Econometrics - Lecture 4
Heteroskedasticity


Contents
nViolations of V{ε} = s2 IN: Illustrations and Consequences
nHeteroskedasticity
nTests against Heteroskedasticity
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
n V{ε} = diag(s12, …, sN2)
n with si2 ¹ sj2 for at least one pair i ¹ j, or using si2 = s2 hi2,
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 (HHs):
n monthly HH-income and expenditures for durable goods
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n
n
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0
400
800
1200
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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
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HH-income
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Typical Situations for Heteroskedasticity
nHeteroskedasticity is typically observed
nin data from cross-sectional surveys, e.g., surveys in households or regions
nin data with variance that depends of one or several explanatory variables, e.g., variance of the
firms’ turnover depends on firm size (in number of staff)
nin data from financial markets, e.g., exchange rates, stock returns
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Example: Household Expenditures
nVariation of expenditures for food, increasing with growing income; 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.: to be expected that successive observations of
economic variables correlate positively
nSeasonal adjustment: application of smoothing and filtering algorithms induces correlation of the
smoothed data
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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: Total Demand
(from AWM-database)
Import function:  MTR = -227320 + 0.36 FDD
R2 = 0.977, tFFD = 74.8
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Import Function: Residuals
MTR: Imports
FDD: Total Demand
(from AWM-database)
RESID: et = MTR - (-227320 + 0.36 FDD)
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Import Function: Residuals, 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 to be suspected!
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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Some Import Functions
nRegression of imports (MTR) on total demand (FDD)
n MTR = -2.27x109 + 0.357 FDD, tFDD = 74.9, R2 = 0.977
n Autocorrelation (of 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 for OLS estimators
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
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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 heteroskedasticity and for autocorrelation are important tools
n
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Contents
nViolations of V{ε} = s2 IN: Illustrations and Consequences
nHeteroskedasticity
nTests against Heteroskedasticity
nGLS Estimation
nAutocorrelation
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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 and Potential Regressors
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Inference under Heteroskedasticity
nCovariance matrix of b:
n V{b} = s2 (X'X)-1 X'YX (X'X)-1
n with Y = diag(h12, …, hN2)
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, i = 1,…,N: 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 modifications are available as options
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Labor Demand Function
nFor Belgian companies, 1996; Verbeek’s data set “labour2”
n
n
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n
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n
n
n
n
n labour = b1 + b2*wage + b3*output + b4*capital
n
n
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Labor Demand Function: Residuals vs output
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Labor Demand Function, cont’d
nCan the error terms be assumed to be homoskedastic?
nThey may vary depending on 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
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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
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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   -4,5905    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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Labor Demand Function, cont’d
nEstimated function
n labour = b1 + b2*wage + b3*output + b4*capital
n OLS estimates and standard errors: without (s.e.) and with White correction (White s.e.) and GLS
estimates with wi = 1/(ei2)
n
n
n
n
n
n
nThe White standard errors are inflated by factors 3.7 (wage), 6.4 (capital), 7.0 (output) with
respect to the OLS 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
321.17
-7.404
15.585
-4.740
s.e.
20.328
 0.506
 0.349
 0.255

An Alternative Estimator for b
nIdea of the estimator
1.Transform the model so that it satisfies the Gauss-Markov assumptions
2.Apply OLS to the transformed model
nResults in an (at least approximately) BLUE
n
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
nThis estimator is an example of the “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 wi proportional to the inverse of the error term variance σ2hi2: 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
qEstimates of hi can be based on assumptions on the form of hi
qE.g., hi2 = zi’a or hi2 = exp(zi’a) for some variables zi
nAnalogous with general weights wi
nWhite’s HCCME uses wi = ei-2
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Labor Demand Function, cont’d
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Regression  of “l_usq1”, i.e., log(ei2), on capital and output
Dependent variable : l_usq1
             coefficient   std. error   t-ratio    p-value
  -------------------------------------------------------------
  const     7,24526      0,0987518    73,37     2,68e-291 ***
  CAPITAL   −0,0210417    0,00375036   −5,611    3,16e-08 ***
  OUTPUT     0,0359122    0,00481392      7,460    3,27e-013 ***
Mean dependent var                      7,531559   S.D. dependent var    2,368701
Sum squared resid 2797,660   S.E. of regression   2,223255
R- squared               0,122138   Adjusted R-squared 0,119036
F(2, 129)               39,37427   P-value (F)               9,76e-17
Log-likelihood                           −1260,487 Akaike criterion       2526,975
Schwarz criterion      2540,006   Hannan-Quinn 2532,060
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Labor Demand Function, cont’d
nEstimated function
n labour = b1 + b2*wage + b3*output + b4*capital
n OLS estimates and standard errors: without (s.e.) and with White correction (White s.e.); and GLS
estimates with wi = ei-2, with fitted values for ei from the regression of  log(ei2) on capital and
output
n
n
n
n
n
n
n
n
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b1
wage
output
capital
OLS coeff
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
FGLS coeff
321.17
-7.404
15.585
-4.740
s.e.
20.328
 0.506
 0.349
 0.255

Contents
nViolations of V{ε} = s2 IN: Illustrations and Consequences
nHeteroskedasticity
nTests against Heteroskedasticity
nGLS Estimation
nAutocorrelation
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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, zi containing an intercept and p-1 variables Z2,
…, Zp
nNull hypothesis
n H0: a = 0
n implies si2 = s2 for all i, i.e., homoskedasticity
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
nFor h(zi‘a) = zi‘a
nAuxiliary regression of the “scaled” squared residuals ui2 = ei2/s2 with  s2 = e’e/N on zi (and
squares of zi);
nTest statistic BP follows approximately the Chi-squared distribution with p -1 d.f.
n
n
n
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Koenker Test
nKoenker test: variant of the BP test which is robust against non-normality of the error terms
nFor testing whether the error term variance is a function of Z2, …, Zp
nAuxiliary regression of the squared OLS residuals ei2 on zi
n ei2 = zi‘a + vi
nTest statistic: N*Rv2 with Rv2 of the auxiliary regression; follows approximately the Chi-squared
distribution with p -1 d.f.
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 Tests of the null hypothesis of homoskedasticity
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n Breusch-Pagan: BP = 5931.8, p-value = 0
n Koenker: NR2 = 569*0.5818 = 331.04, p-value = 2.17E-70
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
nRegimes 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 A and B
2.Separate fittings of the model to the NA and NB observations; sum of squared residuals SA and SB
3.Calculate the 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; generalizes the Breusch-Pagan test
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
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Contents
nViolations of V{ε} = s2 IN: Illustrations and Consequences
nHeteroskedasticity
nTests against Heteroskedasticity
nGLS Estimation
nAutocorrelation
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Transformed Model Satisfying Gauss-Markov Assumptions
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
nThis estimator is an example of the “generalized least squares” (GLS) or “weighted least squares”
(WLS) estimator
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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 hold approximately without normality assumption
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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
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
qTransformation corresponds to the multiplication of each observation with the non-negative factor
hi-1
nThe GLS estimator is a least squares estimator that weights the i-th observation with wi = hi-2,
so that the Gauss-Markov assumptions are satisfied
n
n
n
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Feasible GLS Estimator
nIs a GLS estimator with estimated weights wi = hi-2
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 consistent estimates 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
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 (yi = xi’β + εi):
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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FGLS Estimation in GRETL
nPreparatory steps:
1.Calculate the OLS estimates b for b of yi = xi’β + εi
2.Under the assumption V{εi} = σi2 = σ2hi2, conduct an auxiliary regression for ei2 or log(ei2)
that provides estimates ĥi2
3.Define wtvar  as weight variable with wtvar i = (ĥi2)-1
nFGLS estimation:
4.Model => Other linear models => Weighted least squares
5.Use of variable wtvar  as “Weight variable”: both the dependent and all independent variables are
multiplied with the square roots (wtvar)1/2
n
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Labor Demand Function
nFor Belgian companies, 1996; Verbeek
n
n
n
n
n
n
n
n
n
n
nLog-transformation is expected to reduce heteroskedasticity
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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 and standard errors: without (s.e.) and with White correction
(White s.e.); FGLS estimates and standard errors
n
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b1
wage
output
capital
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
nFor Belgian companies, 1996; Verbeek
n
n
n
n
n
n
n
n
n
n
nBreusch-Pagan test: BP = 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 and standard errors: without (s.e.) and with White correction
(White s.e.); FGLS estimates and standard errors
n
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b1
wage
output
capital
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 to 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 has no intercept, is uncentered
qComparison with that of OLS estimation not appropriate, explained variables are different
n
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Your Homework
1.Use the data set “labour2” of Verbeek for the following analyses:
a)(i) Estimate (OLS) the model for log(labor) with regressors log(output) and log(wage); (ii)
generate a display of the residuals which may indicate heteroskedasticity of the error term.
b)Perform (i) the Koenker test with h(zi‘a) = exp{zi‘a} and the White test (ii) without and (iii)
with interactions; explain the tests and compare the results; use zi = (log(capitali),
log(outputi), log(wagei))’.
c)For the model of a): Compare (i) the OLS and (ii) the White standard errors with HC0 of the
estimated coefficients.
d)Estimate (i) the model of a), using FGLS and weights obtained in the auxiliary regression of the
Koenker test in b); (ii) comment on the estimates of the coefficients, the standard errors, and the
R2 of this model and those of c)(i) and (ii).
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Your Homework, cont’d
2.Transform the variables of the model yi  = xi’β + εi with E{εi} = 0 and V{εi} = σi2 = σ2hi2 for i
= 1, …, N, by dividing each variable through hi: yi -> yi /hi and (xi)’ -> (xi /hi)’. Show that for
the model in transformed variables,
n yi /hi = (xi /hi)’β + εi /hi
n       the Gauss-Markov assumptions A3 and A4 are satisfied.
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