ARX-GARCH模式之預測建構

Constructing Forecasts for the Model ARX-GARCH

曾能芳(Neng-Fang Tseng)

《中國統計學報》 43卷1期 (2005/03) Pp. 33-54

https://doi.org/10.29973/JCSA.200503.0003

摘要

在財務領域中，模型ARX-GARCH常被廣泛運用，此模型不僅具備條件變異數隨時間而變動的特性，且可呈現前後期資料的跨時相關性。現今有關ARX-GARCH的研究中，均將外生變數x(下標 t)視爲已知常數，若是x(下標 t)與內生變數y(下標 t)具有同期效應(contemporaneous relationship)，爲已知常數的假設則不見得合理。本文以ARX(2)-GARCH(1,1)作爲研究對象，假設x(下標 t)與y(下標 t)具有同期效應，且將x(下標 t)視作隨機變數，藉助差分方程的技巧，重新推導預測值和預測誤差變異數。如此一來，無需事先供應x(下標 t)的未來資訊，便可進行預測，且預測誤並變異數也不再低估，更能符合實際狀況。本文針對美國ISM製造業指數對台灣景氣綜合判斷分數之關係，進行實例探討，並建立台灣景氣綜合判斷分數的預測模型；我們發現，美國ISM製造業指數對台灣景氣綜合判斷分數的影響爲正向關係，且景氣綜合判斷分數具有波動持續性的效果。我們也預測未來4期的景氣綜合判斷分數，每月約降低0.5分，顯現景氣逐漸降溫。

關鍵字

ARX-GARCH ； 差分方程 ； 預測值 ； 預測誤差變異數 ； 台灣景氣綜合判斷分數

並列摘要

In financial time series, the ARX—GARCH model have been widely applied. This model not only has the feature that the conditional variance changes over time, but also can present the autocorrelation between non contemporaneous data from some series. In the research concerning ARX-GARCH model, all assume the exogenous variable x(subscript t) to be known constant. If the exogenous variable x(subscript t) and the endogenous variable y(subscript t) have contemporaneous relationship, the assumption of the constant x(subscript t) is then not necessarily reasonable. Our concern is to consider ARX(2)-GARCH(1, 1) model with random variable x(subscript t). To derive the forecast and the forecast error variance of this model, we will carry out the technique of stochastic difference equations. Thus, under the condition that the future information of x(subscript t) can't be supplied in advance, we can still carry out forecasting. And the forecast error variance is no longer underestimated. This text aims at the relation between U.S. ISM Manufacturing Index and the Taiwan Business Cycle Index, and building up the forecasting model of Taiwan Business Cycle Index. We find that the correlation of U.S. ISM Manufacturing Index to Taiwan Business Cycle Index is just positive, and there is persistency of volatility clustering at Taiwan Business Cycle Index. We also predict the future 4 periods of Taiwan Business Cycle Index. The predicted values reduce approximately 0.5 unit monthly. The economic prosperity will go down gradually.

並列關鍵字

ARX-GARCH ； difference equation ； forecast ； forecast error variance ； Index of Taiwan Business Cycle

參考文獻

Berument, H.,Kiymaz, H.(2001).The day of the week effect on stock market volatility.Journal of Economics and Finance.25,181-193.

Bollerslev, T.(1986).Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics.31,307-327.

Bollerslev, T.,Chou, R. Y.,Kroner, K(1992).ARCH modeling in finance: a review of the theory and empirical evidence.Journal of Econometrics.52,5-59.

Enders, W.(1995).Applied Econometric Time Series.New York:John Wiley & Sons.

Engle, R. F.(1982).Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation.Econometrica.50,987-1008.