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Chapter 5: The Generalized Linear Regression Modeland Heteroscedasticity
Advanced Econometrics - HEC Lausanne
Christophe Hurlin
University of Orléans
December 15, 2013
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Section 1
Introduction
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1. Introduction
The outline of this chapter is the following:
Section 2. The generalized linear regression model
Section 3. Ine¢ ciency of the Ordinary Least Squares
Section 4. Generalized Least Squares (GLS)
Section 5. Heteroscedasticity
Section 6. Testing for heteroscedasticity
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1. Introduction
References
Amemiya T. (1985), Advanced Econometrics. Harvard University Press.
Greene W. (2007), Econometric Analysis, sixth edition, Pearson - PrenticeHil (recommended)
Pelgrin, F. (2010), Lecture notes Advanced Econometrics, HEC Lausanne (aspecial thank)
Ruud P., (2000) An introduction to Classical Econometric Theory, OxfordUniversity Press.
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1. Introduction
Notations: In this chapter, I will (try to...) follow some conventions ofnotation.
fY (y) probability density or mass function
FY (y) cumulative distribution function
Pr () probability
y vector
Y matrix
Be careful: in this chapter, I dont distinguish between a random vector(matrix) and a vector (matrix) of deterministic elements (except in section2). For more appropriate notations, see:
Abadir and Magnus (2002), Notation in econometrics: a proposal for astandard, Econometrics Journal.
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Section 2
The generalized linear regression model
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2. The generalized linear regression model
Objectives
The objective of this section are the following:
1 Dene the generalized linear regression model
2 Dene the concept of heteroscedasticity
3 Dene the concept of autocorrelation (or correlation) of disturbances
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2. The generalized linear regression model
Consider the (population) multiple linear regression model:
y = Xβ+ ε
where (cf. chapter 3):
y is a N 1 vector of observations yi for i = 1, ..,N
X is a N K matrix of K explicative variables xik for k = 1, .,K andi = 1, ..,N
ε is a N 1 vector of error terms εi .
β = (β1..βK )> is a K 1 vector of parameters
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2. The generalized linear regression model
In chapter 3 (linear regression model), we assume spherical disturbances(assumption A4):
V (εjX) = σ2IN
In this chapter, we will relax the assumption that the errors areindependent and/or identically distributed and we will study:
1 Heteroscedasticity
2 Autocorrelation or correlation.
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2. The generalized linear regression model
Denition (Generalized linear regression model)The generalized linear regression model is dened as to be:
y = Xβ+ ε
where X is a matrix of xed or random regressors, β 2 RK , and the errorterm ε satises:
E (εjX) = 0N1V (εjX) = Σ = σ2Ω
where Ω and Σ are symmetric positive denite matrices.
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2. The generalized linear regression model
Reminder
V (εjX)| z NN
= E
εε>X| z
NN
=
0BB@V
ε21X Cov ( ε1ε2jX) .. Cov ( ε1εN jX)
E ( ε2ε1jX) V
ε22X .. Cov ( ε2εN jX)
.. .. .. ..Cov ( εN ε1jX) .. .. V
ε2NX
1CCA
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2. The generalized linear regression model
Remark
In the generalized linear regression model, we have
V (εjX) = Σ = σ2Ω
with
Σ =
0BB@σ21 σ12 .. σ1Nσ21 σ22 .. σ2N.. .. .. ..
σN1 .. .. σ2N
1CCA = σ2
0BB@ω11 ω12 .. ω1N
ω21 ω22 .. ω2N
.. .. .. ..ωN1 .. .. ωNN
1CCAand ωij = σij/σ2.
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2. The generalized linear regression model
Denition (Heteroscedasticity)
Disturbances are heteroscedastic when they have di¤erent (conditional)variances:
V ( εi jX) 6= V ( εj jX) for i 6= j
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2. The generalized linear regression model
Remarks
1 Heteroscedasticity often arises in volatile high-frequency time-seriesdata such as daily observations in nancial markets.
2 Heteroscedasticity often arises in cross-section data where the scaleof the dependent variable and the explanatory power of the modeltend to vary across observations. Microeconomic data such asexpenditure surveys are typical
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2. The generalized linear regression model
Example (Heteroscedasticity)If the disturbances are heteroscedastic but they are still assumed to beuncorrelated across observations, so Ω and Σ would be:
Σ =
0BB@σ21 0 .. 00 σ22 .. 0.. .. .. ..0 .. .. σ2N
1CCA = σ2Ω = σ2
0BB@ω1 0 .. 00 ω2 .. 0.. .. .. ..0 .. .. ωN
1CCAwith ωi = σ2i /σ2 for i = 1, ..,N.
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2. The generalized linear regression model
Denition (Autocorrelation)
Disturbances are autocorrelated (or correlated) when:
Cov ( εi , εj jX) 6= 0 for i 6= j
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2. The generalized linear regression model
Example (Autocorrelation)For instance, time-series data are usually homoscedastic, butautocorrelated, so Ω and Σ would be:
Σ =
0BB@σ2 σ12 .. σ1Nσ21 σ2 .. σ2N.. .. .. ..
