Econometrics I

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Econometrics IProfessor William GreeneStern School of BusinessDepartment of Economics

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Econometrics I

Part 22 – Semi- and Nonparametric Estimation

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Cornwell and Rupert DataCornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file areEXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA = 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationLWAGE = log of wage = dependent variable in regressionsThese data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155.  See Baltagi, page 122 for further analysis.  The data were downloaded from the website for Baltagi's text.

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A First Look at the DataDescriptive Statistics

Basic Measures of Location and Dispersion Graphical Devices

Histogram Kernel Density Estimator

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Histogram for LWAGE

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The kernel density estimator is ahistogram (of sorts).

n i mm mi 1

** *x x1 1f̂(x ) K , for a set or points x

n B B

B "bandwidth" chosen by the analystK the kernel function, such as the normal or logistic pdf (or one of several others)x* the point at which the density is approximated.This is essentially a histogram with small bins.

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Computing the KDE1 2 n

* *1

Given the sample observations: x , x , ..., x (x , 1,..., )

Choose a set of points x ,..., x These may be the original data if n is not very large Otherwise, choose an equally spaced se

i

M

i n

min max

** *

1

t of points in [x , x ]

x x1 1ˆ For each point x , f x

K[t] is the kernel function: common choices are the normal pdf, K[t] = (t)

Epanechniko

n i mm x m i

Kn B B

2

1/5

*

v kernel, K[t] = .75(1-.2t ) / 5, if |t| 5, 0 else

B is the bandwidth: e.g., Silverman's Rule of Thumb = .9w/n w = Min(s , /1.349)

ˆ Plot f x agx

x m

IQR

*ainst x and connect points.m

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Kernel Density Estimator

n i mm mi 1

** *x x1 1f̂(x ) K , for a set of points x

n B B

B "bandwidth"K the kernel functionx* the point at which the density is approximated.

f̂(x*) is an estimator of f(x*)1

The curse of dimensionality

nii 1

3/5

Q(x | x*) Q(x*). n

1 1But, Var[Q(x*)] Something. Rather, Var[Q(x*)] * somethingN N

ˆI.e.,f(x*) does not converge to f(x*) at the same rate as a meanconverges to a population mean.

Part 22: Semi- and Nonparametric Estimation22-10/29

Kernel Estimator for LWAGE

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Application: Stochastic Frontier Model

Production Function Regression: logY = b’x + v - u

where u is “inefficiency.” u > 0. v is normally distributed.

Save for the constant term, the model is consistently estimated by OLS.

If the theory is right, the OLS residuals will be skewed to the left, rather than symmetrically distributed if they were normally distributed.

Application: Spanish dairy data used in Assignment 2

yit = log of milk productionx1 = log cows, x2 = log land, x3 = log feed, x4 = log labor

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Regression Results

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Distribution of OLS Residuals

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A Nonparametric Regression y = µ(x) +ε Smoothing methods to approximate µ(x) at

specific points, x* For a particular x*, µ(x*) = ∑i wi(x*|x)yi

E.g., for ols, µ(x*) =a+bx* wi = 1/n +

We look for weighting scheme, local differences in relationship. OLS assumes a fixed slope, b.

2( ) / ( )i i i x x x x

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Nearest Neighbor Approach Define a neighborhood of x*. Points near get high

weight, points far away get a small or zero weight Bandwidth, h defines the neighborhood:

e.g., Silverman h =.9Min[s,(IQR/1.349)]/n.2

Neighborhood is + or – h/2 LOWESS weighting function: (tricube)

Ti = [1 – [Abs(xi – x*)/h]3]3. Weight is wi = 1[Abs(xi – x*)/h < .5] * Ti .

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LOWESS Regression

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OLS Vs. Lowess

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Smooth Function: Kernel Regression

1

1

2

*1

ˆ ( * | , )*1

Kernel Functions:Normal: K(t) = (t)Logistic: K(t) = (t)[1- (t)]

Epanechnikov: K(t)=.75(1-.2t )/ 5, if |t| 5 and 0 otherwise

n iii

n ii

x xK yB Bx B

x xKB B

x

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Kernel Regression vs. Lowess (Lwage vs. Educ)

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Locally Linear Regression

1

1 1

( *) ( *) ' *.

( *) ( *, ) ( *, ) y

( *, ) [( * ) ( * ), ]

n ni i i i i i i ii i

i i i i

w w

w K h

x x x

x x x x x x x x

x x x x x x

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OLS vs. LOWESS

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Quantile Regression

Least squares based on: E[y|x]=ẞ’x

LAD based on: Median[y|x]=ẞ(.5)’x

Quantile regression: Q(y|x,q)=ẞ(q)’x

Does this just shift the constant?

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OLS vs. Least Absolute Deviations----------------------------------------------------------------------Least absolute deviations estimator...............Residuals Sum of squares = 1537.58603 Standard error of e = 6.82594Fit R-squared = .98284 Adjusted R-squared = .98180Sum of absolute deviations = 189.3973484--------+-------------------------------------------------------------Variable| Coefficient Standard Error b/St.Er. P[|Z|>z] Mean of X--------+------------------------------------------------------------- |Covariance matrix based on 50 replications.Constant| -84.0258*** 16.08614 -5.223 .0000 Y| .03784*** .00271 13.952 .0000 9232.86 PG| -17.0990*** 4.37160 -3.911 .0001 2.31661--------+-------------------------------------------------------------Ordinary least squares regression ............Residuals Sum of squares = 1472.79834 Standard error of e = 6.68059 Standard errors are based onFit R-squared = .98356 50 bootstrap replications Adjusted R-squared = .98256--------+-------------------------------------------------------------Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X--------+-------------------------------------------------------------Constant| -79.7535*** 8.67255 -9.196 .0000 Y| .03692*** .00132 28.022 .0000 9232.86 PG| -15.1224*** 1.88034 -8.042 .0000 2.31661--------+-------------------------------------------------------------

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Quantile Regression

Q(y|x,) = x, = quantile Estimated by linear programming Q(y|x,.50) = x, .50 median regression Median regression estimated by LAD (estimates

same parameters as mean regression if symmetric conditional distribution)

Why use quantile (median) regression? Semiparametric Robust to some extensions (heteroscedasticity?) Complete characterization of conditional distribution

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Quantile Regression

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1 1

Model : , ( | , ) , [ , ] 0ˆˆResiduals: u

1Asymptotic Variance:

= E[f (0) ] Estimated by

Asymptotic Theory Based Estimator of Variance of Q - REGx | x

A C A

A xx

i i i i i i i i

i i i

u

y u Q y Q u

y

N

βx βx-βx

1

.2

1 1 1 ˆ1 | | BB 2

Bandwidth B can be Silverman's Rule of Thumb: ˆ ˆ( | .75) ( | .25)1.06 ,

1.349(1- )(1- ) [ ] Estimated by

x x

C = xx

Ni i ii

i iu

uN

Q u Q uMin s

N

EN

12For =.5 and normally distributed u, this all simplifies to .2

But, this is an ideal application for bootstrapping

X

X

.

X

Xus

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= .25

= .50

= .75

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