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Chapter 12

ECON 6002 Econometrics Memorial University of Newfoundland. Nonstationary Time Series Data and Cointegration. Chapter 12. Adapted from Vera Tabakova’s notes. Chapter 12: Nonstationary Time Series Data and Cointegration. 12.1 Stationary and Nonstationary Variables

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Chapter 12

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  1. ECON 6002 Econometrics Memorial University of Newfoundland Nonstationary Time Series Data and Cointegration Chapter 12 Adapted from Vera Tabakova’s notes

  2. Chapter 12: Nonstationary Time Series Data and Cointegration • 12.1 Stationary and Nonstationary Variables • 12.2 Spurious Regressions • 12.3 Unit Root Tests for Stationarity • 12.4 Cointegration • 12.5 Regression When There is No Cointegration Principles of Econometrics, 3rd Edition

  3. 12.1 Stationary and Nonstationary Variables Fluctuates about a rising trend Figure 12.1(a) US economic time series Yt-Y t-1 On the right hand side “Differenced series” Fluctuates about a zero mean Principles of Econometrics, 3rd Edition

  4. 12.1 Stationary and Nonstationary Variables Figure 12.1(b) US economic time series Yt-Y t-1 On the right hand side “Differenced series” Principles of Econometrics, 3rd Edition

  5. 12.1 Stationary and Nonstationary Variables Stationary if: Principles of Econometrics, 3rd Edition

  6. 12.1 Stationary and Nonstationary Variables Principles of Econometrics, 3rd Edition

  7. 12.1.1 The First-Order Autoregressive Model Each realization of the process has a proportion rho of the previous one plus an error drawn from a distribution with mean zero and variance sigma squared It can be generalised to a higher autocorrelation order We just show AR(1) Principles of Econometrics, 3rd Edition

  8. 12.1.1 The First-Order Autoregressive Model We can show that the constant mean of this series is zero Principles of Econometrics, 3rd Edition Slide 12-8

  9. 12.1.1 The First-Order Autoregressive Model We can also allow for a non-zero mean, by replacing yt with yt-mu Which boils down to (using alpha = mu(1-rho)) Principles of Econometrics, 3rd Edition

  10. 12.1.1 The First-Order Autoregressive Model Or we can allow for a AR(1) with a fluctuation around a linear trend (mu+delta times t) The “de-trended” model , which is now stationary, behaves like an autoregressive model: With alpha =(mu(1-rho)+rho times delta) And lambda = delta(1-rho) Principles of Econometrics, 3rd Edition

  11. 12.1.1 The First-Order Autoregressive Model Figure 12.2 (a) Time Series Models Principles of Econometrics, 3rd Edition

  12. 12.1.1 The First-Order Autoregressive Model Figure 12.2 (b) Time Series Models Principles of Econometrics, 3rd Edition

  13. 12.1.1 The First-Order Autoregressive Model Figure 12.2 (c) Time Series Models Principles of Econometrics, 3rd Edition

  14. 12.1.2 Random Walk Models The first component is usually just zero, since it is so far in the past that it has a negligible effect now The second one is the stochastic trend Principles of Econometrics, 3rd Edition

  15. 12.1.2 Random Walk Models • A random walk is non-stationary, although the mean is constant: Principles of Econometrics, 3rd Edition

  16. 12.1.2 Random Walk Models A random walk with a drift both wanders and trends: Principles of Econometrics, 3rd Edition

  17. 12.1.2 Random Walk Models Principles of Econometrics, 3rd Edition

  18. 12.2 Spurious Regressions Both independent and artificially generated, but… Principles of Econometrics, 3rd Edition

  19. 12.2 Spurious Regressions Figure 12.3 (a) Time Series of Two Random Walk Variables Principles of Econometrics, 3rd Edition

  20. 12.2 Spurious Regressions Figure 12.3 (b) Scatter Plot of Two Random Walk Variables Principles of Econometrics, 3rd Edition

  21. 12.3 Unit Root Test for Stationarity • Dickey-Fuller Test 1 (no constant and no trend) Principles of Econometrics, 3rd Edition

  22. 12.3 Unit Root Test for Stationarity • Dickey-Fuller Test 1 (no constant and no trend) Easier way to test the hypothesis about rho Remember that the null is a unit root: nonstationarity! Principles of Econometrics, 3rd Edition

  23. 12.3 Unit Root Test for Stationarity • Dickey-Fuller Test 2 (with constant but no trend) Principles of Econometrics, 3rd Edition

  24. 12.3 Unit Root Test for Stationarity • Dickey-Fuller Test 3 (with constant and with trend) Principles of Econometrics, 3rd Edition

  25. 12.3.4 The Dickey-Fuller Testing Procedure First step: plot the time series of the original observations on the variable. • If the series appears to be wandering or fluctuating around a sample average of zero, use Version 1 • If the series appears to be wandering or fluctuating around a sample average which is non-zero, use Version 2 • If the series appears to be wandering or fluctuating around a linear trend, use Version 3 Principles of Econometrics, 3rd Edition

  26. 12.3.4 The Dickey-Fuller Testing Procedure Principles of Econometrics, 3rd Edition

  27. 12.3.4 The Dickey-Fuller Testing Procedure • An important extension of the Dickey-Fuller test allows for the possibility that the error term is autocorrelated. • The unit root tests based on (12.6) and its variants (intercept excluded or trend included) are referred to as augmented Dickey-Fuller tests. Principles of Econometrics, 3rd Edition

  28. 12.3.5 The Dickey-Fuller Tests: An Example F = US Federal funds interest rate B = 3-year bonds interest rate Principles of Econometrics, 3rd Edition

