Copyright © 2006 The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Functional Forms of Regression Models chapter nine

9-2 Time Trends and Growth Rates Linear Trend Models Time series data Test for trend over time Test for breaks in a trend Absolute changes over time Results for U.S. population from Table 9-4

9-3 Table 9-4 Population of United States (millions of people),

9-4 Modeling Absolute Trends Example: Appellate80-06.xls Number of court of appeals sham litigation decisions by year Linear trend: Y = B 1 + B 2 t + u Non-linear trend: Y = B 1 + B 2 t + B 3 t 2 + u Non-linear trend with break: Y = B 1 + B 2 t + B 3 t 2 +B 4 D + u Non-linear trend with break and interaction (add B 5 Dt) Test among models using F-test for difference in R 2 [(R u 2 - R r 2 )/m]/[(1 - R u 2 )/(n-k)]~F m,n-k

9-5 Compound Growth Rate The Semilog Model Beginning value Y 0 Value at t Y t Compound growth rate r Take natural log (base e) Let B 1 = lnY 0 and B 2 = ln(1+r) B 2 measures the yearly proportional change in Y

9-6 Semilog Model Example Growth rate of US population US population increased at a rate of per year Or a percentage rate of 100x = 0.98% See Fig. 9-3 Note lnY t is linear in t

9-7 Figure 9-3 Semilog model.

9-8 Instantaneous vs. Compound Growth Rate b 2 is estimate of ln(1 + r) where r is the compound growth rate Antilog (b 2 ) = (1 + r) or r = antilog(b 2 ) – 1 For US population: r = antilog(0.0098) – 1 Or r = – 1 = Compound growth rate of 0.948% The instantaneous growth rate is usually reported, unless the compound rate is specifically required.

9-9 Log-linear Models and Elasticities Consider this function for Lotto expenditure that is nonlinear in X Convert to a linear form by taking natural logarithms (base e) The result is a double-log or log-linear model Make a nonlinear model into a linear one by a suitable transformation Logarithmic transformation

9-10 Log-linear Models and Elasticities The slope coefficient B 2 measures the Elasticity of Y with respect to X % change in Y for a % change in X If Y is quantity demanded and X is price, then B 2 is the price elasticity of demand (Fig. 9-1) In log form, Y has a constant slope in X, B 2 So the elasticity is also constant Sometimes called a constant elasticity model

9-11 Figure 9-1 A constant elasticity model.

9-12 Lotto Example Using data in Table 9-1, run OLS to estimate the log- linear model If income increases by one %, expenditure on lotto increases by 0.74 % on average Lotto exp. is inelastic wrt income as 0.74 < 1 See Fig. 9-2

9-13 Table 9-1 Weekly lotto expenditure (Y) in relation to weekly personal disposable income (X) ($).

9-14 Figure 9-2 Log-linear model of Lotto expenditure.

9-15 Example: Electricity Demand See ElectricExcel2.xls. Calculate natural logarithms Estimate the log-linear model by OLS Note: No change in hypothesis testing for log form Only POP and PKWH coefficients are significant R 2 cannot be compared directly between linear and log- linear models How to choose between models? Try not to use R 2 alone

9-16 Example: Cobb-Douglas Production Function See data in Table 9-2 Estimate Ln(GDP) as a function of Ln(Employment) and Ln(Capital) B 2 and B 3 are elasticities wrt output B 2 + B 3 is the returns to scale parameter = 1 constant returns > 1 increasing returns < 1 decreasing returns

9-17 Table 9-2 Real GDP, employment, and real fixed capital, Mexico,

9-18 Polynomial Regression Models Estimating cost functions, when total and average cost must have specific non- linear shapes Table 9-8 and Fig. 9-8 Cubic function or third- degree polynomial B 1, B 2, B 4 >0 B 3 < 0 B 3 2 <3B 2 B 4

9-19 Table 9-8 Hypothetical cost-output data.

9-20 Figure 9-8 Cost-output relationship.

9-21 Example Does smoking have an increasing or decreasing effect on lung cancer? Non-linear relationship between cigarette smoking and lung cancer deaths Table 9-9, data Figure 9-9, regression results Quadratic function or second degree polynomial

9-22 Table 9-9 Cigarette smoking and deaths from various types of cancer.

9-23 Figure 9-9 MINITAB output of regression (9.34).

9-24 Table 9-11 Summary of functional forms.