Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: semilogarithmic models Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

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

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

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

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

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

9 Subtract Y from both sides. SEMILOGARITHMIC MODELS

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

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

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. SEMILOGARITHMIC MODELS negligible not negligible if  X is one unit

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

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

. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | Here is the regression output from a wage equation regression using Data Set 21. The estimate of  2 is As an approximation, this implies that an extra year of schooling increases hourly earnings by a proportion SEMILOGARITHMIC MODELS

. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | 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

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 and the percentage increase 11.6%. 18 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | not negligible if  X is one unit

19 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | 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. not negligible if  X is one unit

20 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | 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. not negligible if  X is one unit

21 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | In general, when  2 is less than 0.1, there is little to be gained by working out the effect exactly. not negligible if  X is one unit

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 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons |

23 SEMILOGARITHMIC MODELS. reg LGEARN S Source | SS df MS Number of obs = F( 1, 538) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGEARN | Coef. Std. Err. t P>|t| [95% Conf. Interval] S | _cons | 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.

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

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

27 SEMILOGARITHMIC MODELS 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.

28 SEMILOGARITHMIC MODELS 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.

Copyright Christopher Dougherty 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 Individuals studying econometrics on their own and 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics