Econometrics Regression Analysis with Time Series …docentes.fe.unl.pt/~azevedoj/Web...
Transcript of Econometrics Regression Analysis with Time Series …docentes.fe.unl.pt/~azevedoj/Web...
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
EconometricsRegression Analysis with Time Series Data:
Serial Correlation and Heteroskedasticity
Joao Valle e Azevedo
Faculdade de EconomiaUniversidade Nova de Lisboa
Spring Semester
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 1 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serially Correlated Errors: Consequences
With assumptions TS.1 through TS.3, OLS estimators are unbiased
With assumptions TS.1’ through TS.3’, OLS estimators areconsistent
Could have serial correlation in the errors (TS.5 or TS.5’ could beviolated)
But for inference, results are NOT valid if TS.5 (or TS.5’) fail
With serial correlation in the errors, usual OLS variances are NOTvalid
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 2 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serially Correlated Errors
But notice, can have consistency even in the presence of serial correlationin the errors and lagged dependent variables as regressors
Example:
yt = β0 + β1yt−1 + utI where E [ut |yt−1] = 0 and {ut} are serially correlated
However, if
yt = β0 + β1yt−1 + ut and ut = ρut−1 + et , t = 2, ..., n
et are i.i.d., |ρ| < 1 and E [et |ut−1, ut−2, ...] = E [et |yt−1, yt−2, ...] = 0
Then,
Cov(yt−1, ut) = E [yt−1(ρut−1 + et)] = ρE (yt−1, ut−1)
= ρE [yt−1(yt−1 − β0 − β1yt−2)] 6= 0 unless ρ = 0
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 3 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serially Correlated Errors (Cont.)
In this case the OLS estimators are not consistent for β0,β1. This is aspecial (although typical) form of autocorrelation
But in the previous case, it turns out that:
yt = β0 + β1yt−1 + ρ(yt−1 − β0 − β1yt−2) + et
= β0(1− ρ) + (β1 + ρ)yt−1 − ρβ1yt−2 + et
= α0 + α1yt−1 + α2yt−2 + et
I where E [et |yt−1, yt−2, ...] = 0 and
E [yt |yt−1, yt−2, ...] = E [yt |yt−1, yt−2] = α0 + α1yt−1 + α2yt−2
Thus, the ”relevant” model is an AR(2) model for y. With furtherconditions on the parameters (that ensure stability) we can estimatethe α′js consistently
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 4 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serially Correlated Errors (Cont.)
Conclusion: Models with lagged dependent variables and serialcorrelation in the errors can often be easily transformed into modelswithout serial correlation in the errors (e.g., by adding more lags ofthe dependent variable as regressors)
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 5 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Testing for AR(1) Serial Correlation (Cont.)
Want to be able to test whether the errors are serially correlated ornot
yt = β0 + β1xt1 + ...+ βkxtk + ut
Want to test the null that ρ = 0 in ut = ρut−1 + et ,t=2,...,nI where ut is the model error term, |ρ| < 1 and et is an i.i.d. sequence
with E [et |ut−1, ut−2, ...] = 0 and Var [et |ut−1] = σ2e
With strictly exogenous regressors, E (ut |X) = 0, t = 1, 2, ..., n, thetest is straightforward
I Obtain the OLS residuals of the original modelI Then, regress the residuals on (one period) lagged residuals (use OLS)I Then, use a typical t-test for H0 : ρ = 0 (don’t need intercept)I We assume TS.1 through TS.4 hold. If TS.4 fails use a
heteroskedasticity robust statistic
Detects correlation between ut and ut−1 but not between ut and ut−2
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 6 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Testing for AR(1) Serial Correlation (Cont.)
An alternative is the Durbin-Watson (DW) statistic, which iscalculated by most Econometric packages. Need TS.1 through TS.6holding
DW =n∑
t=2
(ut − ut−1)2/
n∑t=1
u2t ≈ 2(1− ρ)
where ρ is that of the previous test
If the DW statistic is around 2, then we do not reject H0 (absence ofserial correlation of AR(1) type), while if it is significantly < 2 wereject the null (against the alternative H1 : ρ > 0, the most typical)
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 7 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Testing for AR(q) Serial Correlation
If the regressors are not strictly exogenous, previous t or DW test willNOT be valid. Have alternatives under TS.1’ through TS.4’ (thatare valid also with TS.1 through TS.4). Also, can have more generalforms of serial correlation (e.g., AR(q) serial correlation)
yt = β0 + β1xt1 + ...+ βkxtk + ut
I where ut = ρ1ut−1 + ρ2ut−2 + ...+ ρqut−q + etI et is an i.i.d. sequence with E [et |ut−1, ut−2, ...] = 0 and
Var [et |xt , ut−1, ut−2, ..., ut−q] = σ2e
I The null is H0 : ρ1 = ρ2 = ... = ρq = 0
Regress the OLS residuals (or y) on the q lagged residuals and all ofthe x ’s (this makes the test valid without assuming strict exogeneity.Don’t need to include the x ’s if strict exogeneity is assumed)Then use a usual test of multiple exclusion restrictions (on thecoefficients of the lagged residuals)
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 8 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Testing for AR(q) Serial Correlation (Cont.)
