EC220 - Introduction to econometrics (chapter 4) Christopher Dougherty EC220 - Introduction to econometrics (chapter 4) Slideshow: elasticities and logarithmic models   Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 4). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/130/ 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. http://creativecommons.org/licenses/by-sa/3.0/   http://learningresources.lse.ac.uk/

ELASTICITIES AND LOGARITHMIC MODELS Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: A elasticity Y O X This sequence defines elasticities and shows how one may fit nonlinear models with constant elasticities. First, the general definition of an elasticity. 1

ELASTICITIES AND LOGARITHMIC MODELS Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: A elasticity Y slope of the tangent at A slope of OA O X Re-arranging the expression for the elasticity, we can obtain a graphical interpretation. 2

ELASTICITIES AND LOGARITHMIC MODELS Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: A elasticity Y slope of the tangent at A slope of OA O X The elasticity at any point on the curve is the ratio of the slope of the tangent at that point to the slope of the line joining the point to the origin. 3

ELASTICITIES AND LOGARITHMIC MODELS Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: A elasticity Y slope of the tangent at A slope of OA O X elasticity < 1 In this case it is clear that the tangent at A is flatter than the line OA and so the elasticity must be less than 1. 4

ELASTICITIES AND LOGARITHMIC MODELS Elasticity of Y with respect to X is the proportional change in Y per proportional change in X: elasticity A Y slope of the tangent at A slope of OA O X elasticity > 1 In this case the tangent at A is steeper than OA and the elasticity is greater than 1. 5

ELASTICITIES AND LOGARITHMIC MODELS Y A elasticity slope of the tangent at A slope of OA O X x In general the elasticity will be different at different points on the function relating Y to X. 6

ELASTICITIES AND LOGARITHMIC MODELS Y A elasticity slope of the tangent at A slope of OA O X x In the example above, Y is a linear function of X. 7

ELASTICITIES AND LOGARITHMIC MODELS Y A elasticity slope of the tangent at A slope of OA O X x The tangent at any point is coincidental with the line itself, so in this case its slope is always b2. The elasticity depends on the slope of the line joining the point to the origin. 8

ELASTICITIES AND LOGARITHMIC MODELS Y B A elasticity slope of the tangent at A slope of OA O X x OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the mathematical expression: (b1 / X) + b2 is smaller at B than at A, assuming that b1 is positive.) 9

ELASTICITIES AND LOGARITHMIC MODELS However, a function of the type shown above has the same elasticity for all values of X. 10

ELASTICITIES AND LOGARITHMIC MODELS For the numerator of the elasticity expression, we need the derivative of Y with respect to X. 11

ELASTICITIES AND LOGARITHMIC MODELS For the denominator, we need Y/X. 12

ELASTICITIES AND LOGARITHMIC MODELS elasticity Hence we obtain the expression for the elasticity. This simplifies to b2 and is therefore constant. 13

ELASTICITIES AND LOGARITHMIC MODELS Y X By way of illustration, the function will be plotted for a range of values of b2. We will start with a very low value, 0.25. 14

ELASTICITIES AND LOGARITHMIC MODELS Y X We will increase b2 in steps of 0.25 and see how the shape of the function changes. 15

ELASTICITIES AND LOGARITHMIC MODELS Y X 16

ELASTICITIES AND LOGARITHMIC MODELS Y X When b2 is equal to 1, the curve becomes a straight line through the origin. 17

ELASTICITIES AND LOGARITHMIC MODELS Y X 18

ELASTICITIES AND LOGARITHMIC MODELS Y X 19

ELASTICITIES AND LOGARITHMIC MODELS Y X Note that the curvature can be quite gentle over wide ranges of X. 20

ELASTICITIES AND LOGARITHMIC MODELS Y X This means that even if the true model is of the constant elasticity form, a linear model may be a good approximation over a limited range. 21

ELASTICITIES AND LOGARITHMIC MODELS It is easy to fit a constant elasticity function using a sample of observations. You can linearize the model by taking the logarithms of both sides. 22

ELASTICITIES AND LOGARITHMIC MODELS where You thus obtain a linear relationship between Y' and X', as defined. All serious regression applications allow you to generate logarithmic variables from existing ones. 23

ELASTICITIES AND LOGARITHMIC MODELS where The coefficient of X' will be a direct estimate of the elasticity, b2. 24

ELASTICITIES AND LOGARITHMIC MODELS where The constant term will be an estimate of log b1. To obtain an estimate of b1, you calculate exp(b1'), where b1' is the estimate of b1'. (This assumes that you have used natural logarithms, that is, logarithms to base e, to transform the model.) 25

ELASTICITIES AND LOGARITHMIC MODELS FDHO EXP Here is a scatter diagram showing annual household expenditure on FDHO, food eaten at home, and EXP, total annual household expenditure, both measured in dollars, for 1995 for a sample of 869 households in the United States (Consumer Expenditure Survey data). 26

