Economics 20 - Prof,Anderson 1
Time Series Data
yt = b0 + b1xt1 +,,,+ bkxtk + ut
2,Further Issues
Economics 20 - Prof,Anderson 2
Testing for AR(1) Serial
Correlation
Want to be able to test for whether the
errors are serially correlated or not
Want to test the null that r = 0 in ut = rut-1
+ et,t =2,…,n,where ut is the model error
term and et is iid
With strictly exogenous regressors,the test
is very straightforward – simply regress the
residuals on lagged residuals and use a t-test
Economics 20 - Prof,Anderson 3
Testing for AR(1) Serial
Correlation (continued)
An alternative is the Durbin-Watson (DW)
statistic,which is calculated by many
packages
If the DW statistic is around 2,then we can
reject serial correlation,while if it is
significantly < 2 we cannot reject
Critical values are difficult to calculate,
making the t test easier to work with
Economics 20 - Prof,Anderson 4
Testing for AR(1) Serial
Correlation (continued)
If the regressors are not strictly exogenous,
then neither the t or DW test will work
Regress the residual (or y) on the lagged
residual and all of the x’s
The inclusion of the x’s allows each xtj to
be correlated with ut-1,so don’t need
assumption of strict exogeneity
Economics 20 - Prof,Anderson 5
Testing for Higher Order S.C,
Can test for AR(q) serial correlation in the
same basic manner as AR(1)
Just include q lags of the residuals in the
regression and test for joint significance
Can use F test or LM test,where the LM
version is called a Breusch-Godfrey test and
is (n-q)R2 using R2 from residual regression
Can also test for seasonal forms
Economics 20 - Prof,Anderson 6
Correcting for Serial Correlation
Start with case of strictly exogenous
regressors,and maintain all G-M
assumptions except no serial correlation
Assume errors follow AR(1) so ut = rut-1 +
et,t =2,…,n
Var(ut) = s2e/(1-r2)
We need to try and transform the equation
so we have no serial correlation in the errors
Economics 20 - Prof,Anderson 7
Correcting for S.C,(continued)
Consider that since yt = b0 + b1xt + ut,
then yt-1 = b0 + b1xt-1 + ut-1
If you multiply the second equation by r,
and subtract if from the first you get
yt – r yt-1 = (1 – r)b0 + b1(xt – r xt-1) + et,
since et = ut – r ut-1
This quasi-differencing results in a model
without serial correlation
Economics 20 - Prof,Anderson 8
Feasible GLS Estimation
Problem with this method is that we don’t
know r,so we need to get an estimate first
Can just use the estimate obtained from
regressing residuals on lagged residuals
Depending on how we deal with the first
observation,this is either called Cochrane-
Orcutt or Prais-Winsten estimation
Economics 20 - Prof,Anderson 9
Feasible GLS (continued)
Often both Cochrane-Orcutt and Prais-
Winsten are implemented iteratively
This basic method can be extended to allow
for higher order serial correlation,AR(q)
Most statistical packages will automatically
allow for estimation of AR models without
having to do the quasi-differencing by hand
Economics 20 - Prof,Anderson 10
Serial Correlation-Robust
Standard Errors
What happens if we don’t think the
regressors are all strictly exogenous?
It’s possible to calculate serial correlation-
robust standard errors,along the same lines
as heteroskedasticity robust standard errors
Idea is that want to scale the OLS standard
errors to take into account serial correlation
Economics 20 - Prof,Anderson 11
Serial Correlation-Robust
Standard Errors (continued)
Estimate normal OLS to get residuals,root MSE
Run the auxiliary regression of xt1 on xt2,…,xtk
Form at by multiplying these residuals with ?t
Choose g – say 1 to 3 for annual data,then
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