econometrics for consumption and demandamoneta/lecture_mar08.pdf · introduction • individual...

Econometrics for Consumption and Demand

Alessio Moneta

Doctoral Training Course "Evolutionary Economics"Max Planck Institute of Economics

Jena

6 March 2008

MAX-PLANCK-GESELLSCHAFT

Introduction

• Individual Demand Analysis

• Demand system• Based on Utility Theory

• Cross-section Demand Analysis• Family budgets• Budget allocations

Nonparametric Methods for EstimatingEngel Curves

Outline

• What is an Engel Curve?

• What does Nonparametric Regression mean?

• NP density estimation. Income Distribution.

• NP smoothing

What is an Engel Curve?

Engel curves (cross-sectional)ECs describe how expenditure for commodity g depends onincome.• expg = f (y ,z)

• expg = expenditure on g (e.g. food). Problem ofaggregation across g’s.

• y : income. “Total consumption" proxy.• z: vector of characteristics: e.g. age/household.

compositions.• Prices? Implications of the fixed-prices assumption.

• Budget shares ECs: wg = f (y ,z), wherewg = (expg/y)∗100

• Quantity EC: qg = f (y ,z)

• Price EC: pg = f (y ,z) (cfr. Bils and Klenow AER 2001).

Engel curves (individual)

• qg,i = f (yi ,zi)

• Link with Demand Analysis.• The problem of aggregation (see Stoker 1993):

• Theory: focus on individual maximizing behavior.• Available data: aggregate.

• Individual ECs are not observable.

Historical perspective

• Engel (1857): no functional for pre-specified. Engel’s law:food budget shares decrease with income.

• Allen and Bowley (1935): qg = a+by (fitted by OLS).

• Working (1943): wg = a+b log(y)

• wg = a+b log(y)+cy−1

• large errors

• heterogeneity of tastes

• non-linearities

• nonparametric approaches to EC estimation (Härdle andJerison 1991, Banks et al. RES 1997)

What does Nonparametric Regression mean?

Regression Function

• Estimation of the functional dependence of Y (e.g.expenditure on food) on X (e.g. income).

• Y = m(X )+ ε, where E(ε|x) = 0

• E(Y |X = x) = E(m(x)|x)+E(ε|x) = m(x)

• Linear regression model: m(x) = α +βx

• Log-linear: m(x) = α +β logx .

• OLS: BLUE estimator

• Non parametric approach:

E(Y |X = x) =∫

∞

−∞

yfY |X (y |x)dy

Example (Engel 1857)

Example (UK FES data 2001)

0 200 400 600 800 1000 1200

2040

6080

100

Total Expenditure

Expe

nditu

re o

n To

tal F

ood

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Engel Curve 2001 for Total Food

0.0 0.2 0.4 0.6 0.8 1.0

05

1015

20

Total Expenditure

Expe

nditu

re o

n Ce

real

s

Engel Curve 2001 for Cereals

0.0 0.2 0.4 0.6 0.8 1.0

02

46

810

12

Total Expenditure

Expe

nditu

re o

n Su

gar e

tc.

Engel Curve 2001 for Sugar etc.

0.0 0.2 0.4 0.6 0.8 1.0

020

4060

8010

012

0

Total Expenditure

Expe

nditu

re o

n TV

, vid

eo a

nd a

udio

equ

ip.

Engel Curve 2001 for TV, video and audio equip.

Some definitions

• Conditional mean:

E(Y |X = x) =∫

∞

−∞

yfY |X (y |x)dy =∫

∞

−∞

yfX ,Y (x ,y)

fX (x)dy

• We would like to estimate fX (x) and fXY (x ,y).• Recall that:

• Distribution function: FX ,Y (x ,y) = Pr(X ≤ x ,Y ≤ y)• The joint pdf fX ,Y (x ,y) is defined as:

FX ,Y (x ,y) =∫ x

−∞

∫ y

−∞

fX ,Y (z,w)dzdw

• Marginal pdf: fX (x) =∫

∞

−∞fX ,Y (x ,y)dy

• Conditional df: fY |X (y |x) = fX ,Y (x ,y)/fX (x)

Nonparametric Density Estimation

How to estimate fX (x) and fX ,Y (X ,Y )?

• Parametric approach: ML Estimator

• Simple nonparametric approach. Histogram.

f̂ (x) =1

Nb

N

∑i=1

I(Xi ∈ bin(x)),

where b in the length of each of the bins according towhich the X -axis is divided.

Histogram and Smooth Estimations

Total Consumption (UK 2001)

Dens

ity

0 200 400 600 800 1000 1200 1400

0.00

000.

0005

0.00

100.

0015

0.00

20

KernelML Normal

A slightly more complicated histogram:

f̂ (x) =1

Nb

N

∑i=1

I(x− 12

b ≤ Xi ≤ x +12

b),

f̂ (x) =1

Nb

N

∑i=1

I(−1

2≤ Xi −x

b≤ 1

2

),

f̂ (x) =1

Nb

N

∑i=1

K(

Xi −xb

)where

K (u)

{1 if |u| ≤ 1/20 if |u| ≥ 1/2.

• b: bandwidth parameter; K : kernel (weighting) function

Histogram and naive Kernel


Dens

ity

0 200 400 600 800 1000 1200 1400

0.00

000.

0005

0.00

100.

0015

0.00

200.