σN1 .. .. σ2
1CCA = σ2Ω = σ2
0BB@1 ω12 .. ω1N
ω21 1 .. ω2N
.. .. .. ..ωN1 .. .. 1
1CCAwith ωij = σij/σ2 for i = 1, ..,N denotes the correlation (autocorrelation)
ωij =σijσ2= cor (εi , εj )
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2. The generalized linear regression model
Key Concepts
1 The generalized linear regression model
2 Heteroscedasticity
3 Autocorrelation (or correlation) of disturbances
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Section 3
Ine¢ ciency of the Ordinary Least Squares
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3. Ine¢ ciency of the Ordinary Least Squares
Objectives
The objective of this section are the following:
1 Study the properties of the OLS estimator in the generalized linearregression model
2 Study the nite sample properties of the OLS
3 Study the asymptotic properties of the OLS
4 Introduce the concept of robust / non-robust inference
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3. Ine¢ ciency of the Ordinary Least Squares
Introduction
Assume that the data are generated by the generalized linear regressionmodel:
y = Xβ+ ε
E (εjX) = 0N1V (εjX) = σ2Ω = Σ
Now consider the OLS estimator, denoted bβOLS , of the parameters β:
bβOLS = X>X1 X>yWe will study its nite sample and asymptotic properties.
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Assumption 3: Strict exogeneity of the regressors)The regressors are exogenous in the sense that:
E (εjX) = 0N1
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3. Ine¢ ciency of the Ordinary Least Squares
Finite sample properties of the OLS estimator
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Bias)In the generalized linear regression model, under the assumption A3(exogeneity), the OLS estimator is unbiased:
EbβOLS = β0
where β0 denotes the true value of the parameters.
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3. Ine¢ ciency of the Ordinary Least Squares
Remark
Heteroscedasticity and/or autocorrelation dont induce a bias for theOLS estimator
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3. Ine¢ ciency of the Ordinary Least Squares
Proof
bβOLS = X>X1 X>y = β0 +X>X
1 X>ε
So we have:
EbβOLS X = β0 +
X>X
1 X>E (εjX)
Under assumption A3 (exogeneity), E (εjX) = 0. Then, we get:
EbβOLS X = β0
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3. Ine¢ ciency of the Ordinary Least Squares
Proof (contd)
EbβOLS X = β0
So, we have:
EbβOLS = EX
EbβOLS X = EX (β0) = β0
where EX denotes the expectation with respect to the distribution of X.
The OLS estimator is unbiased:
EbβOLS = β0
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Bias)In the generalized linear regression model, under the assumption A3(exogeneity), the OLS estimator has a conditional variance covariancematrix given by
VbβOLS X = σ20
X>X
1X>ΩX
X>X
1and a variance covariance matrix given by:
VbβOLS = EX
VbβOLS X
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3. Ine¢ ciency of the Ordinary Least Squares
Proof
bβOLS = X>X1 X>y = β0 +X>X
1 X>ε
So we have:
VbβOLS X = E
X>X
1X>εε>X
X>X
1X=
X>X
1X>E
εε>
XX X>X1= σ20
X>X
1X>ΩX
X>X
1
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Variance estimator)An estimator of the variance covariance matrix of the OLS estimatorbβOLS is given by
bV bβOLS = bσ2 X>X1 X> bΩX X>X1where bσ2 bΩ is a consistent estimator of Σ = σ2Ω. This estimator holdswhether X is stochastic or non-stochastic.
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Normality assumption)
Under assumptions A3 (exogeneity) and A6 (normality), the OLSestimator obtained in the generalized linear regression model has an(exact) normal conditional distribution:
bβOLS X N β0, σ
2X>X
1X>ΩX
X>X
1
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3. Ine¢ ciency of the Ordinary Least Squares
Asymptotic properties of the OLS estimator
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3. Ine¢ ciency of the Ordinary Least Squares
Assumptions
plim1NX>X = Q
plim1NX>ΩX = Q
where:
1 Q is a K K nite (non null) denite positive matrix
2 Q is a K K nite (non null) denite positive matrix with
rank (Q) = K
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Consistency of the OLS estimator)
If plim N1X>ΩX and plim N1X>X are both nite positive denitematrices, then bβOLS is a consistent estimator of β:
bβOLS p! β0
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3. Ine¢ ciency of the Ordinary Least Squares
Proof
bβOLS = β0 +X>X
1 X>ε
We know that under assumption A3 (exogeneity):
plim1NX>ε = 0K1
plim1NX>X = Q
So, we haveplim bβOLS = β0
So, the estimator bβ is consistent. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 35 / 153
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Asymptotic distribution of the OLS)If the regressors are su¢ ciently well behaved and the o¤-diagonal terms indiminish su¢ ciently rapidly, then the least squares estimator isasymptotically normally distributed with
pNbβOLS β0
d! N
0, σ2Q1QQ1
where
Q = plim1NX>X Q = plim
1NX>ΩX
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3. Ine¢ ciency of the Ordinary Least Squares
Remark
1 Regularity conditions include the exogeneity conditions, but also (i)the regressors are su¢ ciently well-behaved and (ii) the o¤-diagonalterms of the variance-covariance matrix diminish su¢ ciently rapidly(relative to the diagonal elements).
2 For a formal proof in a general case, see Amemiya (1985, p. 187).
Amemiya T. (1985), Advanced Econometrics. Harvard University Press.
