12.3 correcting for serial correlation w/ strictly exogenous regressors the following...

12.3 Correcting for Serial Correlation w/ Strictly Exogenous

RegressorsThe following autocorrelation correction requires all our regressors to be strictly exogenous-in particular, we should have no lagged explanatory variables

Assume that our error terms follow AR(1) SERIAL CORRELATION :

(12.26) 1 ttt euu -assuming from here on in that everything is

conditional on X, we can calculate variance as:

(12.27) 1

)(

2

2

etuVar


RegressorsIf we consider a single explanatory variable, we can eliminate the correlation in the error term as follows:

2)(t ~)1(~)()()1(

10

11101

10

ttt

tttttt

ttt

exy

uuxxyy

uxy

This provides us with new error terms that are uncorrelated-Note that ytilde and xtilde are called QUASI-DIFFERENCED

DATA


RegressorsNote that OLS is not BLUE yet as the initial y1 is undefined

-to make OLS blue and ensure the first term’s errors are the same as other terms, we set

11102

1

12

12

102

12

~~)1(~

)1()1()1()1(

uxy

uxy

-note that our first term’s quasi-differenced data is calculated differently than all other terms

-note also that this is another example of GLS estimation


RegressorsGiven multiple explanatory variables, we have:

1111102

1

110

1,1

1,11101

~~...~)1(~

2)(t ~...~)1(~)()(...

)()1(

uxxy

exxy

uuxx

xxyy

kk

ttkktt

ttkttkk

tttt

-note that this GLS estimation is BLUE and will generally differ from OLS

-note also that our t and F statistics are now valid and testing can be done


RegressorsUnfortunately, ρ is rarely know, but it can be estimated from the formula:

ttt euu 1ˆˆ We then use ρhat to estimate:

)ˆ1(~

2for t )ˆ1(~

2)(t ~...~~

210

0

1100

x

xwhere

exxxy

t

ttkkttt

Note that in this FEASIBLE GLS (FGLS), the estimation error in ρhat does not affect FGLS’s estimator’s asymptotic distribution

Feasible GLS Estimation of the AR(1) Model

1) Regress y on all x’s to obtain residuals uhat2) Regress uhatt on uhatt-1 and obtain OLS estimates of ρhat

3) Use these ρhat estimates to estimate

2)(t ~...~~1100 ttkkttt exxxy

We now have adjusted slope estimates with valid standard errors for testing

12.3 FGLS Notes-Using ρhat is not valid for small sample sizes-This is due to the fact that FGLS is not

unbiased, it is only consistent if the data is weakly dependent

-while FGLS is not BLUE, it is asymptotically more efficient than OLS (again, large samples)

-two examples of FGLS are the COCHRANE-ORCUTT (CO) ESTIMATION and the PRAIS-WINSTEN (PW) ESTIMATION

-these estimations are similar and differ only in treating the first observation

12.3 Iterated FGLS-Practically, FGLS is often iterated:-once FGLS is estimated once, its residuals

are used to recalculate phat, and FGLS is estimated again

-this is generally repeated until phat converges to a number

-regression programs can automatically perform this iteration

-theoretically, the first iteration satisfies all large sample properties needed for tests

-Note: regression programs can also correct for AR(q) using a complicated FGLS

12.3 FGLS vrs. OLS-In certain cases (such as in the presence

of unit roots), FGLS can fail to obtain accurate estimates; its estimates can vary greatly from OLS

-When FGLS and OLS give similar estimates, FGLS is always preferred if autocorrelation exists

-If FGLS and OLS estimates differ greatly, more complicated statistical estimation is needed

12.5 Serial Correlation-Robust Inference

-FGLS can fail for a variety of reasons:-explanatory variables are not strictly exogenous-sample size is too low-the form of autocorrelation is unknown and more complicated than AR(1)

-in these cases OLS standard errors can be corrected for arbitrary autocorrelation-the estimates themselves aren’t affected, and therefore OLS is inefficient (much like the het-robust correction of simple OLS)

12.5 Autocorrelation-Robust Inference

-To correct standard errors for arbitrary autocorrelation, chose an integer g>0 (generally 1-2 in most cases, 1x-2x where x=frequency greater than annually):

)ˆˆ()ˆˆ()1

1(2)ˆˆ(ˆ111

2htht

n

httt

g

h

n

ttt urur

g

hurv

-where rhat is the residual from regressing x1 on all other x’s and uhat is the residual from the typical OLS estimation

