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Econometrics I

Professor William Greene

Stern School of Business

Department of Economics

Econometrics I

Part 6 – Dummy Variables

and Functional Form

Monet in Large and Small

ln (SurfaceArea)

ln (US$)

6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.

18 17 16 15 14 13 12 11

S 1. R-SqR-Sq(adj) 20.0%19.8%

Fitted Line Plot ln (US$) = 2.825 + 1.725 ln (SurfaceArea)

Log of $price = a + b log surface area + e

Sale prices of 328 signed Monet paintings

How Much for the Signature?

 The sample also contains 102 unsigned

paintings

Average Sale Price

Signed $3,364,

Not signed $1,832,

 Average price of a signed Monet is almost

twice that of an unsigned one.

Monet Multiple Regression

Regression Analysis: ln (US$) versus ln (SurfaceArea), Signed The regression equation is ln (US$) = 4.12 + 1.35 ln (SurfaceArea) + 1.26 Signed Predictor Coef SE Coef T P Constant 4.1222 0.5585 7.38 0. ln (SurfaceArea) 1.3458 0.08151 16.51 0. Signed 1.2618 0.1249 10.11 0. S = 0.992509 R-Sq = 46.2% R-Sq(adj) = 46.0%

Interpretation:

(1) Elasticity of price with respect to surface area is 1.3458 – very large

(2) The signature multiplies the price of a painting by exp(1.2618) (about

3.5), for any given size.

A Conspiracy Theory

for Art Sales at

Auction

Sotheby’s and Christies, 1995 to

about 2000 conspired on

commission rates.

Evidence

The statistical

evidence seems to

be consistent with

the theory.

Effects on Price Unsigned Signed

Not 1995 - 2000 exp(0.0000) =1.0000 exp(1.2777) = 3.

1995 - 2000 exp(0.2009) =1.2225 exp(1.2777 + 0.2009) = 4.

Women appear to assess health satisfaction differently from men.

Dummy Variable for One Observation

A dummy variable that isolates a single

observation. What does this do?

Define d to be the dummy variable in question.

Z = all other regressors. X = [ Z,d ]

Multiple regression of y on X. We know that

X'e = 0 where e = the column vector of

residuals. That means d'e = 0, which says that

ej = 0 for that particular residual. The

observation will be predicted perfectly.

Fairly important result. Important to know.

I have a simple question for you. Yesterday, I was

estimating a regional production function with yearly

dummies. The coefficients of the dummies are usually

interpreted as a measure of technical change with

respect to the base year (excluded dummy variable).

However, I felt that it could be more interesting to

redefine the dummy variables in such a way that the

coefficient could measure technical change from one

year to the next. You could get the same result by

subtracting two coefficients in the original regression but

you would have to compute the standard error of the

difference if you want to do inference.

Is this a well known procedure? YES

Example with 4 Periods

The estimated model with time dummies is

y = a +b 2 * d 2 + b 3 * d 3 + b 4 * d 4 + e (possibly some other variables, not needed now).

Estimated least squares coefficients are

b = a, b 2 , b 3 , b 4

Desired coefficients are

c = a, b 2 , b 3 – b 2 , b 4 – b 3

The original model is y = Xb + e.

The new model would be y = ( XC)(C -1 b) + e = Qc + e

The transformation of the data is Q = XC. c = C -1 b

The transformed X is [1,d 2 +d 3 +d 4 , d 3 +d 4 .d 4 ]

1

1 0 0 0 1 0 0 0

0 1 0 0 0 1 0 0 , 0 1 1 0 0 1 1 0

0 0 1 1 0 1 1 1



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C C

A Categorical Variable

Nonlinear Specification:

Quadratic Effect of Experience

Model Implication: Effect of Experience and

Male vs. Female