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Asymptotic variance)Under suitable regularity conditions, the asymptotic variance covariancematrix of the OLS estimator bβ is given by:
Vasy
bβOLS = σ2
NQ1QQ1
withQ = plim
1NX>X Q = plim
1NX>ΩX
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3. Ine¢ ciency of the Ordinary Least Squares
Fact (Non-robust inference)Because the variance of the least squares estimator is not
σ2X>X
1statistical inference (non-robust inference) based on
bσ2 X>X1 may be misleading. For instance the t-test-statistic:tβk =
bβkbσpmkkwhere mkk is kth diagonal element of X>X do not have a Studentdistribution.
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3. Ine¢ ciency of the Ordinary Least Squares
Robust / Non-robust inference
As a consequence, the familiar inference procedures based on the Fand t distributions will no longer be appropriate.
The question is to know how to estimate VbβOLS in the context
of the linear generalized regression model in order to make robustinference.
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3. Ine¢ ciency of the Ordinary Least Squares
Denition (Estimator of the asymptotic variance covariance matrix)
If Σ = σ2Ω were known, the consistent estimator of the (asymptotic)variance covariance of bβOLS would be:
bVasy
bβOLS = σ2
N
1NX>X
1 1NX>ΩX
1NX>X
1
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3. Ine¢ ciency of the Ordinary Least Squares
Proof
By denition:
Q = plim1NX>X
Q = plim1NX>ΩX
So,
plim bVasy
bβOLS = plimσ2
N
1NX>X
1 1NX>ΩX
1NX>X
1=
σ2
NQ1QQ1
Or equivalently bVasy
bβOLS p! Vasy
bβOLS
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3. Ine¢ ciency of the Ordinary Least Squares
Reminder
X>X =N
∑i=1xix>i
X>ΩX =N
∑i=1
N
∑j=1
ωijxix>i
X>ΣX =N
∑i=1
N
∑j=1
σijxix>i = σ2N
∑i=1
N
∑j=1
ωijxix>i
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3. Ine¢ ciency of the Ordinary Least Squares
Remark
The estimator
bVasy
bβOLS = σ2
N
1NX>X
1 1NX>ΩX
1NX>X
1can also be written as
bVasy
bβOLS = σ2
N
1N
N
∑i=1xix>i
!1 1N
N
∑i=1
N
∑j=1
ωijxix>i
! 1N
N
∑i=1xix>i
!1
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3. Ine¢ ciency of the Ordinary Least Squares
Remark
In the next section, we will give a feasible estimator bVasy
bβOLS in thespecic case of an heteroscedastic model.
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3. Ine¢ ciency of the Ordinary Least Squares
Summary
In the GLR model, under some regularity conditions:
1 The OLS estimator is unbiased
2 The OLS estimator is (weakly) consistent
3 The OLS estimator is asymptotically normally distributed
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3. Ine¢ ciency of the Ordinary Least Squares
Summary
But...
1 The inference based on the estimator σ2X>X
1is misleading.
2 The OLS is ine¢ cient.
VbβOLS I1N (β0) is a positive denite matrix
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3. Ine¢ ciency of the Ordinary Least Squares
Key Concepts
1 OLS estimator in the generalized regression model
2 Finite sample properties
3 Asymptotic variance covariance matrix of the OLS estimator
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Section 4
Generalized Least Squares (GLS)
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4. Generalized Least Squares (GLS)
Objectives
The objective of this section are the following:
1 Dene the Generalized Least Squares (GLS)
2 Dene the Feasible Generalized Least Squares (FGLS)
3 Study the statistical properties of the GLS and FGLS estimators
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4. Generalized Least Squares (GLS)
Consider the generalized linear regression model with
V (εjX) = Σ = σ2Ω
We will distinguish two cases:
Case 1: the variance covariance matrix Σ is known (unrealistic case)
Case 2: the variance covariance matrix Σ is unknown
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4. Generalized Least Squares (GLS)
Case 1: Σ is known
The Generalized Least Squares (GLS) estimator
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4. Generalized Least Squares (GLS)
Denition (Factorisation)Since Ω is a positive denite matrix, it can factored as follows:
Ω = CΛC>
where the columns of C are the characteristics vectors of Ω, thecharacteristic roots of Ω are arrayed in the diagonal matrix Λ, and
C>C = CC> = IN
where I denotes the identity matrix.
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4. Generalized Least Squares (GLS)
DenitionWe dene the matrix P such that
P> = CΛ1/2
so thatΩ1 = P>P
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4. Generalized Least Squares (GLS)
ProofP> = CΛ1/2
Since Λ is diagonal, Λ1/2Λ1/2 = Λ1, and we have:
P>P = CΛ1/2Λ1/2C> = CΛ1C>
Consider the quantity P>PΩ:
P>PΩ = CΛ1C>CΛC>
= CΛ1ΛC>
= CC>
= IN
Since C satises CC> = IN . Then, P>P = Ω1
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4. Generalized Least Squares (GLS)GLS estimator
Premultiply the generalized linear regression model by P to obtain
Py = PXβ+Pε
or equivalentlyy = Xβ+ ε
The conditional variance of ε is
V (εjX) = E
εε>X
= PE
εε>XP>
= σ2PΩP>
= σ2Λ1/2C>CΛC>CΛ1/2
= σ2IN
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4. Generalized Least Squares (GLS)
GLS estimator (contd)
y = Xβ+ ε
V (εjX) = σ2IN
The classical regression model applies to this transformed model.
