Non-Parametric Econometrics

Viewer
Transcript

Non-Parametric Econometrics Emmanuel Flachaire

Emmanuel Flachaire

Non-Parametric Econometrics

Chapter 5

Semi-Parametric Regression Models

Emmanuel Flachaire

Non-Parametric Econometrics

Introduction I

A multiple regression model can be defined as: y = m(x1 , x2 , . . . , xk ) + ε

I

In theory: no problem. Ex: Nadaraya-Watson estimator can be used, with a multivariate kernel: Pn xk −xki x1 −x1i yi i=1 K h1 ; . . . ; hk m(x ˆ 1 , . . . , xk ) = P n xk −xki x1 −x1i i=1 K h1 ; . . . ; hk

I

In practice: the curse of dimensionality → m ˆ unreliable

I

Solution: semiparametric models

Emmanuel Flachaire

Non-Parametric Econometrics

Introduction Semiparametric models

I

Semiparametric models: mixing parametric and nonparametric specifications

I

Partial linear regression model: y = X β + m(z) + ε The relationship between y and X is parametric The relationship between y and z is nonparametric

I

Advantage: β with economic interpretation and robust to the specification at other parts of the model

I

It is a way to reduce the curse of dimensionality

Emmanuel Flachaire

Non-Parametric Econometrics

Introduction Semiparametric models

I

Dimension: 3 y = X β + m(z1 , z2 , z3 ) + ε

I

Dimension: 2 y = X β + m1 (z1 ) + m2 (z2 , z3 ) + ε

I

Dimension: 1 y = X β + m1 (z1 ) + m2 (z2 ) + m3 (z3 ) + ε

Emmanuel Flachaire

Non-Parametric Econometrics

Application Hedonic price

We consider the partial linear model: y = X β + m1 (z1 ) + m2 (z2 , z3 ) + ε I

y is the (log) price - 1135 observations, 1995 in Brest

I

X dummies: Studio, T1, T2, T3, T4, T5, house, parking

I

z1 : distance to the closest green park

I

z2 , z3 : geographical coordinates (location)

Spatial dependence is specified nonparametrically rather than with standard parametric models (spatial autocorrelation models)

Emmanuel Flachaire

Non-Parametric Econometrics

Application Hedonic price I

Results: X Const. T1 T2 T3 T4 T5 Maison Parking

I

coef 4.775 0.079 0.386 0.763 0.992 1.139 0.241 0.242

e.t. (0.052) (0.056) (0.051) (0.049) (0.050) (0.055) (0.032) (0.024)

With similar other characteristics: I I I

In average, a Studio costs 118.510 francs (e 4.775 ) a T2 is expected to cost 38.6% more than a Studio A house is expected to cost 21.1% more than an apartment Emmanuel Flachaire

Non-Parametric Econometrics

Application Green park influence

m ˆ 1 (z1 )

0.4 0.2 0.0 -0.2 -0.4 -0.6

........................................................................................................................................................................................................................................................................................................................................................................ ... .... . . .......... ... ... ... .. ... .... ... ... ... .... ... ... ... ... ... ... ... . ... ... . ... . . . . . ... . ... . ......... . . . ... ... ... .. . . . ... ... . .. ... . .. ... . ... .. . .. . .. .. . . . .. .. ... . . . .. .. . . .. .. . . .. ... . .. . ... .......................... ... . ... .. . . ... .... ... .... ................................. ... ... ... ... ... .......... . ... ... .... ... ... ... . .. . .. ... .... .............. ........ ......... ... ... ........ ...... ..... ... ... ......... ... ......... ... ....... ... ... . . ... . ..... ......... ... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... ... .......... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... . . . .. ... ......... ... ... ...... ... ... ... ... ... ... .......................................................................................................................................................................................................................................................................................................................................................................... .. .. .. ..

... ... ... ... ... ... ... .... .......... ............ ........... ...................... .................................. ........... ........... .......... .......... ......... ......... .....

