A Comparison of Seasonal Adjustment Methods When Forecasting Intraday Volatility

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(1)The Journal of Financial Research • Vol. XXV, No. 2 • Pages 283–299 • Summer 2002. A COMPARISON OF SEASONAL ADJUSTMENT METHODS WHEN FORECASTING INTRADAY VOLATILITY. Martin Martens Erasmus University Rotterdam. Yuan-Chen Chang National Chenchi University. Stephen J. Taylor Lancaster University. Abstract In this article we compare volatility forecasts over a thirty-minute horizon for the spot exchange rates of the Deutsche mark and the Japanese yen against the U.S. dollar. Explicitly modeling the intraday seasonal pattern improves the out-of-sample forecasting performance. We find that a seasonal estimated from the log of squared returns improves with the use of simple squared returns, and that the flexible Fourier form (FFF) is an efficient way of determining the seasonal. The two-step approach that first estimates the seasonal using the FFF and then the parameters of the generalized autoregressive conditional heteroskedasticity (GARCH) model for the deseasonalized returns performs only marginally worse than the computationally expensive periodic GARCH model that includes the FFF. JEL Classifications: C53, G15. I. Introduction The intraday seasonal patterns in the volatility of foreign exchange and stock markets have important implications for modeling the volatility of highfrequency returns. The patterns are so distinctive there is a strong case for taking them into account before attempting to model the dynamics of volatility. Standard time-series models of volatility have proven inadequate when applied to high-frequency returns data. Andersen and Bollerslev (1997, 1998). The authors would like to thank Torben Andersen, Tim Bollerslev, Eric Ghysels, Bruce Lehmann, editor William T. Moore, Diethelm Wurtz, an anonymous referee, and participants of the HFDF-II conference in Zurich, Switzerland, for useful comments. We are also grateful to Olsen & Associates for providing the exchange rate data. All remaining errors are our responsibility.. 283.

(2) 284. The Journal of Financial Research. argue that the reason for this is simply the systematic pattern in average volatility across the trading day. They also present a practical method for the estimation of the intraday seasonal pattern. Until now, however, there are no studies known to us that investigate the effect of the seasonal pattern on forecasting intraday volatility. Although Andersen and Bollerslev (1997) argue that persistence estimates for generalized autoregressive conditional heteroskedasticity (GARCH) models estimated at different frequencies are in conflict with aggregation results for autoregressive conditional heteroskedasticity (ARCH) effects (Drost and Nijman 1993), these models could still be used to forecast the volatility in the next period. Our objective is to test whether filtering out the seasonal pattern—that is, deseasonalizing—improves the out-of-sample forecasting performance of volatility models. We also want to determine the best method for removing seasonality. We compare the forecasting performance of a GARCH model for the original intraday returns with the forecasting performance of a GARCH model for deseasonalized intraday returns. The seasonal is estimated by either simply averaging the volatility over a number of trading days for each intraday period or by using the flexible Fourier form (FFF) advocated by Andersen and Bollerslev (1997, 1998). These approaches are two-step procedures. The seasonal is first estimated and extracted, and a GARCH model is then estimated. The natural question is whether the onestep periodic GARCH (P-GARCH) model of Bollerslev and Ghysels (1996) can improve this. We evaluate volatility forecasts over a thirty-minute horizon for the exchange rates of the Deutsche mark against the U.S. dollar (DEM/USD) and Japanese yen against the U.S. dollar (YEN/USD). Methods are developed from the returns for January to September 1996 and are assessed out-of-sample for October to December 1996. Our conclusions are the same for both currencies. The results show it is worthwhile to consider the seasonal explicitly when forecasting volatility. All three seasonal models outperform the GARCH model for the original returns. It is also better to estimate the FFF than the simple method of averaging the realized volatility over appropriate intervals. The smoothing effect of the FFF can explain this result. The two-step method using the FFF provides results that are only marginally worse than those of the P-GARCH model. Because the two-step method using the FFF is computationally much more efficient than P-GARCH, there is a strong case for giving preference to the FFF method. There is a consensus in the literature that the total volatility is described by the product of two components: the seasonal and the remaining volatility dynamics. We find some support for the strategy of Andersen and Bollerslev (1997, 1998) that estimating the seasonal from the logarithms of squared returns improves over estimating the seasonal directly from squared returns..

