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Chapter 9

ECON 6002 Econometrics Memorial University of Newfoundland. Dynamic Models, Autocorrelation and Forecasting. Chapter 9. Adapted from Vera Tabakova’s notes. Chapter 9: Dynamic Models, Autocorrelation and Forecasting. 9.1 Introduction 9.2 Lags in the Error Term: Autocorrelation

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Chapter 9

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  1. ECON 6002 Econometrics Memorial University of Newfoundland Dynamic Models, Autocorrelation and Forecasting Chapter 9 Adapted from Vera Tabakova’s notes

  2. Chapter 9: Dynamic Models, Autocorrelation and Forecasting • 9.1 Introduction • 9.2 Lags in the Error Term: Autocorrelation • 9.3 Estimating an AR(1) Error Model • 9.4 Testing for Autocorrelation • 9.5 An Introduction to Forecasting: Autoregressive Models • 9.6 Finite Distributed Lags • 9.7 Autoregressive Distributed Lag Models Principles of Econometrics, 3rd Edition

  3. 9.1 Introduction Figure 9.1 Principles of Econometrics, 3rd Edition

  4. The model so far assumes that the observations are not correlated with one another. • This is believable if one has drawn a random sample, but less likely if one has drawn observations sequentially in time • Time series observations, which are drawn at regular intervals, usually embody a structure where time is an important component. Principles of Econometrics, 3rd Edition

  5. If we cannot completely model this structure in the regression function itself, then the remainder spills over into the unobserved component of the statistical model (its error) and this causes the errors be correlated with one another. Principles of Econometrics, 3rd Edition

  6. 9.1 Introduction Three ways to view the dynamics: Principles of Econometrics, 3rd Edition

  7. 9.1 Introduction Figure 9.2(a) Time Series of a Stationary Variable Assume Stationarity For now Principles of Econometrics, 3rd Edition

  8. 9.1 Introduction Figure 9.2(b) Time Series of a Nonstationary Variable that is ‘Slow Turning’ or ‘Wandering’ Principles of Econometrics, 3rd Edition

  9. 9.1 Introduction Figure 9.2(c) Time Series of a Nonstationary Variable that ‘Trends’ Principles of Econometrics, 3rd Edition

  10. 9.2 Lags in the Error Term: Autocorrelation 9.2.1 Area Response Model for Sugar Cane Principles of Econometrics, 3rd Edition

  11. 9.2.2 First-Order Autoregressive Errors Assume also Principles of Econometrics, 3rd Edition

  12. 9.2.2 First-Order Autoregressive Errors OK OK, constant Not zero!!! Principles of Econometrics, 3rd Edition

  13. 9.2.2 First-Order Autoregressive Errors Aka autocorrelation coefficient Principles of Econometrics, 3rd Edition

  14. 9.2.2 First-Order Autoregressive Errors Principles of Econometrics, 3rd Edition

  15. 9.2.2 First-Order Autoregressive Errors Figure 9.3 Least Squares Residuals Plotted Against Time Principles of Econometrics, 3rd Edition

  16. 9.2.2 First-Order Autoregressive Errors Constant variance Null expectation of x and y Principles of Econometrics, 3rd Edition

  17. 9.3 Estimating an AR(1) Error Model The existence of AR(1) errors implies: • The least squares estimator is still a linear and unbiased estimator, but it is no longer best. There is another estimator with a smaller variance. • The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading. Principles of Econometrics, 3rd Edition

  18. 9.3 Estimating an AR(1) Error Model • As with heteroskedastic errors, you can salvage OLS when your data are autocorrelated. • Now you can use an estimator of standard errors that is robust to both heteroskedasticity and autocorrelation proposed by Newey and West. • This estimator is sometimes called HAC, which stands for heteroskedasticityautocorrelated consistent. Principles of Econometrics, 3rd Edition

  19. 9.3 Estimating an AR(1) Error Model • HAC is not as automatic as the heteroskedasticity consistent (HC) estimator. • With autocorrelation you have to specify how far away in time the autocorrelation is likely to be significant • The autocorrelated errors over the chosen time window are averaged in the computation of the HAC standard errors; you have to specify how many periods over which to average and how much weight to assign each residual in that average. • That weighted average is called a kernel and the number of errors to average is called bandwidth. Principles of Econometrics, 3rd Editio

  20. 9.3 Estimating an AR(1) Error Model • Usually, you choose a method of averaging (Bartlett kernel or Parzen kernel) and a bandwidth (nw1, nw2, in GRETL language or some integer). Often your software defaults to the Bartlett kernel and a bandwidth computed based on the sample size, N. • Trade-off: Larger bandwidths reduce bias (good) as well as precision (bad). Smaller bandwidths exclude more relevant autocorrelations (and hence have more bias), but use more observations to increase precision (smaller variance). • Choose a bandwidth large enough to contain the largest autocorrelations. The choice will ultimately depend on the frequency of observation and the length of time it takes for your system to adjust to shocks. Principles of Econometrics, 3rd Edition

  21. 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

  22. 9.3.2 Nonlinear Least Squares Estimation Principles of Econometrics, 3rd Edition

  23. 9.3.2 Nonlinear Least Squares Estimation Now this transformed nonlinear model has errors that are uncorrelated over time Principles of Econometrics, 3rd Edition

