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

NONSTATIONARY PROCESSES

1

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

11 212 ttt XX

0)( 02 XXE tt

222

222

222

11

11

t

X t

222

2

1 and of covariance population

s

stt XX

Stationary process


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

ttt XX 1

Random walk


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It will be assumed, as before, that the innovations are generated independently from a fixed distribution with mean 0 and population variance .2

ttt XX 1

Random walk


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If the process starts at X0 at time 0, its value at time t is given by X0 plus the sum of the innovations in periods 1 to t.

ttt XX 1

ttt XX 110 ...

Random walk


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If expectations are taken at time 0, the expected value at any future time t is fixed at X0 because the expected values of the future innovations are all 0. Thus E(Xt) is independent of t and the first condition for stationarity remains satisfied.

ttt XX 1

ttt XX 110 ...

010 )(...)()( XEEXXE nt

Random walk

2

222

11

1102

...

)...( of variance population


t

X

tt

ttX t


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However, the condition that the variance of Xt be independent of time is not satisfied.

ttt XX 1

ttt XX 110 ...

010 )(...)()( XEEXXE nt

Random walk

2

222

11

1102

...



t

X

tt

ttX t


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The variance of Xt is equal to the variance of X0 plus the sum of the innovations. X0 may be dropped from the expression because it is an additive constant (variance rule 4).

ttt XX 1

ttt XX 110 ...

010 )(...)()( XEEXXE nt

Random walk

2

222

11

1102

...



t

X

tt

ttX t


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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.

ttt XX 1

ttt XX 110 ...

010 )(...)()( XEEXXE nt

Random walk


ttt XX 1

ttt XX 110 ...

2

222

11

1102

...



t

X

tt

ttX t

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The variance of each innovation is equal to , by assumption. Hence the population variance of Xt is directly proportional to t. Its distribution becomes wider and flatter, the further one looks into the future.

010 )(...)()( XEEXXE nt

2

Random walk


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0

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1 11 21 31 41 51 61 71 81 91

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


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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.

11 2121 ttt XX

2021

2

2

11

11)(

XXE t

t

t

222

222

222

11

11

t

X t

222

2

1 and of covariance population

s

stt XX

Stationary process


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If 2 = 1, however, the series becomes a nonstationary process known as a random walk with drift.

ttt XX 11

Random walk with drift


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Xt is now equal to the sum of the innovations, as before, plus the constant 1 multiplied by t.

ttt XtX 1101 ...

ttt XX 11



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As a consequence, the expected value of Xt becomes a function of t and the first condition for nonstationarity is violated.

ttt XtX 1101 ...

tXXE t 10)(

ttt XX 11



ttt XtX 1101 ...

tXXE t 10)(

ttt XX 11

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(The second condition for nonstationarity remains violated since the variance of the distribution of Xt is proportional to t. It is unaffected by the inclusion of the constant 1.)

22 t

tX



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-15

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-5

0

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1 11 21 31 41 51 61 71 81 91

The chart shows a typical random walk. It was generated with 1 equal to 0.2.



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-5

0

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1 11 21 31 41 51 61 71 81 91

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

Stationary process


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-5

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1 11 21 31 41 51 61 71 81 91

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

Stationary process


tt tX 21

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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


tt tX 21

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It is nonstationary because the expected value of Xt is not independent of t. Its population variance is not even defined.

Deterministic trend

tXE t 21


tt tX 21

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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

ttt XtX 1101 ...



tt tX 21

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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

ttt XtX 1101 ...



tt tX 21

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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

ttt XtX 1101 ...



ttt XX 11

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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


ttt XX 11

25


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If we difference the series, the differenced series is just 1 + t.


ttt XX 11

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This is stationary because the expected value of Xt 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.


tttt XXX 11

1 tXE

22 tX

0 and of covariance population stt XX


ttt XX 11

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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).


tttt XXX 11

Xt is I(1)


ttt XX 11

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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).


tttt XXX 11

Xt is I(1)


ttt XX 11

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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.


tttt XXX 11

Xt is I(1)

ttt XX 110 ...


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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 Xt is just the residuals from a regression on time.

Trend-stationarity

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tt tX 21

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~


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

Trend-stationarity

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tt tX 21

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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

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tt tX 21

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Typically the cyclical component was analyzed using detrended versions of the variables in the model.

Trend-stationarity

tbbX t 21ˆ

tt tX 21

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tt tX 21

Deterministic trend

ttt XtX 1101 ...


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.

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

21.08.06

Documents

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