Generate an ARMA(1,1) sequence given \(\phi\), \(\theta\), and \(\sigma^2\).
gen_arma11(N, phi = 0.1, theta = 0.3, sigma2 = 1)A vec containing the MA(1) process.
An integer for signal length.
A double that contains autoregressive.
A double that contains moving average.
A double that contains process variance.
The Autoregressive order 1 and Moving Average order 1 (ARMA(1,1)) process with non-zero parameters \(\phi \in (-1,+1)\) for the AR component, \(\theta \in (-1,+1)\) for the MA component, and \(\sigma^2 \in {\rm I\!R}^{+}\). This process is defined as: $${X_t} = {\phi _1}{X_{t - 1}} + {\theta _1}{\varepsilon_{t - 1}} + {\varepsilon_t}$$, where $${\varepsilon_t}\mathop \sim \limits^{iid} N\left( {0,\sigma^2} \right)$$
The function first generates a vector of white noise using gen_wn and then obtains the
ARMA values under the above equation.
The \(X_0\) (first value of \(X_t\)) is discarded.
The function implements a way to generate the \(x_t\) values without calling the general ARMA function.