LjungBox algorithm for pure MA models

Short description

We consider the following stationary moving average model of order $q$

$$y_t = \Theta \left(B \right) \epsilon_t$$

First, considering the initial innovations $\epsilon^*_t$ defined for $-q \leq t \lt 0$, we can write

$$\begin{pmatrix} y^*_t \\ y_t \end{pmatrix} = A \begin{pmatrix} \epsilon^*_t \\ \epsilon_t \end{pmatrix}$$

where $A_{(n+q)\times (n+q)}$ is defined by

$$\begin{pmatrix} 1 & 0 & \cdots & \cdots & \cdots & \cdots & 0 \\ \theta_1 & 1 & 0 & \cdots & \cdots & \cdots & \vdots \\ \theta_2 & \theta_1 & 1 & 0 & \cdots & \cdots & \vdots\\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \cdots & \theta_q & \cdots & \theta_1 & 1 & 0 & \vdots\\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \theta_q & \cdots & \theta_1 & 1 \end{pmatrix}$$

Using the Pi-Weights of the model, we obtain $G=A^{-1}$

$$G = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ \pi_1 & 1 & \cdots & 0 \\ \pi_2 & \pi_1 & \cdots & \vdots\\ \vdots & \vdots & \vdots & \vdots \\ \pi_{n+q-1} & \pi_{n+q-2} & \cdots & \pi_{n} \end{pmatrix}$$

and we have that

$$\begin{pmatrix} \epsilon^* \\ \epsilon \end{pmatrix} = G \begin{pmatrix} y^* \\ y \end{pmatrix} = G_1 y^* + G_2 y$$

Writing $(\epsilon^, \epsilon)$as$\bar{\epsilon}$and$(y^, y)$as$\bar{y}$ , we have that

$$p(\bar{\epsilon})=p(\bar{y})=p(y)p(y^*|y)$$ $$p(\bar{\epsilon})=(2\pi\sigma^2)^{-(n+q)/2}e^{-(\bar{\epsilon}\bar{\epsilon}')/2\sigma^2}=(2\pi\sigma^2)^{-(n+q)/2}e^{-\bar{y}'G'G\bar{y})/2\sigma^2}$$

Considering the equation $G_2 y = - G_1 y^* +\bar{\epsilon}$, by standard regression theory, we get:

$$\begin{align} \hat{y}^*=E(y^*|y) &= - (G_1'G_1)^{-1}G_1'G_2 y \\ var(y^*|y) &= (G_1'G_1)^{-1}\sigma^2 \end{align}$$

We can now split $p(\bar{\epsilon})$ into

$$\begin{align} p(y^*|y) &= (2\pi\sigma^2)^{-q/2}|G_1'G_1|^{-1/2}e^{-(y^*-\hat{y}^*)'G_1'G_1(y^*-\hat{y}^*)/2\sigma^2} \\ p(y) &= (2\pi\sigma^2)^{-n/2}|G_1'G_1|^{1/2}e^{-(G_1 y^* + G_2 y)'(G_1 y^* + G_2 y)/2\sigma^2} \end{align}$$

So, the different steps of the algoritm for computing the likelihood are:

• Compute $V=G_1’G_1$
• Compute L, the Cholesky factor of $V$

• Solve $V=G’$

• Compute by Cholesky $G’G \hat z_* = G’a \quad and \quad \vert G’G \vert$

• Get by recursion the exact likelihood residuals $\Theta\left(B\right)\begin{pmatrix}\hat a_* \ \hat a_t \end{pmatrix} = \begin{pmatrix}z_* \ z_t \end{pmatrix}$

The processing defines the linear transformation $T z_t = \begin{pmatrix}\hat a_* \ \hat a_t \end{pmatrix}$ and the searched determinant.