0

500 Emmanuel Flachaire

1000

1500

Non-Parametric Econometrics

Application Location influence

5.8

.......................................................................................................................................................................................... ......... . ... .......... ...... .. ...... ...... .... ... ...... ... ...... . .. . . . ...... ... .. ... . . . . ...... . . . ... . ... . ...... . . . . . . ... ...... . ... . . . . . . . . . . . . . . . . . . . . . . . ... ... .............................................................. ... ............................ . . . . . . ....... ..... . . . . . . . . . ... . ................................ ................................................................ ................................ .......... .... . . . . . . . . . . . . . . . . . . . . . . . ...... ... ..... . ................................... ............. ................................................................ .... . . . . . . . . . . . . . . . . . . ..... .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ............................................................................................ .............. .. .. .. .. ... ................................................... .. .. .. .. .. ......... ... ... ................... .. ............. ................. ......................................... ........... ..... ...... ... ... ... .................................................................................................................................................................. ........................................................... ............................................ . . . . . . . . . . . . . . . . ... . ... .. .... . ....... .. .. ... ... .. . .. .. . ... ................................................. .. ......................................................................... .... ........................... ......... .. ......... ....... .. ................ ... ... .................... ............................... .. .................................................................................................................................................................................................................... ..................................................................................... . . . ... . . . ................. .............................................. . . . ....... ..... .. . .. . . .. .. ..... . . ... ... ................................................................................. .. .. .. ....... ............................... ................. ....... .. ...... ..... ... ............................. ........................................... ............................................................................. .. .. .......................................................... ...... .. ................. ... ................ ..................... ...... ..... .. ................................ ......................................................................................... .......................... ..................................................................................................... .. ... .................. .... .... ....................... ................................................................................................ .................. ........ .. ................................... .................................... .. .......................................................................... .......................................................................................................................................................... ...................................................................................................................................................... . . ... . . . . . ............................................ .............................................. .............. ..... ... ..... . .. ... . . . ......... ... . ......... ... ... .............................................................................................. ........................... ......... ........ .... ... ........ .. ..... .... ... .. ..................... ... . ........................................... ............................. .... .. ....... ................ .................. ... ... ...................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .. ... .......................................... ..................................................................................................... ............. ........ ................................. .... ............. .... ............. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ..................................................... ...................................... ............... ....................................... .. ................. ................................................ ...................... ................ ................................... .. ................ ... ... ... .............................................................. ............. ....................... ............... ............. ... ... ... ................................................ ................ .................... ..................... ... .. .................................................................. ... .............................. .............. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................... ... ... ............................................... .......... .... . .................. ... ... ............... ....... .............. ... .. ... ... ...................................... ......... ... .. ... .................... ..... ... .. . . . . . . . . . . . . . . . . . . . . . . ... ... .................... ... .......... ... ... ... ... ... ... .......... ... .. ... .. ... .. .. ... . . . ... ... ... ... ... .. ... ...... ... ........... ....... .... ... .. ... .. ... . . . . . ... .. ... .. ... .. ... ... ... .. ... ... .. .. .... .............................................................................................................................................................................................................................................................................. .. .. . . .. .. ... ... .. .. .. ..

102000

5.6

100000

5.4

z2

z1

98000

92000

94000

Emmanuel Flachaire

96000

98000

Non-Parametric Econometrics

Application Semiparametric to parametric model

I

Green park influence: nonparametric estimation suggests the following parametric model, m1 (z1 ) = γz1 + δ(z1 − 200)+

I

Location: nonparametric estimation suggests strong non-linearity, which is difficult to modelize parametrically: m2 (z2 , z3 ) =?

I

Finally, we consider the following semiparametric model: y = X β + γz1 + δ(z1 − 200)+ + m2 (z2 , z3 ) + ε

Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model I

In practice: arbitrage robustness vs. efficiency

I

Parametric model: efficiency, if the specification is correct

I

Nonparametric model: robustness to misspecification

I

The partial linear model is defined as: y = X β + m(Z ) + ε

I

Advantage 1: This model is robust to any form of the unknown function m √ Advantage 2: βˆ is n-consistent (Robinson 1988)

I

Finally, we have robustness + efficiency !!