(3) Seasonal Adjustment Methods. 285. II. Data Olsen & Associates provided the database HFDF-II, which includes the most recent and subsequent Reuters FXFX quotes around half-hour marks for all of 1996. We study the two major exchange rates: the DEM/USD and the YEN/USD. Because our primary objective is to forecast volatility, we use the last quote before the half-hour mark to reflect the exchange rate at the half-hour mark. The exchange rates on the half-hour marks are calculated as the midpoint of the bid and ask rates. From these half-hour prices we construct the half-hour return series: bid ask rt = ln ptbid + ptask /2 − ln pt−1 + pt−1 /2 . (1) Intervals that have no quote are assigned zero returns. There are only sixteen such zeros during the 260 days used to obtain empirical results. Because of microstructure effects,1 the return series exhibit negative firstorder autocorrelation. For this reason we first filter the returns before starting our analysis, estimating rt = ρ rt−1 + et .. (2). We find ρˆ is −0.088 for the DEM/USD and −0.159 for the YEN/USD. The residual return, Rt , is then defined as Rt = eˆt = rt − ρˆ rt−1 .. (3). In the following text, residual returns are often simply called returns. Next, we follow the adjustment process of Andersen and Bollerslev (1998) by excluding the returns from Friday 21:00 Greenwich mean time (GMT) through Sunday 21:00 GMT. A description of this weekend definition can be found in Bollerslev and Domowitz (1993). The last three hours of Sunday returns are regarded as Friday returns. Because there is a discontinuity in the returns between consecutive weeks, we do not use forecasts for the last three hours on Friday, and we remove the first two hours on Monday to diminish the effect of this discontinuity. We also remove January 1 and December 25 from the sample because these holidays can have very low volatility (Taylor and Xu 1997). As a result, for 1996 we have 260 days each with forty-eight (residual) half-hour returns. Reuters’ data are recorded in GMT, which does not reflect the daylight saving schemes followed by most of the major financial markets. Analysis of the volatility patterns without a suitable adjustment to GMT could be misleading during daylight saving periods. Consequently, we adjust GMT during the American. 1 Because banks are often willing only to sell (buy) but not buy (sell), they will skew the bid-ask quotes. This will result in negative autocorrelation when using midpoints of the bid and ask quotes (e.g., Goodhart and Figliuoli 1991; Bollerslev and Domowitz 1993)..

(4) 286. The Journal of Financial Research. daylight saving period, which is in effect from the first Sunday in April until the last Sunday in October. For 1996 this is from April 7 to October 27. The adjustment procedure is to add one hour to GMT during the American daylight saving period and to leave other times unchanged. We call this adjusted clock GMTadj . The remainder of the analysis is based on this adjusted time. U.S. Eastern time can be obtained by subtracting five hours from GMTadj .. III. Alternative Models and Performance Measures No Seasonal Adjustment A natural question is whether out-of-sample forecasting of volatility is actually improved when using seasonal methods. To provide an answer, we first forecast volatility using the residual returns as defined in equation (3) using a GARCH(1,1) model for the thirty-minute returns. We use this model as our benchmark as follows, Rt = µ + εt εt | t−1 ∼ D(0, h t ) 2 + βh t−1 , h t = ω + αεt−1. (4). where t−1 is the information set at time t − 1, and for D we use the scaled tdistribution.2 The out-of-sample variance forecast, calculated using estimated parameters, is then simply hˆ T +1 = ωˆ + αˆ εˆ T2 + βˆ hˆ T .. (5). Because we are forecasting volatility related to realized volatility represented by |Rt |, it is necessary to adjust the level of the forecast to avoid bias. For GARCH models having conditional scaled t-distributions with ν degrees of freedom, the density of D(0, 1) is −(ν+1)/2 z2 − 12 − 12 [(ν + 1)/2] f (z) = π (ν − 2) 1+ (ν > 2), (6) [ν/2] ν−2 where is the gamma-function, and the expected absolute return is √ ν − 2 [(ν + 1)/2] ˆ E|RT +1 | = 2 √ h T +1 . π [ν/2](ν − 1). (7). 2 We originally used the normal distribution. The t-distribution significantly improves the in-sample fit, with 4 degrees of freedom. However, out-of-sample variance forecasts hardly change, and neither does the forecasting performance. We therefore did not search for improvements on the t-distribution..

