###### 4.6.3 Stationary Stochastic Processes

To fully specify a stochastic process, we must specify—explicitly or implicitly—a joint distribution for all components * ^{t}X_{i}* of all terms

*. This entails specifying infinitely many parameters. To reduce this to a manageable—and finite!—number of distinct parameters, we assume some sort of homogeneity across terms. A simple solution is to assume that terms are IID. This reduces the task of specifying a stochastic process to that of specifying the joint distribution of the components of a single term. Of course, this solution defeats the purpose of time series analysis. We introduced stochastic processes as having non-IID terms specifically because we wanted to model temporal dependencies in time series.*

^{t}**X**An alternative and widely used solution is to consider only processes that have some form of stationarity. A process is said to be **strictly stationary** if the unconditional joint distribution of any segment {* ^{t}X*,

^{t}^{+1}

**, … ,**

*X*

^{t}^{+m}

**} is identical to the unconditional joint distribution of any other segment {**

*X*

^{t}^{+k}

**,**

*X*

^{t}^{+k+1}

**, … ,**

*X*

^{t}^{+k+m}

**} of the same length. Note the similarity of this definition to the IID condition, which we just rejected. That approach requires that unconditional distributions of terms be identical. Strict stationarity requires that unconditional distributions of segments be identical. Strict stationarity is appealing because it affords a form of homogeneity across terms without requiring that terms be independent.**

*X*A more widely used solution is to consider processes that are covariance stationary. A process is said to be **covariance stationary**—or simply **stationary**—if the unconditional distribution of any segment {* ^{t}X*,

^{t}^{+1}

**, … ,**

*X*

^{t}^{+m}

**} has means, standard deviations, and correlations that are identical to the corresponding means, standard deviations and correlations of the unconditional distribution of any other segment {**

*X*

^{t}^{+k}

**,**

*X*

^{t}^{+k+1}

**, … ,**

*X*

^{t}^{+k+m}

**} of the same length. Correlations include autocorrelations and cross correlations.**

*X*Note that covariance stationarity requires that all first and second moments exist whereas strict stationarity does not. In this one respect, covariance stationarity is a stronger condition.

In applications, a stationarity assumption is not always reasonable. For example, a time series may rise over time, making a constant unconditional mean assumption unreasonable. Every situation is unique, but a solution that often works is to transform a time series in some manner to make it compatible with a stationarity assumption. Transformations take various forms to address different departures from stationarity.