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

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

Y X A 3 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. O Definition: The elasticity of Y with respect to X is the proportional change in Y per proportional change in X. elasticity ELASTICITIES AND LOGARITHMIC MODELS slope of the tangent at A slope of OA

4 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. ELASTICITIES AND LOGARITHMIC MODELS Y X A O Definition: The elasticity of Y with respect to X is the proportional change in Y per proportional change in X. elasticity slope of the tangent at A slope of OA elasticity < 1

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

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

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

8 The tangent at any point is coincidental with the line itself, so in this case its slope is always  2. The elasticity depends on the slope of the line joining the point to the origin. ELASTICITIES AND LOGARITHMIC MODELS x A O Y X slope of the tangent at A slope of OA elasticity

9 OB is flatter than OA, so the elasticity is greater at B than at A. (This ties in with the mathematical expression: (  1  / X) +  2 is smaller at B than at A, assuming that  1 is positive.) x A O B Y X ELASTICITIES AND LOGARITHMIC MODELS slope of the tangent at A slope of OA elasticity

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

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

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

13 Hence we obtain the expression for the elasticity. This simplifies to  2 and is therefore constant. elasticity ELASTICITIES AND LOGARITHMIC MODELS

14 By way of illustration, the function will be plotted for a range of values of  2. We will start with a very low value, Y X ELASTICITIES AND LOGARITHMIC MODELS

15 Y X ELASTICITIES AND LOGARITHMIC MODELS We will increase  2 in steps of 0.25 and see how the shape of the function changes.

16 Y X ELASTICITIES AND LOGARITHMIC MODELS  2 = 0.75.

17 Y X ELASTICITIES AND LOGARITHMIC MODELS When  2 is equal to 1, the curve becomes a straight line through the origin.

18 Y X ELASTICITIES AND LOGARITHMIC MODELS  2 = 1.25.

19 Y X ELASTICITIES AND LOGARITHMIC MODELS  2 = 1.50.

20 Y X ELASTICITIES AND LOGARITHMIC MODELS  2 = Note that the curvature can be quite gentle over wide ranges of X.

21 Y X ELASTICITIES AND LOGARITHMIC MODELS 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.

22 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. ELASTICITIES AND LOGARITHMIC MODELS

23 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. ELASTICITIES AND LOGARITHMIC MODELS where

24 The coefficient of X' will be a direct estimate of the elasticity,  2. ELASTICITIES AND LOGARITHMIC MODELS where

25 The constant term will be an estimate of log  1. To obtain an estimate of  1, you calculate exp(b 1 '), where b 1 ' is the estimate of  1 '. (This assumes that you have used natural logarithms, that is, logarithms to base e, to transform the model.) ELASTICITIES AND LOGARITHMIC MODELS where

26 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). FDHO EXP ELASTICITIES AND LOGARITHMIC MODELS

. reg FDHO EXP Source | SS df MS Number of obs = F( 1, 867) = Model | Prob > F = Residual | e R-squared = Adj R-squared = Total | e Root MSE = FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] EXP | _cons | 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. ELASTICITIES AND LOGARITHMIC MODELS

28 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. ELASTICITIES AND LOGARITHMIC MODELS. reg FDHO EXP Source | SS df MS Number of obs = F( 1, 867) = Model | Prob > F = Residual | e R-squared = Adj R-squared = Total | e Root MSE = FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] EXP | _cons |

29 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. ELASTICITIES AND LOGARITHMIC MODELS. reg FDHO EXP Source | SS df MS Number of obs = F( 1, 867) = Model | Prob > F = Residual | e R-squared = Adj R-squared = Total | e Root MSE = FDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] EXP | _cons |

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

31 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. LGFDHO LGEXP ELASTICITIES AND LOGARITHMIC MODELS

. g LGFDHO = ln(FDHO). g LGEXP = ln(EXP). reg LGFDHO LGEXP Source | SS df MS Number of obs = F( 1, 866) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | _cons | Here is the result of regressing LGFDHO on LGEXP. The first two commands generate the logarithmic variables. ELASTICITIES AND LOGARITHMIC MODELS

33 The estimate of the elasticity is Does this seem plausible? ELASTICITIES AND LOGARITHMIC MODELS. g LGFDHO = ln(FDHO). g LGEXP = ln(EXP). reg LGFDHO LGEXP Source | SS df MS Number of obs = F( 1, 866) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | _cons |

34 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. ELASTICITIES AND LOGARITHMIC MODELS. g LGFDHO = ln(FDHO). g LGEXP = ln(EXP). reg LGFDHO LGEXP Source | SS df MS Number of obs = F( 1, 866) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | _cons |

35 The intercept has no substantive meaning. To obtain an estimate of  1, we calculate e 3.16, which is ELASTICITIES AND LOGARITHMIC MODELS. g LGFDHO = ln(FDHO). g LGEXP = ln(EXP). reg LGFDHO LGEXP Source | SS df MS Number of obs = F( 1, 866) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = LGFDHO | Coef. Std. Err. t P>|t| [95% Conf. Interval] LGEXP | _cons |

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

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

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

39 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). EXP ELASTICITIES AND LOGARITHMIC MODELS FDHO

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 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 EC2020 Elements of Econometrics