8 Nonstationary time-series

• As already emphasized in previous section a static model provides spurious results due to common issues when dealing with time-series data

• Very often $3$ problems might appear:

1. Heteroscedasticity -> $Var(u_t)\ne\sigma^2_u~~~~\forall t=1,2,\dots,T$
2. Autocorrelation -> $Cov(u_t,u_{t-j})\ne0~~~\forall j=1,2,\dots,p$
3. Endogeneity -> $Cov(x_t,u_t)\ne0$

• When these problems exist OLS estimates are no longer BLUE

• Heteroscedasticity and autocorrelation problems are expected when a time-series is nonstationary

A time-series is nonstationary if it has time-varying moments, i.e. the mean, variance and autocovariance of a time-series are not constant over time. Nonstationary time-series is defined as a radnom walk (RW).

• Pure random walk is an autoregression of the first order AR$(1)$ with $\alpha=0$ and $\beta=1$ $$\begin{equation} \begin{aligned} y_t&=\alpha+\beta y_{t-1}+u_t \\ y_t&=y_{t-1}+u_t \end{aligned} \tag{8.1} \end{equation}$$

• It can be shown that the variance of a random walk (8.1) is not constant even it’s mean is constant, i.e. the variance of pure random walk increases as a liner function of time

$$\begin{equation} Var(y_t)=t \sigma^2_u \tag{8.2} \end{equation}$$

• If a random walk also exhibits consistent upward or downward trend then a constant term $\alpha\ne0$, and model (8.1) becomes a random walk with drift

$$\begin{equation} y_t=\alpha+y_{t-1}+u_t \tag{8.3} \end{equation}$$

• Both, the mean and the variance of a random walk with drift are not constant

$$\begin{equation} \begin{aligned} E(y_t)&=y_0+t \alpha \\ Var(y_t)&=t \sigma^2_u \end{aligned} \tag{8.4} \end{equation}$$

• No matter if a drift term $\alpha$ is included or not, both random walks (pure RW and RW with drift) are special cases of autoregression AR$(1)$ when $\beta=1$

• If time-series follows any type of RW we say that it has a unit root or exhibits a stochastic trend

• Nonstationary time-series is usually heteroskedastic and highly persistent

• A highly persistent time-series has a long-memory, i.e. autocorrelation function ACF decays very slowly and it takes a long time to converge towards zero

Differencing the data removes the stochastic trend and makes it stationary. Therefore, a time-series can be transformed into first or second differences $$\begin{equation} \begin{aligned} \Delta y_t&=y_t-y_{t-1}~~~~~~~~~~~~~~~~~~~~~t=2,3,\dots,T \\ \Delta^2y_t&=\Delta y_t-\Delta y_{t-1} \\ & =y_t-2y_{t-1}+y_{t-2} ~~~~~~~t=3,4,\dots,T \end{aligned} \tag{8.5} \end{equation}$$

• If the time-series of the first differences is stationary then it is integrated of order one $y_t\sim I(1)$

• If the time-series of the second differences is stationary then it is integrated of order two $y_t\sim I(2)$