.24 S.E. of regression 12.70880 R-squared 0.227113 Adjusted R-squared 0.222787 F(3, 536) 52.50126 P-value(F) 9.18e-30 Log-likelihood -2137.058 Akaike criterion 4282.117 Schwarz criterion 4299.283 Hannan-Quinn 4288.830 What would

6. 5 The measures inform about the level of explanation of the independend variable, with no effect os degrees of freedom. MEASURES OF GOODNESS OF FIT Adjusted determination and indetermination coefficients

7. 6 The average deviation of the theoretical values of the explanatory variables from their real values. MEASURES OF GOODNESS OF FIT Standard deviation: The variance od the regression:

8. 4 MEASURES OF GOODNESS OF FIT Model 1: OLS, using observations 1-540 Dependent variable: EARNINGS CoefficientStd. Errort-ratiop-value const -13.9335 3.21985 -4.32740.00002*** S 2.45532 0.231851 10.5901<0.00001*** Mean dependent var 19.63622S.D. dependent var 14.41566 Sum squared resid 92688.67S.E. of regression 13.12569 R-squared 0.172498Adjusted R-squared 0.170960 F(1, 538) 112.1496p-value(F) 6.22e-24 Log-likelihood -2155.494Akaik criterion 4314.987 Schwarz criterion 4323.570Hannan-Quinn 4318.344

9. 7 MEASURES OF GOODNESS OF FIT Model 1: OLS, using observations 1-540 Dependent variable: EARNINGS Coefficient Std. Error t-ratiop-value const -13.9335 3.21985 -4.32740.00002*** S 2.45532 0.231851 10.5901<0.00001*** Mean dependent var 19.63622S.D. dependent var 14.41566 Sum squared resid 92688.67S.E. of regression 13.12569 R-squared 0.172498Adjusted R-squared 0.170960 F(1, 538) 112.1496p-value(F) 6.22e-24 Log-likelihood -2155.494 Akaik criterion 4314.987 Schwarz criterion 4323.570Hannan-Quinn 4318.344 Coefficient of variation

10. For given level of significance alpha: if F<Falpha then we reject H0 8. MEASURES OF GOODNESS OF FIT Are the variabled Age and Female needed in the model? Model Y =  1 +  2 X + u Null hypothesis: H 0 :  2 = 0 and  3 = 0 Alternative hypothesis: H 1 :  2 ≠ 0 or  3 ≠ 0

11. Test on Model 1: Null hypothesis: the regression parameters are zero for the variables AGE, FEMALE Test statistic: F(2, 536) = 18.9378, p-value 1.12978e-008 Adding variables improved 3 of 3 model selection statistics. Model 2: OLS, using observations 1-540 Dependent variable: EARNINGS coefficient std. error t-ratio p-value --------------------------------------------------------- const -14.6539 10.1243 -1.447 0.1484 S 2.42647 0.224538 10.81 9.24e-025 *** AGE 0.109493 0.235791 0.4644 0.6426 FEMALE -6.73090 1.09465 -6.149 1.53e-09 *** Mean dependent var 19.63622 S.D. dependent var 14.41566 Sum squared resid 86571.24 S.E. of regression 12.70880 R-squared 0.227113 Adjusted R-squared 0.222787 F(3, 536) 52.50126 P-value(F) 9.18e-30 Log-likelihood -2137.058 Akaike criterion 4282.117 Schwarz criterion 4299.283 Hannan-Quinn 4288.830 9. MEASURES OF GOODNESS OF FIT

12. The null hypothesis should be rejected. 10 It was reasonable to include more variables to the model. MEASURES OF GOODNESS OF FIT The critical value: F(2,536)=3.01254 Test statistic: F(2, 536) = 18.9378, p-value 1.12978e-008

13. 21 PROBLEM OF HETEROSKEDASTICITY If the model is correct then the variance of error terms is constant over time.

14. 22 If the value of statistic is greater than the critical value then we reject the null hypothesis. The White test Model Y =  1 +  2 X + u Null hypothesis: H 0 : variance of error terms is constant Alternative hypothesis: H 1 :variance of error terms in not constant

15. 23 The critical value of Chi-square test (5), for alpha 0.05 is 11.0705 and is smaller than the value of the statistic. So we do not reject the null hypothesis. TEST FOR HETEROSKEDASTICITY Another version of the White test: White's test for heteroskedasticity (squares only) OLS, using observations 1-540 Dependent variable: uhat^2 coefficient std. error t-ratio p-value ------------------------------------------------------- const -3637.38 9957.40 -0.3653 0.7150 S -60.6640 112.690 -0.5383 0.5906 FEMALE -89.3715 59.9131 -1.492 0.1364 AGE 178.624 483.266 0.3696 0.7118 sq_S 3.51437 3.92209 0.8960 0.3706 sq_AGE -1.97357 5.88421 -0.3354 0.7375 Unadjusted R-squared = 0.028482 Test statistic: TR^2 = 15.380495, with p-value = P(Chi-square(5) > 15.380495) = 0.008854