I

Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Double residual estimator

I

The partial linear model is defined as y = X β + m(Z ) + ε

I

Apply a conditional expectation to both terms: E (y |Z ) = E (X |Z ) β + m(Z )

I

Substract both equations: y − E (y |Z ) = [X − E (X |Z )] β + ε

I

Conditional expectations known → linear regression model

Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Double residual estimator I

The partial linear regression model can be rewritten: y − E (y |Z ) = [X − E (X |Z )] β + ε

I

Conditional expectations unknown: use consistent estimators

I

E (y |Z ) and E (X |Z ): nonparametric estimators obtained from y = my (Z ) + ε1

I

X = mX (Z ) + ε2

Finally, the “double residual” estimator of β is the OLS estimator from the following model: y −m ˆ y = [X − m ˆX]β + ε εˆ1 = εˆ2 β + ε Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Double residual estimator

I

From the model: y −m ˆ y = [X − m ˆX]β + ε the OLS estimator, or double residual estimator, is βˆres = [(X − m ˆ x )> (X − m ˆ x )]−1 (X − m ˆ x )> (y − m ˆy)

I

From the model: εˆ1 = εˆ2 β + ε the OLS estimator, or double residual estimator is βˆres = (ˆ ε> ˆ2 )−1 εˆ> ˆ1 2ε 2ε

Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Double residual estimator I

The double residual estimator is asymptotically Normal βˆres = (ˆ ε> ˆ2 )−1 εˆ> ˆ1 2ε 2ε

I

Its variance is equal to σ2 V (βˆres ) = ε2 nσε2

I

Estimate of m(Z ) obtained by a nonparametric estimation of y − X βˆres = m(Z ) + ε.

I

Nonparametric estimation: local polynomial, splines, wavelets

I

Confidence interval and test Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Differencing I

Yatchew (1998): we can estimate β in the partial linear y = X β + m(z) + ε

I I

with one OLS estimation . . . no nonparametric estimation!! Sort the data (y , X , z) by z, such that z1 ≤ z2 ≤ · · · ≤ zn Differencing the model yields ys − ys−1 = (Xs − Xs−1 ) β + [m(zs ) − m(zs−1 )] + εs − εs−1

I

Approximately, we have: ys − ys−1 ≈ (Xs − Xs−1 ) β + εs − εs−1 because [m(zs ) − m(zs−1 )]→0 as n increases to ∞

I

OLS estimator of β from this model is consistent Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Differencing I

With sorted data, the partial linear model is approximately ∆y = ∆X β + η,

I

From which, a consistent estimator of β can be obtained βˆdiff = (∆X > ∆X )−1 ∆X > ∆y

I

This estimator is asymptotically Normal with variance σ2 V (βˆdiff ) = 1.5 ε2 nσε2

I

m(z) ˆ requires a nonparametric estimation y − X βˆdiff = m(z) + ε. Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Interpretation of coefficients

Parametric linear regression model I

FWL theorem - OLS βˆ are identical from the 2 models: y = Xβ + Zγ + ε

εˆ01 = εˆ02 β + ε0

whith εˆ01 and εˆ02 ≡ OLS residuals from y on Z and from X on Z I

Interpretation: βˆk ≡ net impact of variation of Xk over variation of y , net of the impact of any others variables Xj6=k

Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Interpretation of coefficients

Nonparametric regression model: double residual estimator I

βˆres identical from the 2 models: y = X β + m(Z ) + ε

εˆ1 = εˆ2 β + ε

where εˆ1 and εˆ2 ≡ NP residuals from y on Z and from X on Z I

Interpretation: βˆ ≡ similar to parametric regression

Emmanuel Flachaire

Non-Parametric Econometrics

Partial linear model Interpretation of coefficients

Nonparametric regression model: differencing I

Assume X and z not independent, such that X = g (z) + u

I

Differencing yields

ys −ys−1 = [g (zs )−g (zs−1 )] β+(us −us−1 ) β+[m(zs )−m(zs−1 )]+εs −εs−1 I

The two terms in brackets → 0 as the sample size increases

I

With n large enough, we have ys − ys−1 ≈ (us − us−1 ) β + εs − εs−1 βˆdiff ≈ βˆols : impact of u (part of X unexplained by z) over y

I

Interpretation: βˆ ≡ similar to parametric regression

Emmanuel Flachaire

Non-Parametric Econometrics

Additive model Definition

I

Nonparametric model y = m(Z1 , Z2 , . . . , Zk ) + ε

I

Additive model y = m1 (Z1 ) + m2 (Z2 ) + · · · + mk (Zk ) + ε

I

Additive separability: we assume m can be decompose as a sum of several functions of dimension one or two (or more)

I

Iterative estimation process

Emmanuel Flachaire

Non-Parametric Econometrics

Additive model Iterative estimation

I

Additive model: y = m1 (z1 ) + m2 (z2 ) + ε

I

Apply conditional expectations to both terms: E [y − m2 (z2 )|z1 ] = m1 (z1 ) E [y − m1 (z1 )|z2 ] = m2 (z2 )

I

With an estimate m ˆ 2 , regress y − m ˆ 2 on z1 −→ m ˆ1

I

With an estimate m ˆ 1 , regress y − m ˆ 1 on z2 −→ m ˆ2

Emmanuel Flachaire

Non-Parametric Econometrics

Additive model Iterative estimation

I

Additive model: y = m1 (z1 ) + m2 (z2 ) + ε

I

Iterative estimation 1. 2. 3. 4.

