Lecture 12. Functional form

Transcription

1 Lecture 12. Functional form Multiple linear regression model β1 + β2 2 + L+ β K K + u Interpretation of regression coefficient k Change in if k is changed by 1 unit and the other variables are held constant. This change β : Does not depend on the value of k Does not depend on the value of other variables Examples Relation between demand for electricity and its price 2. Price effect may depend on temperature 3. Relation between consumption and income 2. Change in consumption due to extra income may decrease with income.

2 Linear regression model seems not able to allow for this. Linear in coefficients or linear in variables Compare the following relations between and (omit error term u ) 1. β β2 β β + β β Both relations are nonlinear (see graphs) Relation 1 is linear in the regression coefficients, i.e. it can be expressed as a linear relation between and independent variables Define 2 2, 3,K. 2, 3 For relation 2 this is not possible (try!).

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5 If a nonlinear relation can be expressed as a linear relation by redefining variables we can estimate that relation using OLS.

6 Review of (natural) logarithms and the exponential function (natural) logarithm log exponential function e See graphs. Note Both functions increasing Logarithm only defined if is positive Some relations e log log e log a log log a log a + log a a+ e e a e

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8 Derivatives d log 1 d de d e

9 Examples of nonlinear relations that can be expressed as a linear relation (independent variables, Z ; define the 2, 3, K that make the relation linear; in some cases the dependent variable changes as well; u is omitted) Quadratic relation β β2 β3 2 Relation with interaction term β 1 + β2 + β3z Reciprocal relation β 1 + β 2 1 Linear log relation β + log 1 β 2

10 Exponential relation e β 1 + β 2 Power relation (here constant must be redefined as well) γ 1 β 2 Z β 3 How do we analyze nonlinear relations? Graphs (only for 1 independent variable) Derivatives: Change in associated with small change in Remember derivative d d in point (, ) on curve is the slope of the straight line that touches the nonlinear relation in that point, i.e. the slope of the linear relation that approximates the nonlinear relation in that point.

11 Example: Quadratic relation β β2 β3 2 Then d d β 2 + 2β 3 Hence effect of small change of depends on (and may even change sign) Other use: Find value of that maximizes, e.g is income and is age if β 2 20, β3.25 the income maximal at age 40.

12 In economics often interested in elasticities: relative change in associated with a small relative change in. Note relative changes do not depend on unit of measurement. Small relative change in variable : d or small relative percentage change d 100 Elasticity (note 100 cancels if defined for percentage change) d d d d In other words: Multiply derivative by

13 In quadratic relation d d d d ( β + 2β3 2 ) Note depends on both and Remember d log 1 d Hence for small change in d log 1 d With this we can compute an elasticity as d d d log d log

14 Convenient if the (in)dependent variables are log s Log-log relation log β 1 + β 2 log In this relation 2 β is the elasticity of is the elasticity of with respect to.

15 Also consider Semi-log relation Here log β + β2 1 β 2 d log d d d This is relative change in associated with 1 unit change in. Note: percentage change is 100β. 2

16 Relation with interaction β 1 + β2 + β3z If relation has more than 1 independent variable, we look at the effect van holding Z constant. Note if Z constant β 2 + β 3 Z (we use instead of d to indicate that some variables are held constant) Note effect of depends on Z (compare electricity demand above).

17 Special exponential relation t e βt 0 with t the value of in period t and 0 the value of in the initial period. Relative change in t βt β ( t 1) t 1 0e 0e β e β ( t 1) t 1 0e Hence the coefficient in the relation 1 log t log0 + βt can be used to obtain the relative change/growth rate in

18 Other relations: Lags in, With time series data we can define new variables by lags, 2,K t 1 t Note in period t the value of t 1 is the value of in period t 1 Linear relation with lagged variables This expresses 1 β 1 + β2t 1 + β3 t + β4 t 1 Delayed response Adjustment over time Habits/reluctance to change

19 Some caveats in estimating nonlinear relations Graphs with one are misleading 2 Do not compare R if dependent variables are different For rest estimation and testing as before Consider β3 3 β + β + β + u β + β + + u Then e.g. test of nonlinearity is test of H 0 : β3 0