ARMA Forecasting and Variance – Covariance based on GARCH 介紹與應用

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ARMA Forecasting and Variance – Covariance based on GARCH 介紹與應用. 主講人 : 柯娟娟. Autoregressive Processes. 自我迴歸模型 AR( p ) An autoregressive model of order p , an AR( p ) can be expressed as Or using the lag operator notation:. Autoregressive Processes. or or where .

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### ARMA Forecasting and Variance – Covariance based on GARCH 介紹與應用

Autoregressive Processes
• 自我迴歸模型 AR(p)
• An autoregressive model of order p, an AR(p) can be expressed as
• Or using the lag operator notation:
Moving Average Processes
• 移動平均MA(q).
• Let ut (t=1,2,3,...) be a sequence of independently and identically distributed (iid) random variables with E(ut)=0 and Var(ut)= , then

yt =  + ut + 1ut-1+ 2ut-2 + ... + qut-q

is a qth order moving average model MA(q).

MovingAverage Processes
• Its properties are

E(yt)=; Var(yt) = 0 = (1+ )2

Covariances

A White Noise Process
• A white noise process is one with (virtually) no discernible structure. A definition of a white noise process is
• (a)期望值為0
• (b)變異數為固定常數
• (c)自我共變數等於0
ARMA Processes
• ARMA模型是一種時間序列的『資料產生過程』(data generating process, DGP)
• 現在的變數和過去的變數的函數或統計『關係』
• ARMA是由兩種DGP,及AR和MA結合而成
• ARMA= AR+MA
ARMA Processes
• By combining the AR(p) and MA(q) models, we can obtain an ARMA(p,q) model:

where

and

or

with

ARMA 模型估計步驟
• ACF和PACF初步判斷ARMA(p,q)的落後期數。
• OLS做初步估計，並檢查估計系數是否顯著。
• LM統計量或Q統計量檢定殘差中是否仍有未納入的ARMA型態。若有，則回到步驟2。
• JB統計量檢查殘差是否符合常態性。
• 若有好幾種p,q的組合都符合步驟3﹑4，則用AIC 或SBC等準則
Variance – Covariance based on GARCH
• GARCH模型是「ㄧ般化的ARCH模型」
• ARCH是將估計迴歸AR模型的概念用在估計條件變異數
• GARCH是同時將AR和MA的觀念用在估計條件變異數
Autoregressive Conditionally Heteroscedastic (ARCH) Models
• So use a model which does not assume that the variance is constant.
• Recall the definition of the variance of ut:

= Var(ut ut-1, ut-2,...) = E[(ut-E(ut))2 ut-1, ut-2,...]

We usually assume that E(ut) = 0

so = Var(ut  ut-1, ut-2,...) = E[ut2 ut-1, ut-2,...].

• This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors:

= 0 + 1

• This is known as an ARCH(1) model.
• The full model would be

yt = 1 + 2x2t + ... + kxkt + ut , ut N(0, )

where = 0 + 1

• We can easily extend this to the general case where the error variance depends on q lags of squared errors:

= 0 + 1+2+...+q

Generalised ARCH (GARCH) Models
• Due to Bollerslev (1986). Allow the conditional variance to be dependent upon previous own lags
• The variance equation is now
• (1)
• This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the variance equation.
Generalised ARCH (GARCH) Models
• We could also write
• Substituting into (1) for t-12 :
Generalised ARCH (GARCH) Models
• Now substituting into (2) for t-22
• An infinite number of successive substitutions would yield
Generalised ARCH (GARCH) Models
• So the GARCH(1,1) model can be written as an infinite order ARCH model.
• We can again extend the GARCH(1,1) model to a GARCH(p,q):

### 論文導讀Energy risk management and value at risk modeling

Introduction
• The importance of oil price risk in managing price risk in energy markets
• 石油價格風險管理
• The application of VaR in quantifying oil price risk
• 風險值衡量
• 風險管理策略
• Avoid big losses due to price fluctuations or changing energy consumption patterns
• Reduce volatility in earnings while maximizing return on investment
• Meet regulatory requirements that limit exposure to risk
VaR quantification methods
• Historical simulation Approach
• Monte Carlo Simulation Method
• Variance-Covariance methods
Eviews軟體運用及操作
• Historical simulation ARMA forecasting approach
• The variance-covariance approach for VaR estimation