12.5 Autocorrelation-Robust Inference

-After obtaining vhat, our standard errors are adjusted using:

vse

se OLS ˆ)ˆ)ˆ(

()ˆ( 211

-note that this transformation is applied to all variables (as any can be listed as x1)

-these standard errors are also robust to arbitrary heteroskedasticity-this transformation is done using the OLS subcommand /autcov=1 in SHAZAM,

but can also be done step by step:

Serial Correlation-Robust Standard Error for B1hat

1) Regress y on all x’s to obtain residuals uhat, OLS standard errors, and σhat2) Regress x1 on all other x’s and obtain residuals rhat

3) Use these estimates to estimate vhat as seen previously4) Using vhat, obtain new standard errors through:

vse

se OLS ˆ)ˆ)ˆ(

()ˆ( 211

12.5 SC-Robust Notes-Note that these Serial correlation (SC) robust standard errors are poorly behaved for small sample sizes (even as

large as 100)-note that g must be chosen, making this correction less than automatic-if serial correlation is severe, this correction leaves OLS very inefficient, especially in small sample sizes-use this correction only if forced to (some variables not strictly exogenous, lagged dependent variables)-correction is like a hand grenade, not a sniper

12.6 Het in Time Series-Like in cross sectional studies, heteroskedasticity in time series studies doesn’t cause unbiasedness or inconsistency

-it does invalidate standard errors and tests-while robust solutions for autocorrelation may correct Het, the opposite is NOT true

-Heteroskedasticity-Robust Statistics do NOT correct for autocorrelation-note also that Autocorrelation is often more damaging to a model than Het (depending on the amount of auto (ρ) and

amount of Het)

12.6 Testing and Fixing Het in Time SeriesIn order to test for Het:

1) Serial correlation must be tested for and corrected first2) Dynamic Heteroskedasticity (see next section) must not existFixing Het is the same as the cross secitonal case:1) WLS is BLUE if correctly specified2) FGLS is asymptotically valid in large samples3) Het-robust corrections are better than nothing (they don’t correct estimates, only s.e.’s)

12.6 Dynamic Het-Time series adds the complication that the variance of the error term may depend

on explanatory variables of other periods (and thus errors of other periods)-Engle (1982) suggested the AUTOREGRESSIVE CONDITIONAL HETEROSKEDASTICITY

(ARCH) model. A first-order Arch (ARCH(1)) model would look like:

21101

221

2 ),|(),...,,|( tttttt uXuuEXuuuE

12.6 ARCH-The ARCH(1) model can be rewritten as:

(12.50) 2110

2ttt vuu

-Which is similar to the autoregressive model and has the similar stability condition that α1<1

-While ARCH does not make OLS biased or inconsistent, if it exists a WLS or maximum likelihood (ML) estimation are asymptotically more efficient (better estimates)

-note that the usual het-robust standard errors and test statistics are still valid under ARCH

12.6 Het and Autocorrelation…end of the world?

-typically, serial correlation is a more serious issue than Het as it affects standard errors and estimation efficiency more

-however, a low ρ value may cause Het to be more serious-we’ve already seen that het-robust autocorrelation tests are straightforward

12.6 Het and Autocorrelation…is there hope?

If Het and Autocorrelation are found, one can:1) Fix autocorrelation using the CO or PW method (Auto command in Shazam)2) Apply heteroskedastic-robust standard errors to the regression (Not possible through a simple Shazam command)

As a last resort, SC-robust standard errors are also heteroskedastic-robust

12.6 Het and Autocorrelation…is there hope?

Alternately, het can be corrected through a combined WLS AR(1) procedure:

1|| ,v

(12.50) vh

...

1t

ttt

110

tt

ttkktt

ev

u

uxxy

t

t

t

tkk

t

t

tt

t

tvt

h

u

h

x

h

x

hh

y

huVar

...

ng)conditioni g(supressin )(

110

2

-Since ut/ht1/2 is homoskedastic, the above

equation can be estimated using CO or PW

FGLS with Heteroskedasticity and AR(1) Serial Correlation:

1) Regress y on all x’s to obtain residuals uhat2) Regress log(uhatt

2) on all xt’s (or ythat and ythat2) and obtain fitted values, ghatt

3) Estimate ht: hthat=exp(ghatt)

4) Estimate the equation

t

t

t

tkk

t

t

tt

t

h

u

h

x

h

x

hh

y

ˆˆ...

ˆˆˆ110

By Cochrane-Orcutt (CO) or Prais-Winsten (PW) methods. (This corrects for serial correlation.)

12.3 correcting for serial correlation w/ strictly exogenous regressors the following...

Documents