How to do regression for this stochastic process? I wonder if there is some good way to determine the parameters for a random process. In my research work, I have a random process of the following form:
$$
X_t=\alpha_{1}G_{t-1}+\alpha_{2}G_{t-2}+...+\alpha_{k}G_{t-k}
$$
Where $\alpha_i$ are unknown parameters, $G_j$ are i.i.d. Gaussian variables. $k$ is known fixed number.
We can sample the random process $X_t$ for a long time and we are trying to obtain the parameter $\alpha_1, \alpha_2, ...\alpha_k$ as well as the mean and variance of the random variable $G_j$?
Does anyone know a good way to determine this regression problem? And what area are these problems belong to?
 A: The process you have written down is an $MA(q)$ process, which falls under the general $ARMA$ processes theory. Let me write your model in a more standard way:
$$X_t = \epsilon_t + \theta_1 \epsilon_{t-1} + \ldots + \theta_q\epsilon_{t-q}$$
where $\epsilon_t$ is an iid sequence with $\sigma_\epsilon^2 = \mathbb{E}(\epsilon_t^2) < \infty$ and $\mathbb{E}(\epsilon_t) = 0$. One observes that this process is stationary and the autocorrelation function is given by:
$$\gamma (h) = \mathbb{E}(X_t X_{t+h}) = \frac{-\theta_h + \theta_1 \theta_{h+1} + \theta_2 \theta_{h+2} + \ldots + \theta_{q-h} \theta_q}{1 + \theta_1^2 + \ldots + \theta_q^2}$$
for $h = 1, 2, \ldots, q$. By considering the empirical version of this function, i.e. $\hat{\gamma}$, one obtains $q$ non-linear equations in $q$ unknowns, which can then be placed into a solver. To estimate $\sigma_\epsilon^2$, one may use the variance formula for $X_t \sim MA(q)$: $$\sigma_X^2 = \sigma_\epsilon^2 (1 + \theta_1^2 + \ldots + \theta_q^2)$$
where $\sigma_X^2 = Var(X_t)$.
Alternatively, if one posits that $\epsilon_t \sim N(0, \sigma^2)$, one may write down the likelihood function of $(X_1, \ldots , X_n)$ and find estimates via maximum likelihood estimation. See e.g. Maximum Likelihood Estimation for MA Processes.
For more information about MA and ARMA processes, you may consult the time series texts by Box and Jenkins, Shumway and Stoffer, or Brockwell and Davis, to name just three.
