## 17.1 Stochastic processes

A stochastic process can be defined as a set of random variables associated with different instants of time.

• At each time point we have a variable that has its corresponding probability distribution; for example, if we consider the process $$(Y_t)$$, for $$t=1$$, we will have a random variable, $$Y_1$$, which will take different values with different probabilities.

• A time series is a sample or realization of a stochastic process, consisting of a single observation of each of the random variables that make up the process.

• The analyst’s objective is to infer the shape of the stochastic process from the time series it generates.

• A stochastic process, $$(Y_t)$$, is usually described by the following characteristics: mathematical expectation, variance, autocovariances and autocorrelation coefficients.
• The mathematical expectation or mean of $$(Y_t)$$ is the succession of the mathematical expectations of the variables that compose the process over time, such that:

$E(Y_t)=\mu_t,\ \ t=1,2,3...$

• The variance of a process $$(Y_t)$$ is a succession of variances, one for each process variable:

$Var(Y_t)=E(Y_t–\mu_t)^2,\ \ t=1,2,3...$

• The autocovariances are the covariances between each pair of process variables, such that:

$\gamma_k=Cov(Y_t,Y_{t+k})=E[(Y_t–\mu_t)(Y_{t+k}–\mu_{t+k})]=\gamma_{t,t+k},\ \ t=1,2,3...$

• Finally, the autocorrelation coefficients are the linear correlation coefficients between each pair of variables that make up the process:

$\rho_{t,t+k}=\frac{\gamma_{t,t+k}}{\sqrt{Var(Y_t)Var(Y_{t+k})}},\ \ t=1,2,3...,\ \ -1 \leq \rho_{t,t+k} \leq 1$

Examples

Name Expression
White Noise $$\{\omega_{t}\}$$
Random Walk $$\{Y_{t}\} \quad Y_{t}=Y_{t-1}+\omega_{t}$$
Autorregressive (AR) $$\{Y_{t}\} \quad Y_{t}=\varphi_{1} Y_{t-1}+\ldots+\varphi_{p} Y_{t-p}$$
Moving Averages (MA) $$\{Y_{t}\} \quad Y_{t}=\omega_{t}+\theta_{1} \omega_{t-1}+\ldots+\theta_{q} \omega_{t-q}$$
ARMA $$\{Y_{t}\} \quad Y_{t}=\varphi_{1} Y_{t-1}+ \ldots+\varphi_{p}Y_{t-p}+\theta_{1}\omega_{t-1}+\ldots+\theta_{q} \omega_{t-q}$$