Stationary process NONSTATIONARY PROCESSES 1 In the last sequence, the process shown at the top was...

Stationary process

NONSTATIONARY PROCESSES

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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.

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The condition –1 < b2 < 1 was crucial for stationarity. Suppose b2 = 1, as above. Then the value of X in one time period is equal to its value in the previous time period, plus a random adjustment. This is known as a random walk.

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Random walk


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The figure shows an example realization of a random walk for the case where et has a normal distribution with zero mean and unit variance.

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This figure shows the results of a simulation with 50 realizations. It is obvious that the ensemble distribution is not stationary. The distribution changes as t increases, becoming increasingly spread out. We will confirm this mathematically.

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If the process is valid for time t, it is valid for time t – 1.


Random walk

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Hence Xt can be expressed in terms of Xt–2 and the innovations et–1 and et.


Random walk

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Thus, continuing to lag and substitute, Xt is equal to its value at time 0, X0, plus the sum of the innovations in periods 1 to t.


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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.


Random walk

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This can be seen from the 50 realizations. The distribution of the values of Xt spreads out as t increases, but there is no tendency for the mean of the distribution to change. (In this example X0 = 0, but this is unimportant. It would be true for any value of X0.)


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However, it is also clear from the figure that the ensemble distribution is not constant over time, and therefore that the process is nonstationary. The distribution of the values of Xt spreads out as t increases, so the variance of the distribution is an increasing function of t.


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We will demonstrate this mathematically. We have seen that Xt is equal to X0 plus the sum of the innovations e1, ..., et. X0 is an additive constant, so it does not affect the variance.


Random walk

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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 zero because the innovations are assumed to be generated independently.


Random walk

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The variance of each innovation is equal to s e , by assumption. Hence the population variance of Xt is directly proportional to t. As we have seen from the figure, its distribution spreads out as t increases.


Random walk

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A second process considered in the last sequence is shown above. The presence of the intercept b1 on the right side gave the series a nonzero mean but did not lead to a violation of the conditions for stationarity.


Stationary process

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If b2 = 1, however, the series becomes a nonstationary process known as a random walk with drift.


Random walk with drift

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If the process is valid for time t, it is valid for time t – 1.



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Hence Xt can be expressed in terms of Xt–2, the innovations et–1 and et, and an intercept. The intercept is 2b1. Irrespective of whatever else is happening to the process. a fixed quantity b1 is added in every time period.



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Thus, lagging and substituting t times, Xt is now equal X0 plus the sum of the innovations, as before, plus the constant b1 multiplied by t.



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As a consequence, the mean of the process becomes a function of time, violating the first condition for stationarity.



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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 b1.)



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This process is known as a random walk with drift, the drift referring to the systematic change in the expectation from one time period to the next.



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The figure shows 50 realizations of such a process. The underlying drift line is highlighted in yellow. It can be seen that the ensemble distribution changes in two ways with time.

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The mean changes. In this case it is drifting upwards because b1 has been taken to be positive. If b1 were negative, it would be drifting downwards.


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And, as in the case of the random walk with no drift, the distribution spreads out around its mean.


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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

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This type of trend is described as a deterministic trend, to differentiate it from the trend found in a model of a random walk with drift.


Deterministic trend

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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

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The figure shows 50 realizations of a variation where the disturbance term is the stationary process ut = 0.8ut–1 + et. The underlying trend line is shown in white.

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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.

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The key difference between a deterministic trend and a random walk with drift is that in the former, the series must keep coming back to a fixed trend line.


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Deterministic trend

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In any given observation, Xt will be displaced from the trend line by an amount ut, but, provided that this is stationary, it must otherwise adhere to the trend line.


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By contrast, in a random walk with drift, the displacement from the underlying trend line at time t is the random walk . Since the displacement is a random walk, there is no reason why Xt should ever return to its trend line.

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It is important to make a distinction between the concepts of difference-stationarity and trend-stationarity.


Difference stationarity and trend stationarity

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


Difference stationarity

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The first difference, DXt, is simply equal to the sum of b1 and et.



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This is a stationary process with population mean b1 and variance se2, both independent of time. It is actually iid and the covariance between DXt and DXt+s is zero.



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If a nonstationary time series can be transformed into a stationary process by differencing once, as in this case, it is described as integrated of order 1, or I(1).



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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.



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If a series can be made stationary by differencing twice, it is known as I(2), and so on. To complete the picture, 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).



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The stochastic component et is iid. More generally, the stationary process reached after differencing may be ARMA(p, q): auto-regressive of order p and moving average of order q.



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The original series is then characterized as an ARIMA(p, d, q) time series, where d is the number of times it has to be differenced to render it stationary.

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A nonstationary time series is described as being trend-stationary if it can be transformed into a stationary process by extracting a time trend.


Trend stationarity

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For example, the very simple model given by the first equation can be detrended by fitting it (second equation) and defining a new variable with the third equation. The new, detrended, variable is of course just the residuals from the regression of X on t.


Trend stationarity

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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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At one time it was conventional to assume that macroeconomic time series could be decomposed into trend and cyclical components.


Trend stationarity

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It was thought that the trend components were determined by real factors, such as the growth of GDP, and the cyclical components were determined by transitory factors, such as monetary policy.


Trend stationarity

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

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Trend stationarity

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However, this approach is inappropriate if the process is difference-stationary. 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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Deterministic trend


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As will be seen in the next slideshow, this gives rise to problems of estimation and inference.


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Further, because the approach ignores the contribution of real shocks to economic fluctuations, it causes the role of transitory factors in the cycle to be overestimated.


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Deterministic trend


Copyright Christopher Dougherty 2013.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 13.1 of C. Dougherty,

Introduction to Econometrics, fourth edition 2011, Oxford University Press.

Additional (free) resources for both students and instructors may be

downloaded from the OUP Online Resource Centre

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own who feel that they might benefit

from participation in a formal course should consider the London School of

Economics summer school course

EC212 Introduction to Econometrics

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or the University of London International Programmes distance learning course

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2013.08.17

http://www.oup.com/uk/orc/bin/9780199567089/

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Stationary process NONSTATIONARY PROCESSES 1 In the last sequence, the process shown at the top was...

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