8.1 Stationarity

• Generally, a time-series is stationary if its probabilistic properties do not change over time

• Unlike a nonstationary time-series, a stationary one has time-invariant characteristics:

1. Mean (expected value) $$\begin{equation} E(y_t)=\mu~~~~~~~~~~~~~\forall~t \tag{8.6} \end{equation}$$

2. Variance $$\begin{equation} Var(y_t)=\sigma^2~~~~~~~~~\forall~t \tag{8.7} \end{equation}$$

3. Autocovariance $$\begin{equation} Cov(y_t,y_{t-k})=E[(y_t-\mu)(y_{t-k}-\mu)]=\gamma_k \tag{8.8} \end{equation}$$

• The covariance between shifted values of the same time-series is called autocovariance

• Autocovariance (8.8) is a function of the time lag $k$, but does not depend on time $t$

• Commonly, autocovariance is being standardized, and thus an autocorrelation is derived

$$\begin{equation} \rho(y_t,y_{t-k})=\frac{Cov(y_t,y_{t-k})}{Var(y_t)}=\frac{\gamma_k}{\sigma^2}=\rho_k \tag{8.9} \end{equation}$$

• A sequence of autocorrelation coefficients up to and including $p$ lags $$\rho_0,~~\rho_1,~~\rho_2,\dots,~\rho_k,\dots,\rho_p$$ forms the autocorrelation function ACF

• A graphical representation of the autocorrelation coefficients $\rho_k$ for non-negative lags $k=0,1,2,\dots,p$ is called correlogram

• The ACF is a decreasing function of the lag $k$

• The properties of the autocorrelation function ACF are: $$\begin{equation} \begin{aligned} \rho_0&=1 \\ -1 & \leq\rho_k\leq+1~~~~~~~\forall~k>0 \\ \rho_k &=\rho_{-k}~~~~~~~~~~~~~~~\forall~k>0 \end{aligned} \tag{8.10} \end{equation}$$

• Based on the time-series values from the sample, the autocorrelation coefficient is estimated using the formula

$$\begin{equation} \widehat{\rho}_k=\frac{\displaystyle\sum_{t=k+1}^T (y_t-\bar{y})(y_{t-k}-\bar{y})}{\displaystyle\sum_{t=1}^T (y_t-\bar{y})^2}~~;~~~~~\bar{y}=\frac{\displaystyle\sum_{t=1}^T y_t}{T} \tag{8.11} \end{equation}$$

• For stationary time-series, the ACF decays quickly

• The quick decay of the ACF occurs at the first few lags

• The stationarity conditions in a broader sense will be fulfilled if a time-series does not contain a deterministic trend (clear long-term trending), nor stochastic trend (radnom walk), if the variance of the time-series is constant (stable), or if does not contain any periodic or systematic variations (such as seasonality).