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

4 MEASURES OF GOODNESS OF FIT Model 1: OLS, using observations Dependent variable: EARNINGS CoefficientStd. Errort-ratiop-value const *** S < *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 538) p-value(F) 6.22e-24 Log-likelihood Akaik criterion Schwarz criterion Hannan-Quinn What would happen if we add more variables?

4 MEASURES OF GOODNESS OF FIT Model 2: OLS, using observations Dependent variable: EARNINGS coefficient std. error t-ratio p-value const S e-025 *** FEMALE e-09 *** AGE Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(3, 536) P-value(F) 9.18e-30 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn What would

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

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:

4 MEASURES OF GOODNESS OF FIT Model 1: OLS, using observations Dependent variable: EARNINGS CoefficientStd. Errort-ratiop-value const *** S < *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 538) p-value(F) 6.22e-24 Log-likelihood Akaik criterion Schwarz criterion Hannan-Quinn

7 MEASURES OF GOODNESS OF FIT Model 1: OLS, using observations Dependent variable: EARNINGS Coefficient Std. Error t-ratiop-value const *** S < *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 538) p-value(F) 6.22e-24 Log-likelihood Akaik criterion Schwarz criterion Hannan-Quinn Coefficient of variation

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

Test on Model 1: Null hypothesis: the regression parameters are zero for the variables AGE, FEMALE Test statistic: F(2, 536) = , p-value e-008 Adding variables improved 3 of 3 model selection statistics. Model 2: OLS, using observations Dependent variable: EARNINGS coefficient std. error t-ratio p-value const S e-025 *** AGE FEMALE e-09 *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(3, 536) P-value(F) 9.18e-30 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn MEASURES OF GOODNESS OF FIT

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)= Test statistic: F(2, 536) = , p-value e-008

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

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

23 The critical value of Chi-square test (5), for alpha 0.05 is 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 Dependent variable: uhat^2 coefficient std. error t-ratio p-value const S FEMALE AGE sq_S sq_AGE Unadjusted R-squared = Test statistic: TR^2 = , with p-value = P(Chi-square(5) > ) =