1. Econ 140 Lecture 131 Multiple Regression Models Lecture 13

2. Econ 140 Lecture 132 Today’s plan Functional form (what model do we estimate?) Testing the explanatory power of the model Adjustment to R 2

3. Econ 140 Lecture 133 Reading coefficients With a bi-variate model we could easily determine how a change in X affects Y With a multivariate model you must hold X 1 constant to determine the effect of a change in X 2 on Y –For this reason we call the slope coefficients in a multivariate regression the partial regression coefficients

4. Econ 140 Lecture 134 Functional form To return to arguments about functional form. How do we know what model to estimate? Our example on earnings and years of education has some economic theory in its foundation - but basically an ‘ad- hoc’ specification. We know we want to test the relationship between earnings and years of schooling. An example based on economic theory: the Cobb-Douglas production function Y = AL  K  May want to test for constant returns to scale  +  = 1

5. Econ 140 Lecture 135 Functional form (2) We can transform the Cobb-Douglas equation into a form we can estimate by taking logs: ln Y = ln A +  ln L +  ln K –This is called log linear form since all the variables are in logs –The model is now linear in parameters so we can use least squares to estimate it –The log linear form gives us estimated coefficients that are elasticities: the estimates of  and  give us the elasticities of labor and capital with respect to output

6. Econ 140 Lecture 136 Example with longitudinal data L13-1.xls on the webpage. It contains information on companies in the UK private sector. Data from DATASTREAM; for US: COMPUSTAT Note that this is a longitudinal (panel) data set - we are analyzing the same agents (the companies) over time I have calculated the true output elasticity with respect to labor for a 100% change in labor and the true output elasticity with respect to labor for a 10% change in labor –Note that the larger the increase in the independent variable, the further the approximation is from the coefficient

7. Econ 140 Lecture 137 Example with longitudinal data (2) If we want to calculate the true change, we need to calculate: If we want to estimate the Cobb-Douglas production function, we use the partial slope coefficients We can calculate the partial slope coefficients :

8. Econ 140 Lecture 138 Example with longitudinal data (3) Adding our estimates together we find: Later on we’ll test the constraint that  +  = 1

9. Econ 140 Lecture 139 Phillips Curve The Phillips Curve is an example of ad-hoc variable inclusion (model based on economic reasoning) UnUn W The equation representing this relationship between unemployment and wage inflation is:

10. Econ 140 Lecture 1310 Phillips Curve (2) Estimation of Phillips Curve: transform the independent variable into an inverse (X into 1/X). Note, model is non- linear in variables, but linear in parameters. With ad-hoc specification we don’t know what other variables are relevant –we need to make informed guesses determined by what we know of economic theory

11. Econ 140 Lecture 1311 The story so far Functional form Omitted variable bias Examples of types of data & types of modeling approach (note, they are not exclusive): –Cross section: Return on education –Panel/longitudinal: Cobb-Douglas –Time-series: Phillips Curve

12. Econ 140 Lecture 1312 Variation in multivariate models Let our model be For test statistics we still want to calculate: –How to calculate these values.

13. Econ 140 Lecture 1313 Variation in multivariate models (2) It still holds that the variance of the regression line is It also still holds that:

14. Econ 140 Lecture 1314 Test statistics in multivariate models We will start with the sum of squares identity, where: Total = Explained + Residual or But, the composition of the ESS will be different - our sum of squares identity will look like this: As you add more independent variables to the model, more terms get added to the ESS

15. Econ 140 Lecture 1315 Test statistics in multivariate models (2) Now let’s look back to an example from an earlier lecture (L12-1.xls) –we looked at the returns to earnings of education (b 1 ) and age (b 2 ) –calculate the test statistics and consider model problems Our R 2 is:

16. Econ 140 Lecture 1316 Test statistics in multivariate models (3) You will also need these these values: Remember the matrix of products and cross-products looks like this:

17. Econ 140 Lecture 1317 Test statistics in multivariate models (4) We can start our calculations with: The regression line we calculated earlier is: Taking the square root, we find the root mean square error:

18. Econ 140 Lecture 1318 Test statistics in multivariate models (5) Taking the square root gives us We can then calculate:

19. Econ 140 Lecture 1319 Test statistics in multivariate models (6) Taking the square root gives We can then calculate:

20. Econ 140 Lecture 1320 Hypothesis test on education The t-ratio is calculated: Form a null hypothesis For a significance level of 5% we have a table t value of t  /2,33 = 2.035 Since |t| < t  /2, we cannot reject the null hypothesis Recall that the purpose of the model was to examine whether or not education has an effect on earnings (allowing for age effects). Should we conclude that additional years of education have no effect on earnings?