If Ω is assumed to be known, y = Py and X = PX are observed data.
So, we can apply the ordinary least squares to this transformed model:
bβ = X>X1 X>y
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4. Generalized Least Squares (GLS)
GLS estimator (contd)
bβ =X>X
1 X>y
=
X>P>PX
1 X>P>Py
=
X>Ω1X
1 X>Ω1y
This estimator is the generalized least squares (GLS) estimator of β.
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4. Generalized Least Squares (GLS)
Denition (GLS estimator)
The Generalized Least Squares (GLS) estimator of β is dened as to be:
bβGLS = X>Ω1X1
X>Ω1y
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4. Generalized Least Squares (GLS)
Denition (Bias)
Under the exogeneity assumption (A3), the estimator bβGLS is unbiased:EbβGLS = β0
where β0 denotes the true value of the parameters.
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4. Generalized Least Squares (GLS)
Proof
We have:
bβGLS = X>Ω1X1
X>Ω1y= β0 +
X>Ω1X
1 X>Ω1ε
So,
EbβGLS X = β0 +
X>Ω1X
1 X>Ω1E (εjX)
Under the exogeneity assumption A3, E (εjX) = 0, so we have
EbβGLS X = β0
andEbβGLS = EX
EbβGLS X = EX (β0) = β0
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4. Generalized Least Squares (GLS)
Denition (Variance covariance matrix)
The conditional variance covariance matrix of the estimator bβGLS isdened as to be:
VbβGLS X = σ2
X>Ω1X
1The variance covariance matrix is given by
VbβGLS = σ2EX
X>Ω1X
1
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4. Generalized Least Squares (GLS)Proof
Consider the denition of bβGLS in the transformed model:bβGLS = β0 +
X>X
1 X>ε
VbβGLS X = X>X1 X>E
εε>
XX X>X1Since E
εε>
X = σ2IN , we have
VbβGLS X = σ2
X>X
1X>X
X>X
1= σ2
X>X
1= σ2
X>P>PX
1= σ2
X>Ω1X
1
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4. Generalized Least Squares (GLS)
Denition (Consistency)
Under the exogeneity assumption A3, the GLS estimator bβGLS is (weakly)consistent: bβGLS p! β0
as soon asplim
1NX>X = Q
where Q is a nite positive denite matrix.
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4. Generalized Least Squares (GLS)
Proof
bβGLS = β0 +X>Ω1X
1 X>Ω1ε
Under the assumption A3 (exogeneity):
plim1NX>Ω1ε = 0K1
plim1NX>Ω1X = Q
So, we haveplim bβGLS = β0
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4. Generalized Least Squares (GLS)
Denition (Asymptotic distribution)
Under some regularity conditions, the GLS estimator bβGLS isasymptotically normally distributed:
pNbβGLS β0
d! N
0, σ2Q1
where
Q = plim1NX>X = plim
1NX>Ω1X
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4. Generalized Least Squares (GLS)
Denition (Asymptotic variance covariance matrix)
The asymptotic variance covariance matrix of the estimator bβGLS is:Vasy
bβGLS = σ2
NQ1
If Σ = σ2Ω is known, a consistent estimator is given by:
bVasy
bβGLS = σ2
N
X>Ω1X
1This estimator holds whether X is stochastic or non-stochastic.
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4. Generalized Least Squares (GLS)
Theorem (BLUE estimator)
The GLS estimator bβGLS is the minimum variance linear unbiasedestimator (BLUE estimator) in the semi-parametric generalized linearregression model. In particular, the matrix dened by:
Vasy
bβOLSVasy
bβGLSis a positive semi denite matrix.
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4. Generalized Least Squares (GLS)
Theorem (E¢ ciency)Under suitable regularity conditions, in a parametric generalized linearregression model, the GLS estimator bβGLS is e¢ cient
VbβGLS = I1N (β0)
where I1N (β0) denotes the FDCR or Cramer-Rao bound.
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4. Generalized Least Squares (GLS)
Remark
In a Gaussian generalized linear regression model (under assumption A6),the likelihood of the sample is given by:
LN (θ; y j x) =2πσ2
N/2 jΩjN/2
exp 12σ2
(yXβ)> Ω1 (yXβ)
The log-likelihood is dened as to be:
`N (θ; y j x) = N2ln2πσ2
N2log (jΩj)
12σ2
(yXβ)> Ω1 (yXβ)
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4. Generalized Least Squares (GLS)
Remark
For testing hypotheses, we can apply the full set of results in Chapter 4 tothe transformed model. For instance, for testing the p linear constraintsH0 : Rβ = q, the appropriate test-statistic is:
F =1p
Rbβ
GLS q
> σ2R
X>Ω1X
1R>1
RbβGLS q
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4. Generalized Least Squares (GLS)
FactTo summarize, all the results for the classical model, including the usualinference procedures, apply to the transformed model.
