Help us understand the problem. What is going on with this article?

『経済・ファイナンスデータの計量時系列分析』章末問題を解く-第1章時系列分析の基礎概念-

More than 1 year has passed since last update.

経済・ファイナンスデータの軽量時系列分析
の章末問題を解いていきます。

必要に応じ『統計学のための数学入門30講を参照しています。

Rの実施は『経済・ファイナンスデータの計量時系列分析』章末問題をRで解く-第1章時系列分析の基礎概念-にあります。

1.1

E(x)=\sum{kPr(x=k)} ... 30講p19\\

つまり和は分解できるので

\begin{align}
\gamma_k &= E(y_t y_{t-k}) - \mu{E(y_{t-k}) + E(y_t)} + E(\mu^2) \\ 
&= E(y_t)E(y_{t-k}) - \mu{E(y_{t-k}) + E(y_t)} + E(\mu^2) \\ 
&= \mu\mu - \mu(2\mu) + E(\mu^2) \\ 
&= -\mu^2 + E(\mu^2) \\ 
\gamma_{-k} &= E(y_t y_{t+k}) - \mu{E(y_{t+k} + E(y_t)} + E(\mu^2) \\ 
&= E(y_t)E(y_{t+k}) - \mu{E(y_{t+k} + E(y_t)} + E(\mu^2) \\ 
&= \mu^2 - 2\mu^2 +E(\mu^2) \\ 
&= -\mu^2 + E(\mu^2) \\ 
ゆえに \gamma_k &= \gamma_{-k}
\end{align}

1.2

y_t = \mu + \epsilon_t, \epsilon \sim W.N.(\sigma^2)

定義1.4より、

\begin{align}
E(\epsilon_t) &= 0 \\ 
\gamma_k &= E(\epsilon_t \epsilon_{t-k}) = 
   \begin{cases} \sigma^2 & k = 0 \\ 
                 0 & k \neq 0
   \end{cases} \\
よって \\
E(y_t) &= E(\mu + \epsilon_t) \\ 
&= E(\mu) + E(\epsilon_t) \\ &= \mu \\ Cov(y_t, y_{t-k}) &= E[(y_t - \mu) (y_{t-k} - \mu)] \\ &= E[(\mu + \epsilon_t - \mu) (\mu + \epsilon_{t-k} - \mu)] \\ &= E[\epsilon_t \epsilon_{t-k}] \\ &= \gamma_k \\ 
\end{align}

ゆえに(1.8)のモデルは弱定常過程

1.4

わからず、、、


次章『経済・ファイナンスデータの計量時系列分析』章末問題を解く-第2章ARMA過程-

Why not register and get more from Qiita?
  1. We will deliver articles that match you
    By following users and tags, you can catch up information on technical fields that you are interested in as a whole
  2. you can read useful information later efficiently
    By "stocking" the articles you like, you can search right away
Comments
No comments
Sign up for free and join this conversation.
If you already have a Qiita account
Why do not you register as a user and use Qiita more conveniently?
You need to log in to use this function. Qiita can be used more conveniently after logging in.
You seem to be reading articles frequently this month. Qiita can be used more conveniently after logging in.
  1. We will deliver articles that match you
    By following users and tags, you can catch up information on technical fields that you are interested in as a whole
  2. you can read useful information later efficiently
    By "stocking" the articles you like, you can search right away
ユーザーは見つかりませんでした