Econometrics - Lecture 4
Heteroskedasticity and Autocorrelation


Contents
nViolations of V{ε} = s2 IN
nHeteroskedasticity
nGLS Estimation
nAutocorrelation
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Gauss-Markov Assumptions
A1
E{εi} = 0 for all i
A2
all εi are independent of all xi (exogeneous xi)
A3
V{ei} = s2 for all i (homoskedasticity)
A4
Cov{εi, εj} = 0 for all i and j with i ≠ j (no autocorrelation)
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Observation yi is a linear function
yi = xi'b + εi
of observations xik, k =1, …, K, of the regressor variables and the error term εi
for i = 1, …, N; xi' = (xi1, …, xiK); X = (xik)
n
n
n
n
n
n
n
In matrix notation: E{ε} = 0, V{ε} = s2 IN

OLS Estimator: Properties
nUnder assumptions (A1) and (A2):
n1. The OLS estimator b is unbiased: E{b} = β
n
nUnder assumptions (A1), (A2), (A3) and (A4):
n2. The variance of the OLS estimator is given by
n V{b} = σ2(Σi xi xi’)-1 = σ2(X‘ X)-1
n3. The sampling variance s2 of the error terms εi,
n s2 = (N – K)-1 Σi ei2
n is unbiased for σ2
n4. The OLS estimator b is BLUE (best linear unbiased estimator)
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Violations of V{e} = s2IN
nImplications of the Gauss-Markov assumptions for ε:
n V{ε} = σ2IN
nViolations:
nHeteroskedasticity: V{ε} = diag(s12, …, sN2) or
n V{ε} = s2Y = s2 diag(h12, …, hN2)
nAutocorrelation: V{εi, εj} ¹ 0 for at least one pair i ¹ j or
n V{ε} = s2Y
n with non-diagonal elements different from zero
n
n
n
n
n
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Example: Household Income and Expenditures
n70 households (HH):
n monthly HH-income and expenditures for durable goods
n
n
n
n
n
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0
400
800
1200
1600
2000
2400
0
2000
4000
6000
8000
10000
12000
HH-income

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Household Income and Expenditures, cont‘d
Residuals e = y- ŷ from
   Ŷ = 44.18 + 0.17 X
X: monthly HH-income
Y: expenditures for durable goods
the larger the income, the more scattered are the residuals
-600
-400
-200
0
200
400
600
0
2000
4000
6000
8000
10000
12000


HH-income
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Typical Situations for Heteroskedasticity
nHeteroskedasticity is typically observed
nIn data from cross-sectional surveys, e.g., in households or regions
nData with variance which depends of one or several explanatory variables, e.g., firm size
nData from financial markets, e.g., exchange rates, stock returns
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Example: Household Expenditures
nWith growing income increasing variation of expenditures; from Verbeek, Fig. 4.1
n
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Autocorrelation of Economic Time-series
nConsumption in actual period is similar to that of the preceding period; the actual consumption
„depends“ on the consumption of the preceding period
nConsumption, production, investments, etc.: it is to be expected that successive observations of
economic variables correlate positively
nSeasonal adjustment: application of smoothing and filtering algorithms induces correlation
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Example: Imports
Scatter-diagram of by one period lagged imports [MTR(-1)] against actual imports [MTR]
Correlation coefficient between MTR und MTR(-1): 0.9994
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Example: Import Function
MTR: Imports
FDD: Demand
(from AWM-database)
Import function:  MTR = -227320 + 0.36 FDD
R2 = 0.977, tFFD = 74.8
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Import Function, cont‘d
MTR: Imports
FDD: Demand
(from AWM-database)
RESID: et = MTR - (-227320 + 0.36 FDD)
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Import Function, cont‘d
Scatter-diagram of by one period lagged residuals [Resid(-1)] against actual residuals [Resid]
Serial correlation!
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Typical Situations for Autocorrelation
nAutocorrelation is typically observed if
na relevant regressor with trend or seasonal pattern is not included in the model: miss-specified
model
nthe functional form of a regressor is incorrectly specified
nthe dependent variable is correlated in a way that is not appropriately represented in the
systematic part of the model
nWarning! Omission of a relevant regressor with trend implies autocorrelation of the error terms;
in econometric analyses autocorrelation of the error terms is always possible!
nAutocorrelation of the error terms indicates deficiencies of the model specification
nTests for autocorrelation are the most frequently used tool for diagnostic checking the model
specification
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Import Functions
nRegression of imports (MTR) on demand (FDD)
n MTR = -2.27x109 + 0.357 FDD, tFDD = 74.9, R2 = 0.977
n Autocorrelation (order 1) of residuals:
q Corr(et, et-1) = 0.993
nImport function with trend (T)
n MTR = -4.45x109 + 0.653 FDD – 0.030x109 T
n tFDD = 45.8, tT = -21.0, R2 = 0.995
n Multicollinearity? Corr(FDD, T) = 0.987!