(0)

(0)

Select initial estimates m1 and m2 (i) (i−1) Obtain m ˆ 1 by regressing y − m ˆ2 on z1 (i) (i−1) Obtain m ˆ 2 by regressing y − m ˆ1 on z2 Repeat steps 2 an 3 until no significant changes

I

Initial estimates can be equal to 0 or obtained by OLS

I

Extension to more than 2 functions

Emmanuel Flachaire

Non-Parametric Econometrics

Additive model Additive separability

Additive separability hypothesis: what does it mean? I

we can decompose m as a sum of several functions

I

implications: a more restrictive function

I

what kind of restrictions ?

I

consequences ?

Emmanuel Flachaire

Non-Parametric Econometrics

Additive model Additive separability: m(z2 , z3 )

5.9

.......................................................................................................................................................................................... ......... . ... .......... ...... .. ...... ...... .... ... ...... ... ...... . .. . . . ...... ... .. ... . . . . ...... . . . ... . ... . ...... . . . . . . ... ...... . ... . . . . . . . . . . . . . ... . ... ............. ................... . . . . . . . . . . . ........ ..... . . . . . . . . . . . . ... ................ . ................ .............. ....................... .......... . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... ... .............................................. .................. . ................... .... .... . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..................................................... ........ .. .. .. .. .. .. ... ......................................... .. .. .. .. .. .......... . . . . . . . . . . . . . . . . . . . . . . . ... . . ..... ........ ..... . ...... ... ................. .. .. ... ............ ................................... ............ ... ... ................... ......................... .............. ............................................................................. ............. .............................. . . ... . . . . . ... ... . .. ........... ......... .. ... ... .......................................... ........... ........ ....................... ..................... .. ... ... .. ................................................................................................................ .... ............................................ .... ............................... ...................... . . ... ..... ..... ... .. ............. . ... . . . . . ................................................................ ............... ............................... ....... .. .... .... ......... .. ...... ...... . ..... ... ... ... ... .. ... ....................................................................................................... ......... ...... ........................ .. ........ ....... .. ... ................ ... ... ...... ............... ........................................................................... .. .. ....... ................ .................................... ......... ....... ................ .. ............... ........ .... ........... ............. ............................................................................. ... .. ........................................................ .. ........... ..... ... .. ............. ........ ......... .................................................................................................................................................................. .................................... .... .... .................................................. ............................................. . . . . . ... ........ ..... .... .. ... ................. .... ............................ .. .... .. .......... ................................................... ... .... ........................... .................. ... ... .......... .. ........................ .. .. ..... .............. ..... .... .............. ........... ............................ .. ............ .. ................ ... ... ....... . ................................. .. ...... .. .. .... .... .. ................. ............. .. .......... ............ ... .............................................................................................................................................................................................................................................. ... .. .............................................................. .... ............ ............. ..... ........ ..... ........ .... .... .................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ......... .................... ...... ......................................................................... ..................... ....... ..................................................... ........... ...... .............. ... ... ... ............................................. ................................. .................. ............................ ...... .. .......... ... ...... ..... ................. ..... .... ... .. ................ .......................... .. .. ...... ... ... ... ........... ....... ............ .... .. .. ............. .. ............ ............. .................... ... ......................... ........... ... ................. ... .. .................... .... .. .. ............ . ..................................................... ... . ... ........ .............. .............. ........................ ................ ... ... ............ ................. ........ ..... . ............ ... ... ......... .... ........... .......... ...... ................................ ... ... ... .. ....... ..... ....... ... .. ... ... ...... ........... . .. . . ... ................. ... ... ... ............. ... .. .......... ... ... ... ...... ... ... ... ... ........ .... ... .. .. ... . . ... ... ... ... ... ... ... ... ... . ... ... ... ..... ... ... ........ .... .... ... ... ... .. ... .. ... ... ... ... ... ... .... ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ .. .. . . ....... .. .. ... ... .. .. .. ..

