serial correlation 1 - transtutors...pure versus impure serial correlation consequences and...

Pure Versus Impure Serial Correlation Consequences and Detection Remedies

Serial Correlation 1

Pierre Nguimkeu — BUEC 333

Summer 2011

1Based on P. Lavergne, Lectures notes


Outline

Pure Versus Impure Serial Correlation

Consequences and Detection

Remedies


Pure Serial Correlation

Serial correlation Correlation across the error terms: Violates ClassicalAssumption 4.Happens in time-series, or if the order of observations has some meaningor importance.From now on,

Yt = β0 + β1X1t + β2X2t + εt t = 1, . . .T

We always haveIE (εt |Xt) = 0 ,

and then classical Assumption 2 holds. But

Cov (εt , εt−1) != 0


Impure Serial CorrelationCaused by misspecification in the model, i.e. Classical Assumption 1 doesnot hold.If Yt = β0 + β1X1t + β2X2t + εt but we omit X2t , then

Yt = β0 + β1X1t + ε∗t where ε

∗t = εt + β2X2t

The new error term will be serially correlated if

• X2 is relevant, i.e. β2 != 0• X2 is serially correlated, and in particular has a trend• High autocorrelation if “size” of β2X2t large compared to “size” of εt

Remark We should actually write

Yt = β∗0 +β

∗1X1t +ε

∗t where ε

∗t = εt +β0+β1X1t +β2X2t−(β∗0 + β∗1X1t)

because the values of the coefficients may not be the same in the twoequations.If Yt = β0 + β1X1t + β2X 21t + εt but we specify a linear equation, then

Yt = β0 + β1X1t + ε∗t where ε

∗t = εt + f (X1t)


Consequences

• Estimates remain unbiased• OLS is not BLUE in general, then OLS has not minimum variance• The standard errors are biased, typically too small• t-scores are typically too large, they don’t have a t-distribution, so

confidence intervals and hypothesis tests are unreliable.


Preliminary Checks

• Are there any obvious specification errors? Delay testing for serialcorrelation until you are confident with your specification.

• Is the dependent variable likely afflicted with serial correlation?

11

12

13

14

15

16

80 82 84 86 88 90 92 94 96 98 00 02

U.S. poverty rate

• Is there any autocorrelation in the OLS residuals?


Autocorrelation Coefficient

Autocorrelation coefficient of order k for a time series Zt is

rk =

∑Tt=k+1

(Zt − Z̄

) (Zt−k − Z̄

)

∑Tt=1

(Zt − Z̄

)2 .

rk measures the correlation between the Z that are k-periods away.Note the difference with the correlation coefficient

r =

∑ni=1

[(Xi − X̄

) (Yi − Ȳ

)]√∑n

i=1

(Xi − X̄

)2√∑ni=1

(Yi − Ȳ

)2


First Order Autocorrelation

Key example: First-order autocorrelation of errors

εt = ρεt−1 + ut t = 1, . . . ,T

εt = error term at t,ut = a “classical error” term, not serially correlated,

and uncorrelated with εt−1ρ = the first-order autocorrelation coefficient.

ρ measures the strength of correlation between εt and εt−1.Then

r1 =

∑Tt=2 (εt − ε̄) (εt−1 − ε̄)∑T

t=1 (εt − ε̄)2

=

∑Tt=2 (ρεt−1 + ut − ε̄) (εt−1 − ε̄)∑T

t=1 (εt − ε̄)2

= ρ

∑Tt=2 ε

2t−1∑T

t=1 (εt − ε̄)2

+

∑Tt=2 (ut − ε̄) (εt−1 − ε̄)∑T

t=1 (εt − ε̄)2

≈ ρ .


First Order Autocorrelation (ct’d)

• The sign of ρ indicates the nature of serial correlation• Positive serial correlation: successive values tend to have the same

sign.Common situation in economics if we look at level variables.

• Negative serial correlation: successive values tend to alternate sign.Can occur when we look at variables in first-differences.

• As a correlation −1 < ρ< 1.• |ρ| ≥ 1 not reasonable, since it implies that the error term has a

tendency to “explode”


Other Autocorrelation Patterns

Other possible forms of autocorrelation

Second-order serial correlation

εt = ρ1εt−1 + ρ2εt−2 + ut

Fourth-order for Quarterly Data

εt = ρεt−4 + ut

Fourth-orderεt = ρ1εt−1 + ρ2εt−2 + ρ3εt−3 + ρ4εt−4 + ut


Correlogramm

We cannot compute the autocorrelations of the errors since we don’tobserve them.Rather we compute the autocorrelations of the residuals.Correlogram: graph of autocorrelation coefficients of the residuals.

Dependent Variable: POVERTYMethod: Least SquaresDate: 11/16/09 Time: 13:58Sample: 1980 2003Included observations: 24

Variable Coefficient Std. Error t-Statistic Prob.

