simts (version 0.1.1)

gen_arma11: Generate an ARMA(1,1) sequence

Description

Generate an ARMA(1,1) sequence given $\varphi$, $\theta$, and ${\sigma }^2$.

Usage

gen_arma11(N, phi = 0.1, theta = 0.3, sigma2 = 1)

Arguments

N

An integer for signal length.

phi

A double that contains autoregressive.

theta

A double that contains moving average.

sigma2

A double that contains process variance.

Value

A vec containing the MA(1) process.

Process Definition

The Autoregressive order 1 and Moving Average order 1 (ARMA(1,1)) process with non-zero parameters $\varphi \in (-1,+1)$ for the AR component, $\theta \in (-1,+1)$ for the MA component, and ${\sigma }^2\in {\mathrm{I}\mathrm{R}}^{+}$. This process is defined as: $$X_t={\varphi }_1{X}_{t-1}+{\theta }_1{\epsilon }_{t-1}+{\epsilon }_t$$, where $${\epsilon }_t\stackrel{iid}{\sim }N\left(0,{\sigma }^2\right)$$

Generation Algorithm

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.

Details

The function implements a way to generate the $x_t$ values without calling the general ARMA function.