Stationary of Moving Average Process

Question

I have studied about moving average process $MA$ of first and second orders, and I need the values of parameters that make the process $MA(1)$ and $MA(2)$ are stationary. Thanks

Answer 1

'Contrary to the AR model, the finite MA model is always stationary.' https://en.wikipedia.org/wiki/Moving-average_model

Indeed, if $$X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q}$$ where $(\varepsilon_t)$ is a white noise, then the mean and the variance are constant since $X_t$ is essentially a linear combination of present and past values of $(\varepsilon_t)$ which has constant zero mean and constant variance $\sigma^2$ and whose terms are uncorrelated so that $$ \text{Var}(X_t) = \text{Var}(\varepsilon_t) + \sum_{i=1}^q \theta_i^2\text{Var}(\varepsilon_{t-i}) = \sigma^2.\left(\sum_{i=1}^q \theta_i^2\right).$$