nImport function with lagged imports as regressor
n MTR = -0.124x109 + 0.020 FDD + 0.956 MTR-1
n tFDD = 2.89, tMTR(-1) = 50.1, R2 = 0.999
n
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Consequences of V{e} ¹ s2IN
nOLS estimators b for b
nare unbiased
nare consistent
nhave the covariance-matrix
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
nare not efficient estimators, not BLUE
nfollow – under general conditions – asymptotically the normal distribution
nThe estimator s2 = e'e/(N-K) for s2 is biased
n
n
n
n
nand
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Consequences of V{e} ¹ s2IN for Applications
nOLS estimators b for b are still unbiased
nRoutinely computed standard errors are biased; the bias can be positive or negative
nt- and F-tests may be misleading
nRemedies
nAlternative estimators
nCorrected standard errors
nModification of the model
nTests for identification of
nheteroskedasticity
nautocorrelation
n are important tools
n
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Contents
nViolations of V{ε} = s2 IN
nHeteroskedasticity
nGLS Estimation
nAutocorrelation
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Inference under Heteroskedasticity
nCovariance matrix of b:
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
nUse of s2 (X'X)-1 (the standard output of econometric software) instead of V{b} for inference on b
may be misleading
nRemedies
nUse of correct variances and standard errors
nTransformation of the model so that the error terms are homoskedastic
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The Correct Variances
nV{εi} = σi2 = σ2hi2: each observation has its own unknown parameter hi
nN observation for estimating N unknown parameters?
nTo estimate σ2i – and V{b}
nKnown form of the heteroskedasticity, specific correction
qE.g., hi2 = zi’a for some variables zi
qRequires estimation of a
nWhite’s heteroskedasticity-consistent covariance matrix estimator (HCCME)
n Ṽ{b} = s2(X'X)-1(Siĥi2xixi’) (X'X)-1
n with ĥi2=ei2
qDenoted as HC0
qInference based on HC0: heteroskedasticity-robust inference
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White’s Standard Errors
nWhite’s standard errors for b
nSquare roots of diagonal elements of HCCME
nUnderestimate the true standard errors
nVarious refinements, e.g., HC1 = HC0[N/(N-K)]
nIn GRETL: HC0 is the default HCCME, HC1 and other refinements are optionally available
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An Alternative Estimator for b
nIdea of the estimator
nTransform the model so that it satisfies the Gauss-Markov assumptions
nApply OLS to the transformed model
nShould result in a BLUE
nTransformation often depends upon unknown parameters that characterizing heteroskedasticity:
two-step procedure
1.Estimate the parameters that characterize heteroskedasticity and transform the model
2.Estimate the transformed model
nThe procedure results in an approximately BLUE
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An Example
nModel:
n yi  = xi’β + εi  with V{εi} = σi2 = σ2hi2
nDivision by hi results in
n yi /hi = (xi /hi)’β + εi /hi
n with a homoskedastic error term
n V{εi /hi} = σi2/hi2 = σ2
nOLS applied to the transformed model gives
n
nIt is called a generalized least squares (GLS) or weighted least squares (WLS) estimator
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Weighted Least Squares Estimator
nA GLS or WLS estimator is a least squares estimator where each observation is weighted by a
non-negative factor wi > 0:
n
n
nWeights proportional to the inverse of the error term variance: Observations with a higher error
term variance have a lower weight; they provide less accurate information on β
nNeeds knowledge of the hi
qIs seldom available
qIs mostly provided by estimates of hi based on assumptions on the form of hi
qE.g., hi2 = zi’a for some variables zi
nAnalogous with general weights wi
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Example: Labor Demand
nVerbeek’s data set “labour2”: Sample of 569 Belgian companies (data from 1996)
nVariables
qlabour: total employment (number of employees)
qcapital: total fixed assets
qwage: total wage costs per employee (in 1000 EUR)
qoutput: value added (in million EUR)
nLabour  demand function
n labour = b1 + b2*wage + b3*output + b4*capital
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Labor Demand Function
nFor Belgian companies, 1996; Verbeek
n
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Labor Demand Function, cont’d
nCan the error terms be assumed to be homoskedastic?
nThey may vary depending of the company size, measured by, e.g., size of output or capital
nRegression of squared residuals on appropriate regressors will indicate heteroskedasticity
n
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Labor Demand Function, cont’d
nAuxiliary regression of squared residuals, Verbeek
n
n
n
n
n
n
n
n
n
nIndicates dependence of error terms on output, capital, not on wage
n
n
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Labor Demand Function, cont’d
nEstimated function
n labour = b1 + b2*wage + b3*output + b4*capital
n OLS estimates without (s.e.) and with White standard errors (White s.e.), and GLS estimates with
wi = 1/ei
n
n
n
n
n
n
nThe standard errors are inflated by factors 3.7 (wage), 6.4 (capital), 7.0 (output) wrt the White
s.e.
n
n
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b1
b2
b3
b4
Coeff OLS
287.19
-6.742
15.400
-4.590
s.e.
