The confidence interval for the PACF: the confidence interval for the coefficient, The null hypothesis is iid and thus the line is constant.

MA(q) model (Moving average model):

Recall AR: $x_t = \phi_0 + \phi_1 x_{t-1} + \dots +\phi_q x_{t-q} + \epsilon_t$

$\epsilon_t$ iid mean-zero and $\text{var}(\epsilon_t)=\sigma^2$.

MA(1) is a one-parameter family of $AR(\infty)$ model with $\phi_i = - \theta_1^i$ for $i\geq 1$.

\[x_t = \phi_0 - \theta_1 x_{t-1} - \theta_1^2 x_{t-2} -\dots\]
with $ \theta_1 <1$.

This is equivalent to

\[x_t + \theta_1x_{t-1} + \theta_1^2 x_{t-2} + \dots = \phi_0 + \epsilon_t\]

For $x_{t-1}$, we have

\[x_{t-1} + \theta_1 x_{t-2} + \theta_1^2x_{t-3} + \dots = \phi_1 + \epsilon_{t-1}\]

combine them, we find

\[\begin{aligned} &x_t + \theta_1(\phi_1+\epsilon_{t-1}) = \phi_0 + \epsilon_t\\ &\Rightarrow x_t = (1-\theta_1)\phi_0 + \epsilon_t - \theta_1\epsilon_{t-1} \end{aligned}\]

Generally speaking, $MA(q)$ model becomes

\[\begin{aligned} x_t &= c_0 + \epsilon_t - \theta_1\epsilon_{t-1} - \dots -\theta_q \epsilon_{t-q} \\ &=c_0 + (1-\theta_1B -\theta_2 B^2 -\dots -\theta_qB^q) \epsilon_t \end{aligned}\]

if $q<\infty$, the $x_t$ has finite memory. This is stationary and thus, $E[x_t] = c_0 = (1-\theta_1)\phi_0$. By indpendence of $\epsilon_t$

\[\text{Var}(x_t) = \sigma^2(1+\theta_1^2 + \dots+ \theta_q^2)\]

Setting $c_0 = 0$, consider $MA(1)$,

\[x_t = \epsilon_t-\theta_1\epsilon_{t-1}\]


\[\gamma_l = \mathbb{E}(x_{t-l}x_t) = \mathbb(X_{t-l}\epsilon_t) - \theta_1 \mathbb{E}(x_{t-l} \epsilon_{t-1})\]

The last term is $0$ if $l\geq 2$. when $l=1$, we find

\[\mathbb{E}(x_{t-1}\epsilon_{t-1}) = \mathbb{E}(\epsilon_{t-1}^2-\theta_2\epsilon_{t-2}\epsilon_{t-1}) = \sigma^2\]

Thus, the ACF becomes

\[\rho_0 = 1,\quad \rho_1 = \frac{-\theta_1\sigma^2}{(1+\theta_1^2)\sigma^2} = -\frac{\theta_1}{1+\theta_1^2}\]

Thus, ACF of MA(1) cuts off at lag $1$.

For MA(2) models:

\[\rho_0 = 1 \quad \rho_1 = \frac{-\theta_1 + \theta_1\theta_2}{1+\theta_1^2+\theta_2^2},\quad \rho_2 = -\frac{\theta_2}{1+\theta_1^2+\theta_2^2}\]

Estimation : use MLE.

for $MA(1)$, $x_t = c_0 +\epsilon_t -\theta_1\epsilon_{t-1}$. we have data $x_1,\dots, x_T$ and parameter to estimate $c_0, \theta, \sigma^2$.

write down the joint likelihood function

\[\begin{aligned} &l(c_0,\theta,\sigma^2|x_1,\dots x_T) \\ &=\log \prod_{i=1}^T \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{\epsilon_i^2}{2\sigma^2}\right)\\ &=-\frac{T}{2} \log(2\pi\sigma^2) - \frac{1}{2\sigma^2}\sum_{i=1^T}\left[ \sum_{j=1}^i\theta_1^j(x_{t-j}-c_0)\right]^2 \end{aligned}\]


\[\begin{aligned} x_1 = c_0 + \epsilon_1 - \theta_1\epsilon_0\\ x_2 = c_0 + \epsilon_2 - \theta_1\epsilon_1 \end{aligned}\]

We assume $\epsilon_t=0$ for $t\leq0$ and this is conditional log likelihood.

This shows $\epsilon_2 = x_2 -c_0 + \theta_1\epsilon_1 = (x_2-c_0) + \theta_1(x_1-c_0)$ . recursively, we find

\[\begin{aligned} \epsilon_t &= (x_t-c_0) + \theta_1\epsilon_{t-1}\\ &=(x_t-c_0) + \theta_1(x_{t-1}-c_0) + \theta_1^2(x_{t-2}-c_0)+\dots +\theta_1^{t-1} (x_1-c_0) \end{aligned}\]

The exact likelihood method whould treat $\epsilon_{t}$ for $t\leq 0$ as unknow. Using asymptotic behaviour instead of the second derivative of the loglikelihood for the confidence interval.

ARMA Model

ARMA(1,1) model: $x_t - \phi_1x_{t-1} = \phi_0 + \epsilon_t - \theta_1\epsilon_{t-1}$. Here $\epsilon_t$ iid white noise.

need $\phi_1\neq \theta_1$. If they are the same, then we have $\epsilon_t -\epsilon_t = \phi_0 + \phi_1(x_{t-1}-\epsilon_{t-1})$. This means we can solve $x_t-\epsilon_t$ in terms of $x_0$ and $\epsilon_0$. Also, $\epsilon_t$ are independent. This shows $x_t$ does not have time serieal structure.


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