Introduction to Econometrics, 5th edition Type author name/s here Dougherty Introduction to Econometrics, 5th edition Chapter heading Chapter 4: Nonlinear Models and Transformations of Variables © Christopher Dougherty, 2016. All rights reserved.

SEMILOGARITHMIC MODELS 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. 1

SEMILOGARITHMIC MODELS The differential of Y with respect to X simplifies to b2Y. 2

SEMILOGARITHMIC MODELS Hence the proportional change in Y per unit change in X is equal to b2. It is therefore independent of the value of X. 3

SEMILOGARITHMIC MODELS Strictly speaking, this interpretation is valid only for small values of b2. When b2 is not small, the interpretation may be a little more complex. 4

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

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

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

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

SEMILOGARITHMIC MODELS Subtract Y from both sides. 9

SEMILOGARITHMIC MODELS negligible We now consider two cases: where b2 and DX are so small that (b2 DX)2 is negligible, and the alternative. 10

SEMILOGARITHMIC MODELS negligible If (b2 DX)2 is negligible, we obtain the same interpretation of b2 as we did using the calculus, as one would expect. 11

SEMILOGARITHMIC MODELS not negligible If (b2 DX)2 is not negligible, the proportional change in Y given a DX change in X has an extra term. (We are assuming that b2 and DX are small enough that terms with higher powers of DX can be neglected.) 12

SEMILOGARITHMIC MODELS not negligible if DX is one unit Usually we talk about the effect of a one-unit change in X. If DX = 1, the proportional change in Y is as shown. The issue now becomes whether b2 is so small that the second and subsequent terms can be neglected. 13

SEMILOGARITHMIC MODELS b1 is the value of Y when X is equal to zero (note that e0 is equal to 1). 14

SEMILOGARITHMIC MODELS 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. 15

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 Here is the regression output from a wage equation regression using Data Set 21. The estimate of b2 is 0.066. As an approximation, this implies that an extra year of schooling increases hourly earnings by a proportion 0.066. 16

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 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 6.6%. 17

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 not negligible If DX is one unit, 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 0.068 and the percentage increase 6.8%. 18

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 not negligible If DX is one unit, In general, if a unit change in X is genuinely marginal, the estimate of b2 will be small and one can interpret it directly as an estimate of the proportional change in Y per unit change in X. 19

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 not negligible If DX is one unit, 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, but evem so the refinement makes only a small difference. 20

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 not negligible If DX is one unit, In general, when b2 is less than 0.1, there is little to be gained by working out the effect exactly. 21

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 . 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 The intercept in the regression is an estimate of log b1. From it, we obtain an estimate of b1 equal to e1.836, which is 6.27. 22

SEMILOGARITHMIC MODELS . reg LGEARN S ---------------------------------------------------------------------------- Source | SS df MS Number of obs = 500 -----------+------------------------------ F( 1, 498) = 60.71 Model | 16.5822819 1 16.5822819 Prob > F = 0.0000 Residual | 136.016938 498 .273126381 R-squared = 0.1087 -----------+------------------------------ Adj R-squared = 0.1069 Total | 152.59922 499 .30581006 Root MSE = .52261 LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] -----------+---------------------------------------------------------------- S | .0664621 .0085297 7.79 0.000 .0497034 .0832207 _cons | 1.83624 .1289384 14.24 0.000 1.58291 2.089571 . 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 Literally this implies that a person with no schooling would earn $6.27 per hour. However it is dangerous to extrapolate so far from the range for which we have data. 23

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

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

SEMILOGARITHMIC MODELS The linear specification predicts that hourly earnings will increase by a fixed amount, $1.27, with each additional year of schooling. This is implausible for high levels of education. The semi-logarithmic specification allows the increment to increase with level of education. 27

SEMILOGARITHMIC MODELS Second, the linear specification predicts very low earnings for an individual with no schooling. The semilogarithmic specification predicts hourly earnings of $6.27, which at least is not obvious nonsense. 28

Copyright Christopher Dougherty 2016. 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, fifth edition 2016, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre www.oxfordtextbooks.co.uk/orc/dougherty5e/. 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 or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lse. 2016.05.02