(5) Seasonal Adjustment Methods. 287. Thus, the volatility forecast from the GARCH model is scaled to remove bias from the expected absolute return, which is our measure for realized volatility. Seasonal Adjustment Using Averages of Squared Returns Taylor and Xu (1997) use for each intraday period the appropriate average of the squared returns over all trading days. Let Rd,n be the nth intraday return on day d, and suppose we have D days and N intraday periods. Then the seasonal variance estimate is given by sn2 =. D 1 R2 (n = 1 . . . N ). D d=1 d ,n. (8). Note that throughout this article returns are either denoted by Rt or Rd,n with t = 48(d – 1) + n. A second way to estimate the seasonal pattern takes account of the day of the week. Let Sd be the set of daily time indexes that share the same day of the week as time index d. Let Nd be the number of time indexes to be found in Sd . Then the seasonal variance is given by 1 2 2 sd,n (1) = R . (9) Nd s∈Sd s,n Andersen and Bollerslev (1997, 1998) use the logarithm of the squared returns to help estimate seasonal patterns, following the popular assumption that volatility is the product of the seasonal volatility and the time-varying nonseasonal component. Hence, alternatively, . 1 2 2 ¯ ) , sd,n (2) = exp ln((Rs,n − R) (10) Nd s∈Sd where R¯ is the overall mean taken over all returns. Using the seasonal terms we filter the returns: Rd,n R˜ t ≡ R˜ d,n ≡ sd,n. (11). and estimate a GARCH(1,1) model for the filtered returns: R˜ t = µ + εt ˜ t−1 ∼ D(0, h˜ t ) εt | 2 + β h˜ t−1 , h˜ t = ω + αεt−1. (12). ˜ t−1 the returns history t−1 augmented by the seasonal terms. To transform with the volatility forecasts for the filtered returns into volatility forecasts for the original returns, we multiply the volatility forecasts by the appropriate seasonal term, sd,n ..

(6) 288. The Journal of Financial Research. FFF Andersen and Bollerslev (1997, 1998) advocate the FFF to model the intraday volatility pattern. They assume the following representation for the intraday returns: σd sd,n Z d,n Rd,n = E(Rd,n ) + , (13) N 1/2 with the terms Z d,n being independent and identically distributed (i.i.d.) with zero mean and variance one. They then define 2 2 xd,n ≡ 2 ln[|Rd,n − E(Rd,n )|] − ln σd2 + ln N = ln sd,n + ln Z d,n .. (14). Replacing E(Rd,n ) by the average of all intraday returns, and σ d by an estimate from a daily volatility model, xˆ d,n is obtained. The seasonal pattern is estimated by ordinary least squares (OLS) using J I n n2 j σd µ0 j + µ1 j + µ2 j + λij It=di xˆˆ d,n = N1 N2 j=0 i=1 p 2πin 2πin + δij sin , (15) γij cos + N N i=1 where N1 = (N + 1)/2 and N2 = (N + 1) (N + 2)/6 are normalizing constants, and p is set equal to four.3 Each of the corresponding J + 1 FFFs are parameterized by a quadratic component (terms with µ coefficients) and a number of sinusoids. Moreover, it may be advantageous to include time-specific dummies for applications in which some intraday intervals do not fit well within the overall regular periodic pattern (the λ coefficients). We follow a similar specification. Because we want to forecast volatility, we cannot use σd based on future information. We therefore assume there is no interaction between the daily volatility level and the seasonal volatility pattern; hence, J = 0. We do, however, estimate a separate FFF for each day of the week. Ignoring σd also means that the term ln(σd2 ) is removed from the definition of xˆ d,n . Andersen and Bollerslev (1998) show that this has little effect on the seasonal. For the dummy variables we use the three most important U.S. macroeconomic news announcements (Payne 1996): unemployment, retail sales, and international trade. These dummies reflect higher volatility for the thirty minutes following news announcements. The news announcements are collected from. 3 Several specifications are possible, all based on the same idea; even Andersen and Bollerslev (1997, 1998) use different specifications. All specifications lead to a similar smooth fit of the average of squared returns for each interval. See the results in section IV and Figures Ia–Id..