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4. Generalized Least Squares (GLS)
Case 2: Σ is unknown
The Feasible Generalized Least Squares (FGLS) estimator
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4. Generalized Least Squares (GLS)
Introduction
1 If Σ contains unknown parameters that must be estimated, thengeneralized least squares is not feasible.
2 With an unrestricted matrix Σ = σ2Ω, there are N (N + 1) /2additional parameters (since Σ is symmetric) to estimate
3 This number is far too many to estimate with N observations.
4 Obviously, some structure must be imposed on the model if we areto proceed.
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4. Generalized Least Squares (GLS)
Denition (Structure of variance covariance matrix)We assume that the conditional variance covariance matrix of thedisturbances can be expressed as a function of a small set of parameters α:
V (εjX) = σ2Ω (α)
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4. Generalized Least Squares (GLS)
Example (Time series)For instance, a commonly used formula in time-series settings is
Ω (ρ) =
0BBBBBB@
1 ρ ρ2 ρ3 .. ρN1
ρ 1 ρ ρ2 .. ρN2
ρ2 ρ 1 ρ .. ρN3
ρ3 ρ2 ρ 1 .. .... .. .. .. .. ..
ρN1 ρN2 ρN3 .. .. 1
1CCCCCCA
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4. Generalized Least Squares (GLS)
Example (Heteroscedascticity)If we consider a heteroscedastic model, where the variance of εi dependson a variable zi , with
V ( εi jX) = σ2zθi
we have
Ω (θ) =
0BBBB@zθ1 0 0 .. 00 zθ
2 0 .. 00 0 zθ
3 .. 0.. .. .. .. ..0 0 0 .. zθ
N
1CCCCA
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4. Generalized Least Squares (GLS)
Denition (Feasible Generalized Least Squares (FGLS))
Consider a consistent estimator bα of α, then the Feasible Least GeneralizedSquares (FGLS) estimator of β is dened as to be:
bβFGLS = X> bΩ1X1
X> bΩ1y
where bΩ = Ω (bα) is a consistent estimator of Ω (α) .
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4. Generalized Least Squares (GLS)
Remark
If
plim
1NX> bΩ1
X1NX>Ω1X
= 0
plim
1NX> bΩ1
y1NX>Ω1y
= 0
Then the GLS and FGLS estimators are asymptotically equivalent
bβFGLS bβGLS p! 0K1
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4. Generalized Least Squares (GLS)
Theorem (E¢ ciency)An asymptotically e¢ cient FGLS estimator does not require that we havean e¢ cient estimator of α; only a consistent one is required to achieve fulle¢ ciency for the FGLS estimator.
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4. Generalized Least Squares (GLS)
Remark
If the estimator bα is consistentbα p! α
then the FGLS estimator has the same asymptotic properties (consistency,e¢ ciency, asymptotic distribution etc.) than the GLS estimator.
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4. Generalized Least Squares (GLS)
Key Concepts
1 Factorisation of the variance covariance matrix
2 Generalized Least Squares (GLS) estimator
3 Feasible Generalized Least Squares (FGLS) estimator
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Section 5
Heteroscedasticity
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5. Heteroscedasticity
Objectives
The objective of this section are the following:
1 To determine the properties of the OLS in presence ofheteroscedasticity
2 To estimate the asymptotic variance covariance matrix of the OLSestimator in presence of heteroscedasticity
3 To introduce the concept of robust inference (to heteroscedasticity)
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5. Heteroscedasticity
Introduction
In the rest of this chapter, we will focus on the case of heteroscedasticdisturbances.
V ( εi jX) = σ2i for i = 1, ..,N
Heteroscedasticity arises in numerous applications, in both cross-sectionand time-series data.
For example, even after accounting for rm sizes, we expect to observegreater variation in the prots of large rms than in those of small ones.
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5. Heteroscedasticity
Assumption: We assume that the disturbances are pairwiseuncorrelated and heteroscedastic:
V (εjX) = Σ = σ2Ω
with
Σ =
0BB@σ21 0 .. 00 σ22 .. 0.. .. .. ..0 .. .. σ2N
1CCA = σ2Ω = σ2
0BB@ω1 0 .. 00 ω2 .. 0.. .. .. ..0 .. .. ωN
1CCAwith ωi = σ2i /σ2 for i = 1, ..,N.
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5. Heteroscedasticity
Denition (Scaling)The fact to scale the variances as
σ2i = σ2ωi for i = 1, ..,N
allows us to use a normalisation on Ω
trace (Ω) =N
∑i=1
ωi = N
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5. Heteroscedasticity
Introduction (contd)
We will consider three cases:
Case 1: the heteroscedasticity form (structure) is unknown: OLSestimator and robust inference
Case 2: the variance covariance matrix Σ is known: GLS or WeightedLeast Square (WLS)
Case 3: the variance covariance matrix Σ is unknown but its form(structure) is known: two-steps or iterated FGLS estimator
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5. Heteroscedasticity
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5. Heteroscedasticity
Case 1: Heteroscedasticity of unknown form
OLS and robust inference
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5. Heteroscedasticity
Assumption: We assume that the variances σ2i are unknown for i = 1, ..Nand no particular form (structure) is imposed on Ω (or Σ).
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5. Heteroscedasticity
Introduction
1 The GLS cannot be implemented since Σ is unknown.
2 The FGLS estimator requires to estimate (in a rst step) Nparameters σ21, .., σ2N . With N observations, the FGLS is not feasible.
3 The only solution to estimate β consists in using the OLS.
4 Under suitable regularity conditions, the OLS estimator is unbiased,consistent, asymptotically normally distributed but... ine¢ cient.
