12 Autocorrelation

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12 Autocorrelation

• Serial Correlation exists when errors are
  correlated across periods
• -One source of serial correlation is
  misspecification of the model (although correctly
  specified models can also have autocorrelation)
• -Serial correlation does not make OLS biased or
  inconsistent
• -Serial correlation does ruin OLS standard errors
  and all significance tests
• -Serial correlation must therefore be corrected
  for any regression to give valid information

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12. Serial Correlation and Heteroskedasticity in
Time Series Regressions

• 12.1 Properties of OLS with Serial Correlation
• 12.2 Testing for Serial Correlation
• 12.3 Correcting for Serial Correlation with
  Strictly Exogenous Regressors
• 12.5 Serial Correlation-Robust Inference after
  OLS
• 12.6 Het in Time Series Regressions

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12.1 Serial Correlation and se

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

-where et are uncorrelated random variables with
mean zero and constant variance -assume that
?lt1 (stability condition) -if we assume the
average of x is zero, in the model with one
independent variable, OLS estimates
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12.1 Serial Correlation and se

• Computing the variance of OLS requires us to take
  into account serial correlation in ut

-Evidently this is much different than typical
OLS variance unless ?0 (no serial
correlation)
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12.1 Serial Correlation Notes

• -Typically, the usual OLS formula for variance
  underestimates the true variance in the presence
  of serial correlation
• -this variance bias leads to invalid t and F
  statistics
• -note that if the data is stationary and weakly
  dependent, R2 and adjusted R2 are still valid
  measures of goodness of fit
• -the argument is the same as for cross sectional
  data with heteroskedasticity

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12.2 Testing for Serial Correlation

• -We first test for serial correlation when the
  regressors are strictly exogenous (ut is
  uncorrelated with all regressors over time)
• -the simplest and most popular serial correlation
  to test for is the AR(1) model
• -in order to the strict exogeneity assumption, we
  need to assume that

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12.2 Testing for Serial Correlation

• -We adopt a null hypothesis for no serial
  correlation and set up an AR(1) model

-We could estimate (12.13) and test if ?hat is
zero, but unfortunately we dont have the true
errors -luckily, due to the strict exogeneity
assumption, the true errors can be replaced with
OLS residuals
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Testing for AR(1) Serial Correlation with
Strictly Exogenous Regressors

• Regress y on all xs to obtain residuals uhat
• Regress uhatt on uhatt-1 and obtain OLS estimates
  of ?hat
• Conduct a t-test (typically at the 5 level) for
  the hypotheses
• Ho ?0 (no serial correlation)
• Ha ??0 (AR(1) serial correlation)
• Remember to report p-value

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12.2 Testing for Serial Correlation

• -If one has a large sample size, serial
  correlation could be found with a small ?hat.
• -in this case typical OLS inference will not be
  far off
• -note that this test can detect ANY serial
  correlation that causes adjacent error terms to
  be correlated
• -correlation between ut and ut-4 would not be
  picked up however
• -if the AR(1) formula suffers from HET,
  Heteroskedastic-robust t statistics are used

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12.2 Durbin-Watson Test

• Another classic test for AR(1) serial correlation
  is the Durbin-Watson test. The Durbin-Watston
  (DW) statistic is calculated from OLS residuals

-It can be shown that the DW statistic is linked
to the previous test for AR(1) serial correlation
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12.2 DW Test

• Even with moderate sample sizes, (12.16) is
  relatively close
• -the DW test does, however, depend on ALL CLM
  assumptions
• -typically the DW test is computed for the
  alternative hypothesis Ha?gt0 (since rho is
  usually positive and rarely negative)
• -from (12.16) the null hypothesis is rejected if
  DW is significantly less than 2
• -unfortunately the null distribution is difficult
  to determine for DW

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12.2 DW Test

• -The DW test produces two sets of critical
  values, dU (for upper), and dL (for lower)
• -if DWltdL, reject H0
• -if DWgtdU, do not reject Ho
• -otherwise the tests is inconclusive
• -the DW test has an inconclusive region and
  requires all CLM assumptions
• -the t test can be used asymptotically and can be
  corrected for heteroskedasticity
• -Therefore t tests are generally preferred to DW
  tests

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12.2 Testing without Strictly Exogenous Regressors

• -it is often the case that explanatory variables
  are NOT strictly exogenous
• -one or more xtj are correlated with ut-1
• -ie when yt-1 is an explanatory variable
• -in these cases typical t or DW tests are invalid
• -Durbins h statistic is one alternative, but
  cannot always be calculated
• -the following test works for both strictly
  exogenous and not strictly exogenous regressors

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Testing for AR(1) Serial Correlation without
Strictly Exogenous Regressors

• Regress y on all xs to obtain residuals uhat
• Regress uhatt on uhatt-1 and all xt variables
  obtain OLS estimates of ?hat (coefficient of
  uhatt-1)
• Conduct a t-test (typically at the 5 level) for
  the hypotheses
• Ho ?0 (no serial correlation)
• Ha ??0 (AR(1) serial correlation)
• Remember to report p-value

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12.2 Testing without Strictly Exogenous Regressors

• -the different in this testing sequence is uhatt
  is regressed on
• 1) uhatt-1
• 2) all independent variables
• -a heteroskedasticity-robust t statistic can also
  be used if the above regression suffers from
  heteroskedasticity

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12.2 Higher Order Serial Correlation

• Assume that our error terms follow AR(2) SERIAL
  CORRELATION

-here we test for second order serial
correlation, or
As before, we run a typical OLS regression for
residuals, and then regress uhatt on all
explanatory (x) variables, uhatt-1 and
uhatt-2 -an F test is then done on the joint
significance of the coefficients of uhatt-1 and
uhatt-2 -we can test for higher order serial
correlation
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Testing for AR(q) Serial Correlation

• Regress y on all xs to obtain residuals uhat
• Regress uhatt on uhatt-1, uhatt-2,, uhatt-q and
  all xt variables obtain OLS estimates of ?hat
  (coefficient of uhatt-1)
• Conduct an F-test (typically at the 5 level) for
  the hypotheses
• Ho ?1 ?2 ?q0 (no serial correlation)
• Ha Not H0 (AR(1) serial correlation)
• Remember to report p-values

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12.2 Testing for Higher Order Serial Correlation

• -if xtj is strictly exogenous, it can be removed
  from the second regression
• -this test requires the homoskedasticity
  assumption

-but if heteroskedasticity exists in the second
equation a heteroskedastic-robust transformation
can be made as described in Chapter 8
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12.2 Seasonal forms of Serial Correlation

• Seasonal data (ie quarterly or monthly), might
  exhibit seasonal forms of serial correlation

-our test is similar to that for AR(1) serial
correlation, only the second regression includes
ut-4 or the seasonal lagged variable instead of
ut-1