19.642
0.501
0.356
0.269
White s.e.
64.877
1.852
2.482
1.713
Coeff GLS
282.06
-6.609
15.235
-4.197
s.e.
1.808
0.042
0.094
0.141

Labor Demand Function, cont’d
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With White standard errors: Output from GRETL
Dependent variable : LABOR
Heteroskedastic-robust standard errors, variant HC0,
             coefficient   std. error   t-ratio    p-value
  -------------------------------------------------------------
  const          287,719                  64,8770          4,435     1,11e-05 ***
  WAGE    -6,7419      1,8516        -3,641      0,0003    ***
  CAPITAL -4,59049    1,7133       -2,679     0,0076    ***
  OUTPUT   15,4005      2,4820          6,205      1,06e-09 ***
Mean dependent var                   201,024911   S.D. dependent var    611,9959
Sum squared resid                   13795027   S.E. of regression   156,2561
R- squared               0,935155   Adjusted R-squared 0,934811
F(2, 129)               225,5597   P-value (F)               3,49e-96
Log-likelihood          455,9302   Akaike criterion       7367,341
Schwarz criterion                        -3679,670   Hannan-Quinn 7374,121
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Tests against Heteroskedasticity
nDue to unbiasedness of b, residuals are expected to indicate heteroskedasticity
nGraphical displays of residuals may give useful hints
nResidual-based tests:
nBreusch-Pagan test
nKoenker test
nGoldfeld-Quandt test
nWhite test
n
n
n
n
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Breusch-Pagan Test
nFor testing whether the error term variance is a function of Z2, …, Zp
nModel for heteroskedasticity
n si2/s2 = h(zi‘a)
n with function h with h(0)=1, p-vectors zi und a, an intercept and p-1 variables Z2, …, Zp
nNull hypothesis
n H0: a = 0
n implies si2 = s2 for all i, i.e., homoskedasticity
nAuxiliary regression of the standardized squared OLS residuals gi = ei2/s2 - 1, s2 = e’e/N, on
zi (and squares of zi)
nTest statistic: BP = N*ESS with the explained sum of squares ESS = N*V(ĝ), of the auxiliary
regression; ĝ are the fitted values for g. BP follows approximately the Chi-squared distribution
with p d.f.
n
n
n
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Breusch-Pagan Test, cont‘d
nTypical functions h for h(zi‘a)
nLinear regression: h(zi‘a) = zi‘a
nExponential function h(zi‘a) = exp{zi‘a}
qAuxiliary regression of the log (ei2) upon zi
q“Multiplicative heteroskedasticity”
qVariances are non-negative
nKoenker test: variant of the BP test which is robust against non-normality of the error terms
nGRETL: The output window of OLS estimation allows the execution of the Breusch-Pagan test with
h(zi‘a) = zi‘a
qOLS output => Tests => Heteroskedasticity => Breusch-Pagan
qKoenker test: OLS output => Tests => Heteroskedasticity => Koenker
n
n
n
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Labor Demand Function, cont’d
nAuxiliary regression of squared residuals, Verbeek
n
n
n
n
n
n
n
n
nNR2 = 331.04, p-value = 2.17E-70; reject null hypothesis of homoskedasticity
n
n
n
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Goldfeld-Quandt Test
nFor testing whether the error term variance has values sA2 and sB2 for observations from regime A
and B, respectively, sA2 ¹ sB2 regimes can be urban vs rural area, economic prosperity vs
stagnation, etc.
nExample (in matrix notation):
qyA = XAbA + eA,  V{eA} = sA2INA  (regime A)
qyB = XBbB + eB,  V{eB} = sB2INB  (regime B)
nNull hypothesis: sA2 = sB2
nTest statistic:
n
n
n with Si: sum of squared residuals for i-th regime; follows under H0 exactly or approximately the
F-distribution with NA-K and NB-K d.f.
n
n
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Goldfeld-Quandt Test, cont‘d
nTest procedure in three steps:
1.Sort the observations with respect to the regimes
2.Separate fittings of the model to the NA and NB observations; sum of squared residuals SA and SB
3.Calculation of test statistic F
n
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White Test
nFor testing whether the error term variance is a function of the model regressors, their squares
and their cross-products
nAuxiliary regression of the squared OLS residuals upon xi’s, squares of xi’s and cross-products
nTest statistic: NR2 with R2 of the auxiliary regression; follows the Chi-squared distribution with
the number of coefficients in the auxiliary regression as d.f.
