1. NONSTATIONARY PROCESSES 1 In the last sequence, the process shown at the top was shown to be stationary. The expected value and variance of X t were shown to be (asymptotically) independent of time and the covariance between X t and X t+s was also shown to be independent of time. Stationary process

2. NONSTATIONARY PROCESSES 2 The condition –1 <  2 < 1 was crucial for stationarity. If  2 = 1, the series becomes a nonstationary process known as a random walk. Random walk

3. NONSTATIONARY PROCESSES 3 It will be assumed, as before, that the innovations  are generated independently from a fixed distribution with mean 0 and population variance  . 2 Random walk

4. NONSTATIONARY PROCESSES 4 If the process starts at X 0 at time 0, its value at time t is given by X 0 plus the sum of the innovations in periods 1 to t. Random walk

5. NONSTATIONARY PROCESSES 5 If expectations are taken at time 0, the expected value at any future time t is fixed at X 0 because the expected values of the future innovations are all 0. Thus E(X t ) is independent of t and the first condition for stationarity remains satisfied. Random walk

6. NONSTATIONARY PROCESSES 6 However, the condition that the variance of X t be independent of time is not satisfied. Random walk

7. NONSTATIONARY PROCESSES 7 The variance of X t is equal to the variance of X 0 plus the sum of the innovations. X 0 may be dropped from the expression because it is an additive constant (variance rule 4). Random walk

8. NONSTATIONARY PROCESSES 8 The variance of the sum of the innovations is equal to the sum of their individual variances. The covariances are all 0 because the innovations are assumed to be generated independently. Random walk

9. NONSTATIONARY PROCESSES 9 The variance of each innovation is equal to  , by assumption. Hence the population variance of X t is directly proportional to t. Its distribution becomes wider and flatter, the further one looks into the future. 2 Random walk

10. NONSTATIONARY PROCESSES 10 The chart shows a typical random walk. If it were a stationary process, there would be a tendency for the series to return to 0 periodically. Here there is no such tendency. Random walk

11. NONSTATIONARY PROCESSES 11 A second process considered in the last sequence is shown above. The presence of the constant  1 on the right side gave the series a nonzero mean but did not lead to a violation of the conditions for stationarity. Stationary process

12. NONSTATIONARY PROCESSES 12 If  2 = 1, however, the series becomes a nonstationary process known as a random walk with drift. Random walk with drift

13. NONSTATIONARY PROCESSES 13 X t is now equal to the sum of the innovations, as before, plus the constant  1 multiplied by t. Random walk with drift

14. NONSTATIONARY PROCESSES 14 As a consequence, the expected value of X t becomes a function of t and the first condition for nonstationarity is violated. Random walk with drift

15. NONSTATIONARY PROCESSES 15 (The second condition for nonstationarity remains violated since the variance of the distribution of X t is proportional to t. It is unaffected by the inclusion of the constant  1.) Random walk with drift

16. NONSTATIONARY PROCESSES 16 The chart shows a typical random walk. It was generated with  1 equal to 0.2. Random walk with drift

17. NONSTATIONARY PROCESSES 17 The chart shows three series for comparison, all generated with the same set of random numbers. The middle series is a stationary autoregressive process, the first process considered in the last sequence, with  2 equal to 0.7. Random walk with drift Random walk Stationary process

18. NONSTATIONARY PROCESSES 18 In the bottom series, a random walk,  2 was changed to 1. The top series is the random walk with drift just discussed. Random walk with drift Random walk Stationary process

19. NONSTATIONARY PROCESSES 19 Random walks are not the only type of nonstationary process. Another common example of a nonstationary time series is one possessing a time trend. Deterministic trend

20. NONSTATIONARY PROCESSES 20 It is nonstationary because the expected value of X t is not independent of t. Its population variance is not even defined. Deterministic trend

21. NONSTATIONARY PROCESSES 21 Superficially, this model looks similar to the random walk with drift, when the latter is written in terms of its components from time 0. Deterministic trend Random walk with drift

22. NONSTATIONARY PROCESSES 22 The difference is that, with a deterministic trend, the deviations from the trend are short- lived. Even if the shocks are autocorrelated, the series sticks to its trend in the long run. Deterministic trend Random walk with drift

23. NONSTATIONARY PROCESSES 23 However, in the case of a random walk with drift, the divergence from the trend line is random walk and the variance around the trend increases without limit. Deterministic trend Random walk with drift

24. NONSTATIONARY PROCESSES 24 If a nonstationary process can be transformed into a stationary one by differencing, it is said to be difference-stationary. A random walk, with or without drift, is an example. Difference-stationarity

25. NONSTATIONARY PROCESSES 25 Difference-stationarity If we difference the series, the differenced series is just  1 +  t.

26. NONSTATIONARY PROCESSES 26 This is stationary because the expected value of  X t at time t,  1, and its variance,   2, are independent of time and the covariance between its value at time t and its value at time t + s is 0. Difference-stationarity

27. NONSTATIONARY PROCESSES 27 A nonstationary time series that can be transformed into a stationary process by differencing once, as in this case, is described as integrated of order 1, I(1). Difference-stationarity X t is I(1)

28. NONSTATIONARY PROCESSES 28 If a time series can be made stationary by differencing twice, it is known as I(2), and so on. A stationary process, which by definition needs no differencing, is described as I(0). In practice most series are I(0), I(1), or, occasionally, I(2). Difference-stationarity X t is I(1)

29. NONSTATIONARY PROCESSES 29 The reason that the series is described as 'integrated' is that the shock in each time period is permanently incorporated in it. There is no tendency for the effects of the shocks to attenuate with time, as in a stationary process or in a model with a deterministic trend. Difference-stationarity X t is I(1)

30. NONSTATIONARY PROCESSES 30 A trend-stationary model is one that can be made stationary by removing a deterministic trend. In the case of the model shown, the de-trended series X t is just the residuals from a regression on time. Trend-stationarity ~

31. NONSTATIONARY PROCESSES 31 The distinction between difference-stationarity and trend-stationarity is important for the analysis of time series. Trend-stationarity

32. NONSTATIONARY PROCESSES 32 It used to be assumed that time series could be decomposed into trend and cyclical components, the former being determined by real factors, such as the growth of GDP, and the latter being determined by transitory factors, such as monetary policy. Trend-stationarity

33. NONSTATIONARY PROCESSES 33 Typically the cyclical component was analyzed using detrended versions of the variables in the model. Trend-stationarity

34. NONSTATIONARY PROCESSES Deterministic trend Random walk with drift However this approach is inappropriate if the process is difference- stationary, for although detrending may remove any drift, it does not affect the increasing variance of the series, and so the detrended component remains nonstationary. 34

35. Copyright Christopher Dougherty 2000–2006. This slideshow may be freely copied for personal use. 21.08.06