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If autocorrelation in a model is detected by the Breusch-Godfrey test for r-th order autocorrelation, what is the GLS procedure for "fixing" the autocorrelation problem? And is Cochrane-Orcutt procedure a part of the GLS method, or is it a separate "remedy" for autocorrelation?

Richard Hardy
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Badalyan
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2 Answers2

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GLS is the model that takes autocorrelated residuals into account, while Cochrane-Orcutt is one of the many procedures to estimate such GLS model.

Strictly speaking, the GLS model requires the true value of $\rho$ in $\varepsilon_t = \rho\varepsilon_{t-1} + w_t$ to be known. However, apparently $\rho$ is unobservable. Cochrane-Orcutt is one (of many) iterative procedures to estimate $\rho$ and the GLS model. (a random reference/further reading)

GLS is not the only way to fix the autocorrelated residual problem. For example, adding lagged terms of dependent/independent variables as predictors to the OLS may also fix the problem, depends on the true relationship among the variables.

Probably
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The solution is known generally as a Transfer Function sometimes as a Dynamic Regression How to forecast a time series which is dependent on different time series? . You might want to also look at ARIMAX model's exogenous components? and An example of autocorrelation in residuals causing misinterpretation.

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IrishStat
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    Really? Is that considered a GLS procedure? I was expecting something else under that title. – Richard Hardy Jan 04 '17 at 15:09
  • GLS raises two possible issues .. – IrishStat Jan 04 '17 at 15:23
  • Yes, it also appeared to me that making the regression dynamic with adding lags to the dependent variable is a way to fix autocorrelation, but isn't it a separate procedure from the GLS? – Badalyan Jan 04 '17 at 15:28
  • The idea behind GLS raises two possible issues .. 1) non-constancy of diagonal elements suggesting weighted least squares or some sort of power transform & 2) non-zero elements in the off-diagonal elements suggesting possibly omitted lag structure for one of the user-specified stochastic X's or ARIMA structure effectively dealing with omitted lag structure for the Y series . https://www.google.com/?gws_rd=ssl#q=generalized+least+squares and http://www.econ.uiuc.edu/~wsosa/econ507/gls.pdf – IrishStat Jan 04 '17 at 15:30
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    I think Transfer function modelling is not a special case of GLS as it requires maximum likelihood estmation rather than feasible GLS (which is a different estimation method). So I think you answer does not address the question that is being asked. – Richard Hardy Jan 04 '17 at 21:26
  • you can estimate a tf using exact lilelihood – IrishStat Jan 04 '17 at 21:30
  • Exactly. The question is about GLS, your answer deals with maximum likelihood. I fail to see the connection... – Richard Hardy Jan 05 '17 at 13:04