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

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

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,...].

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’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 • 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

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 • 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

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