nThe number of coefficients in the auxiliary regression may become large, maybe conflicting with
size of N, resulting in low power of the White test
n
n
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Labor Demand Function, cont’d
nWhite's test for heteroskedasticity
nOLS, using observations 1-569
nDependent variable: uhat^2
n
n              coefficient   std. error       t-ratio       p-value
n  --------------------------------------------------------------
n  const         -260,910     18478,5          -0,01412   0,9887
n  WAGE        554,352         833,028         0,6655    0,5060
n  CAPITAL  2810,43          663,073        4,238      2,63e-05  ***
n  OUTPUT                    -2573,29         512,179       -5,024      6,81e-07  ***
n  sq_WAGE                      -10,0719         9,29022   -1,084      0,2788
n  X2_X3              -48,2457       14,0199     -3,441      0,0006    ***
n  X2_X4                 58,5385        8,11748    7,211      1,81e-012 ***
n  sq_CAPITAL            14,4176        2,01005    7,173      2,34e-012 ***
n  X3_X4               -40,0294        3,74634 -10,68       2,24e-024 ***
n  sq_OUTPUT             27,5945        1,83633   15,03      4,09e-043 ***
n
n  Unadjusted R-squared = 0,818136
n
nTest statistic: TR^2 = 465,519295,
nwith p-value = P(Chi-square(9) > 465,519295) = 0,000000
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Contents
nViolations of V{ε} = s2 IN
nHeteroskedasticity
nGLS Estimation
nAutocorrelation
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Generalized Least Squares Estimator
nA GLS or WLS estimator is a least squares estimator where each observation is weighted by a
non-negative factor wi > 0
nExample:
n yi  = xi’β + εi  with V{εi} = σi2 = σ2hi2
qDivision by hi results in a model with homoskedastic error terms
n V{εi /hi} = σi2/hi2 = σ2
qOLS applied to the transformed model results in the weighted least squares (GLS) estimator with
wi = hi-2:
q
q
nThe concept of transforming the model so that Gauss-Markov assumptions are fulfilled is used also
in more general situations, e.g., for autocorrelated error terms
n
n
n
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Properties of GLS Estimators
nThe GLS estimator
n
n
n is a least squares estimator; standard properties of OLS estimator apply
nThe covariance matrix of the GLS estimator is
n
n
nUnbiased estimator of the error term variance
n
n
nUnder the assumption of normality of errors, t- and F-tests can be used; for large N, these
properties apply approximately without normality assumption
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Feasible GLS Estimator
nIs a GLS estimator with estimated weights wi
nSubstitution of the weights wi = hi-2 by estimates ĥi-2
n
n
nFeasible (or estimated) GLS or FGLS or EGLS estimator
nFor consistent estimates ĥi, the FGLS and GLS estimators are asymptotically equivalent
nFor small values of N, FGLS estimators are in general not BLUE
nFor consistently estimated ĥi, the FGLS estimator is consistent and asymptotically efficient with
covariance matrix (estimate for s2: based on FGLS residuals)
n
nWarning: the transformed model is uncentered
n
n
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Multiplicative Heteroskedasticity
nAssume V{εi} = σi2 = σ2hi2 = σ2exp{zi‘a}
nThe auxiliary regression
n log ei2 = log σ2 + zi‘a + vi  with vi = log(ei2/σi2)
n provides a consistent estimator a for α
nTransform the model yi  = xi’β + εi  with V{εi} = σi2 = σ2hi2 by dividing through ĥi from ĥi2 =
exp{zi‘a}
nError term in this model is (approximately) homoskedastic
nApplying OLS to the transformed model gives the FGLS estimator for β
n
n
n
n
n
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FGLS Estimation
nIn the following steps:
1.Calculate the OLS estimates b for b
2.Compute the OLS residuals ei = yi – xi‘b
3.Regress log(ei2) on zi and a constant, obtaining estimates a for α
n log ei2 = log σ2 + zi‘a + vi
4.Compute ĥi2 = exp{zi‘a}, transform all variables and estimate the transformed model to obtain the
FGLS estimators:
n yi /ĥi = (xi /ĥi)’β + εi /ĥi
5.The consistent estimate s² for σ2, based on the FGLS-residuals, and the consistently estimated
covariance matrix
n
n are part of the standard output when regressing the transformed model
5.
n
n
n
n
n
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Labor Demand Function
nFor Belgian companies, 1996; Verbeek
n
n
n
n
n
n
n
n
n
n
nLog-tranformation is expected to reduce heteroskedasticity
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Labor Demand Function, cont’d
nFor Belgian companies, 1996; Verbeek
n
n
n
n
n
n
n
n
n
n
nBreusch-Pagan test: NR2 = 66.23, p-value: 1,42E-13
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Labor Demand Function, cont’d
nFor Belgian companies, 1996; Verbeek
n Weights estimated assuming multiplicative heteroskedasticity
n
n
n
n
n
n
n
n
n
n
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Labor Demand Function, cont’d
nEstimated function
n   log(labour) = b1 + b2*log(wage) + b3*log(output) + b4*log(capital)
n The table shows: OLS estimates without (s.e.) and with White standard errors (White s.e.) as well
as FGLS estimates and standard errors
n
n
n
n
n
n
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b1
b2
b3
b4
OLS coeff
6.177
-0.928
0.990
-0.0037
s.e.