Can use F test or LM test (or even t test if q=1), where the LMversion is called the Breusch-Godfrey test: LM = (n − q)R2 ∼ χ2
q
(approx.) under the null where R2 is from the residuals regression.
If there is heteroskedasticity in the error term ut , can useheteroskedasticity-robust statistics
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 9 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Correcting Serial Correlation with strictly exogenousregressors
If there is serial correlation in the errors, inference ”as usual” withOLS is Not valid
Let us assume TS.1 through TS.4 hold in our model, but TS.5 (noserial correlation) fails to hold. Also, assume stationarity and weakdependence as in TS.1’
Assume errors follow an AR(1) so ut = ρut−1 + et , t = 2, ..., n, where|ρ| < 1 and et is an i.i.d. sequence with E [et |X ] = 0 andVar [et |X ] = σ2
e
I Var [ut |X ] = σ2e/(1− ρ2)
We can transform the equation so we have no serial correlation in theerrors
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 10 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Correcting Serial Correlation with strictly exogenousregressors (Cont.)
Consider the case of one regressor (similar analysis with more)
yt = β0 + β1xt + ut
Then,
ρyt−1 = ρβ0 + ρβ1xt−1 + ρut−1
Subtract this from previous equation to get:
yt − ρyt−1 = (1− ρ)β0 + β1(xt − ρxt−1) + et
I since et = ut − ρut−1 for t ≥ 2
In this quasi-differenced model for t ≥ 2, TS.1 through TS.5 hold!
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 11 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
GLS Estimationyt − ρyt−1 = (1− ρ)β0 + β1(xt − ρxt−1) + et for t ≥ 2
Can transform equation for t=1, y1 = β0 + β1x1 + u1, so that TS.1through TS.5 hold for t ≥ 1 (not yet the case since u1 and et have adifferent variance)
Var(ut |X ) = σ2e/(1− ρ2), so can multiply equation by (1− ρ2)1/2 to
have TS.1 through TS.5 holding for t ≥ 1
(1− ρ2)1/2y1 = (1− ρ2)1/2β0 + (1− ρ2)1/2β1x1 + (1− ρ2)1/2u1
In this quasi-differenced model, TS.1 through TS.5 hold!
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 12 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
GLS Estimation (Cont.)
Transformed model is:
yt = β0xt0 + β1xt1 + et , t ≥ 1
yt = yt − ρyt−1, t ≥ 2 and (1− ρ2)1/2y1 for t = 1
xt1 = xt − ρxt−1, t ≥ 2 and (1− ρ2)1/2x1 for t = 1
xt0 = (1− ρ), t ≥ 2 and (1− ρ2)1/2 for t = 1
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 13 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
GLS Estimation (Cont.)
If ρ is known, can estimate the transformed regression by OLS
This is called the GLS estimator of the original model
GLS is BLUE if TS.1 through TS.5 hold in the transformed model
Can use t and F tests from the transformed equation to conductinference on the parameters of the original equation
These tests are valid (asymptotically) if TS.1 through TS.5 hold inthe transformed model (along with stationarity and weak dependencein the original variables) and distributions (conditional on X) areexact (and with minimum variance) if TS.6 holds for the et
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 14 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Feasible GLS
Problem: don’t know ρ, need to get an estimate first
I Run OLS on the original model and then regress residuals on laggedresiduals (with OLS)
I Use OLS on the transformed regression (using the estimated ρ). This isa feasible GLS (FGLS) estimator (or Prais-Winsten estimator)
I If we just forget the first equation (t=1) we have theCochrane-Orcutt estimator (also a FGLS estimator)
I These FGLS estimators are not unbiased, but are consistent if TS.1through TS.5 hold in the transformed model (along with stationarityand weak dependence in the original model)
I There is no difference, asymptotically, between the two procedures
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 15 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Feasible GLS
t and F tests from the transformed equations are valid(asymptotically). If TS.6 holds for the et , relevant distributions alsohold only asymptotically. FGLS is asymptotically more efficient thanOLS
This basic method can be extended to allow for higher order serialcorrelation, AR(q), in the error term. Econometrics packages dealautomatically with estimation of AR serial correlation
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 16 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serial Correlation-Robust Standard Errors
FGLS asymptotic distributions rely on strict exogeneity of theregressors
If strict exogeneity does not hold, we can still use OLS (it isconsistent with only contemporaneous exogeneity, or TS.1’, alongwith TS.2’ and TS.3’). Further, we can calculate serialcorrelation-robust standard errors
Idea is to scale the OLS standard errors to take into account serialcorrelation
Actually, with only TS.1’ through TS.3’ we can deriveHeteroskedasticity and Serial Correlation robust standard errorsand corresponding t-statistics
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Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serial Correlation-Robust Standard Errors (Cont.)
