1. SEMILOGARITHMIC MODELS 1 This sequence introduces the semilogarithmic model and shows how it may be applied to an earnings function. The dependent variable is linear but the explanatory variables, multiplied by their coefficients, are exponents of e.

2. 2 The differential of Y with respect to X simplifies to  2 Y. SEMILOGARITHMIC MODELS

3. 3 Hence the proportional change in Y per unit change in X is equal to  2. It is therefore independent of the value of X. SEMILOGARITHMIC MODELS

4. 4 Strictly speaking, this interpretation is valid only for small values of  2. When  2 is not small, the interpretation may be a little more complex. SEMILOGARITHMIC MODELS

5. 5 Suppose that X increases by an amount  X and that as a consequence Y increases by an amount  Y. SEMILOGARITHMIC MODELS

6. 6 We can rewrite the right side of the equation as shown. SEMILOGARITHMIC MODELS

7. 7 We can simplify the right side of the equation as shown. SEMILOGARITHMIC MODELS

8. 8 Now expand the exponential term using the standard expression for e to some power. SEMILOGARITHMIC MODELS

9. 9 Subtract Y from both sides. SEMILOGARITHMIC MODELS

10. 10 We now consider two cases: where  2 and  X are so small that (  2  X) 2 is negligible, and the alternative. SEMILOGARITHMIC MODELS negligible

11. 11 If (  2  X) 2 is negligible, we obtain the same interpretation of  2 as we did using the calculus, as one would expect. SEMILOGARITHMIC MODELS negligible

12. 12 SEMILOGARITHMIC MODELS If (  2  X) 2 is not negligible, the proportional change in Y given a  X change in X has an extra term. (We are assuming that  2 and  X are small enough that terms with higher powers of  X can be neglected.) not negligible

13. 13 Usually we talk about the effect of a one-unit change in X. If  X = 1, the proportional change in Y is as shown. The issue now becomes whether  2 is so small that the second and subsequent terms can be neglected. if  X is one unit SEMILOGARITHMIC MODELS not negligible

14. 14  1 is the value of Y when X is equal to zero (note that e 0 is equal to 1). SEMILOGARITHMIC MODELS

15. 15 To fit a function of this type, you take logarithms of both sides. The right side of the equation becomes a linear function of X (note that the logarithm of e, to base e, is 1). Hence we can fit the model with a linear regression, regressing log Y on X. SEMILOGARITHMIC MODELS

16. . reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------ Here is the regression output from a wage equation regression using Data Set 21. The estimate of  2 is 0.110. As an approximation, this implies that an extra year of schooling increases hourly earnings by a proportion 0.110. 16 SEMILOGARITHMIC MODELS

17. In everyday language it is usually more natural to talk about percentages rather than proportions, so we multiply the coefficient by 100. It implies that an extra year of schooling increases hourly earnings by 11.0%. 17 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------

18. If we take account of the fact that a year of schooling is not a marginal change, and work out the effect exactly, the proportional increase is 0.116 and the percentage increase 11.6%. 18 SEMILOGARITHMIC MODELS If  X is one unit, not negligible. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------

19. 19 In general, if a unit change in X is genuinely marginal, the estimate of  2 will be small and one can interpret it directly as an estimate of the proportional change in Y per unit change in X. SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------ If  X is one unit, not negligible

20. 20 However if a unit change in X is not small, the coefficient may be large and the second term might not be negligible. In the present case, a year of schooling is not marginal and the refinement does make a small difference. SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------ If  X is one unit, not negligible

21. 21 In general, when  2 is less than 0.1, there is little to be gained by working out the effect exactly. SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------ If  X is one unit, not negligible

22. The intercept in the regression is an estimate of log  1. From it, we obtain an estimate of  1 equal to e 1.29, which is 3.64. 22 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------

23. 23 Literally this implies that a person with no schooling would earn $3.64 per hour. However it is dangerous to extrapolate so far from the range for which we have data. SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 1, 538) = 140.05 Model | 38.5643833 1 38.5643833 Prob > F = 0.0000 Residual | 148.14326 538.275359219 R-squared = 0.2065 -------------+------------------------------ Adj R-squared = 0.2051 Total | 186.707643 539.34639637 Root MSE =.52475 ------------------------------------------------------------------------------ LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- S |.1096934.0092691 11.83 0.000.0914853.1279014 _cons | 1.292241.1287252 10.04 0.000 1.039376 1.545107 ------------------------------------------------------------------------------

24. Here is the scatter diagram with the semilogarithmic regression. 24 SEMILOGARITHMIC MODELS

25. Here is the semilogarithmic regression line plotted in a scatter diagram with the untransformed data, with the linear regression shown for comparison. 25 SEMILOGARITHMIC MODELS

26. 26 There is not much difference in the fit of the regression lines, but the semilogarithmic regression is more satisfactory in two respects. SEMILOGARITHMIC MODELS

27. 27 The linear specification predicts that earnings will increase by about $2 per hour with each additional year of schooling, which is implausible for high levels of education. The semi- logarithmic specification allows the increment to increase with level of education. SEMILOGARITHMIC MODELS

28. 28 Second, the linear specification predicts negative earnings for an individual with no schooling. The semilogarithmic specification predicts hourly earnings of $3.64, which at least is not obvious nonsense. SEMILOGARITHMIC MODELS

29. Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 4.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.03