1. Dynamic Models, Autocorrelation and Forecasting
Modified JJ
Vera Tabakova, East Carolina University

2. Chapter 9: Dynamic Models, Autocorrelation and Forecasting
9.1 Introduction
9.2 Autocorrelated errors
9.4 Testing for Autocorrelation
9.5 Estimation : NLS
Autoregressive model with explanatory variables
9.6 Finite Distributed Lags
9.7 Autoregressive Distributed Lag Models
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3. 9.2 Lags in the Error Term: Autocorrelation
9.2.1 Area Response Model for Sugar Cane
(9.4)
(9.5)
(9.6)
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4. 9.2.2 First-Order Autoregressive Errors
(9.7)
(9.8)
(9.9)
(9.10)
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5. 9.2.2 First-Order Autoregressive Errors
(9.11)
(9.12)
(9.13)
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6. 9.2.2 First-Order Autoregressive Errors
(9.14)
(9.15)
(9.16)
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7. 9.2.2 First-Order Autoregressive Errors
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8. 9.2.2 First-Order Autoregressive Errors
Figure Least Squares Residuals Plotted Against Time
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9. Detecting autocorrelated errors
The presence of autocorrelated errors can be detected by the Durbin-Watson Statistic (see Appendix) It is provided in standard OLS output. A value of D-W close to 2 indicates no autocorrelation If D-W statistic is low, you may als0 want to calculate the sample ACF of the OLS residuals and plot it to see if their autocorrelations are significant.
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10. Durbin Watson test D-W test for autocorrelation at lag 1: (9B.1)
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Slide 9-10

11. Test of residual autocorrelation
Autocorrelations of residuals can be tested
(9.31)
(9.32)
Slide 9-11
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12. ACF of OLS Residuals
Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example
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13. 9.3 Estimating an AR(1) Error Model
The existence of AR(1) errors implies:
The OLS estimator is still a linear and unbiased estimator, but it is no longer best, i.e. OLS is not efficient. There exist other estimators which provide correct variances (standard errors (s.e.)) and “t-ratios” of estimated parameters: NLS, GLS or MLE.
The standard errors usually computed from the OLS estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading.
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14. 9.3 Estimating an AR(1) Error Model
Sugar cane example The two sets of standard errors, along with the estimated equation are: The 95% confidence intervals for β2 are:
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15. 9.3.2 Nonlinear Least Squares Estimation
(9.19)
(9.20)
(9.21)
(9.22)
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16. 9.3.2 Nonlinear Least Squares Estimation
(9.23)
(9.24)
(9.25)
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17. 9.3.2a Generalized Least Squares Estimation
The parameters are estimated by NLS by minimizing the sum of squared v_t numerically. NLS is an asymptotic estimator, i.e. it is valid in large samples. It can be shown that NLS nonlinear least squares estimator of (9.24) is equivalent to using an iterative GLS (generalized least squares) estimator called the Cochrane-Orcutt procedure. Details are provided in Appendix
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18. NLS Residual Autocorrelation
Diagnostic check of NLS Residual Correlogram
(9.29)
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19. 9.4 Testing for Autocorrelation
(9.31)
(9.32)
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20. NLS Residual Correlogram
Figure 9.5 Correlogram for Nonlinear Least Squares Residuals from Sugar Cane Example
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21. AR(1) with explanatory variables
The autocorrelated error model can be replaced by the lagged dependent variable model, i.e. an AR(1) with explanatory variables. It can be estimated by OLS: OLS is consistent IFF the residuals of the model are White Noise, and it is BIASED otherwise.
The AR(1) with X_t and its first lag is estimated from the same data as follows:
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22. AR(1) with Explanatory variables
(9.26)
(9.28)
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23. 9.6 Finite Distributed Lags
(9.44)
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24. 9.6 Finite Distributed Lags
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25. 9.6 Finite Distributed Lags
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26. 9.7 Autoregressive Distributed Lag Models
(9.45)
(9.46)
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27. 9.7 Autoregressive Distributed Lag Models
Figure 9.7 Correlogram for Least Squares Residuals from Finite Distributed Lag Model
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28. 9.7 Autoregressive Distributed Lag Models
(9.47)
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29. 9.7 Autoregressive Distributed Lag Models
Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model
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30. 9.7 Autoregressive Distributed Lag Models
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31. 9.7 Autoregressive Distributed Lag Models
Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model
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32. Chapter 9 Appendices Appendix 9A Generalized Least Squares Estimation
Appendix 9B The Durbin Watson Test
Appendix 9C Deriving ARDL Lag Weights
Appendix 9D Forecasting: Exponential Smoothing
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33. Appendix 9A Generalized Least Squares Estimation
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34. Appendix 9A Generalized Least Squares Estimation
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35. Appendix 9A Generalized Least Squares Estimation
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36. Appendix 9A Generalized Least Squares Estimation
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37. Appendix 9B The Durbin-Watson Test
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38. Appendix 9B The Durbin-Watson Test
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39. Appendix 9B The Durbin-Watson Test
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40. Appendix 9B The Durbin-Watson Test
Figure 9A.1:
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41. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
Figure 9A.2:
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42. Appendix 9B 9B.1 The Durbin-Watson Bounds Test
if the test is inconclusive.
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43. Appendix 9C Deriving ARDL Lag Weights
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44. Appendix 9C 9C.1 The Geometric Lag
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45. Appendix 9C 9C.1 The Geometric Lag
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46. Appendix 9C 9C.1 The Geometric Lag
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47. Appendix 9C 9C.1 The Geometric Lag
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48. Appendix 9C 9C.2 Lag Weights for More General ARDL Models
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49. Appendix 9D Forecasting: Exponential Smoothing
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50. Appendix 9D Forecasting: Exponential Smoothing
Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α
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