gen_arma: Generate Autoregressive Order $p$ - Moving Average Order $q$ (ARMA($p$,$q$)) Model

Description

Generate an ARMA($p$,$q$) process with supplied vector of Autoregressive Coefficients ($\phi$), Moving Average Coefficients ($\theta$), and $\sigma^2$.

Usage

gen_arma(N, ar, ma, sigma2 = 1.5, n_start = 0L)

Value

A vec that contains the generated observations.

Arguments

N: An integer for signal length.
ar: A vec that contains the AR coefficients.
ma: A vec that contains the MA coefficients.
sigma2: A double that contains process variance.
n_start: An unsigned int that indicates the amount of observations to be used for the burn in period.

Process Definition

The Autoregressive order $p$ and Moving Average order $q$ (ARMA($p$,$q$)) process with non-zero parameters $\phi_i \in (-1,+1)$ for the AR components, $\theta_j \in (-1,+1)$ for the MA components, and $\sigma^2 \in {\rm I\!R}^{+}$. This process is defined as:

$${X_t} = \sum\limits_{i = 1}^p {{\phi _i}{X_{t - i}}} + \sum\limits_{i = 1}^q {{\theta _i}{\varepsilon _{t - i}}} + {\varepsilon _t}$$ where $${\varepsilon_t}\mathop \sim \limits^{iid} N\left( {0,\sigma^2} \right)$$

Generation Algorithm

The innovations are generated from a normal distribution. The $\sigma^2$ parameter is indeed a variance parameter. This differs from R's use of the standard deviation, $\sigma$.

Details

For AR(1), MA(1), and ARMA(1,1) please use their functions if speed is important as this function is designed to generate generic ARMA processes.