5.8

5.7

5.6

5.5

5.4

5.3

92000

94000

Emmanuel Flachaire

96000

98000

Non-Parametric Econometrics

Additive model Additive separability: m2 (z2 ) + m3 (z3 )

6.0

.......................................................................................................................................................................................... ......... . ... .......... ...... .. ...... ...... .... ... ...... ... ...... . .. . . . ...... ... .. ... . . . . ...... . . . ... . ... . ..... . . . . . . ... .. . ... . . . . . ................ . . . ... . ... . . . .... .. ...... . . . . ... . ......... ..... .......... .... . . . . . . . . . ... ...... . ......... ..... ... . .... . . . . . . . . . . . . . ..... .. .. .. .. .. .. .. .. .. .. .. .. ....... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ........ .......... .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .......... .. .... .. .. .. .. ......... ... ... .. ..... .............. ..... ... ... ... ... .. . ....... .... ................................................... . . . . . . ... . . . ... . ... .......... ... ... . .. ................. ............................................. ..... .... ... .............. .... ..... .. . . . . ..................................................................... . .................... . . . . ... . . . . . . . . . . . . . . . . . . ... . . . ... ....... ... . . . ..... . . ... .. .. . ... ... ... ............................. ..................................... ... ... ... ... .... ............ ................. .... .... ......................... ........... ... ... ... ............. .............. .. ..... ................. ............................. ... . ... ..................... ...... ... ............ ... .. ... .. ........ ......................................... . . .. ... .................. ... .... ...... ..................... ........... . ................ ........................................................................................................................................................ .... ...... ......................... ....... ... .. ............ .......... ... ..... ... ............. ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ........... .. .. ........ .. ... ..... ... ............ ............. .......... ........................................ .................................... ..... .. ................. .. ... . .. ... ... ........... ... ..... .. . .. .................... ... .. ... ..... .... ..... ....... .................................................... ................. ....... ...... ... ... ... .... ... ................................................................................................. ... ............. ................................ ............ ......... .................................................................................................................................................. ................. ... ... ... .. ... ... ................................ ....... ........................................................................................................................................................................................................................................................................................................................................................................................................................................ .... ..... .. ................ .......... .... ..... ...................... .... ............. ........................ ........ ... ............................ .... ... .. ...... .. ..... ..................... ... ... . ... .... . . . . .. . . . . . . .. . . ... ........ .................................................................................................... ... ... ... .................... ................. ............... ............................................................ .......................................................................... ...... .......... .... ... ... ............. .................................................................................................. .... ..... ........................... ............ ............... .............. ........................................................................................................................ ............... ... .. ..................................... ... ............................. ... ......... .... ........................................... ....... ........................... .... ...... ...... ... ... .. ... ............. ..................... .. ... . ... .......... .... . .... .................. ............................................................................... ................................................. ............ ... .. ... ... ............ ... ... ..... ............. .. .... ........... .......... ... ............. ... ....... ..................... ............ ... ... ... ... ... .......... .. . ...... .. .. ... ....... ... .. ....... ................... ... ... ... ............................................ ... ... ... .. ... ......... ... ...... ................................ .............................................. ................ ............... ... ............ ........... .. ... ............ ...... ... ... ............ .......................... .............................. .. . ............ ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .............. .. ..... .... ... ........ ............. ..... . ............................................................. ... ................ .. ........... ... ... .................. ........ ... .. ... .. ............... ............... ... ... .... ... .... ..... ..... ... ... ... .......................... ... .. ........ ........ .... .............. ........ ..... ................... .. ... ............................. ... . . . . . . . . . . ... ............. ... ... ... ... ... ... ... ... ... .. ... .. ... .. .. . . . . . . ... .. ... .. ... .. ... .. ... ... . .. .. ..... .. ....... ............................................................................................................................................................................................................................................................................. .. .. . . .. .. ... ... .. .. .. ..

