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:

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


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 \(\{Y_{t}\} \quad Y_{t}=\varphi_{1} Y_{t-1}+ \ldots+\varphi_{p}Y_{t-p}+\theta_{1}\omega_{t-1}+\ldots+\theta_{q} \omega_{t-q}\)