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