Introduction

Testing for a unit root is a fundamental aspect of analyzing non-stationary time series data. A unit root implies that a time series possesses a stochastic trend, leading to challenges in accurate modeling and forecasting. In this chapter, we will delve into the techniques and methodologies used to test whether a time series contains a unit root. By understanding these tests, you’ll be equipped to assess the stationarity properties of your data and make informed decisions in time series analysis.

Testing for Unit Roots:

There are several well-established tests to determine if a time series contains a unit root. One common test is the Augmented Dickey-Fuller (ADF) test. The ADF test assesses the null hypothesis that a unit root is present in the time series data. The test equation is as follows:

$$\mathrm{\Delta }{y}_t=\alpha +\beta t+\gamma y_{t-1}+\sum _{i=1}^p{\delta }_i\mathrm{\Delta }{y}_{t-i}+{\epsilon }_t$$

Where:

• $\mathrm{\Delta }{y}_t$ is the first difference of the time series at time $t$.
• $\alpha$ represents a constant.
• $\beta$ represents the coefficient of the time trend.
• $\gamma$ is the coefficient associated with the lagged level of the time series.
• ${\delta }_i$ represents the coefficients of the lagged first differences.
• ${\epsilon }_t$ is the error term.

Example: Consider a stock price time series. We want to determine if the stock price contains a unit root. We run the ADF test, and if the test statistic is less negative than the critical values, we fail to reject the null hypothesis, indicating the presence of a unit root. Conversely, if the test statistic is more negative than the critical values, we reject the null hypothesis, implying the absence of a unit root.