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5. Heteroscedasticity
Introduction (contd)
Consider the OLS estimator:
bβOLS = X>X1 X>yWe know that bβOLS asy N
β0,
σ2
NQ1QQ1
Vasy
bβOLS = σ2
NQ1QQ1
withQ = plim
1NX>X Q = plim
1NX>ΩX
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5. Heteroscedasticity
Problem (Robust inference with OLS)The conventionally estimated covariance matrix for the least squares
estimator σ2X>X
1is inappropriate; the appropriate matrix is
σ2X>X
1 X>ΩX
1 X>X
1. It is unlikely that these two would
coincide, so the usual estimators of the standard errors are likely to be
erroneous. The inference (test-statistics) based σ2X>X
1is
misleading.
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5. Heteroscedasticity
Question
How to estimate Vasy
bβOLS and to make robust inference?Vasy
bβOLS = σ2
NQ1QQ1
Q = plim1NX>X Q = plim
1NX>ΩX
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5. Heteroscedasticity
We seek an estimator for
Q = plim1NX>ΩX = plim
1N
N
∑i=1
ωixix>i = EX
ωixix>i
or equivalently of
Q = plim1NX>ΣX = plim
1N
N
∑i=1
σ2i xix>i = EX
σixix>i
with
Q = σ2Q
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5. Heteroscedasticity
Q = plim1NX>ΣX = plim
1N
N
∑i=1
σ2i xix>i
White (1980) shows that under very general condition, the estimator
S0 =1N
N
∑i=1
bε2i xix>iwhere bεi = yi x>i bβOLS , converges to Q = σ2Q
S0p! Q = σ2Q
White, H. A Heteroscedasticity-Consistent Covariance Matrix Estimator anda Direct Test for Heteroscedasticity.Econometrica, 48, 1980, pp. 817838.
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5. Heteroscedasticity
Vasy
bβOLS = σ2
NQ1QQ1
We know that:
S0 =1N
N
∑i=1
bε2i xix>i p! σ2Q
1NX>X
1=
1N
N
∑i=1xix>i
!1p! Q1
So,1N
1NX>X
1S0
1NX>X
1p! Vasy
bβOLS
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5. Heteroscedasticity
Denition (White heteroscedasticity consistent estimator)The White consistent estimator of the asymptotic variance-covariancematrix of the ordinary least squares estimator bβOLS in the generalizedlinear regression model is dened to be:
bVasy
bβOLS = N X>X1 S0 X>X1bVasy
bβOLS p! Vasy
bβOLSwith
S0 =1N
N
∑i=1
bε2i xix>i
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5. Heteroscedasticity
Corollary (White heteroscedasticity consistent estimator)The White consistent estimator can written as:
bVasy
bβOLS = 1N
1N
N
∑i=1xix>i
!1 1N
N
∑i=1
bε2i xix>i!
1N
N
∑i=1xix>i
!1
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5. Heteroscedasticity
Remarks
1 This result is extremely important and useful. It implies that withoutactually specifying the type of heteroscedasticity, we can still makeappropriate inferences based on the results of least squares.
2 This implication is especially useful if we are unsure of the precisenature of the heteroscedasticity (which is probably most of the time).
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5. Heteroscedasticity
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5. Heteroscedasticity
Remark
Given the normalisation trace(Ω) = N, we have:
σ2 =1N
N
∑i=1
σ2i
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5. Heteroscedasticity
Denition (SSR)
The least squares estimator bσ2 dened by:bσ2 = bε>bε
N K =1
N KN
∑i=1
bε2iconverges to the probability limit of the average variance of thedisturbances bσ2 p! lim
N!∞σ2 = lim
N!∞
1N
N
∑i=1
σ2i
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5. Heteroscedasticity
Example (White robust estimator. Source: Greene (2012))Consider the generalized linear regression model:
AVGEXPi = β1 + β2AGEi + β3Ownrenti + β4Incomei + β5Income2i + εi
where AVGEXP denotes the Avg. monthly credit card expenditure,Ownrent denotes a binary variable (individual owns (1) or rents (0) home),Age denotes the age in years, Income denotes the income divided by10,000. The data are available in le Chapter5_data.xls. Question:write a Matlab code to (1) estimate the parameters by OLS, (2) computethe standard errors and the robust standard errors and (3) compare yourresults with Eviews.
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5. Heteroscedasticity
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5. Heteroscedasticity
1 2 3 4 5 6 7 8 9 10500
0
500
1000
1500
2000
Income
OLS
resi
dual
s
This graph is the sign of heteroscedasticity.. the variance of the residualsseems to depend on the income.
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5. Heteroscedasticity
The values are the same.. perfect
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5. Heteroscedasticity
The values are di¤erent... Why?
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5. Heteroscedasticity
Remark
This di¤erence is due to the fact that Eviews uses a nite samplecorrection for S0 (Davidson and MacKinnon, 1993)
S0 =1
N KN
∑i=1
bε2i xix>iDavidson, R. and J. MacKinnon. Estimation and Inference in Econometrics.New York: Oxford University Press, 1993.
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5. Heteroscedasticity
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5. Heteroscedasticity
The values are now identical.