0.246
0.071
0.026
0.0188
White s.e.
0.293
0.086
0.047
0.0377
FGLS coeff
5.895
-0.856
1.035
-0.0569
s.e.
0.248
0.072
0.027
0.0216

Labor Demand Function, cont’d
nSome comments:
nReduction of standard errors in FGLS estimation as compared with heteroskedasticity-robust
estimation, efficiency gains
nComparison with OLS estimation not appropriate
nFGLS estimates differ slightly from OLS estimates; effect of capital is indicated to be relevant
(p-value: 0.0086)
nR2 of FGLS estimation is misleading
qModel is uncentered, no intercept
qComparison with that of OLS estimation not appropriate, explained variable differ
n
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Contents
nViolations of V{ε} = s2 IN
nHeteroskedasticity
nGLS Estimation
nAutocorrelation
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Autocorrelation
nTypical for time series data such as consumption, production, investments, etc., and models for
time series data
nAutocorrelation of error terms is typically observed if
qa relevant regressor with trend or seasonal pattern is not included in the model: miss-specified
model
qthe functional form of a regressor is incorrectly specified
qthe dependent variable is correlated in a way that is not appropriately represented in the
systematic part of the model
nAutocorrelation of the error terms indicates deficiencies of the model specification such as
omitted regressors, incorrect functional form, incorrect dynamic
nTests for autocorrelation are the most frequently used tool for diagnostic checking the model
specification
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Example: Demand for Ice Cream
nTime series of 30 four weekly observations (1951-1953)
nVariables
qcons: consumption of ice cream per head (in pints)
qincome: average family income per week (in USD, red line)
qprice: price of ice cream (in USD per pint, blue line)
qtemp: average temperature (in Fahrenheit); tempc: (green, in °C)
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Demand for Ice Cream, cont’d
nTime series plot of
n Cons: consumption of ice cream per head (in pints); mean: 0.36
qTemp/100: average temperature (in Fahrenheit)
qPrice (in USD per pint); mean: 0.275 USD
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Demand for Ice Cream, cont’d
nDemand for  ice cream, measured by cons, explained by price, income, and temp
n
n
n
n
n
n
n
n
n
n
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Demand for Ice Cream, cont’d
nDemand for ice cream explained from income and price index
n
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Demand for Ice Cream, cont’d
nIce cream model: Scatter-plot of residuals et vs et-1 (r = 0.401)
n
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A Model with AR(1) Errors
nLinear regression
n yt = xt‘b + et 1)
n with
n et = ret-1 + vt with -1 < r < 1 or |r| < 1
n where vt are uncorrelated random variables with mean zero and constant variance sv2
nFor ρ ¹ 0, the error terms et are correlated; the Gauss-Markov assumption V{e} = se2IN is violated
nThe other Gauss-Markov assumptions are assumed to be fulfilled
nThe sequence et, t = 0, 1, 2, … which follows et = ret-1 + vt is called an autoregressive process
of order 1 or AR(1) process
n_____________________
n1) In the context of time series models, variables are indexed by „t“
n
n
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Properties of AR(1) Processes
nRepeated substitution of et-1, et-2, etc. results in
n et = ret-1 + vt = vt + rvt-1 + r2vt-2 + …
nwith vt being uncorrelated and having mean zero and variance sv2:
nE{et} = 0
nV{et} = se2 = sv2(1-r2)-1
nThis results from V{et} = sv2 + r2sv2 + r4sv2 + … = sv2(1-r2)-1 for |r|<1 as the geometric series
1 + r2 + r4 + … has the sum (1-r2)-1 given that |r| < 1
qfor |r| > 1, V{et} is undefined
nCov{et, et-s } = rs sv2 (1-r2)-1 for s > 0
n all error terms are correlated; covariances – and correlations Corr{et, et-s } = rs (1-r2)-1 –
decrease with growing distance s in time
n
n
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AR(1) Process, cont’d
nThe covariance matrix V{e}:
n
n
n
n
n
n
n
nV{e} has a band structure
nDepends only of two parameters: r and sv2
n
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Consequences of V{e} ¹ s2IT
nOLS estimators b for b
nare unbiased
nare consistent
nhave the covariance-matrix
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
nare not efficient estimators, not BLUE
nfollow – under general conditions – asymptotically the normal distribution
nThe estimator s2 = e'e/(T-K) for s2 is biased
nFor an AR(1)-process et with r > 0, s.e. from s2 (X'X)-1 underestimates the true s.e.