C 9.792052 0.611186 16.02138 0.0000UNEMPLOY 0.586614 0.094726 6.192734 0.0000

R-squared 0.635460 Mean dependent var 13.47917Adjusted R-squared 0.618890 S.D. dependent var 1.095437S.E. of regression 0.676259 Akaike info criterion 2.135173Sum squared resid 10.06116 Schwarz criterion 2.233344Log likelihood -23.62207 F-statistic 38.34995Durbin-Watson stat 0.323725 Prob(F-statistic) 0.000003

Correlogram of RESID01

Date: 11/16/09 Time: 14:07Sample: 1980 2003Included observations: 24

Autocorrelation Partial Correlation AC PAC Q-Stat Prob

1 0.756 0.756 15.506 0.0002 0.579 0.018 25.024 0.0003 0.353 -0.212 28.721 0.0004 0.068 -0.333 28.866 0.0005 -0.110 -0.020 29.260 0.000

The partial correlation of order k is the correlation of the εt with theεt−k , once we take into account the autocorrelation of order 1, 2, . . . k-1.


The Durbin-Watson Test

• The regression includes an intercept• Potential serial correlation is first-order autocorrelation• The regression model does not include a lagged dependent variable

as an independent variable

The Durbin-Watson statistic is

d =

∑Tt=2 (et − et−1)

2

∑Tt=1 e

2t

• If there is extreme positive correlation, d ≈ 0. Imagine indeed thatet = et−1, then et − et−1 = 0 and d = 0.

• If there is extreme negative correlation, d ≈ 4. Since if et = −et−1,then et − et−1 = 2et and d ≈ 4.

• If there is no first-order autocorrelation, d ≈ 2.


The Durbin-Watson Test (ct’d)

Indeed,

d =

∑Tt=2 (et − et−1)

2

∑Tt=1 e

2t

=

∑Tt=2 e

2t − 2

∑Tt=2 etet−1 +

∑Tt=2 e

2t−1∑T

t=1 e2t

≈∑T

t=2 e2t +

∑Tt=2 e

2t−1∑T

t=1 e2t

≈ 2 .


Using the DW Test

• Obtain the OLS residuals and compute d• Determine the lower critical value dL and the upper critical value dU

from Tables B-4, B-5 or B-6.

For testing H0 : ρ ≤ 0 against HA : ρ > 0

Reject H0 if d < dLDo not reject H0 if d > dU

Inconclusive if dL ≤ d ≤ dU

Mostly apply if economic variables in levels.For testing H0 : ρ = 0 against HA : ρ != 0

Reject H0 if d < dL or d > 4− dLDo not reject H0 if 4− dU > d > dU

Inconclusive if dL ≤ d ≤ dU or 4− dU ≤ d ≤ 4− dL

Could apply to economic variables in first differences.


Poverty and Unemployment



C 9.792052 0.611186 16.02138 0.0000UNEMPLOY 0.586614 0.094726 6.192734 0.0000


We expect positive first-order autocorrelation. Here dL = 1.27 so wereject the null hypothesis of no first-order autocorrelation.Same conclusion as with the autocorrelogram.


The Breusch-Godfrey Test

The idea of the Breusch-Godfrey test is similar to the White test: use theresiduals and run an auxiliary regression. It allows to test potentialhigh-order autocorrelation.

1. Estimate your equation Yt = β0 + β1X1t + β2X2t + εt by OLS andget the residuals et

2. Run the OLS “auxiliary regression”

et = α0 + α1X1t + α2X2t + γ1et−1 + . . . + γket−k + ut

That is regress the residuals on all independent variables and alllagged residuals up to order k.

3. Test the significance of all coefficients γ1, . . . γk with an F-test.

H0 : γ1 = γ2 = . . . = γk = 0 against HA : at least one is not 0

The test also works if there is dependent lagged variables as independentvariables.



Dependent Variable: RESID01Method: Least SquaresDate: 11/16/09 Time: 14:08Sample (adjusted): 1982 2003Included observations: 22 after adjustments


C -0.773518 0.386543 -2.001119 0.0607UNEMPLOY 0.121246 0.060607 2.000527 0.0608RESID01(-1) 0.709750 0.219957 3.226764 0.0047RESID01(-2) 0.264418 0.246816 1.071313 0.2982


Wald Test:Equation: EQ02

Test Statistic Value df Probability

F-statistic 29.03862 (2, 18) 0.0000Chi-square 58.07724 2 0.0000

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

C(3) 0.709750 0.219957C(4) 0.264418 0.246816

Restrictions are linear in coefficients.

Not only the F-test indicates that there is autocorrelation, but t-scoressuggest that only first-order autocorrelation is important.


Generalized Least-Squares

Yt = β0 + β1X1t + εt with εt = ρεt−1 + ut .

Yt = β0 + β1X1t + ρεt−1 + ut

ρYt−1 = ρβ0 + ρβ1X1t−1 + ρεt−1

Yt − ρYt−1 = β0 (1− ρ) + β1 (X1t − ρX1t−1) + ut .