5.8

5.6

5.4

92000

94000

Emmanuel Flachaire

96000

98000

Non-Parametric Econometrics

Additive model Additive separability: z2 β2 + z3 β3

5.65

.......................................................................................................................................................................................... ......... . ... .......... ...... .. ...... ...... .... ... ...... ... ...... . .. . . . ...... ... .. ... . . . . ...... . . . ... . ... . . . . . . . ....... ... . ... . . . ........... ..... . . . . . ... . ... ................ .......... . . . . . . . . . . . . . . ...... ... ..... .............. . . .... . . . . . . . . . . . . . ...... . . . . . . . . . . . . . . ... . ... .. ... ......... ......... .................. ...... ......... .. .. .. .. .. .. .. .. .. .. .. ........ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ......................................................................... .. .. .. .. .. .. .. .. .. ......... ... ... .... ............ ..................... . . . . . . . ... . . . . . . . ... ... ......................... .......................................... ... . . . . . . . . . . . . . . ... . ... ... .. .......... ........ ........ ................... ..... .. ... ... ....... ....................................................................................... ... .. . . . . . . ... . . . . ... . . . .. . ... ... ... ... .. ... .. ... .......... ........ ......... ....... .......... ............... .... ... ... ...................... .. ....... ........ ....... ........ .............. ......... ... ... ... ................. ............... ....... ........ ........ .......... ........ ................... ............. .. ... .. ... ........... ........ ........... ........ .......... ........ ...................... ....... . .. .. ... ............. .................. ........................................................................................................................... . . ... . . .. ........ ......... ............ ........... .................................................................... ... .. ... .. ..... ......... ..... .. .... ..... ...... ..... ..... ..... ..... ... ... ....... ............................................................................................................................................................................................................................................................................... ... .. ... . . . . . .. ........ .. ... .. ....... ................. ..................................................................................................................... . . . . . . . . . . ... .. ............................... .......................................................................................... ... ... ....... . . . . . .. ... ... ... .............. ............... .............. ........................................................................... ... ........... ........ ....... .......... ........... ............ .. .. ... ... ......................... .......... ................................... . . .............. ............................ ... ... ................ ... ... ... ... .. ... ... ... ... ... ... ... .. .. .... . . ... .. ... ... ... ........ .... ... ... ... ... ... ... ... ... ... .. .. ... . . ... ... ... ... ... ... ... ... ... . .. .. ... .. ........ .. . . . . . . . . . . . . . . .. . ... .. ... ... ... ... ... .. ... .. ...... .. .. ............................................................................................................................................................................................................................................................................. .. .. . . .. .. ... ... .. .. .. ..

5.60

5.55

5.50

92000

94000

Emmanuel Flachaire

96000

98000

Non-Parametric Econometrics

Heteroskedasticity Efficient estimator in parametric model

I

Linear regression model, with Ω = diag{σ12 , σ22 , . . . , σn2 }: y = Xβ + ε

V (ε|X ) = Ω

I

Consequences: OLS parameter consistent but not efficient

I

Efficient estimator: the GLS estimator is blue, βˆgls = (X > Ω−1 X )−1 X > Ω−1 y

V (βˆgls ) = (X > Ω−1 X )−1

I

A consistent estimate of Ω is required: σi2 = Wi γ

I

Feasible GLS: use σ ˆi , fitted values from εˆ2i = Wi γ + i

Emmanuel Flachaire

Non-Parametric Econometrics

Heteroskedasticity Efficient estimator in nonparametric model

I

The double residual estimator is given by an OLS estimation of y −m ˆ y = (X − m ˆx) β + ε

I

Nonparametric estimator m ˆ y and m ˆ x are still consistent

I

Efficient estimator: compute the FGLS estimator ˆ −1 (X − m ˆ −1 (y − m βˆgres = [(X − m ˆ x )> Ω ˆ x )]−1 (X − m ˆ x )> Ω ˆy) ˆ −1 (X − m V (βˆgres ) = [(X − m ˆ x )> Ω ˆ x )]−1 ,

I

Note that we can use a nonparametric model εˆ2i = g (Wi ) + i

Emmanuel Flachaire

Non-Parametric Econometrics

Heteroskedasticity Efficient estimator in nonparametric model

I

The differencing estimator is given by an OLS estimation of ∆y = ∆X β + η

I

It is a (parametric) linear regression model

I

With heteroskedasticity error terms, a GLS estimator is blue: βˆdiff = (∆X > Ω−1 ∆X )−1 ∆X > Ω−1 ∆y

I

In practice: replace Ω by a consistent estimator (FGLS)