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5. Heteroscedasticity
Case 2: Heteroscedasticity with known Σ
GLS and Weighted Least Squares
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5. Heteroscedasticity
Assumption: We assume that the disturbances are heteroscedastic with
V (εjX) = Σ = σ2Ω
with
Σ =
0BB@σ21 0 .. 00 σ22 .. 0.. .. .. ..0 .. .. σ2N
1CCA = σ2Ω = σ2
0BB@ω1 0 .. 00 ω2 .. 0.. .. .. ..0 .. .. ωN
1CCAwhere the parameters σ2i and ωi are known for i = 1, ..N.
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5. Heteroscedasticity
Denition (GLS estimator)
In presence of heteroscedasticity, the Generalized Least Squares (GLS)estimator of β is dened as to:
bβGLS =
N
∑i=1
xix>iωi
!1 N
∑i=1
xiyiωi
!
or equivalently by
bβGLS =
N
∑i=1
xix>iσ2i
!1 N
∑i=1
xiyiσ2i
!
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5. HeteroscedasticityProof
In general, whatever the form of Σ = σ2Ω, we have:
bβGLS = X>Ω1X1
X>Ω1y
Since Ω is diagonal:
X>Ω1X =N
∑i=1
xix>iωi
X>Ω1y =N
∑i=1
xiyiωi
As a consequence:
bβGLS =
N
∑i=1
xix>iωi
!1 N
∑i=1
xiyiωi
!
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5. Heteroscedasticity
Remark
bβGLS =
N
∑i=1
xix>iωi
!1 N
∑i=1
xiyiωi
!This formula is similar to that obtained for a Weighted Least Squares(WLS).
bβWLS =
N
∑i=1
δixix>i
!1 N
∑i=1
δixiyi
!
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5. Heteroscedasticity
Fact (GLS and WLS)In presence of heteroscedasticity, the GLS estimator is a particular case ofthe Weighted Least Squares (WLS) estimator.
bβWLS =
N
∑i=1
δixix>i
!1 N
∑i=1
δixiyi
!
where δi is an arbitrary weight. For δi = 1/ωi , we have bβWLS = bβGLS .
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5. Heteroscedasticity
Remark
1 The WLS estimator is consistent regardless of the weights used, aslong as the weights are uncorrelated with the disturbances.
2 In general, we consider a weight which is proportional to oneexplicative variable (the income in the last example):
σ2i = σ2x2ik () δi =1x2ik
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5. Heteroscedasticity
Case 3: Heteroscedasticity for a given structure
FGLS and two-step or iterated estimators
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5. Heteroscedasticity
Assumption: We assume that the disturbances are heteroscedastic with
V (εjX) = Σ (α) = σ2Ω (α)
where α denotes a set of parameters.
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5. Heteroscedasticity
Example (Restriction)We assume that
V ( εi jX) = σ2i (α) = σ2z>i α
2where α = (α1 : .. : αH )
> is a H 1 vector of parameters and zi is H 1of explicative variables (not necessarily the same as in xi ).
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5. Heteroscedasticity
Example (Harveys (1976) restriction)
Harvey (1976) considers a restriction of the form:
V ( εi jX) = σ2i (α) = expx>i α
where α = (α1 : .. : αH )
> is a H 1 vector of parameters and zi is H 1of explicative variables (not necessarily the same as in xi ).
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5. Heteroscedasticity
We know that the GLS estimator is dened by:
bβGLS =
N
∑i=1
xix>iσ2i (α)
!1 N
∑i=1
xiyiσ2i (α)
!
S, the feasible GLS (FGLS) estimator is:
bβFGLS =
N
∑i=1
xix>iσ2i (bα)
!1 N
∑i=1
xiyiσ2i (bα)
!
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5. Heteroscedasticity
If we assume for instance that
V ( εi jX) = σ2i (α) = expz>i α
where zi is a vector of H variables, a way to estimate α consists inconsidering the model:
lnbε2i = z>i α+ vi
and to estimate α by OLS. The OLS is consistent even it is ine¢ cient (dueto the heteroscedasticity). Given bα, we have a consistent estimator for σ2i :
bσ2i = expz>i bα p! σ2i (α)
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5. Heteroscedasticity
ProblemIn order to estimate β by the GLS, we need bα, and to estimate α, we needthe residuals bεi = yi x>i bβGLS ...
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5. Heteroscedasticity
Two solutions
1 A two steps FGLS estimator
2 An iterative FGLS estimator
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5. Heteroscedasticity
Denition (Two-steps FGLS estimator)
First step: estimate the parameters β by OLS. Compute the residualsbεi = yi x>i bβOLS and estimate the parameters α according to theappropriate model. Second step: compute the estimated variances σ2i (bα)and compute the FGLS estimator:
bβFGLS =
N
∑i=1
xix>iσ2i (bα)
!1 N
∑i=1
xiyiσ2i (bα)
!
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5. Heteroscedasticity
Denition (Iterated FGLS estimator)
Estimate the parameters β by OLS. Compute the residualsbεi = yi x>i bβOLS and estimate the parameters α according to theappropriate model. Compute the estimated variances σ2i (bα) and computethe FGLS estimator:
bβ(1)FGLS =
N
∑i=1
xix>iσ2i (bα)
!1 N
∑i=1
xiyiσ2i (bα)
!
