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chapter nine

chapter nine. Functional Forms of Regression Models. 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 1970-1999 from Table 9-4. Table 9-4.

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chapter nine

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  1. chapter nine Functional Formsof Regression Models

  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 1970-1999 from Table 9-4

  3. Table 9-4 Population of United States (millions of people),1970-1999.

  4. Modeling Absolute Trends • Example: Appellate80-06.xls • Number of court of appeals sham litigation decisions by year 1980-2006 • Linear trend: Y = B1 + B2t + u • Non-linear trend: Y = B1 + B2t + B3t2 + u • Non-linear trend with break: Y = B1 + B2t + B3t2 +B4D + u • Non-linear trend with break and interaction (add B5Dt) • Test among models using F-test for difference in R2 • [(Ru2 - Rr2)/m]/[(1 - Ru2)/(n-k)]~Fm,n-k

  5. Compound Growth Rate • The Semilog Model • Beginning value Y0 • Value at t Yt • Compound growth rate r • Take natural log (base e) • Let B1 = lnY0 and B2 = ln(1+r) • B2 measures the yearly proportional change in Y

  6. Semilog Model Example • Growth rate of US population 1970-1999 • US population increased at a rate of 0.0098 per year • Or a percentage rate of 100x0.0098 = 0.98% • See Fig. 9-3 • Note lnYt is linear in t

  7. Figure 9-3 Semilog model.

  8. Instantaneous vs. Compound Growth Rate • b2 is estimate of ln(1 + r) where r is the compound growth rate • Antilog (b2) = (1 + r) or r = antilog(b2) – 1 • For US population: r = antilog(0.0098) – 1 • Or r = 1.00948 – 1 = 0.00948 • Compound growth rate of 0.948% • The instantaneous growth rate is usually reported, unless the compound rate is specifically required.

  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

  10. Log-linear Models and Elasticities • The slope coefficient B2 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 B2 is the price elasticity of demand (Fig. 9-1) • In log form, Y has a constant slope in X, B2 • So the elasticity is also constant • Sometimes called a constant elasticity model

  11. Figure 9-1 A constant elasticity model.

  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

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

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

  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 • R2 cannot be compared directly between linear and log-linear models • How to choose between models? • Try not to use R2 alone

  16. Example: Cobb-Douglas Production Function • See data in Table 9-2 • Estimate Ln(GDP) as a function of Ln(Employment) and Ln(Capital) • B2 and B3 are elasticities wrt output • B2 + B3 is the returns to scale parameter • = 1 constant returns • > 1 increasing returns • < 1 decreasing returns

  17. Table 9-2 Real GDP, employment, and real fixed capital, Mexico, 1955-1974.

  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 • B1, B2, B4 >0 • B3 < 0 • B32<3B2B4

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

  20. Figure 9-8 Cost-output relationship.

  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

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

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

  24. Table 9-11 Summary of functional forms.

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