0025

b=100b=20

Kernel Density Estimator:

f̂ (x) =1

Nb

N

∑i=1

K(

Xi −xb

)• K satisfies:

• K (u) = K (−u)•

∫K (u)du = 1

• K (u)≥ 0 everywhere• b −→ 0 as N −→ ∞

• Nb −→ ∞ as N −→ ∞

• Triangular kernel: (1−|u|)I(|u| ≤ 1).

• Epanechnikov kernel: 34(1−u2)I(|u| ≤ 1).

• Gaussian kernel: (2π)−1/2 exp(−1

2u2) .

Comparison of Kernel Estimators


Dens

ity

0 200 400 600 800 1000 1200 1400

0.00

000.

0005

0.00

100.

0015

0.00

200.

0025

Epanechnikov b= 100Epanechnikov b= 20Triangular b=100Gaussian b=100

Bias and Variance of the Kernel Density Estimator:

• Mean Square Error: MSE(θ̂) = E(θ̂ −θ0)2

• MSE(θ̂) = [Bias(θ̂)]2 +Var(θ̂)

Bias(f̂ (x))≈ b2

2f (2)(x)

∫K (u)u2du

Var(f̂ (x))≈ f (x)Nb

∫K (u)2du

• Trade-off between Bias and Variance:• ↑ b,↑ Bias,↓ Var• ↓ b,↓ Bias,↑ Var

Consistency and Asymptotic Normality:

• MS Consistency: limN→∞ E( ˆf (x)− f (x))2 = 0.

• Asymptotic distribution:

√Nb( ˆf (x)− f (x))→ N

(0, f (x)

∫K (u2)du

)• For d-dimensional densities the convergence rate is

√Nbd

(curse of dimensionality).

Choice of bandwidth:

• Bandwidth which minimizes MISE =∫

MSE( ˆf (x))dx .

• It turns out that dMISEdb = 0 for:

b∗ =( ∫

K (u)2du∫f (2)(x)dx [

∫K (u)u2du]2

)N−1/5

• Note that f (2)(x)2 is unknown!• Possible solutions:

• Use arbitrary b to estimate f (2)(x)2 and then plug in itsvalue in b∗.

• Silverman’s rule of thumb: estimate f (2)(x)2 using a normaldensity function:

bs = 1.06σN−1/5.

Cross-validation method:

• Recall:

f̂b(x) =1

Nb

N

∑i=1

K(

Xi −xb

)• Consider:

L(b) =N

∏j=1

f̂b(Xj) =N

∏j=1

1Nb

[K (0)+∑

i 6=jK

(Xi −Xj

b

)],

which is unfortunately maximized for b = 0.

• Cross validation bandwidth:

b̂CV = argmaxb

N

∏j=1

1(N−1)b

N

∑i 6=j

K(

Xi −Xj

b

)

Multivariate Kernel Density Estimations:

• Product Kernel estimation:

f̂X1,...,Xd (x1, . . . ,xd) =1

Nbd

N

∑i=1

Kd

(X1i −x1

b, . . . ,

Xdi −xd

b

)where Kd(u1, . . . ,ud) = K (u1), . . . ,K (ud).

Nonparametric Regression

Nadaraya-Watson Estimator

• Recall:

E(Y |X = x) = m(x) =1

fX (x)

∫∞

−∞

yfX ,Y (x ,y)dy (1)

• After substituting k. estimators for f (x) and f (x ,y) in (1):

m̂NW (x) =∑

Ni YiK

(Xi−x

b

)∑

Ni K

(Xi−x

b

)• Note that m̂NW (x) is a weighted average: ∑

Ni Yiwi .

• Special case: rectangular K (moving average).• Analogies:

• histogram and regressogram;• moving histogram and moving average;• kernel density and kernel regression.

Other Kernel Smoothers:

• Suppose 0 < x1 < x2 < .. . < xn < 1

• Priestely and Chao:

m̂PC(x) =1b

N

∑i=1

(Xi −Xi−1)YiK(

Xi −xb

)• Gasser-Müller estimator:

m̂GM(x) =1b

N

∑i=1

Yi

∫ si

si−1

K(

Xi −ub

)du,

where s0 = 0, si = (Xi +Xi+1)/2, sn = 1.

Other Issues:

• Bias and Variance.

• Heteroscedasticity.

• Correlated errors.

• Variances curves.

Summary and Conclusions:• Advantages of the nonparametric approach to estimate:

• Income distribution• Engel curves

• Nonparametric density estimation. Two choices:• Kernel function;• Bandwidth.

• Nonparametric regression. Three choices:• Kernel function• Bandwidth• Smoothing estimator (weighting system).

Links

• CRAN web page: http://cran.r-project.org

• TINN: http://www.sciviews.org/Tinn-R

• R codes used here:https://mail.sssup.it/∼amoneta/codelecture.txt

References:

Engel, J. and A. Kneip (1996), Recent Approaches toEstimating Engel Curves, Journal of Economics, 63(2),http://www.springerlink.com/content/v671xj3p6402mg74

Lewbel, A. (2006), Engel Curves, New Palgrave Dictionaryof Economics, 2nd Edition,http://www2.bc.edu/∼lewbel/palengel.pdf

Wand, M.P. and M.C. Jones (1995), Kernel Smoothing,Chapman & Hall.

Hart, J.D. (1997), Nonparametric Smoothing andLack-of-Fit Tests, Springer Verlag.

THANK YOU

Contacts:[email protected]

https://mail.sssup.it/∼amoneta

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