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Autocorrelation in Time Series

Autocorrelation in Time Series. KNNL – Chapter 12. Issues in Autocorrelated Data. When error terms are correlated (not independent), problems occur when using ordinary least squares (OLS) estimates Regression Coefficients are Unbiased, but not Minimum Variance MSE underestimates s 2

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Autocorrelation in Time Series

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  1. Autocorrelation in Time Series KNNL – Chapter 12

  2. Issues in Autocorrelated Data • When error terms are correlated (not independent), problems occur when using ordinary least squares (OLS) estimates • Regression Coefficients are Unbiased, but not Minimum Variance • MSE underestimates s2 • Standard errors of regression coefficients based on OLS underestimate the true standard error •  Inflated t and F statistics and artificially narrow confidence intervals

  3. First-Order Model - I

  4. First-Order Model - II

  5. Test For Independence - Durbin-Watson Test

  6. Autocorrelation - Remedial Measures • Determine whether a missing predictor variable can explain the autocorrelation in the errors • Include a linear (trend) term if the residuals show a consistent increasing or decreasing pattern • Include seasonal dummy variables if data are quarterly or monthly and residuals show cyclic behavior • Use transformed Variables that remove the (estimated) autocorrelation parameter (Cochrane-Orcutt and Hildreth-Lu Procedures) • Use First Differences • Estimated Generalized Least Squares

  7. Transformed Variables

  8. Cochrane-Orcutt Method • Start by estimating r in Model: et = ret-1 + utby regression through the origin for residuals (see below) • Fit transformed regression model (previous slide) • Check to see if new residuals are uncorrelated (Durbin-Watson test), based on the transformed model • If uncorrelated, stop and keep current model • If correlated, repeat process with new estimate r based on current regression residuals from the original (back transformed) model

  9. Hildreth-Lu and First Difference Methods • Hildreth-Lu Method • Find value of r (between 0 and 1) that minimizes the SSE for the transformed model by grid search • Apply the transformed analysis based on the estimated r • First Differences Method • Uses r = 1 in transformed model (Yt’ = Yt – Yt-1 Xt’ = Xt – Xt-1) • Set b0’ = 0 and fits regression through origin of Y’ on X’ • When back-transforming:

  10. Forecasting with Autocorrelated Errors

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