In coatless/ITS: A Tour of Time Series Analysis with R

PACF

Show that $\phi_{hh} = 0$ if $h > p$ for an AR(p).

PACF: $$\begin{aligned} \phi_{11} &= \corr(X_{t+1},X_{t}) = \rho(1) \ \phi_{hh} &= \corr(X_{t+h} - \hat{X}{t+h}, X{t} - \hat{X}_{t}), h \ge 2 \end{aligned}$$

where $\hat{X}t = \widetilde{\beta}_1 X{t+h-1} + \cdots + \widetilde{\beta}{p-1} X{t+1}$

therefore we have: $\widetilde{\beta} = \mathop {\arg \min }\limits_\beta \underbrace {E\left[ {{{\left( {{X_{t + h}} - \sum\limits_{j = 1}^{h - 1} {{\beta j}{X{t + j}}} } \right)}^2}} \right]}_{MS{E_L}\left( \beta \right)}$

Consider a more general case:

$ \widetilde{\beta}^{*} = \mathop {\arg \min }\limits_\beta \underbrace {E\left[ {{{\left( {{X_{t + h}} - m\left( {{X_{t + 1}}, \cdots ,{X_{t + h}}} \right)} \right)}^2}} \right]}_{MSE\left( \beta \right)}$

It is clear that:

[MS{E_L}\left( {\tilde \beta } \right) \ge MSE\left( {{{\tilde \beta }^*}} \right)]

Let's now consider

[{\left( {{X_t} - m} \right)^2} = {\left[ {\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right) + \left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)} \right]^2}]

Where we have: $m = m\left( {{X_{t + 1}}, \cdots ,{X_{t + h}}} \right)$, ${\Omega t} = \left( {{X{t + 1}}, \cdots ,{X_{t + h}}} \right)$.

Therefore,

[{\left( {{X_t} - m} \right)^2} = {\left( {{X_t} - E\left[ {{X_t}|{\Omega _T}} \right]} \right)^2} + {\left( {E\left[ {{X_t}|{\Omega _T}} \right] - m} \right)^2} + 2\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right)\left( {E\left[ {{X_t}|{\Omega _T}} \right] - m} \right)]

Focusing on only the last term, we have that:

[\underbrace {\left( {{X_t} - E\left[ {{X_t}|{\Omega t}} \right]} \right)}{ = {\varepsilon _t}}\left( {E\left[ {{X_t}|{\Omega _T}} \right] - m} \right)]

[E\left[ {{\varepsilon _t}|{\Omega _t}} \right] = E\left[ {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]|{\Omega _t}} \right] = 0]

So, by the decomposition property, we have that:

[E\left[ {{\varepsilon t}\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)} \right] = E\left[ {E\left[ {{\varepsilon _t}\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)|{\Omega _t}} \right]} \right] = E\left[ {\underbrace {E\left[ {{\varepsilon _t}|{\Omega _t}} \right]}{ = 0}\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)} \right] = 0]

By the previous discussion, we have that

[{\left( {{X_t} - m} \right)^2} = {\left( {{X_t} - E\left[ {{X_t}|{\Omega _t}} \right]} \right)^2} + {\left( {E\left[ {{X_t}|{\Omega _t}} \right] - m} \right)^2}]

is therefore minimized for $m = E\left[ {{X_t}|{\Omega _t}} \right]$.

Note that $X < Y \Rightarrow E[X] < E[Y]$. In the case of an AR(p): $E\left[ {{X_t}|{\Omega t}} \right] = \sum\limits{j = 1}^{h - 1} {{\beta j}{X{t + j}}}$

As a result, we have that

[MS{E_L}\left( {\tilde \beta } \right) = MSE\left( {{{\tilde \beta }^*}} \right)]

Therefore, for $h > p$, we have:

[\begin{aligned} {\phi {hh}} &= corr\left( {{X{t + h}} - E\left[ {{X_{t + h}}|{X_{t + 1}}, \cdots ,{X_{t + h - 1}}} \right],{X_t} - E\left[ {{X_t}|{X_{t - h}}, \cdots ,{X_{t - 1}}} \right]} \right) \hfill \ &= corr\left( \begin{gathered} {X_{t + h}} - E\left[ {{\phi 1}{X{t + h - 1}} + \cdots {\phi p}{X{t + h - p}} + {W_{t + h}}|{X_{t + 1}}, \cdots ,{X_{t + h - 1}}} \right], \hfill \ {X_t} - E\left[ {{X_t}|{X_{t - h}}, \cdots ,{X_{t - 1}}} \right] \hfill \ \end{gathered} \right) \hfill \ &= corr\left( {{W_{t + h}},{W_t}} \right) = 0 \hfill \ \end{aligned} ]

coatless/ITS documentation built on May 13, 2019, 8:45 p.m.