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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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autoregressive processes
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
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).

moving average processes1
MovingAverage Processes
  • Its properties are

E(yt)=; Var(yt) = 0 = (1+ )2

Covariances

a white noise process
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 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 processes1
ARMA Processes
  • By combining the AR(p) and MA(q) models, we can obtain an ARMA(p,q) model:

where

and

or

with

slide9
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
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
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,...].

autoregressive conditionally heteroscedastic arch models cont d
Autoregressive Conditionally Heteroscedastic (ARCH) Models (cont’d)
  • This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors:

= 0 + 1

  • This is known as an ARCH(1) model.
autoregressive conditionally heteroscedastic arch models cont d1
Autoregressive Conditionally Heteroscedastic (ARCH) Models (cont’d)
  • 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
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 models1
Generalised ARCH (GARCH) Models
  • We could also write
  • Substituting into (1) for t-12 :
generalised arch garch models2
Generalised ARCH (GARCH) Models
  • Now substituting into (2) for t-22
  • An infinite number of successive substitutions would yield
generalised arch garch models3
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):
introduction
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
price volatility and price risk management in energy markets
Price volatility and price risk management in energy markets
  • 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
VaR quantification methods
  • Historical simulation Approach
  • Monte Carlo Simulation Method
  • Variance-Covariance methods
eviews
Eviews軟體運用及操作
  • Historical simulation ARMA forecasting approach
  • The variance-covariance approach for VaR estimation