Question

Can we transform a non-stationary process into a 1st order weakly stationary or 2nd order weakly stationary/covariance-stationary process?

Answer 1

Non – stationary data as a rule are un predictable and can not be modeled or forecasted. The result obtained by using non stationary time series may be spurious in that they may indicate a relationship between two variable where one does not exist. In order to receive consistent, reliable result, 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 long term mean and has a constant variance independent of time .

The point of transformation for the non stationary financial time series data , we should distinguish between the different types of the non stationary process. This will provide us with a better understanding of the process and allow us to apply the correct transformation. Examples of non stationary processes are random walk with or without a drift (a slow study change) and deterministic trends (trends that are constant , positive or negative, independent of time for the whole life of the series).

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 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.

Weakly stationary process commonly employed in single processing is known as weakly stationary process. , wide sense stationary or covariance stationary. A sequence of random variables is covariance stationary if all the terms of the sequence have the same mean an d if the covariance between any two terms of the sequence depends only on the relative positions of the two terms that is on how for apart they are located from each other , and not on their absolute position that is on where they are located in the sequence.

A stochastic process is called a 2nd order process if it is 2nd order movement is finite for all. Covariance function it is denoted by C(s,t) given by

C(s,t) = cov (X(s), X(t))

= E(X(s)X(t)) - E(X(s) E(X(t))

A stochastic process has to be a second order process for covariance function to exist. The covariance functions satisfies the properties.