# Chapter 12

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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 suggested 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 • If you have unit roots in the time series in your model, you risk the problem of spurious regressions • However, spuriousness will not arise if those series are cointegrated, so that determining whether cointegration exists is also key • The series are cointegrated if they follow the same stochastic trend or share an underlying common factor • In that case you can find a linear combination of your nonstationaryvariables that is itself stationary Principles of Econometrics, 3rd Edition

40. 12.4 Cointegration • You must make sure that you have a balanced (potentially) cointegrating regression, so you want to find out the level of integration of your series (usually they are all I(1)) • The coefficients in that linear combination form the cointegrating vector, which should have one of its elements normalized to one, because the cointegratingvector is only defined up to a factor of proportionality • The cointegrating vector may include a constant, in order to allow for unequal means of the two series Principles of Econometrics, 3rd Edition

41. 12.4 Cointegration • The estimator from a cointegrating regression is superconsistent Principles of Econometrics, 3rd Edition

42. 12.4 Cointegration Two main approaches can be used to check if there is cointegration: • The residual approach • The error correction approach Principles of Econometrics, 3rd Edition

43. 12.4 Cointegration Two main approaches can be used to check if there is cointegration: • The residual approach. The classic Engle-Granger approach, based on testing whether the error of the (potentially) cointengrating regression is itself stationary • The error correction approach, which test whether the error correction term is significant Principles of Econometrics, 3rd Edition

44. 12.4 Cointegration Not the same as for dfuller, since the residuals are estimated errors no actual Ones (also no constant!) Note: unfortunately STATA dfulleras such will not notice and give you erroneous critical values! They would lead to an overoptimistic conclusion  Principles of Econometrics, 3rd Edition

45. 12.4.1 An Example of a Cointegration Test Check: These are wrong!

46. 12.4.1 An Example of a Cointegration Test Using egranger with option regress  Now these are right!

47. 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

48. 12 Error Correction Models • Let us consider the simple form of a dynamic model: • Here the SR and LR effects are measured respectively by: • Rearranging terms, we obtain the usual ECM:

49. 12 Error Correction Models Where the LR effect will be given by: And is a partial correction term for the extent to which Yt-1 deviated from its Equilibrium value associated with Xt-1

50. 12 Error Correction Models This representation assumes that any short-run shock to Y that pushes it off the long-run equilibrium growth rate will gradually be corrected, and an equilibrium rate will be restored is the residual of the long-run equilibrium relationship between X and Y and its Coefficient can be seen as the “speed of adjustment”