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Regression Models

Regression Models. Professor William Greene Stern School of Business IOMS Department Department of Economics. Regression and Forecasting Models . Part 5 – Elasticities and Functional Form. Linear Regression Models. Model building Linear models – cost functions

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Regression Models

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  1. Regression Models Professor William Greene Stern School of Business IOMS Department Department of Economics

  2. Regression and Forecasting Models Part 5 – Elasticities and Functional Form

  3. Linear Regression Models • Model building • Linear models – cost functions • Semilog models – growth models • Logs and elasticities • Analyzing residuals • Violations of assumptions • Unusual data points • Hints for improving the model

  4. Using and Interpreting the Model • Interpreting the linear model • Semilog and growth models • Log-log model and elasticities

  5. Statistical Cost Analysis The units of the LHS and RHS must be the same. $M cost = b0+ b1MKWH Y = $ cost b0= $ cost = 2.444 $M b1= $M /MKWH = 0.005291 $M/MKWH So,….. b0=fixed cost= total cost if MKWH = 0 b1=marginal cost= dCost/dMKWH b1* MKWH = variable cost Generation cost ($M) and output (Millions of KWH) for 123 American electric utilities. (1970).

  6. In the millennial edition of its World Health Report, in 2000, the World Health Organization published a study that compared the successes of the health care systems of 191 countries. The results notoriously ranked the United States a dismal 37th, between Costa Rica and Slovenia. The study was widely misrepresented , universally misunderstood and was, in fact, unhelpful in understanding the different outcomes across countries. Nonetheless, the result remains controversial a decade later, as policy makers argue about why the world’s most expensive health care system isn’t the world’s best.

  7. Application: WHO • WHO data on 191 countries in 1995-1999. • Analysis of Disability Adjusted Life Expectancy = DALE • EDUC = average years of education • DALE = β0 + β1EDUC + ε

  8. The (Famous) WHO Data

  9. The slope is the interesting quantity.Each additional year of education is associated with an increase of 3.611 in disability adjusted life expectancy.

  10. Increase of per capita GDP of 1,000 PPP units is associated with an increase of ‘happy’ of 1000(.0018566) = 1.86. Happy ranges from 0 to 100.

  11. Semilog Models and Growth Rates LogSalary = 9.84 + 0.05 Years + e

  12. Semilog Model for Fuel Bills Each increase of 1 room raises the fuel bill by about 21%. [Actually closer to exp(.215)-1 = 24%.]

  13. Using Semilog Models for Trends Frequent Flyer Flights for 72 Months. (Text, Ex. 11.1, p. 508)

  14. Regression Approach logFlights = β0+ β1Months + ε b0= 2.770, b1= 0.03710, s = 0.06102

  15. Elasticity and Loglinear Models • logy = β0+ β1logx + ε • The “responsiveness” of one variable to changes in another • E.g., in economics demand elasticity = (%ΔQ) / (%ΔP) • Math: Ratio of percentage changes • %ΔQ / %ΔP = {100%[(ΔQ )/Q] / {100%[(ΔP)/P]} • Units of measurement and the 100% fall out of this eqn. • Elasticity = (ΔQ/ΔP)*(P/Q) • Elasticities are units free

  16. Monet Regression

  17. Using the Residuals • How do you know the model is “good?” • Various diagnostics to be developed over the semester. • But, the first place to look is at the residuals.

  18. Residuals Can Signal a Flawed Model • Standard application: Cost function for output of a production process. • Compare linear equation to a quadratic model (in logs) • (123 American Electric Utilities)

  19. Electricity Cost Function

  20. Candidate Model for Cost Log c = a + b log q + e

  21. A Better Model? Log Cost = α + β1 logOutput + β2 [logOutput]2 + ε

  22. Candidate Models for Cost The quadratic equation is the appropriate model. Logc = b0+ b1 logq + b2 log2q + e

  23. Missing Variable Included Residuals from the quadratic cost model Residuals from the linear cost model

  24. Unusual Data Points Outliershave (what appear to be) very large disturbances, ε The 500 most successful movies

  25. Outliers Remember the empirical rule, 99.5% of observations will lie within mean ± 3 standard deviations? We show (b0+b1x) ± 3se below.) Titanic is 8.1 standard deviations from the regression! Only 0.86% of the 466 observations lie outside the bounds. (We will refine this later.) These points might deserve a closer look.

  26. Prices paid at auction for Monet paintings vs. surface area (in logs) logPrice = b0+ b1logArea + e Not an outlier: Monet chose to paint a small painting. Possibly an outlier: Why was the price so low?

  27. What to Do About Outliers (1) Examine the data (2) Are they due to mismeasurement error or obvious “coding errors?” Delete the observations. (3) Are they just unusual observations? Do nothing. (4) Generally, resist the temptation to remove outliers.Especially if the sample is large. (500 movies islarge.) (5) Question why you think it is an outlier. Is it really?

  28. Regression Options

  29. Minitab’s Opinions Minitab uses ± 2S to flag “large” residuals.

  30. On Removing Outliers Be careful about singling out particular observations this way. The resulting model might be a product of your opinions, not the real relationship in the data. Removing outliers might create new outliers that were not outliers before. Statistical inferences from the model will be incorrect.

  31. Correlation?

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