n
n
n
n
nand
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Inference under Autocorrelation
nCovariance matrix of b:
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
nUse of s2 (X'X)-1 (the standard output of econometric software) instead of V{b} for inference on b
may be misleading
nIdentification of autocorrelation:
nStatistical tests, e.g., Durbin-Watson test
nRemedies
nUse of correct variances and standard errors
nTransformation of the model so that the error terms are uncorrelated
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Estimation of r
nAutocorrelation coefficient r: parameter of  the AR(1) process
n et = ret-1 + vt
nEstimation of ρ
nby regressing the OLS residual et on the lagged residual et-1
n
n
n
nestimator is
qbiased
qbut consistent under weak conditions
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Autocorrelation Function
nAutocorrelation of order s:
n
n
n
nAutocorrelation function assigns rs to s
nCorrelogram: graphical representation of the autocorrelation function
n
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Example: Ice Cream Demand
nAutocorrelation function (ACF) of cons
n
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LAG      ACF          PACF         Q-stat. [p-value]
    1   0,6627  ***   0,6627 ***     14,5389  [0,000]
    2   0,4283  **   -0,0195         20,8275  [0,000]
    3   0,0982       -0,3179 *       21,1706  [0,000]
    4  -0,1470       -0,1701         21,9685  [0,000]
    5  -0,3968  **   -0,2630         28,0152  [0,000]
    6  -0,4623  **   -0,0398         36,5628  [0,000]
    7  -0,5145  ***  -0,1735         47,6132  [0,000]
    8  -0,4068  **   -0,0299         54,8362  [0,000]
    9  -0,2271        0,0711         57,1929  [0,000]
   10  -0,0156        0,0117         57,2047  [0,000]
   11   0,2237        0,1666         59,7335  [0,000]
   12   0,3912  **    0,0645         67,8959  [0,000]

Example: Ice Cream Demand
nCorrelogram of cons
n
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Tests for Autocorrelation of Error Terms
nDue to unbiasedness of b, residuals are expected to indicate autocorrelation
nGraphical display, correlogram of residuals may give useful hints
nResidual-based tests:
nDurbin-Watson test
nBox-Pierce test
nBreusch-Godfrey test
n
n
n
n
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Durbin-Watson Test
nTest of H0: r = 0 against H1: r ¹ 0
nTest  statistic
n
n
n
n
nFor r > 0, dw is expected to have a value in (0,2)
nFor r < 0, dw is expected to have a value in (2,4)
ndw close to the value 2 indicates no autocorrelation of error terms
nCritical limits of dw
qdepend upon xt’s
qexact critical value is unknown, but upper and lower bounds can be derived, which depend only of
the number of regression coefficients
nTest can be inconclusive
n
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Durbin-Watson Test: Bounds for Critical Limits
nDerived by Durbin and Watson
nUpper (dU) and lower (dL) bounds for the critical limits and a = 0.05
n
n
n
n
n
n
ndw < dL: reject H0
ndw > dU: do not reject H0
ndL < dw < dU: no decision (inconclusive region)
n
n
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T
K=2
K=3
K=10
dL
dU
dL
dU
dL
dU
15
1.08
1.36
0.95
1.54
0.17
3.22
20
1.20
1.41
1.10
1.54
0.42
2.70
100
1.65
1.69
1.63
1.71
1.48
1.87

Durbin-Watson Test: Remarks
nDurbin-Watson test gives no reference to causes of rejection of the null hypothesis and how the
model to modify
nVarious types of misspecification may cause the rejection of the null hypothesis
nDurbin-Watson test is a test against first-order autocorrelation; a test against autocorrelation
of other orders may be more suitable, e.g., order four if the model is for quarterly data
nUse of tables unwieldy
qLimited number of critical bounds (K, T, a) in tables
qInconclusive region
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Asymptotic Tests
nAR(1) process for error terms
n et = ret-1 + vt
nAuxiliary regression of et on xt‘b and et-1: produces
nRe2
nTest of H0: r = 0
1.Breusch-Godfrey test (GRETL: OLS output => Tests => Autocorr.)