We have a new equation that satisfies the classical assumptions:

Y ∗t = β∗0 + β1X

∗1t + ut (1)

where Y ∗t = Yt − ρYt−1 and X ∗1t = (X1t − ρX1t−1), calledquasi-differences.

• The error term is not serially correlated. Thus OLS are BLUE.• The slope β1 is the same as in the original equation.• The dependent variable has changed, so don’t compare the R2.

But we don’t know ρ!


Feasible Generalized Least-Squares

Cochrane-Orcutt method

1. Estimate the original equation by OLS2. Estimate by OLS the auxiliary regression

et = ρet−1 + ut

3. Use ρ̂ to create quasi-differences and estimate Equation (1) by OLSto get β̂∗0 and β̂1.

4. Compute β̂0 = β̂∗0/ (1− ρ̂) and new residualset = Yt −

(β̂0 + β̂1Xt

).

5. Iterate the procedure until estimates do not change.



The AR(1) methodAR(1) stands for AutoRegressive error term of order 1.Estimates simultaneously β0, β1 and ρ from

Yt − ρYt−1 = β0 (1− ρ) + β1 (X1t − ρX1t−1) + ut .

OLS cannot be used since the equation in nonlinear in the parameters.But Generalized Least-Squares (GLS) can be used. How?

minT∑

t=2

{Yt − ρYt−1 − β0 (1− ρ)− β1 (X1t − ρX1t−1)}2

with respect to β0, β1 and ρ.





C 9.792052 0.611186 16.02138 0.0000UNEMPLOY 0.586614 0.094726 6.192734 0.0000


Dependent Variable: POVERTYMethod: Least SquaresDate: 11/16/09 Time: 14:19Sample (adjusted): 1981 2003Included observations: 23 after adjustmentsConvergence achieved after 31 iterations


C 9.890041 0.759313 13.02498 0.0000UNEMPLOY 0.583454 0.107377 5.433708 0.0000

AR(1) 0.807484 0.133612 6.043504 0.0000


Inverted AR Roots .81



Remarks

• Cochrane-Orcutt is a pioneering method. It is good to know it, butAR(1) is mostly used now.

• AR(1) may not work well in small samples.• Caution: When using AR(1), the R2, standard error of the

regression, and DW are computed by Eviews in a specific way. Don’tpay attention to them!

• We may want to allow for autocorrelation of order 2: AR(2)!


HAC Robust Standard Errors

In place of using another estimation method, we use OLS (which isunbiased and consistent) and correct the computation of the standarderrors. Heteroscedasticity and AutoCorrelation robust standard errors:also called Newey-West standard errors

• Estimate the standard deviation of the OLS coefficients whetherthere is heteroscedasticity and autocorrelation or not

• Are often larger than the OLS standard errors• Can be used to construct tests and confidence intervals in the usual

way

• Works best in “large” samples• Are given by Eviews, see Options/Heteroscedasticty consistent

coefficient covariance.





C 9.792052 0.611186 16.02138 0.0000UNEMPLOY 0.586614 0.094726 6.192734 0.0000


Dependent Variable: POVERTYMethod: Least SquaresDate: 11/16/09 Time: 14:20Sample: 1980 2003Included observations: 24Newey-West HAC Standard Errors & Covariance (lag truncation=2)


C 9.792052 0.744832 13.14665 0.0000UNEMPLOY 0.586614 0.105420 5.564560 0.0000



Impure Correlation and The Philips Curve

Dependent Variable: INFLMethod: Least SquaresDate: 10/13/09 Time: 22:30Sample: 1958 2004Included observations: 47


C 0.008736 0.018306 0.477214 0.6355UNEMPLOY 0.542127 0.300264 1.805503 0.0777

R-squared 0.067548 Mean dependent var 0.040902Adjusted R-squared 0.046827 S.D. dependent var 0.029548S.E. of regression 0.028848 Akaike info criterion -4.211930Sum squared resid 0.037449 Schwarz criterion -4.133201Log likelihood 100.9804 F-statistic 3.259841Durbin-Watson stat 0.448168 Prob(F-statistic) 0.077688

Dependent Variable: INFLMethod: Least SquaresDate: 10/13/09 Time: 22:34Sample (adjusted): 1959 2004Included observations: 46 after adjustments


C 0.038913 0.010023 3.882169 0.0004UNEMPLOY -0.677368 0.194650 -3.479923 0.0012

INFL(-1) 1.021775 0.093617 10.91446 0.0000

R-squared 0.753863 Mean dependent var 0.041197Adjusted R-squared 0.742415 S.D. dependent var 0.029805S.E. of regression 0.015127 Akaike info criterion -5.481707Sum squared resid 0.009839 Schwarz criterion -5.362448Log likelihood 129.0793 F-statistic 65.84976Durbin-Watson stat 1.533848 Prob(F-statistic) 0.000000

Pure Versus Impure Serial CorrelationConsequences and DetectionRemedies

serial correlation 1 - transtutors...pure versus impure serial correlation consequences and...

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