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

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 Principles of Econometrics, 3rd Edition

9.2 Lags in the Error Term: Autocorrelation 9.2.1 Area Response Model for Sugar Cane (9.4) (9.5) (9.6) Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors (9.7) (9.8) (9.9) (9.10) Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors (9.11) (9.12) (9.13) Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors (9.14) (9.15) (9.16) Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors Principles of Econometrics, 3rd Edition

9.2.2 First-Order Autoregressive Errors Figure 9.3 Least Squares Residuals Plotted Against Time Principles of Econometrics, 3rd Edition

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. Principles of Econometrics, 3rd Edition

Durbin Watson test D-W test for autocorrelation at lag 1: (9B.1) Principles of Econometrics, 3rd Edition Slide 9-10

Test of residual autocorrelation Autocorrelations of residuals can be tested (9.31) (9.32) Slide 9-11 Principles of Econometrics, 3rd Edition

ACF of OLS Residuals Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example Principles of Econometrics, 3rd Edition

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. Principles of Econometrics, 3rd Edition

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: Principles of Econometrics, 3rd Edition

9.3.2 Nonlinear Least Squares Estimation (9.19) (9.20) (9.21) (9.22) Principles of Econometrics, 3rd Edition

9.3.2 Nonlinear Least Squares Estimation (9.23) (9.24) (9.25) Principles of Econometrics, 3rd Edition

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 Principles of Econometrics, 3rd Edition

NLS Residual Autocorrelation Diagnostic check of NLS Residual Correlogram (9.29) Principles of Econometrics, 3rd Edition

9.4 Testing for Autocorrelation (9.31) (9.32) Principles of Econometrics, 3rd Edition

NLS Residual Correlogram Figure 9.5 Correlogram for Nonlinear Least Squares Residuals from Sugar Cane Example Principles of Econometrics, 3rd Edition

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: Principles of Econometrics, 3rd Edition

AR(1) with Explanatory variables (9.26) (9.28) Principles of Econometrics, 3rd Edition

9.6 Finite Distributed Lags (9.44) Principles of Econometrics, 3rd Edition

9.6 Finite Distributed Lags Principles of Econometrics, 3rd Edition

9.6 Finite Distributed Lags Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models (9.45) (9.46) Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models Figure 9.7 Correlogram for Least Squares Residuals from Finite Distributed Lag Model Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models (9.47) Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models Figure 9.8 Correlogram for Least Squares Residuals from Autoregressive Distributed Lag Model Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models Principles of Econometrics, 3rd Edition

9.7 Autoregressive Distributed Lag Models Figure 9.9 Distributed Lag Weights for Autoregressive Distributed Lag Model Principles of Econometrics, 3rd Edition

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 Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation Principles of Econometrics, 3rd Edition

Appendix 9A Generalized Least Squares Estimation Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test Principles of Econometrics, 3rd Edition

Appendix 9B The Durbin-Watson Test Figure 9A.1: Principles of Econometrics, 3rd Edition

Appendix 9B 9B.1 The Durbin-Watson Bounds Test Figure 9A.2: Principles of Econometrics, 3rd Edition

Appendix 9B 9B.1 The Durbin-Watson Bounds Test if the test is inconclusive. Principles of Econometrics, 3rd Edition

Appendix 9C Deriving ARDL Lag Weights Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag Principles of Econometrics, 3rd Edition

Appendix 9C 9C.1 The Geometric Lag Principles of Econometrics, 3rd Edition

Appendix 9C 9C.2 Lag Weights for More General ARDL Models Principles of Econometrics, 3rd Edition

Appendix 9D Forecasting: Exponential Smoothing Principles of Econometrics, 3rd Edition

Appendix 9D Forecasting: Exponential Smoothing Figure 9A.3: Exponential Smoothing Forecasts for two alternative values of α Principles of Econometrics, 3rd Edition