qRe2 of the auxiliary regression: close to zero if r = 0
q(T-1) Re2 follows approximately the Chi-square distribution with 1 d.f. if r = 0
qLagrange multiplier F (LMF) statistic: F-test for explanatory power of et-1; follows approximately
the F(1, T-K-1) distribution if r = 0
qGeneral case of the Breusch-Godfrey test: Auxiliary regression based on higher order
autoregressive process
q
n
n
n
n
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Asymptotic Tests, cont’d
2.Box-Pierce test
qThe corresponding t-statistic
q t = √(T) r
q follows approximately the t-distribution if r = 0
qTest based on √(T) r is a special case of the Box-Pierce test which uses the test statistic Qm = T
Σsm rs2
qSimilar the Ljung-Box test, based on
q
q
q which follows the Chi-square distribution with m d.f. if r = 0
qLjung-Box test in GRETL: OLS output => Graphs => Residual correlogram
n
n
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Asymptotic Tests, cont’d
nRemarks
nIf the model of interest contains lagged values of y the auxiliary regression should also include
all explanatory variables (just to make sure the distribution of the test is correct)
nIf heteroskedasticity is suspected, White standard errors may be used in the auxiliary regression
n
Dec 9, 2011
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73

Demand for Ice Cream, cont’d
nDemand for  ice cream, measured by cons, explained by price, income, and temp
n
n
n
n
n
n
n
n
n
n
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Demand for Ice Cream, cont’d
Hackl,  Econometrics, Lecture 3
75
OLS estimated demand function: Output from GRETL
Dependent variable : CONS
             coefficient   std. error   t-ratio    p-value
  -------------------------------------------------------------
  const       0.197315     0.270216       0.7302   0.4718
 INCOME 0.00330776   0.00117142     2.824    0.0090 ***
 PRICE -1.04441      0.834357      -1.252    0.2218
 TEMP 0.00345843   0.000445547    7.762    3.10e-08 ***
Mean dependent var    0.359433  S.D. dependent var    0,065791
Sum squared resid      0,035273   S.E. of regression   0,036833
R- squared               0,718994   Adjusted R-squared 0,686570
F(2, 129)               22,17489   P-value (F)                 2,45e-07
Log-likelihood          58,61944   Akaike criterion       -109,2389
Schwarz criterion      -103,6341   Hannan-Quinn -107,4459
rho                     0,400633   Durbin-Watson      1,021170
Nov 18, 2011

Demand for Ice Cream, cont’d
nTest for autocorrelation of error terms
nH0: r = 0, H1: r ¹ 0
ndw = 1.02 < 1.21 = dL for T = 30, K = 4
nGRETL also shows the autocorrelation coefficient: r = 0.401
nPlot of actual (o) and fitted (polygon) values
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Demand for Ice Cream, cont’d
nAuxiliary regression et = ret-1 + vt: OLS estimation gives
n et = 0.401 et-1
n with s.e.(r) = 0.177, R2 =  0.154
nTest of H0: r = 0 against H1: r > 0
1.Box-Pierce test:
qt ≈ √(30) 0.401 = 2.196, p-value: 0.018
qt-statistic: 2.258, p-value: 0.016
2.Breusch-Godfrey test
q(T-1) R2 = 4.47, p-value: 0.035
nBoth reject the null hypothesis
Dec 9, 2011
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Inference under Autocorrelation
nCovariance matrix of b:
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
nUse of s2 (X'X)-1 (the standard output of econometric software) instead of V{b} for inference on b
may be misleading
nRemedies
nUse of correct variances and standard errors
nTransformation of the model so that the error terms are uncorrelated
Dec 9, 2011
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HAC-estimator for V{b}
nSubstitution of Y in
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
n by a suitable estimator
nNewey-West: substitution of Sx = s2(X'WX)/T = (StSsstsxtxs‘)/T by
n
n
n
n with wj = j/(p+1); p, the truncation lag, is to be chosen suitably
nThe estimator
n T (X'X)-1 Ŝx (X'X)-1
n for V{b} is called heteroskedasticity and autocorrelation consistent (HAC) estimator, the
corresponding standard errors are the HAC s.e.
n
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Demand for Ice Cream, cont’d
nDemand for  ice cream, measured by cons, explained by price, income, and temp, OLS and HAC
standard errors
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coeff
s.e.