Emmanuel Flachaire

Non-Parametric Econometrics

Heteroskedasticity Robust estimator in parametric model

I

Linear regression model, with Ω = diag{σ12 , σ22 , . . . , σn2 }: y = Xβ + ε

V (ε|X ) = Ω

I

Consequences: OLS parameter consistent, not OLS variance

I

Robust estimator: OLS parameter estimator with variance OK V (βˆols ) = (X > X )−1 (X > ΩX )−1 (X > X )−1

I I

ˆ = diag{ˆ Consistent estimate of X > ΩX : use Ω ε21 , εˆ22 , . . . , εˆ2n } ˆ is a consistent estimator of X > ΩX , but Ω ˆ Note that X > ΩX is not a consistent estimator of Ω

Emmanuel Flachaire

Non-Parametric Econometrics

Heteroskedasticity Robust estimator in nonparametric model

I

The double residual estimator is given by an OLS estimation of y −m ˆ y = (X − m ˆx) β + ε

I

Nonparametric estimator m ˆ y and m ˆ x are still consistent

I

Robust estimator: a consistent covariance matrix is given by

V (βˆres ) = [(X −m ˆ x )> (X −m ˆ x )]−1 (X −m ˆ x )> Ω (X −m ˆ x )[(X −m ˆ x )> (X −m ˆ x )]−1 I

ˆ = diag{ˆ Replace Ω by Ω ε21 , εˆ22 , . . . , εˆ2n }

I

Idem for the differencing estimator

Emmanuel Flachaire

Non-Parametric Econometrics

Autocorrelation Parametric model with autocorrelation I

AR(1) model: yt = Xt β + ε

εt = ρεt−1 + ηt

I

The covariance matrix of error terms V (ε|X ) is equal to   1 ρ ρ2 . . . ρn−1 1 ρ . . . ρn−2  ση2   ρ  Ω=  .. .. .. .. ..  2 1−ρ  . . . . .  n−1 n−2 n−3 ρ ρ ρ ... 1

I

Consequences: βˆols consistent but not efficient With consistent estimates of ση , ρ a consistent estimator of Ω can be obtained and used to get a GLS (efficient) estimator

I

βˆgls = (X > Ω−1 X )−1 X > Ω−1 y Emmanuel Flachaire

V (βˆgls ) = (X > Ω−1 X )−1

Non-Parametric Econometrics

Autocorrelation Nonparametric model with autocorrelation I

The double residual estimator is given by an OLS estimation of y −m ˆ y = (X − m ˆx) β + ε

I

Nonparametric estimator m ˆ y and m ˆ x are still consistent

I

Efficient estimator: compute the FGLS estimator ˆ −1 (X − m ˆ −1 (y − m βˆgres = [(X − m ˆ x )> Ω ˆ x )]−1 (X − m ˆ x )> Ω ˆy) ˆ −1 (X − m V (βˆgres ) = [(X − m ˆ x )> Ω ˆ x )]−1 ,

I

ˆ is a consistent estimator of Ω, based on σ Ω ˆη , ρˆ

I

Problem: m ˆ y and m ˆ x are consistent but kernel estimation may be poor because bandwidth selection is more problematic with autocorrelated error terms in nonparametic regression

I

Robust estimator (Newey-West): unreliable in finite sample Emmanuel Flachaire

Non-Parametric Econometrics

Autocorrelation Parametric dynamic model I

Partial autoregressive linear model: yt = β yt−1 + m(yt−2 ) + εt ,

I

The double residual estimator is the OLS estimator from yt − E (yt |yt−2 ) = β [yt−1 − E (yt−1 |yt−2 )] + εt if we replace the expectations by consistent estimates

I

The double residual estimator is asymptotically Normal

I

Its variance is σε2 /(nσε22 ), with σε22 ← regress yt−1 on yt−2

I

We can use other models, as the additive model: yt = m1 (yt−1 ) + m2 (yt−2 ) + εt

Emmanuel Flachaire

Non-Parametric Econometrics

Endogeneity Parametric linear model

I

The linear regression model is defined as, y = Xβ + ε

I

If E (ε|X ) 6= 0, OLS estimation is not valid, we should use IV

I

Use instruments W , correlated to X , not to ε

I

IV estimation: regress y on PW X by OLS βˆiv = (X > PW X )−1 X > PW y

I

V (βˆiv ) = σ 2 (X > PW X )−1 ,

PW X is the part of X explained by W : OLS fitted values from X = Wπ + ν

Emmanuel Flachaire

Non-Parametric Econometrics

Endogeneity Partial linear model

I

The partial linear regression model is defined as, y = X β + m(Z ) + ε

I

Correlation between ε et regressors can be: I

with X : in the parametric part of the model

I

with Z : in the nonparametric part of the model

Emmanuel Flachaire

Non-Parametric Econometrics

Endogeneity in the parametric part of the partial linear model I

The partial linear regression model is defined as, y = X β + m(Z ) + ε

I

E (ε|X ) 6= 0 E (ε|Z ) = 0

The double residual estimator is given by an OLS estimation of y −m ˆ y = (X − m ˆx) β + ε

