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) 自我迴歸模型 AR(p) An autoregressive model of order p, an AR(p) can be expressed as An autoregressive model of order p, an AR(p) can be expressed as Or using the lag operator notation: Or using the lag operator notation:

Autoregressive Processes or or where where

移動平均 MA(q). 移動平均 MA(q). Let u t (t=1,2,3,...) be a sequence of independently and identically distributed (iid) random variables with E(u t )=0 and Var(u t )=, then Let u t (t=1,2,3,...) be a sequence of independently and identically distributed (iid) random variables with E(u t )=0 and Var(u t )=, then y t =  + u t +  1 u t-1 +  2 u t  q u t-q is a q th order moving average model MA(q). Moving Average Processes

Its properties are Its properties are E(y t )=  ; Var(y t ) =  0 = (1+ )  2 Covariances Moving Average Processes

A white noise process is one with (virtually) no discernible structure. A definition of a white noise process is 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 (a) 期望值為 0 (b) 變異數為固定常數 (b) 變異數為固定常數 (c) 自我共變數等於 0 (c) 自我共變數等於 0 A White Noise Process

ARMA Processes ARMA 模型是一種時間序列的『資料產生過 程』 (data generating process, DGP) 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』 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 是由兩種 DGP, 及 AR 和 MA 結合而成 ARMA= AR+MA ARMA= AR+MA

By combining the AR(p) and MA(q) models, we can obtain an ARMA(p,q) model: By combining the AR(p) and MA(q) models, we can obtain an ARMA(p,q) model:whereandorwith ARMA Processes

ARMA 模型估計步驟 ACF 和 PACF 初步判斷 ARMA(p,q) 的落後期數。 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。 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 。 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。 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 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」 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 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 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. So use a model which does not assume that the variance is constant. Recall the definition of the variance of u t : Recall the definition of the variance of u t : = Var(u t  u t-1, u t-2,...) = E[(u t -E(u t )) 2  u t-1, u t-2,...] = Var(u t  u t-1, u t-2,...) = E[(u t -E(u t )) 2  u t-1, u t-2,...] We usually assume that E(u t ) = 0 so = Var(u t  u t-1, u t-2,...) = E[u t 2  u t-1, u t-2,...].

Autoregressive Conditionally Heteroscedastic (ARCH) Models (cont’d) This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors: This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors: =  0 +  1 This is known as an ARCH(1) model. This is known as an ARCH(1) model.

Autoregressive Conditionally Heteroscedastic (ARCH) Models (cont’d) The full model would be The full model would be y t =  1 +  2 x 2t  k x kt + u t, u t  N(0, ) y t =  1 +  2 x 2t  k x kt + u t, u t  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: We can easily extend this to the general case where the error variance depends on q lags of squared errors: =  0 +  1 +   q =  0 +  1 +   q

Generalised ARCH (GARCH) Models Due to Bollerslev (1986). Allow the conditional variance to be dependent upon previous own lags Due to Bollerslev (1986). Allow the conditional variance to be dependent upon previous own lags The variance equation is now The variance equation is now (1) (1) This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the variance equation. 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 We could also write Substituting into (1) for  t-1 2 : Substituting into (1) for  t-1 2 :

Generalised ARCH (GARCH) Models Now substituting into (2) for  t-2 2 Now substituting into (2) for  t-2 2 An infinite number of successive substitutions would yield 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. 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): 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 The importance of oil price risk in managing price risk in energy markets 石油價格風險管理 石油價格風險管理 The application of VaR in quantifying oil price risk The application of VaR in quantifying oil price risk 風險值衡量 風險值衡量

Price volatility and price risk management in energy markets 風險管理策略 風險管理策略 Avoid big losses due to price fluctuations or changing energy consumption patterns Avoid big losses due to price fluctuations or changing energy consumption patterns Reduce volatility in earnings while maximizing return on investment Reduce volatility in earnings while maximizing return on investment Meet regulatory requirements that limit exposure to risk Meet regulatory requirements that limit exposure to risk

VaR quantification methods Historical simulation Approach Historical simulation Approach Monte Carlo Simulation Method Monte Carlo Simulation Method Variance-Covariance methods Variance-Covariance methods

Eviews 軟體運用及操作 Historical simulation ARMA forecasting approach Historical simulation ARMA forecasting approach The variance-covariance approach for VaR estimation The variance-covariance approach for VaR estimation