考量內分位數變幅的CARR 模型之實證分析

Abstract

CARR(Conditional Autoregressive Range)模型在固定的時間區間對於資產價量的變幅(range)提供一個動態模式。然而採用變幅做為波動性之代理變數仍隱含一些問題，包括變幅對於離群值具高度敏感性，以及採用變幅做為波動性代理變數之CARR模型，所估計出的波動性可能高估實現波動性的變異程度。因此本研究基於適度規避離群值的衝擊與適當刻畫波動性的動態行程此兩目的，針對變幅採取穩健性的衡量方式，意即採用內分位數變幅(interquantile range)做為衡量波動性的代理變數，並與CARR模型做一適切搭配，試圖獲取較佳的樣本內與樣本外之預測結果。本研究主要研究對象為期貨之時間序列，包括NY Light Crude(CL)、Dow Futures(DJ)、Nasdaq 100 Futures(ND)、NY Natural Gas(NG)以及S&P 500 Futures(SP)，而在預測能力的衡量指標上，樣本內採用Mincer-Zarnowitz(MZ)迴歸式，樣本外採用二次損失函數、比例損失函數以及t檢定做為主要衡量指標。實證結果顯示，在樣本資料為實現波動性之波動程度較大且包含數值較大之離群值的資料型態下，如NY Light Crude及NY Natural Gas，內分位數變幅之CARR模型將優於標準變幅之CARR模型。在樣本資料為實現波動性之波動程度較小且離群值較小的資料型態下，如Dow Futures、Nasdaq 100 Futures及S&P 500 Futures，內分位數低估了實現波動性之波動程度，因此標準變幅之CARR模型在預測能力的表現上，優於多數的內分位數變幅之CARR模型。另外，除了樣本Nasdaq 100 Futures外，另外四種期貨資料之樣本內與樣本外預測結果具一致性。

The conditional autoregressive range (CARR) model was proposed a dynamic model for the high/low range of asset prices within fixed time intervals. However, adopting range as the proxy of volatility has some problems. Firstly, range is highly sensitive to outliers. In addition, the CARR model with range will probably overestimate the variance of realized volatility. Based on the purpose of avoiding the effects of outliers and that of properly characterizing the dynamic structure of volatility, we utilize the robust measure of range. In other words, we adopt interquantile range as the proxy of volatility and compare the forecasting performance of the CARR model with either interquantile range or standard range. The forecasting performance measures include Mincer-Zarnowitz (MZ) regression in in-sample forecasts, quadratic loss functions, proportional loss functions and t test in out-of-sample forecasts. The samples include NY Light Crude (CL)、Dow Futures (DJ)、Nasdaq 100 Futures (ND)、NY Natural Gas (NG) and S&P 500 Futures(SP). The empirical results reveal that in the sample which has more volatile realized volatility and many extreme outliers, like NY Light Crude (CL) and NY Natural Gas (NG), the CARR model with interquantile range outperforms the CARR model with standard range, in terms of both in-sample forecasts and out-of-sample forecasts. On the contrary, in the sample which has less volatile realized volatility and small outliers, like Dow Futures (DJ), Nasdaq 100 Futures (ND) and S&P 500 Futures (SP), the results are opposite.