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Econometrics I

Professor William Greene

Stern School of Business

Department of Economics

Econometrics I

Part 5 – Finite

Sample Properties

Application: Health Care Panel Data

German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods

Data downloaded from Journal of Applied Econometrics Archive. There are altogether 27,326 observations. The number of observations ranges from 1 to 7.

(Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987).

Variables in the file are

DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year DOCTOR = 1(Number of doctor visits > 0) HOSPITAL = 1(Number of hospital visits > 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / 10000. (4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years MARRIED = marital status

For now, treat this sample as if it were a cross section, and as if it were the full population.

Population Regression

This is the true value of .

How should we interpret this variation in the regression slope?

The centering suggests the estimator is unbiased.

We will have only one sample. We could have drawn any one of

the possible samples.

The Statistical Context

of Least Squares Estimation

The sample of data from the population: Data

generating process is y = x  + 

The stochastic specification of the regression

model: Assumptions about the random .

Endowment of the stochastic properties of the

model upon the least squares estimator. The

estimator is a function of the observed

(realized) data.

Deriving the Properties

b = a parameter vector + a linear combination of

the disturbances, each times a vector.

Therefore, b is a vector of random variables. We

analyze it as such.

The assumption of nonstochastic regressors. How

it is used at this point.

We do the analysis conditional on an X , then show

that results do not depend on the particular X in

hand, so the result must be general – i.e.,

independent of X.

Properties of the LS Estimator:

b is unbiased

Expected value and the property of unbiasedness.

E[ b|X ] = E[ + ( X  X )

-1X  |X ]

=  + ( X  X ) -1X E[ |X ]

=  + 0

E[ b ] = E X {E[ b | X ]}

= E[ b ].

(The law of iterated expectations.)

Means of Repetitions b|x

Partitioned Regression

A Crucial Result About Specification:

y = X 1 1 + X 2 2 + 

Two sets of variables. What if the regression is computed without the second set of

variables?

What is the expectation of the "short" regression estimator? E[ b 1|( y = X 1 1 + X 2 2 +

)]

b 1 = ( X1  X1 )

-1 X 1  y

Historical Application: Left Out Dummy Variable

in a Keynesian Consumption Function

0 1

C     Y   W  

Application

The (truly) short regression estimator is biased.

Application:

Quantity = 1Price + 2Income + 

If you regress Quantity on Price and leave out Income. What do you get?

Estimated ‘Demand’ Equation

Shouldn’t the Price Coefficient be Negative?

Multiple Regression of G on Y and PG.

The Theory Works!

----------------------------------------------------------------------

Ordinary least squares regression ............

LHS=G Mean = 226. Standard deviation = 50. Number of observs. = 36

Model size Parameters = 3

Degrees of freedom = 33 Residuals Sum of squares = 1472.

Standard error of e = 6.

Fit R-squared =.

Adjusted R-squared =.

Model test F[ 2, 33] (prob) = 987.1(.0000) --------+-------------------------------------------------------------

Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X

--------+-------------------------------------------------------------

Constant| -79.7535* 8.67255 -9.196. Y| .03692*** .00132 28.022 .0000 9232. PG| -15.1224*** 1.88034 -8.042 .0000 2.**

--------+-------------------------------------------------------------