Compute the residuals bεi = yi x>i bβ(1)FGLS and estimate the parameters α
according to the appropriate model. Compute the FGLS bβ(2)FGLS and soon...The procedure stop when
supj=1,..,K
bβ(i )j ,FGLS bβ(i1)j ,FGLS
< threshold (ex: 0.001)Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 129 / 153
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5. Heteroscedasticity
Example (Harveys (1976) multiplicative model of heteroscedasticity)Consider the generalized linear regression model:
AVGEXPi = β1 + β2AGEi + β3Ownrenti + β4Incomei + β5Income2i + εi
where the heteroscedasticity satises the Harveys (1976) specication
V ( εi jX) = σ2i = exp (α1 + α2Incomei )
The data are available in le Chapter5_data.xls. Question: write aMatlab code to estimate the parameters by FGLS by using a two-step andan iterative estimator.
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5. Heteroscedasticity
Remark
A way to get the estimates of the parameters α1 and α2 is to consider theregression:
lnbε2i = α1 + α2Incomei + vi
and to estimate the parameters by OLS.
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5. Heteroscedasticity
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5. Heteroscedasticity
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5. Heteroscedasticity
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5. Heteroscedasticity
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5. Heteroscedasticity
Key Concepts
1 OLS and robust inference
2 White heteroscedasticity consistent estimator
3 GLS and Weighted Least Squares (WLS)
4 FGLS: two-steps and iterated estimators
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Section 6
Testing for Heteroscedasticity
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6. Testing for heteroscedasticity
Objectives
The objective of this section are to introduce the following tests forheteroscedasticity:
1 White general test
2 The Breusch-Pagan / Godfrey LM test
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6. Testing for heteroscedasticity
Denition (White test for heteroscedasticity)The White test for heteroscedasticity is based on:
H0 : σ2i = σ2 for i = 1, ..,N
H1 : σ2i 6= σ2j for at least one pair (i , j)
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6. Testing for heteroscedasticity
The intuition of the test is based on the following idea:
1 If there is no heteroscedasticity (under the null H0):
Vasy
bβOLS = σ2Q1
bVasy
bβOLS = σ2X>X
12 Under the alternative (heteroscedasticity):
Vasy
bβOLS = σ2Q1QQ1
bVasy
bβOLS = σ2X>X
1X>ΩX
X>X
1
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6. Testing for heteroscedasticity
White (1980) proposes the following procedure and test-statistic:
Step 1: Estimation of the model using the OLS estimator of β.
Step 2: Determine the residuals bεi = yi x>i bβOLS .Step 3: Regress bε2i on a constant and all unique columns vectors containedin X and all the squares and cross-products of the column vectors in X.
Step 4: Determine the coe¢ cient of determination, R2, of the previousregression.
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6. Testing for heteroscedasticity
Denition (White test for heteroscedasticity)
Under the null, the White test-statistic NR2 converges:
N R2 d!H0
χ2 (m 1)
where m is the number of explanatory variables in the regression of bε2i .The critical region of size α is
W =y : N R2 > χ21α
where χ21α denotes the 1-α critical value of the χ2 (m 1) distribution.
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6. Testing for heteroscedasticity
Example (Whites (1980) test for heteroscedasticity)Consider the generalized linear regression model:
AVGEXPi = β1 + β2AGEi + β3Ownrenti + β4Incomei + β5Income2i + εi
The data are available in le Chapter5_data.xls. Question: write aMatlab code to compute the White test-statistic for heteroscedasticity andits p-value. What is you conclusion for a signicance level of 5%?Compare your results with Eviews.
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6. Testing for heteroscedasticity
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6. Testing for heteroscedasticity
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6. Testing for heteroscedasticity
Denition (Breusch and Pagan test)
Breusch and Pagan (1979) have devised a Lagrange multiplier test ofthe hypothesis that
σ2i = σ2f
α0 + z>i α
where zi = (zi1..zip)> is a p 1 vector of independent variables. The test
is:H0 : α = 0p1 (homoscedasticity)
H1 : α 6= 0p1 (heteroscedasticity)
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6. Testing for heteroscedasticity
The test can be carried out with a simple regression of
gi = Nbε2ibε>bε 1 = N bε2i
∑Ni=1bε2i 1
on the variables zik for k = 1, .,N and a constant term.
gi = α0 + α1zi1 + ...+ αpzip + vi
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6. Testing for heteroscedasticity
Denition (Breusch and Pagan test-statistic)
Dene Z the N (p + 1) matrix of observations on (1, zi ) and let g bethe N 1 vector of observations
gi = Nbε2ibε>bε 1
Then, the Breusch and Pagans test-statistic is dened by:
LM =12g>Z
Z>Z
1Z>g
Under the null, we have:LM
d!H0
χ2 (p)
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6. Testing for heteroscedasticity
Example (Breusch and Pagans (1979) test for heteroscedasticity)Consider the generalized linear regression model:
AVGEXPi = β1 + β2AGEi + β3Ownrenti + β4Incomei + β5Income2i + εi
The data are available in le Chapter5_data.xls. Question: write aMatlab code to compute the Breusch and Pagan test-statistic forheteroscedasticity with zi = xi and its p-value. What is you conclusion fora signicance level of 5%?
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6. Testing for heteroscedasticity
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6. Testing for heteroscedasticity
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6. Testing for heteroscedasticity
Key Concepts
1 White test for heteroscedasticity
2 Breusch and Pagan test for heteroscedasticity
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End of Chapter 5
Christophe Hurlin (University of Orléans)
Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 153 / 153