21. Econ 140 Lecture 1321 Hypothesis test on age The t-ratio is calculated: We construct the hypothesis test: For a significance level of 5% we have a table t value of t  /2,33 = 2.035 Since |t| > t  /2, we reject the null hypothesis

22. Econ 140 Lecture 1322 Looking at R 2 Coefficient of determination: R 2 : This is a rather low R 2 –This means that the regression equation doesn’t explain the variation well –The regression equation only explains about 1/5 of the variation in Y

23. Econ 140 Lecture 1323 Looking at R 2 (2) What should we do about the form of our estimated equation when years of education are shown to be statistically insignificant at our chosen significance level? We chose a 5% significance level for our test, but we might have been able to reject the null at a different significance level Remember: with hypothesis test we want to reduce the number of type I errors where we falsely reject a null

24. Econ 140 Lecture 1324 Testing explanatory power What if we examined the regression equation as a whole? We want to keep years of education for our model. Also want to retain the conditioning on age - it appears to be important! Are there other tests we can undertake? Could examine the null hypothesis: H 0 : b 1 = b 2 = 0 –This says that neither of the independent variables has any explanatory power –To test this, we will use an F test

25. Econ 140 Lecture 1325 Testing explanatory power (2) The F statistic that we’re looking at can be found on the LINEST output The F test comes from the ANOVA table for the multivariate case, which looks like this:

26. Econ 140 Lecture 1326 Testing explanatory power (3) The F statistic will look like: Using the F table, you choose a significance level and use the degrees of freedom in the numerator and denominator, or F 0.05, 2, 33 –The 1st row in the table is df in the numerator –The 1st column is the df in the denominator –The 2nd column is the significance level ^ ^^

27. Econ 140 Lecture 1327 Testing explanatory power (4) If our calculated F statistic is greater than (to the right of) our F table value, we reject the null If our calculated F statistic is less than (to the left of) our F table value, we cannot reject the null F table value H 0 : Cannot reject the null H 1 : Reject the null F

28. Econ 140 Lecture 1328 Testing explanatory power (5) Looking at the F table, we find that there is no value for exactly 33 df –We have to approximate using 30 df instead –Our approximated F value is F 0.05, 2, 33  3.29 We reject the null because F > F 0.05, 2, 33 Had we picked a 1% significance level, or F table value would be F 0.01, 2, 33  5.27 –and we could not have rejected the null because F < F 0.01, 2, 33

29. Econ 140 Lecture 1329 Testing explanatory power (6) In summary, we’re more likely to reject the null at a greater significance level In this case, we rejected at a 5% significance level and could not reject the null at a 1% level Graphically: F* value F 1%5% 3.29 3.815.27

30. Econ 140 Lecture 1330 Testing explanatory power (7) The t-test suggests that we should remove years of education from our regression An F-test on the joint hypothesis rejects the null, but the test is weak. At a lower significance level (1 percent), we could not have rejected the null. In this instance, we want to keep the years of education variable in the equation because of what we know of economic theory What to do? Conclude that the economic theory is weak? Give up school? Or, obtain more data and try again! Look at results from other studies.

31. Econ 140 Lecture 1331 Adjustment to R 2 The more variables added to a regression, the higher R 2 will be –R 2 is important, but it isn’t the sole criteria for judging a model’s explanatory power Adjusted R 2 adjusts for the loss in degrees of freedom associated with adding independent variables to the regression

32. Econ 140 Lecture 1332 Adjustment to R 2 (2) Adjusted R 2 is written as Adj R 2 = 1 - (1 - R 2 )((n - 1)/(n - k)) n : sample size k : number of parameters in the regression

33. Econ 140 Lecture 1333 What’s next Restricted least squares and the Cobb Douglas Production function Including qualitative indicators into the regression equation (e.g. race, gender, marital status).