# What is the distribution of $y_t$ in ARMA(2,2)?

So, I'm currently having a graduate time series analysis course and we have this theory question in the exam prep which asks the following:

"Given an ARMA(2,2) model with white noise terms following a normal distribution (mean = 0, variance = $$\sigma^2$$), what's the distribution of $$y_t$$ in the model?"

Intuitively, I thought since the white noise terms are driven by a normal distribution, automatically the ARMA(2,2) model is normally distributed too. Can I give more specifics regarding the distribution with the given information?

AR coefficients: $$\phi_1 = -1/6$$ ; $$\phi_2 = 1/6$$

MA coefficients: $$\theta_1 = 1$$; $$\theta_2 = 1/4$$

I've proven stationarity, causality and invertibility. Due to parameter redundancy, the model could be simplified to an ARMA(1,1) model, but does that help to answer the question?

I'd be thankful for any hints.

• What does the "distribution of the model" even mean? Are you talking about the distribution of the response variable, i.e. $y_t$ in $\phi(L)y_t=\theta(L)\epsilon_t$? – hejseb Nov 8 '18 at 15:53
• @hejseb yes the distribution of y_t is asked. – RazorLazor Nov 8 '18 at 16:01
• @RichardHard Yeah, you're right. I've edited it now. – RazorLazor Nov 8 '18 at 22:45
• You may get the coefficients of the infinite moving average representation. See for example this answer. Then, it is easier to determine the mean and variance of the process. As you say, as the disturbances are Gaussian, the distribution will be Gaussian. – javlacalle Nov 9 '18 at 12:37