Part 5: Functional Form 5-1/36 Regression Models Professor William Greene Stern School of Business IOMS Department Department of Economics

Part 5: Functional Form 5-2/36 Regression and Forecasting Models Part 5 – Elasticities and Functional Form

Part 5: Functional Form 5-3/36 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

Part 5: Functional Form 5-4/36 Using and Interpreting the Model  Interpreting the linear model  Semilog and growth models  Log-log model and elasticities

Part 5: Functional Form 5-5/36 Statistical Cost Analysis Generation cost ($M) and output (Millions of KWH) for 123 American electric utilities. (1970). The units of the LHS and RHS must be the same. $M cost = b 0 + b 1 MKWH Y = $ cost b 0 = $ cost = $M b 1 = $M /MKWH = $M/MKWH So,….. b 0 = fixed cost = total cost if MKWH = 0 marginal cost b 1 = marginal cost = dCost/dMKWH b 1 * MKWH = variable cost

Part 5: Functional Form 5-6/36 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 37 th, 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.

Part 5: Functional Form 5-7/36 Application: WHO  WHO data on 191 countries in Analysis of Disability Adjusted Life Expectancy = DALE EDUC = average years of education  DALE = β 0 + β 1 EDUC + ε

Part 5: Functional Form 5-8/36 The (Famous) WHO Data

Part 5: Functional Form 5-9/36 The slope is the interesting quantity. Each additional year of education is associated with an increase of in disability adjusted life expectancy.

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Part 5: Functional Form 5-13/36 Increase of per capita GDP of 1,000 PPP units is associated with an increase of ‘happy’ of 1000( ) = Happy ranges from 0 to 100.

Part 5: Functional Form 5-14/36 Semilog Models and Growth Rates LogSalary = Years + e

Part 5: Functional Form 5-15/36

Part 5: Functional Form 5-16/36 Semilog Model for Fuel Bills Each increase of 1 room raises the fuel bill by about 21%. [Actually closer to exp(.215)-1 = 24%.]

Part 5: Functional Form 5-17/36 Using Semilog Models for Trends Frequent Flyer Flights for 72 Months. (Text, Ex. 11.1, p. 508)

Part 5: Functional Form 5-18/36 Regression Approach logFlights = β 0 + β 1 Months + ε b 0 = 2.770, b 1 = , s =

Part 5: Functional Form 5-19/36 Elasticity and Loglinear Models  logy = β 0 + β 1 logx + 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

Part 5: Functional Form 5-20/36 Monet Regression

Part 5: Functional Form 5-21/36

Part 5: Functional Form 5-22/36

Part 5: Functional Form 5-23/36 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.

Part 5: Functional Form 5-24/36 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)

Part 5: Functional Form 5-25/36 Electricity Cost Function

Part 5: Functional Form 5-26/36 Candidate Model for Cost Log c = a + b log q + e

Part 5: Functional Form 5-27/36 A Better Model? Log Cost = α + β 1 logOutput + β 2 [logOutput] 2 + ε

Part 5: Functional Form 5-28/36 Candidate Models for Cost The quadratic equation is the appropriate model. Logc = b 0 + b 1 logq + b 2 log 2 q + e

Part 5: Functional Form 5-29/36 Missing Variable Included Residuals from the quadratic cost model Residuals from the linear cost model

Part 5: Functional Form 5-30/36 Unusual Data Points Outliers have (what appear to be) very large disturbances, ε The 500 most successful movies

Part 5: Functional Form 5-31/36 Outliers Remember the empirical rule, 99.5% of observations will lie within mean ± 3 standard deviations? We show (b 0 +b 1 x) ± 3s e 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.

Part 5: Functional Form 5-32/36 logPrice = b 0 + b 1 logArea + e Prices paid at auction for Monet paintings vs. surface area (in logs) Not an outlier: Monet chose to paint a small painting. Possibly an outlier: Why was the price so low?

Part 5: Functional Form 5-33/36 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 is large.) (5) Question why you think it is an outlier. Is it really?

Part 5: Functional Form 5-34/36 Regression Options

Part 5: Functional Form 5-35/36 Minitab’s Opinions Minitab uses ± 2S to flag “large” residuals.

Part 5: Functional Form 5-36/36 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.

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Part 5: Functional Form 5-42/36 Correlation?

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