Estimate the model with OLS to get the residuals ut , the standarddeviation of the regression, σ, and the ”usual” standard errors”se(β1)”Run the auxiliary regression of xt1 on xt2, ..., xtk , get the residuals, rt ,and form at = rt utChoose a g - typically the integer part of n1/4
Compute
ν =n∑
t=1
a2t + 2
g∑h=1
[1− h/(g + 1)]
( n∑t=h+1
at at−h
)Then,
Robust se(β1) = [”se(β1)”/σ]2√ν
I and similarly for any βj
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 18 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Serial Correlation-Robust Standard Errors (Cont.)
These Heteroskedasticity and Serial Correlation robust standarderrors can be poorly behaved in small samples (typical in Macro timeseries) in the presence of ”heavy” serial correlation
Also, standard errors can be very large (OLS is far from efficiencywith ”heavy” serial correlation and small samples)
If you believe the regressors are strictly exogenous and errors followan AR(1) (or more generally an AR(q)), use FGLS
Even if you think the errors do not follow an AR(1) (but regressors arestill strictly exogenous) apply FGLS (as if errors followed an AR(1))and compute robust standard errors in the quasi-differenced model
So, use only Heteroskedasticity and Serial Correlation robuststandard errors if strict exogeneity fails
Will not deal with F statistics robust to serial correlation
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 19 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Heteroskedasticity in Time Series Regressions
With assumptions TS.1 through TS.3, OLS estimators are unbiased(no role of homoskedasticity, TS.4)
With assumptions TS.1’ through TS.3’, OLS estimators areconsistent (no role of homoskedasticity, TS.4’)
But for inference, ”usual” OLS results are NOT valid if TS.4 (orTS.4’) fail
With assumptions TS.1’ through TS.3’, we can deriveHeteroskedasticity and Serial Correlation robust standard errors
We can do better than this if TS.1’, TS.2’, TS.3’ and TS.5’ hold(need only heteroskedasticity robust standard errors)
Can compute heteroskedasticity robust standard errors and userobust t and LM statistics, exactly as in the cross sectional case
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 20 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Testing for Heteroskedasticity in Time Series Regressions
However, in small samples we know these robust standard errors canbe large. Therefore, we may want to test for heteroskedasticity
It turns out that we can also use the tests for heteroskedasticity ofthe cross-sectional case (Chapter 8)
But for this, need TS.1’, TS.2’, TS.3’ and TS.5’ holding. So, cannothave serial correlation in the errors
Also, e.g., for the Breusch-Pagan test where we specifyu2 = δ0 + δ1x1 + ...+ δkxk + ν and test H0 : δ1 = δ2 = ... = δk = 0,we need ν to be homoskedastic and serially uncorrelated (which isdifferent from serial correlation in the u’s)
Later, we will see how to deal with dynamic heteroskedasticity (wheresomething like the ν’s above can be correlated across time)
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 21 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Testing for Heteroskedasticity in Time Series Regressions
So, if we find heteroskedasticity can use heteroskedasticity robuststatistics. If we know the form of heteroskedasticity can also useWeighted Least Squares (WLS), along the same lines of thecross-sectional case. But still, TS.5’ (No serial Correlation) must bein place!
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 22 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Autoregressive Conditional heteroskedasticity
Now assume TS.1 through TS.5 hold, so we have Conditional (onX!) Homoskedasticity and no Serial Correlation in the Error term
Let yt = β0 + β1xt1 + ...+ βkxtk + ut
We can have E [u2t |X] = Var(ut |X) = Var(ut) = σ2 for t=1,2,...,n but
still:
E [u2t |X, ut−1, ut−2, ...] = E [u2
t |X, ut−1] = α0 + α1u2t−1 or
u2t = α0 + α1u
2t−1 + νt where E [ν|X, ut−1, ut−2, ...] = 0
I with α0 > 0 and α1 ≥ 0
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 23 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Autoregressive Conditional heteroskedasticity (Cont.)
νt are not independent of past u since νt ≥ −α0 − α1u2t−1 (variance
must be positive)
Also α1 < 1. If this model holds, we have Autoregressive Conditionalheteroskedasticity (ARCH) in the errors
So, even if the errors are not correlated (TS.5 holds), its squares canbe correlated
OLS is still BLUE with ARCH errors. Usual OLS inference is valid ifTS.6 (Normality) holds
Even if Normality does not hold we know that usual OLS inference isvalid asymptotically with TS.1’ through TS.5’ (so can have ARCHerrors in this setting as well)
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 24 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Autoregressive Conditional heteroskedasticity (Cont.)