  29. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: use usa, clear gen date = q(1985q1) + _n - 1 format %tq date tsset date TESTING UNIT ROOTS “BY HAND”: * Augmented Dickey Fuller Regressions regress D.F L1.F L1.D.F regress D.B L1.B L1.D.B Principles of Econometrics, 3rd Edition Slide 12-29

  30. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: TESTING UNIT ROOTS “BY HAND”: * Augmented Dickey Fuller Regressions regress D.F L1.F L1.D.F regress D.B L1.B L1.D.B Principles of Econometrics, 3rd Edition Slide 12-30

  31. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) dfuller B, regress lags(1) Choice of lags if we want to allow For more than a AR(1) order Principles of Econometrics, 3rd Edition Slide 12-31

  32. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) Principles of Econometrics, 3rd Edition Slide 12-32

  33. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) Alternative: pperron uses Newey-West standard errors to account for serial correlation, whereas the augmented Dickey-Fuller test implemented in dfuller uses additional lags of the first-difference variable. Also consider now using DFGLS (Elliot Rothenberg and Stock, 1996) to counteract problems of lack of power in small samples. It also has in STATA a lag selection procedure based on a sequential t test suggeste by Ng and Perron (1995) Principles of Econometrics, 3rd Edition Slide 12-33

  34. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: Augmented Dickey Fuller Regressions with built in functions dfuller F, regress lags(1) Alternatives: use tests with stationarity as the null KPSS (Kwiatowski, Phillips, Schmidt and Shin. 1992) which also has an automatic bandwidth selection tool or the Leybourne & McCabe test . Principles of Econometrics, 3rd Edition Slide 12-34

  35. 12.3.6 Order of Integration The first difference of the random walk is stationary It is an example of a I(1) series (“integrated of order 1” First-differencing it would turn it into I(0) (stationary) In general, the order of integration is the minimum number of times a series must be differenced to make it stationarity Principles of Econometrics, 3rd Edition

  36. 12.3.6 Order of Integration So now we reject the Unit root after differencing once: We have a I(1) series Principles of Econometrics, 3rd Edition Slide 12-36

  37. 12.3.5 The Dickey-Fuller Tests: An Example In STATA: ADF on differences dfuller D.F, noconstant lags(0) dfuller D.B, noconstant lags(0) Slide 12-37

  38. 12.4 Cointegration Principles of Econometrics, 3rd Edition

  39. 12.4 Cointegration Not the same as for dfuller, since the residuals are estimated errors no actual ones Note: unfortunately STATA dfuller will not notice and give you erroneous critical values! Principles of Econometrics, 3rd Edition

  40. 12.4.1 An Example of a Cointegration Test Check: These are wrong!

  41. 12.4.1 An Example of a Cointegration Test The null and alternative hypotheses in the test for cointegration are: Principles of Econometrics, 3rd Edition

  42. 12.5 Regression When There Is No Cointegration • 12.5.1 First Difference Stationary The variable yt is said to be a first difference stationary series. Then we revert to the techniques we saw in Ch. 9 Principles of Econometrics, 3rd Edition

  43. 12.5.1 First Difference Stationary Manipulating this one you can construct and Error Correction Model to investigate the SR dynamics of the relationship between y and x Principles of Econometrics, 3rd Edition

  44. 12.5.2 Trend Stationary where and Principles of Econometrics, 3rd Edition

  45. 12.5.2 Trend Stationary To summarize: • If variables are stationary, or I(1) and cointegrated, we can estimate a regression relationship between the levels of those variables without fear of encountering a spurious regression. • Then we can use the lagged residuals from the cointegrating regression in an ECM model • This is the best case scenario, since if we had to first-differentiate the variables, we would be throwing away the long-run variation • Additionally, the cointegrated regression yields a “superconsistent” estimator in large samples Principles of Econometrics, 3rd Edition

  46. 12.5.2 Trend Stationary Principles of Econometrics, 3rd Edition Slide 12-46 To summarize: If the variables are I(1) and not cointegrated, we need to estimate a relationship in first differences, with or without the constant term. If they are trend stationary, we can either de-trend the series first and then perform regression analysis with the stationary (de-trended) variables or, alternatively, estimate a regression relationship that includes a trend variable. The latter alternative is typically applied.

  47. Keywords • Augmented Dickey-Fuller test • Autoregressive process • Cointegration • Dickey-Fuller tests • Mean reversion • Order of integration • Random walk process • Random walk with drift • Spurious regressions • Stationary and nonstationary • Stochastic process • Stochastic trend • Tau statistic • Trend and difference stationary • Unit root tests Principles of Econometrics, 3rd Edition

  48. Further issues Kit Baum has really good notes on these topics that can be used to learn also about extra STATA commands to handle the analysis: http://fmwww.bc.edu/ec-c/s2003/821/ec821.sect05.nn1.pdf http://fmwww.bc.edu/ec-c/s2003/821/ec821.sect06.nn1.pdf For example, some of you should look at seasonal unit root analysis (command HEGY in STATA) Panel unit roots would be here http://fmwww.bc.edu/ec-c/s2003/821/ec821.sect09.nn1.pdf Principles of Econometrics, 3rd Edition Slide 12-48

  49. Further issues You may want to some time consider unit root tests that allow for structural Breaks You can also take a look at the literature review in this working paper: http://ideas.repec.org/p/wpa/wuwpot/0410002.html Principles of Econometrics, 3rd Edition Slide 12-49

  50. Further issues When you deal with more than 2 regressors you should consider the Johansen’s method to examine the cointegration relationships Principles of Econometrics, 3rd Edition Slide 12-50

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