Understanding Non-Stationary Time Series

A time series is a sequence of data points indexed in time order. Stationarity is a key property in time series analysis. A stationary time series has statistical properties (like mean, variance, and autocorrelation) that do not change over time. Many standard statistical methods for time series require stationarity.

Conversely, a non-stationary time series has properties that change over time. Common examples of non-stationarity include a trend (mean changing over time) or changing variance.

When working with multiple time series, we often look at their relationships. Sometimes, even if individual time series are non-stationary, a specific linear combination of them might be stationary. This indicates a special kind of long-run relationship between the series.

Examining the Options

Let's look at the given options and see how they relate to the scenario described:

1. Random walk: A random walk is a specific type of non-stationary time series where each step is independent of the previous steps (plus some drift). While the question involves non-stationary series, 'random walk' is a description of a single series' behaviour, not a relationship between two series that makes their combination stationary.
2. Spurious regression: A spurious regression occurs when two or more independent non-stationary time series appear to be statistically related in a regression analysis, even though there is no genuine long-run relationship between them. This happens because the standard assumptions for regression do not hold for non-stationary data. This is the opposite of the scenario described, where a genuine long-run relationship makes the combination stationary.
3. Cointegration: Cointegration describes a situation where two or more non-stationary time series have a stable long-run relationship. Specifically, if individual series $X_t$ and $Y_t$ are non-stationary, but a linear combination of them, say $aX_t + bY_t$, is stationary, then $X_t$ and $Y_t$ are said to be cointegrated. The stationary linear combination is often referred to as the cointegrating relationship or the equilibrium error. This perfectly matches the description in the question.
4. Trend stationarity: A trend-stationary process is a non-stationary time series whose non-stationarity is caused by a deterministic trend (e.g., a linear trend). If this deterministic trend is removed (e.g., by detrending), the remaining series is stationary. While the individual series in the question could potentially be trend-stationary (or other types of non-stationary), 'trend stationarity' describes the property of a single series or how its non-stationarity is characterized, not the relationship between two non-stationary series that leads to a stationary combination.

What is Cointegration?

Cointegration is a key concept in time series analysis, particularly in econometrics. It addresses the issue of analyzing relationships between non-stationary variables without falling victim to spurious regression.

Consider two time series, $X_t$ and $Y_t$, both of which are integrated of order 1, denoted as I(1). An I(1) series is one that becomes stationary after differencing once (e.g., $\Delta X_t = X_t - X_{t-1}$ is stationary). If these two I(1) series are cointegrated, it means there exist constants $a$ and $b$ (not both zero) such that the linear combination $Z_t = aX_t + bY_t$ is stationary, denoted as I(0). The vector $(a, b)$ is called the cointegrating vector.

The existence of such a stationary linear combination implies that the two non-stationary series do not drift too far apart from each other in the long run. They maintain a stable relationship around a long-run equilibrium, represented by the stationary linear combination.

Why Cointegration Fits the Description

The question explicitly states: "Though two time series are individually non-stationary, their linear combination is stationary." This is the precise definition of cointegration between two time series.

Let's represent the two time series as $X_t$ and $Y_t$. The question says $X_t$ is non-stationary and $Y_t$ is non-stationary. It also says that for some constants $a$ and $b$, the series $Z_t = aX_t + bY_t$ is stationary.

This scenario is exactly what the concept of Cointegration describes. Therefore, the example provided is an example of cointegration.

Comparing Cointegration with Other Concepts

Conclusion

Based on the definitions and analysis, the phenomenon where two individually non-stationary time series have a stationary linear combination is known as Cointegration. This signifies a stable, long-run equilibrium relationship between the series, preventing them from diverging indefinitely.

Revision Table: Key Time Series Concepts

Additional Information on Cointegration

Cointegration is a fundamental concept for modeling relationships between non-stationary economic variables, such as consumption and income, or prices in different markets. If variables are cointegrated, standard regression techniques applied to their levels can yield meaningful results concerning the long-run relationship, provided the model is specified correctly (e.g., using an Error Correction Model - ECM).

Testing for cointegration is important before running regressions with non-stationary data. Common tests include the Engle-Granger test, the Johansen test, and the Phillips-Ouliaris test. The presence of cointegration suggests that while the series may deviate in the short run, they are bound together by an economic force or equilibrium in the long run.