ELASTICITIES AND LOGARITHMIC MODELS . reg FDHO EXP Source | SS df MS Number of obs = 869 ---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000 Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055 ---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5 ------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634 Here is a regression of FDHO on EXP. It is usual to relate types of consumer expenditure to total expenditure, rather than income, when using household data. Household income data tend to be relatively erratic. 27

ELASTICITIES AND LOGARITHMIC MODELS . reg FDHO EXP Source | SS df MS Number of obs = 869 ---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000 Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055 ---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5 ------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634 The regression implies that, at the margin, 5 cents out of each dollar of expenditure is spent on food at home. Does this seem plausible? Probably, though possibly a little low. 28

ELASTICITIES AND LOGARITHMIC MODELS . reg FDHO EXP Source | SS df MS Number of obs = 869 ---------+------------------------------ F( 1, 867) = 381.47 Model | 915843574 1 915843574 Prob > F = 0.0000 Residual | 2.0815e+09 867 2400831.16 R-squared = 0.3055 ---------+------------------------------ Adj R-squared = 0.3047 Total | 2.9974e+09 868 3453184.55 Root MSE = 1549.5 ------------------------------------------------------------------------------ FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- EXP | .0528427 .0027055 19.531 0.000 .0475325 .0581529 _cons | 1916.143 96.54591 19.847 0.000 1726.652 2105.634 It also suggests that $1,916 would be spent on food at home if total expenditure were zero. Obviously this is impossible. It might be possible to interpret it somehow as baseline expenditure, but we would need to take into account family size and composition. 29

ELASTICITIES AND LOGARITHMIC MODELS FDHO EXP Here is the regression line plotted on the scatter diagram 30

ELASTICITIES AND LOGARITHMIC MODELS LGFDHO LGEXP We will now fit a constant elasticity function using the same data. The scatter diagram shows the logarithm of FDHO plotted against the logarithm of EXP. 31

ELASTICITIES AND LOGARITHMIC MODELS . g LGFDHO = ln(FDHO) . g LGEXP = ln(EXP) . reg LGFDHO LGEXP Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000 Residual | 184.579612 866 .213140429 R-squared = 0.3138 ---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754 Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables. 32

ELASTICITIES AND LOGARITHMIC MODELS . g LGFDHO = ln(FDHO) . g LGEXP = ln(EXP) . reg LGFDHO LGEXP Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000 Residual | 184.579612 866 .213140429 R-squared = 0.3138 ---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754 The estimate of the elasticity is 0.48. Does this seem plausible? 33

ELASTICITIES AND LOGARITHMIC MODELS . g LGFDHO = ln(FDHO) . g LGEXP = ln(EXP) . reg LGFDHO LGEXP Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000 Residual | 184.579612 866 .213140429 R-squared = 0.3138 ---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754 Yes, definitely. Food is a normal good, so its elasticity should be positive, but it is a basic necessity. Expenditure on it should grow less rapidly than expenditure generally, so its elasticity should be less than 1. 34

ELASTICITIES AND LOGARITHMIC MODELS . g LGFDHO = ln(FDHO) . g LGEXP = ln(EXP) . reg LGFDHO LGEXP Source | SS df MS Number of obs = 868 ---------+------------------------------ F( 1, 866) = 396.06 Model | 84.4161692 1 84.4161692 Prob > F = 0.0000 Residual | 184.579612 866 .213140429 R-squared = 0.3138 ---------+------------------------------ Adj R-squared = 0.3130 Total | 268.995781 867 .310260416 Root MSE = .46167 ------------------------------------------------------------------------------ LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] ---------+-------------------------------------------------------------------- LGEXP | .4800417 .0241212 19.901 0.000 .4326988 .5273846 _cons | 3.166271 .244297 12.961 0.000 2.686787 3.645754 The intercept has no substantive meaning. To obtain an estimate of b1, we calculate e3.16, which is 23.8. 35

ELASTICITIES AND LOGARITHMIC MODELS LGFDHO LGEXP Here is the scatter diagram with the regression line plotted. 36

ELASTICITIES AND LOGARITHMIC MODELS FDHO EXP Here is the regression line from the logarithmic regression plotted in the original scatter diagram, together with the linear regression line for comparison. 37

ELASTICITIES AND LOGARITHMIC MODELS FDHO EXP You can see that the logarithmic regression line gives a somewhat better fit, especially at low levels of expenditure. 38

ELASTICITIES AND LOGARITHMIC MODELS FDHO EXP However, the difference in the fit is not dramatic. The main reason for preferring the constant elasticity model is that it makes more sense theoretically. It also has a technical advantage that we will come to later on (when discussing heteroscedasticity). 39

Copyright Christopher Dougherty 2011. 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/. 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 http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lse. 11.07.25