# Introduction to Non-Stationary Processes

Individual investors and academics, as well as financial institutions and organizations, often employ financial time series data (such as asset prices, exchange rates, GDP, inflation, and other macroeconomic indicators) in economic projections, stock market analysis, or data studies.

However, refining data is essential for applying it to stock research. We’ll teach you how to extract the data items that are important to your stock reports in this post.

### Key Takeaways

- A time series analysis in statistics includes assessing how things change over time with regard to specific variables of interest.
- A stationary time series is one in which certain characteristics of the data do not vary over time.
- Some time series, however, are non-stationary, meaning that the values and connections between and among variables change with time.
- Many processes in finance are non-stationary and must be addressed accordingly.

#### Intro to Stationary and Non-Stationary Processes

## Non-Stationary Time Series Data

Data points are often non-stationary or have changing means, variances, and covariances. Trends, cycles, random walks, and combinations of the three are examples of non-stationary behaviour.

Non-stationary data, by definition, is unexpected and hence cannot be modeled or projected. The findings generated by employing non-stationary time series may be erroneous, indicating a link between two variables that do not exist. Non-stationary data must be turned into stationary data in order to get consistent, dependable findings.

In contrast to the non-stationary process, which has a variable variance and a mean that does not stay near or return to a long-run mean over time, the stationary process reverts to a constant long-term mean and has a constant variance regardless of time.

## Types of Non-Stationary Processes

Before we reach to the point of transformation for non-stationary financial time series data, we first define the various forms of non-stationary processes. This will help us to have a deeper knowledge of the processes and apply the appropriate change.

Non-stationary processes include random walks with and without drift (slow steady change) and deterministic trends (trends that are constant, positive, or negative, independent of time for the whole life of the series).

- Walk at Random (Yt = Yt-1 + εt) The random walk predicts that the value at time “t” will be equal to the previous period value plus a stochastic (non-systematic) component of white noise, implying that t is independent and identically distributed with mean “0” and variance “2.” A random walk is also known as a process integrated of some order, a process with a unit root, or a stochastic trend. It is a non-mean-reverting process that may deviate from the mean either positively or negatively. Another feature of a random walk is that the variance grows over time and approaches infinity as time passes; hence, a random walk cannot be anticipated.
- Drift’s Random Walk (Yt = α + Yt-1 + εt) The process is random walk with a drift if the random walk model predicts that the value at time “t” will equal the previous period’s value plus a constant, or drift (), and a white noise term (t). It also does not return to the long-run mean and exhibits time-dependent volatility.
- Trend Deterministic (Yt = + t + t) A random walk with a drift is often mistaken for a deterministic trend. Both have a drift and a white noise component, but in the case of a random walk, the value at time “t” is regressed on the previous period’s value (Yt-1), but in the case of a deterministic trend, it is regressed on a temporal trend (t). A non-stationary process with a deterministic trend has a mean that increases around a constant and time-independent trend.
- (Yt = + Yt-1 + t + t) Random Walk with Drift and Deterministic Trend A non-stationary process that combines a random walk with a drift component () and a deterministic trend (t) is another example. It determines the value at time “t” by using the value from the previous period, a drift, a trend, and a stochastic component.

## Trend and Difference Stationary

A random walk, with or without drift, may be changed to a stationary process by differencing (subtracting Yt-1 from Yt and taking the difference Yt – Yt-1) to Yt – Yt-1 = t or Yt – Yt-1 = + t, and the process becomes difference-stationary. The downside of differencing is that each time the difference is taken, the procedure loses one observation.

After eliminating the trend, or detrending, a non-stationary process with a deterministic trend becomes stationary. As seen in the image below, Yt = + t + t is turned into a stationary process by removing the trend t: Yt – t = + t. When detrending is employed to convert a non-stationary process to a stationary one, no observations are lost.

Detrending can eliminate the deterministic trend and the drift from a random walk with a drift and a deterministic trend, but the variance will continue to grow indefinitely. As a consequence, differencing must be used to eliminate the stochastic trend.

## The Bottom Line

When non-stationary time series data is used in financial models, the findings are inaccurate and erroneous, resulting in poor comprehension and predictions. The issue is solved by transforming the time series data such that it becomes stationary. If the non-stationary process is a random walk with or without drift, differencing converts it to a stationary process. Detrending, on the other hand, may prevent false findings if the time series data being studied has a deterministic trend.

Non-stationary series may combine a stochastic and deterministic trend at the same time, and in order to prevent misleading findings, both differencing and detrending should be used, since differencing removes the trend in the variance and detrending removes the deterministic trend.

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