When is time series stationary

For this reason you should be cautious about trying to extrapolate regression models fitted to nonstationary data. Most business and economic time series are far from stationary when expressed in their original units of measurement, and even after deflation or seasonal adjustment they will typically still exhibit trends, cycles, random-walking, and other non-stationary behavior.

If the series has a stable long-run trend and tends to revert to the trend line following a disturbance, it may be possible to stationarize it by de-trending e. Such a series is said to be trend-stationary. If the mean, variance, and autocorrelations of the original series are not constant in time, even after detrending, perhaps the statistics of the changes in the series between periods or between seasons will be constant. Such a series is said to be difference-stationary.

Sometimes it can be hard to tell the difference between a series that is trend-stationary and one that is difference-stationary, and a so-called unit root test may be used to get a more definitive answer. We will return to this topic later in the course. Return to top of page. The first difference of a time series is the series of changes from one period to the next.

If Y t denotes the value of the time series Y at period t, then the first difference of Y at period t is equal to Y t -Y t Another, more rigorous approach, to detecting stationarity in time series data is using statistical tests developed to detect specific types of stationarity, namely those brought about by simple parametric models of the generating stochastic process see my previous post for details.

The Dickey-Fuller Test The Dickey-Fuller test was the first statistical test developed to test the null hypothesis that a unit root is present in an autoregressive model of a given time series and that the process is thus not stationary.

The original test treats the case of a simple lag-1 AR model. The test has three versions that differ in the model of unit root process they test for;. The choice of which version to use — which can significantly affect the size and power of the test — can use prior knowledge or structured strategies for series of ordered tests, allowing the discovery of the most fitting version. Extensions of the test were developed to accommodate more complex models and data; these include the Augmented Dickey-Fuller ADF using AR of any order p and supporting modeling of time trends , the Phillips-Perron test PP adding robustness to unspecified autocorrelation and heteroscedasticity and the ADF-GLS test locally de-trending data to deal with constant and linear trends.

Python implementations can be found in the statsmodels and ARCH packages. I will touch on how to interpret such combined results in a future post. The Zivot and Andrews Test The tests above do not allow for the possibility of a structural break — an abrupt change involving a change in the mean or other parameters of the process.

Assuming the time of the break as an exogenous phenomenon, Perron showed that the power to reject a unit root decreases when the stationary alternative is true and a structural break is ignored. Hence, to test for a unit root against the alternative of a one-time structural break, Zivot and Andrews use the following regression equations corresponding to the above three models [Waheed et al. Variance Ratio Test [Breitung, ] suggested a non-parametric test for the presence of a unit root based on a variance ratio statistic.

The null hypothesis is a process I 1 integrated of order one while the alternative is I 0. I list this test as semi-parametric because it tests for a specific, model-based, notion of stationarity. In the wake of the limitations of parametric tests, and the recognition they cover only a narrow sub-class of possible cases encountered in real data, a class of non-parametric tests for stationarity has emerged in time series analysis literature. Naturally, these tests open up a promising avenue for investigating time series data: you no longer have to assume very simple parametric models happen to apply to your data to find out whether it is stationary or not, or risk not discovering a complex form of the phenomenon not captured by these models.

Instead, these tests limit themselves to specific types of data or processes. Non-stationary behaviors can be trends, cycles, random walks , or combinations of the three. Non-stationary data, as a rule, are unpredictable and cannot be modeled or forecasted. The results obtained by using non-stationary time series may be spurious in that they may indicate a relationship between two variables where one does not exist. In order to receive consistent, reliable results, the non-stationary data needs to be transformed into stationary data.

In contrast to the non-stationary process that has a variable variance and a mean that does not remain near, or returns to a long-run mean over time, the stationary process reverts around a constant long-term mean and has a constant variance independent of time. Before we get to the point of transformation for the non-stationary financial time series data, we should distinguish between the different types of non-stationary processes.

This will provide us with a better understanding of the processes and allow us to apply the correct transformation. Examples of non-stationary processes are random walk with or without a drift a slow steady change and deterministic trends trends that are constant, positive, or negative, independent of time for the whole life of the series. The disadvantage of differencing is that the process loses one observation each time the difference is taken.

A non-stationary process with a deterministic trend becomes stationary after removing the trend, or detrending. No observation is lost when detrending is used to transform a non-stationary process to a stationary one. In the case of a random walk with a drift and deterministic trend, detrending can remove the deterministic trend and the drift, but the variance will continue to go to infinity.

As a result, differencing must also be applied to remove the stochastic trend. Using non-stationary time series data in financial models produces unreliable and spurious results and leads to poor understanding and forecasting.

The solution to the problem is to transform the time series data so that it becomes stationary. If the non-stationary process is a random walk with or without a drift, it is transformed to stationary process by differencing. On the other hand, if the time series data analyzed exhibits a deterministic trend, the spurious results can be avoided by detrending. Sometimes the non-stationary series may combine a stochastic and deterministic trend at the same time and to avoid obtaining misleading results both differencing and detrending should be applied, as differencing will remove the trend in the variance and detrending will remove the deterministic trend.

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