OLS
HAC
constant
0.197
0.270
0.288
price
-1.044
0.834
0.876
income*10-3
3.308
1.171
1.184
temp*10-3
3.458
0.446
0.411

Cochrane-Orcutt Estimator
nGLS estimator
nWith transformed variables yt* =  yt – ryt-1 and xt* = xt – rxt-1, also called quasi-differences,
the model yt = xt‘b + et with et = ret-1 + vt can be written as
n yt – ryt-1 = yt* = (xt – rxt-1)‘b + vt = xt*‘b + vt (A)
nThe model in quasi-differences has error terms which fulfill the Gauss-Markov assumptions
nGiven observations for t = 1, …, T, model (A) is defined for t = 2, …, T
nEstimation of r using, e.g., the auxiliary regression et = ret-1 + vt gives the estimate r;
substitution of r in (A) for r results in FGLS estimators for b
nThe FGLS estimator is called Cochrane-Orcutt estimator
n
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Cochrane-Orcutt Estimation
nIn following steps
1.OLS estimation of b for b from yt = xt‘b + et, t = 1, …, T
2.Estimation of r for r from the auxiliary regression et = ret-1 + vt
3.Calculation of quasi-differences yt* =  yt – ryt-1 and xt* = xt – rxt-1
4.OLS estimation of b from
n yt* = xt*‘b + vt, t = 2, …, T
n resulting in the Cochrane-Orcutt estimators
nSteps 2. to 4. can be repeated: iterated Cochrane-Orcutt estimator
qGRETL provides the iterated Cochrane-Orcutt estimator:
q Model => Time series => Cochrane-Orcutt
n
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Demand for Ice Cream, cont’d
nIterated Cochrane-Orcutt estimator
n
n
n
n
n
n
n
n
n
n
nDurbin-Watson test: dw = 1.55; dL=1.21 < dw < 1.65 = dU
Dec 9, 2011
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Demand for Ice Cream, cont’d
nDemand for ice cream, measured by cons, explained by price, income, and temp, OLS and HAC standard
errors, and Cochrane-Orcutt estimates
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coeff
s.e.
Cochrane-Orcutt
OLS
HAC
coeff
se
constant
0.197
0.270
0.288
0.157
0.300
price
-1.044
0.834
0.881
-0.892
0.830
income
3.308
1.171
1.151
3.203
1.546
temp
3.458
0.446
0.449
3.558
0.555

Demand for Ice Cream, cont’d
nModel extended by temp-1
n
n
n
n
n
n
n
n
n
n
nDurbin-Watson  test: dw = 1.58; dL=1.21 < dw < 1.65 = dU
Dec 9, 2011
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Demand for Ice Cream, cont’d
nDemand for ice cream, measured by cons, explained by price, income, and temp, OLS and HAC standard
errors, Cochrane-Orcutt estimates, and OLS estimates for the extended model
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OLS
Cochrane-Orcutt
OLS
coeff
HAC
coeff
se
coeff
se
constant
0.197
0.288
0.157
0.300
0.189
0.232
price
-1.044
0.881
-0.892
0.830
-0.838
0.688
income
3.308
1.151
3.203
1.546
2.867
1.053
temp
3.458
0.449
3.558
0.555
5.332
0.670
temp-1
-2.204
0.731

General Autocorrelation Structures
nGeneralization of model
n yt = xt‘b + et
n with et = ret-1 + vt
nAlternative dependence structures of error terms
nAutocorrelation of higher order than 1
nMoving average pattern
n
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Higher Order Autocorrelation
nFor quarterly data, error terms may develop according to
n et = get-4 + vt
n or - more generally - to
n et = g1et-1 + … + g4et-4 + vt
net follows an AR(4) process, an autoregressive process of order 4
nMore complex structures of correlations between variables with autocorrelation of order 4 are
possible than with that of order 1
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Moving Average Processes
nMoving average process  of order 1, MA(1) process
n et = vt + avt-1
nεt is correlated with εt-1, but not with εt-2, εt-3, …
nGeneralizations to higher orders
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Remedies against Autocorrelation
nChange functional form, e.g., use log(y) instead of y
nExtend the model by including additional explanatory variables, e.g., seasonal dummies, or
additional lags
nUse HAC standard errors for the OLS estimators
nReformulate the model in quasi-differences (FGLS) or in differences
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Your Homework
1.Use the data set “labour2” of Verbeek for the following analyses:
a.Estimate (OLS) the model where log labor is explained by log output and log wage; generate a
display of the residuals which may indicate heteroskedasticity of the error term
b.Perform the Breusch-Pagan (i) with h(zi‘a) = exp(zi‘a) and (ii) with h(zi‘a) = zi‘a, and the
White test (iii) with and (iv) without interactions; explain the tests and compare and interpret
the results
c.Compare (i) the OLS and the White standard errors with (ii) HC0 and (iii) HC1 of the estimated
coefficients; interpret the results
d.Estimate the model of a., using FGLS and weights obtained in the auxiliary regression of the
Breusch-Pagan test (ii) in b.; compare the results with that of a.
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Your Homework, cont’d
2.Use the data set “icecream” of Verbeek for the following analyses:
a.Estimate the model where cons is explained by income and temp; generate two displays of the
residuals which may indicate autocorrelation of the error terms
b.Use the Durbin-Watson and the Breusch-Godfrey test against autocorrelation; interpret the result
c.Repeat a., using (i) the iterative Cochrane-Orcutt estimation and (ii) OLS estimation of the
model in differences; interpret the result.
3.Durbin-Watson: (a) Explain the meaning of the statement “The Durbin-Watson test is a
misspecification test”; (b) show that dw ≈ 2 – 2r; (c) which of the following tests is a
generalization of the DW test? (i) Box-Pierce test; (ii) Breusch-Godfrey test; explain why.
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