  24. 9.3.2a Generalized Least Squares Estimation It can be shown that nonlinear least squares estimation of (9.24) is equivalent to using an iterative generalized least squares estimator called the Cochrane-Orcutt procedure. Details are provided in Appendix 9A. See also Baum’s textbook (Ch. 6) Principles of Econometrics, 3rd Edition

  25. 9.3.2a Generalized Least Squares Estimation • The nonlinear least squares estimator only requires that the errors be stable (not necessarily stationary). • Other methods commonly used make stronger demands on the data, namely that the errors be covariance stationary. • Furthermore, the nonlinear least squares estimator gives you an unconditional estimate of the autocorrelation parameter, and yields a simple t-test of the hypothesis of no serial correlation. • Monte Carlo studies show that it performs well in small samples as well. Principles of Econometrics, 3rd Edition

  26. 9.3.2a Generalized Least Squares Estimation • But nonlinear least squares requires more computational power than linear estimation, though this is not much of a constraint these days. • Nonlinear least squares (and other nonlinear estimators) use numerical methods rather than analytical ones to find the minimum of your sum of squared errors objective function. The routines that do this are iterative. • Learn too about praiscommand in STATA and its variants in order to learn how to estimate these models Principles of Econometrics, 3rd Edition

  27. 9.3.3 Estimating a More General Model We should then test the above restrictions Principles of Econometrics, 3rd Edition

  28. 9.4 Testing for Autocorrelation 9.4.1 Residual Correlogram Principles of Econometrics, 3rd Edition

  29. 9.4 Testing for Autocorrelation 9.4.1 Residual Correlogram (more practically…) Principles of Econometrics, 3rd Edition

  30. 9.4.1 Residual Correlogram Figure 9.4 Correlogram for Least Squares Residuals from Sugar Cane Example Principles of Econometrics, 3rd Edition

  31. 9.4.1 Residual Correlogram For this nonlinear model, then, the residuals should be uncorrelated Principles of Econometrics, 3rd Edition

  32. 9.4.1 Residual Correlogram Figure 9.5 Correlogram for Nonlinear Least Squares Residualsfrom Sugar Cane Example Principles of Econometrics, 3rd Edition

  33. 9.4.2 A Lagrange Multiplier Test The other way to determine whether or not your residuals are autocorrelated is to use an LM (Lagrange multiplier) test. For autocorrelation, this test is based on an auxiliary regression where lagged OLS residuals are added to the original regression equation. If the coefficient on the lagged residual is significant then you conclude that the model is autocorrelated. Principles of Econometrics, 3rd Edition

  34. 9.4.2 A Lagrange Multiplier Test We can derive the auxiliary regression for the LM test: Principles of Econometrics, 3rd Edition

  35. 9.4.2 A Lagrange Multiplier Test Centered around zero, so the power of this test now would come from The lagged error, which is what we care about… Principles of Econometrics, 3rd Edition

  36. 9.4.2 A Lagrange Multiplier Test This was the Breusch-Godfrey LM test for autocorrelation and it is distributed chi-sq In STATA: regress la lp estatbgodfrey Principles of Econometrics, 3rd Edition

  37. 9.4.2 A Lagrange Multiplier Test There are other variants of this test Note that it is only valid in large samples (theoretically infinitely big ones) Principles of Econometrics, 3rd Edition

  38. 9.5 An Introduction to Forecasting: Autoregressive Models In general, p should be large enough so that vt is white noise Adding lagged values of y can serve to eliminate the error autocorrelation Principles of Econometrics, 3rd Edition

  39. 9.5 An Introduction to Forecasting: Autoregressive Models Figure 9.6 Correlogram for Least Squares Residuals fromAR(3) Model for Inflation Helps decide How many lags To include in the model Principles of Econometrics, 3rd Edition

  40. 9.5 An Introduction to Forecasting: Autoregressive Models Principles of Econometrics, 3rd Edition

  41. 9.5 An Introduction to Forecasting: Autoregressive Models Principles of Econometrics, 3rd Edition

  42. 9.5 An Introduction to Forecasting: Autoregressive Models Principles of Econometrics, 3rd Edition

  43. 9.5 An Introduction to Forecasting: Autoregressive Models Principles of Econometrics, 3rd Edition

  44. 9.5 An Introduction to Forecasting: Autoregressive Models Principles of Econometrics, 3rd Edition

  45. 9.6 Finite Distributed Lags This model is just a generalization of the ones previously discussed. In this model you include lags of the dependent variable (autoregressive) and the contemporaneous and lagged values of independent variables as regressors (distributed lags). The acronym is ARDL(p,q) where p is the maximum distributed lag and q is the maximum autoregressive lag Principles of Econometrics, 3rd Edition

  46. 9.6 Finite Distributed Lags Principles of Econometrics, 3rd Edition

  47. 9.6 Finite Distributed Lags Principles of Econometrics, 3rd Edition

  48. 9.6 Finite Distributed Lags Principles of Econometrics, 3rd Edition

  49. 9.7 Autoregressive Distributed Lag Models Principles of Econometrics, 3rd Edition

  50. 9.7 Autoregressive Distributed Lag Models Figure 9.7 Correlogram for Least Squares Residuals fromFinite Distributed Lag Model Principles of Econometrics, 3rd Edition

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