I

Nonparametric estimator m ˆ y and m ˆ x are still consistent, since Z is not endogeneous

I

We can compute IV estimator from the model above: βˆvires = [(X − m ˆ x )> PW (X − m ˆ x )]−1 (X − m ˆ x )> PW (y − m ˆ y ). V (βˆivres ) = σε2 [(X − m ˆ x )> PW (X − m ˆ x )]−1 Emmanuel Flachaire

Non-Parametric Econometrics

Endogeneity in the nonparametric part of the partial linear model

I

The partial linear regression model is defined as, y = X β + m(Z ) + ε

I

E (ε|X ) = 0

E (ε|Z ) 6= 0

The double residual estimator is given by an OLS estimation of y −m ˆ y = (X − m ˆx) β + ε

I

Nonparametric estimator m ˆ y and m ˆ x are not consistent, since Z is endogeneous

I

We cannot compute IV estimator from the model above!

Emmanuel Flachaire

Non-Parametric Econometrics

Endogeneity IV estimation

I

The partial linear regression model is defined as, y = X β + m(Z ) + ε

I

E (ε|X ) = 0

E (ε|Z ) 6= 0

If we apply the double residual principle, we obtain E (y |W ) = E (X |W )β + E [m(Z )|W ].

I

The nonparametric term does not disappear!

I

Solution: ill-posed problem → difficult to solve

Emmanuel Flachaire

Non-Parametric Econometrics

Endogeneity Control function I

Assume that W is correlated to Z , not to ε, such that Z = Wπ + ν

E (ν|W ) = 0

I

Additional hypothesis: E (ε|Z , ν) = ρν

I

Thus we have ε = ρν + η, the partial linear model becomes y = X β + m(Z ) + νρ + η with η uncorrelated to regressors

I

If we apply the double residual principle, we obtain y − E (y |Z ) = [X − E (X |Z )]β + [ν − E (ν|Z )]ρ + η.

I

Replace ν by OLS residuals from the first model above

I

Note that an exogeneous test can be done with H0 : ρ = 0 Emmanuel Flachaire

Non-Parametric Econometrics

Discrete choice model Parametric model I

A binary discrete choice model is defined as y◦ = Xβ + ε

ε ∼ F (0, 1),

I

We do not observe y ◦ but a binary variable y

I

The expectation of y with respect to X is defined as E (y |X ) = P(y = 1|X ) = F (X β).

I

That is, y ∼ Bernouilli with parameter P(y = 1|X ) = F (X β)

I

Maximum likelihood estimation (ML) `(y , β) =

n X

yt log F (Xt β) + (1 − yt ) log 1 − F (Xt β)

t=1 I

F is a distribution chosen a priori - Logistic or Normal Emmanuel Flachaire

Non-Parametric Econometrics

Discrete choice model Partial linear model I

The partial linear model is defined as E (y |X , Z ) = F [X β + m(Z )]

I

Iterative estimation: with m ˆ (i) we estimate β by ML `(y , β) =

n X

yt log F [Xt β+m ˆ (i) ]+(1−yt ) log 1−F [Xt β+m ˆ (i) ]

t=1 I

With βˆ(i) , we can estimate m by ML

`(y , m) =

n X

yt log F [Xt βˆ(i) +m(Z )]+(1−yt ) log 1−F [Xt βˆ(i) +m(Z )]

t=1 I

Maximization requires to solve nonlinear optimisation problem Emmanuel Flachaire

Non-Parametric Econometrics

Discrete choice model Index model

I

If F is misspecified, ML estimator is inconsistent!

I

The index model is defined as E (y |X ) = F (X β),

I

where F is an unknown function estimated nonparametrically ˆ we can estimate F from the nonparametric model With β, ˆ +η y = F (X β)

I

Estimation of β is not easy

Emmanuel Flachaire

Non-Parametric Econometrics