Forecasting Exchange Rate Volatility

Transcription

1 Forecasting Exchange Rate Volatility The Superior Performance of Conditional Combinations of Time Series and Option Implied Forecasts Guillermo Benavides Carlos Capistrán Banco de México November 2008 Benavides and Capistrán (Banxico) Conditional Combinations November / 13

2 Motivation 1.8E E E E 04 Realized Volatility Implied Volatility 3 w eeks GARCH(1,1) 1.0E E E E E E+00 02/01/ /03/ /05/ /07/ /09/ /11/ /01/ /04/ /06/ /08/ /10/ /12/ /02/ /04/ /06/ /08/ /10/ /01/ /03/ /05/ /07/ /09/ /11/2007 Benavides and Capistrán (Banxico) Conditional Combinations November / 13

3 Results This paper develops a method to combine forecasts that allows for weights to change according to current information, and applies it to combine volatility forecasts implied in options and from time series models. We show that the Giacomini-White (2006) two-step procedure to select forecasting methods can be extended to select forecast combinations. Our empirical results carried out using Peso Dollar intraday exchange rate data indicate that superior out-of-sample accuracy can be achieved by dynamically combining GARCH and IV exchange rate volatility forecasts. Benavides and Capistrán (Banxico) Conditional Combinations November / 13

4 Data The daily data for the spot exchange rate MXN USD consists of daily spot prices obtained from Banco de México s web page. These are daily averages of quotes o ered by major Mexican banks and other nancial intermediaries. The sample period is from January 2nd, 1998 to December 31st, 2007 for a total of 2,499 observations. The intraday data are realizations of the MXN USD exchange rate with a frequency of ve minutes. The transactions were carried out through the Reuters electronic platform. The sample size for intraday data is from January 2nd, 2004 to December 31st, The option implied data is calculated from daily over-the-counter (OTC) options for 1-month to maturity contracts of the MXN-USD exchange rate. The hard data on IV was downloaded from UBS. The sample period for the option data is from January 2nd, 2004 to December 31st, 2007, which consists of 993 daily observations. Benavides and Capistrán (Banxico) Conditional Combinations November / 13

5 Methodology Simple Combination Granger and Ramanathan (1984) propose combining forecasts through an OLS regression: y t+1 = α 0 + α 1 f t,1 + α 2 f t,2 + ε t+1. This procedure has two desirable characteristics (Deutch et al. (1994)): 1 It yields a combined forecast, which is usually better than either of the individual forecasts. 2 The method is easy to implement. However, this simple combination may not be exible enough if the perfomance of each forecast changes conditional on current and past information. Benavides and Capistrán (Banxico) Conditional Combinations November / 13

6 Methodology Combination using switching regime models Deutch et al. (1994) propose combining forecasts using changing weights: f t,c = I () (α 0 + α 1 f t,1 + α 2 f t,2 ) + 1 I () (α 3 + α 4 f t,1 + α 5 f t,2 ). They examine several choices for I, such as past forecast errors or relevant economic variables. They show that time-varying methods can result in a substantial reduction in MSE. However, they do not propose a way to select the variables used to determine the regimes, nor do they have a way to test if the time-varying combination is worth looking at. Benavides and Capistrán (Banxico) Conditional Combinations November / 13

7 Methodology Conditional predictive ability (Giacomini and White (2006)) Giacomini and White (2006) propose a two-step decision rule that uses current information to select the best forecast for the future date of interest. The rst step performs a conditional predictive ability test: I The null hypothesis is: H 0 : E h(y t+1 f t,1 ) 2 (y t+1 i f t,2 ) 2 j F t h i = E et+1,1 2 et+1,2 2 j F t = 0 t = 1, 2,... I I Which gives the orthogonality condition: i E hh t et+1,1 2 et+1,2 2 = 0 for all F t measurable functions h t. The test is performed using the (out-of-sample) regression: d t+1 = et+1,1 2 et+1,2 2 = δ0 h t + ɛ t H 0 : δ = 0. Benavides and Capistrán (Banxico) Conditional Combinations November / 13

8 Methodology Conditional predictive ability (Giacomini and White (2006)) In case of rejection (h t contains information to predict which forecast will perform better for the future date of interest), the second step entails the following decision rule: I use f 2 if bd t+1 > 0 and use f 1 if bd t+1 < 0. Giacomini and White call the resulting forecast hybrid. But this procedure renders a way to select the variables used to determine the regimes, and a way to test if the time-varying combination is worth looking at! Benavides and Capistrán (Banxico) Conditional Combinations November / 13

9 Methodology Our proposal We modify the second step in Giacomini and White (2006): I use α 0 + α 1 f t,1 + α 2 f t,2 if bd t+1 > 0 and use α 3 + α 4 f t,1 + α 5 f t,2 if bd t+1 < 0. We would expect f t,2 to receive relatively more weight when bd t+1 > 0 and f 1 to receive relatively more weight when bd t+1 < 0. We estimate the time-varying combination using OLS: y t+1 = I ( bd t+1 >0) (α 0 + α 1 f t,1 + α 2 f t,2 ) + 1 I ( bd t+1 >0) (α 3 + α 4 f t,1 + α 5 f t,2 ) + η t+1. Benavides and Capistrán (Banxico) Conditional Combinations November / 13

10 Application to Peso-Dollar exchange rate volatility Individual forecast evaluation Table 1. Individual Forecast Evaluation (January 2, December 31, 2007) Forecast Mean Error MSFE Ratio GARCHroll6-2.00E GARCHroll3 1.24E iv1week -7.05E iv3weeks -6.79E ivmb -5.37E Benavides and Capistrán (Banxico) Conditional Combinations November / 13

11 Application to Peso-Dollar exchange rate volatility Conditional predictive ability tests Table 4. Giacomini-White Tests (January 2, December 30, 2005) P-Values. Instruments are: Constant n Lag realized volatility, lag d GARCHroll6 GARCHroll3 iv1week iv3weeks ivmb GARCHroll GARCHroll iv1week iv3weeks ivmb Benavides and Capistrán (Banxico) Conditional Combinations November / 13

12 Application to Peso-Dollar exchange rate volatility Conditional Predictive Ability test Figure: d t = e 2 t,garch e 2 t,iv Benavides and Capistrán (Banxico) Conditional Combinations November / 13

13 Application to Peso-Dollar exchange rate volatility Evaluation of time-varying combinations Table 5. Evaluation of Conditional Combinations (January 2, December 31, 2007) Forecasts Combination Mean Error MSFE Ratio DMW DMW GARCHroll6 Hybrid (GW) -4.18E and Conditional -2.31E iv1w GR1-2.02E X GARCHroll3 Hybrid (GW) -9.64E and Conditional -1.63E iv3w GR1-1.05E X GARCHroll6-2.29E X Benavides and Capistrán (Banxico) Conditional Combinations November / 13