So, why do we care about ARCH errors?
Because we can obtain asymptotically more efficient estimators thanOLS (albeit non-linear), with a sort of WLS obtained by firstestimating u2
t = α0 + α1u2t−1 + νt (by OLS, using squared residuals).
No more details in this course
ARCH behavior is interesting in itself, can analyze behavior ofvolatility
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 25 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 26 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon (Cont.)
BVLG Lisbon daily returns, January 2, 1995 to November 23, 2001
Just take the index
Apply logs
Take first differences
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 27 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon (Cont.)
Under the efficient market hypothesis, stock returns should beuncorrelated with past returns. Also, should not have calendareffects...But this is (was?) Portugal
I Dependent variable: Daily ReturnsI Regressors: Day of the week dummies, Tuesday* for Tuesdays from Jan 88 to Apr
89 and three lags of returnsI F test is for the hypothesis that all Day of the week dummies are equalI *, ** and *** denote significance at the 10%, 5% and 1% respectively
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 28 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon (Cont.)
Dependent variable: Daily Returns
Regressors: Day of the week dummies, Tuesday* for Tuesdays from Jan 88 to Apr 89 andthree lags of returns
F test is for the hypothesis that all Day of the week dummies are equal
*, ** and *** denote significance at the 10%, 5% and 1% respectively
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 29 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon (Cont.)
Besides, there is volatility clustering in the returns series
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 30 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon (Cont.)
White Heteroskedasticity Test:
F-statistic 63.32189 Probability 0.000000
Obs*R-squared 112.7329 Probability 0.000000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Sample: 3 1004
Included observations: 1002
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000144 1.82E-05 7.933013 0.0000
Y(-1) -0.000222 0.001125 -0.197479 0.8435
Y(-1)^2 0.299471 0.026700 11.21598 0.0000
R-squared 0.112508 Mean dependent var 0.000213
Adjusted R-squared 0.110731 S.D. dependent var 0.000573
S.E. of regression 0.000541 Akaike info criterion -12.20501
Sum squared resid 0.000292 Schwarz criterion -12.19031
Log likelihood 6117.708 F-statistic 63.32189
Durbin-Watson stat 2.075790 Prob(F-statistic) 0.000000y denotes returns
Assume there are no calendar effect but y , y(−1), are correlated, yt = β0 + β1yt−1 + utFind evidence of heteroskedasticity
But if we accept that TS.4’ holds, the significant coefficient on y(−1)2 can be due toARCH effects
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 31 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Example - BVLG index, Lisbon (Cont.)
ARCH effects
F-statistic 114.8229 Probability 0.000000
Obs*R-squared 103.1921 Probability 0.000000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Sample(adjusted): 4 1004
Included observations: 1001 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
C 0.000145 1.83E-05 7.903650 0.0000
RESID^2(-1) 0.321081 0.029964 10.71555 0.0000
R-squared 0.103089 Mean dependent var 0.000213
Adjusted R-squared 0.102191 S.D. dependent var 0.000573
S.E. of regression 0.000543 Akaike info criterion -12.19544
Sum squared resid 0.000295 Schwarz criterion -12.18564
Log likelihood 6105.819 F-statistic 114.8229
Durbin-Watson stat 2.064939 Prob(F-statistic) 0.000000
Find evidence of ARCH effects
Remember, can have serial correlation in the squared errors even if there is no serialcorrelation in the errors
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 32 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Heteroskedasticity and Serial Correlation
Can have violation of TS.4 (Homoskedasticity) and TS.5 (No serialcorrelation) simultaneously. Assume still that TS.1 through TS.3 hold(along with stationarity and weak dependence)
yt = β0 + β1xt1 + ...+ βkxtk + ut
ut = νt√
ht
νt = ρνt−1 + et
I where X are independent of et , ht is a function of the regressorsI Also the process {et} has mean zero, constant variance and is serially
uncorrelated
yt√ht
=β0√ht
+β1xt1√
ht+ ...+
βkxtk√ht
+ut√ht
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 33 / 34
Serial Correlation Heteroskedasticity Example Heteroskedasticity and Serial Correlation
Time Series Analysis
Heteroskedasticity and Serial Correlation (Cont.)
We can estimate the function h exactly as in chapter 8
This transformed equation has still serial correlation but noheteroskedasticity
Use the estimated h in the equation above and applyCochrane-Orcutt or Prais-Winsten
This leads to a feasible GLS estimator that is asymptotically efficient.Test statistics from Cochrane-Orcutt or Prais-Winsten areasymptotically valid
Joao Valle e Azevedo (FEUNL) Econometrics Lisbon, May 2011 34 / 34