Stationary Stochastic Process

Stationary Stochastic Process

A stochastic process is said to be stationary if its mean and variance are constant over time and the value of the covariance between the two time periods depends only on a distance or gap or lag between the two time periods and not the actual time at which the covariance is computed. Such a stochastic process is also known as weak stationary, covariance stationary, second-order stationary or wide sense stochastic process.

In other words, a sequence of random variables {$y_t$} is covariance stationary if there is no trend, and if the covariance does not change over time.

Strictly Stationary (Covariance Stationary)

A time series is strictly stationary if all the moments of its probability distribution are invariance over time but not for first two (mean and variance).

Let $y_t$ be a stochastic time series with

$E(y_t) = \mu $    $\Rightarrow$ Mean
$V(y_t) = E(y_t -\mu)^2=\sigma^2 $  $\Rightarrow$ Variance
$\gamma_k = E[(y_t-\mu)(y_{t+k}-\mu)]$  $\Rightarrow$ Covariance = $Cov(y_t, y_{t-k})$

$\gamma_k$ is covariance or autocovariance at lag $k$.

If $k=0$ then $Var(y_t)=\sigma^2$ i.e. $Cov(y_t)=Var(y_t)=\sigma^2$

If $k=1$ then we have covariance between two adjacent value of $y$.

If $y_t$ is to be stationary, the mean, variance and autocovariance of $y_{t+m}$ (shift or origin of $y=m$) must be the same as those of $y_t$. OR

If a time series is stationary, its mean, variance and autocovariance remain the same no matter at what point we measure them, i.e, they are time invariant.

Non-Stationary Time Series

A time series having a time-varying mean or a time-varying variance or both is called non-stationary time series.

Purely Random/ White Noise Process

A stochastic process having zero mean and a constant variance ($\sigma^2$) and serially uncorrelated is called purely random/ white noise process.

If it is independent also then such a process is called strictly white noise.

White noise denoted by $\mu_t$ as $\mu_t \sim N(0, \sigma^2)$ i.e. $\mu_t$ is independently and identically distributed as a normal distribution with zero mean and constant variance.

Stationary time series is important because if a time series is non-stationary, we can study its behaviour only for the time period under consideration. Each set of time series data will, therefore, be for a particular episode. As consequence, it is not possible to generalize it to other time periods. Therefore, for the purpose of forecasting, such (non-stochastic) time series may be of little practical value. Our interest is in stationary time series.

Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

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