(7) Seasonal Adjustment Methods. 289. Business Weekly. To provide a fair comparison, the averages in equations (8) and (9) are calculated separately for intervals with and without news. The intraday seasonal volatility pattern is determined using sd,n (3) = exp(xˆˆ d,n /2).. (16). Instead of using the logarithm of the squared returns, we estimate the FFF directly from the squared returns. Either estimated seasonal pattern can then be used to filter the returns and to estimate the GARCH(1,1) model in equation (12). P-GARCH Bollerslev and Ghysels (1996) introduce the P-GARCH model explicitly designed to capture the repetitive seasonal time variation in the second-order moments. P-GARCH includes all GARCH models in which hourly dummies, for example, are used in the variance equation. Extending the information set t−1 with a process defining the stage of s the periodic cycle at each point, say to t−1 , the class of P-GARCH(1,1) models is defined as Rt s εt | t−1. = µ + εt ∼ D(0, h t ). 2 + βs(t) h t−1 , h t = ωs(t) + αs(t) εt−1. (17). where s(t) refers to the stage of the periodic cycle at time t. The periodic cycle of interest here is a repetitive cycle covering one week. The P-GARCH model is potentially more efficient than the methods described earlier. These methods first estimate the seasonal, and after deseasonalizing the returns they estimate the volatility of these adjusted returns. Although this is a two-step method, the P-GARCH model allows a simultaneous estimation of the seasonal effects and the remaining time-varying volatility. On the other hand, however, we need a parsimonious model. Otherwise, estimating the P-GARCH model will become too difficult and time consuming. Two extreme cases are to use parameters ωs(t) in (17) in such a way that they represent the role of: (a) the average absolute/squared returns (e.g., 240 dummies), or (b) the FFF. We estimate the second case where we only have one FFF for the entire week instead of a separate FFF for each day of the week:4 4 n n2 2πin 2πin 2 γi cos sn = ω0 + ω1 + ω2 + + δi sin (18) N1 N2 N N i=1 4 As is shown in section IV, the FFF performs better than the simple R 2 method. The latter would also require more parameters in the P-GARCH model as opposed to the FFF. We therefore focus on including the FFF in the P-GARCH model..

(8) 290. The Journal of Financial Research. and Rt s εt | t−1. = µ + εt ∼ D(0, h t ). 2 h t = sn2 + αεt−1 + βh t−1 .. (19). An alternative specification for the variance equation is h t = ωsn2 +. sn2 2 sn−1. 2 (αεt−1 + βh t−1 ),. (20). with ω0 set to one. The motivation for this specification is the GARCH(1,1) model for deseasonalized returns in equation (12), with Rt = sn R˜ t and h t = sn2 h˜ t . Forecasting Performance Measures We compare the volatility forecasts from the four models described earlier with the realized volatility. Realized volatility is simply measured as the absolute realized return in the forecasted period.5 In the previous literature various methods are provided to assess the performance of out-of-sample forecasts. For example, Day and Lewis (1992) and Canina and Figlewski (1993) use the squared correlation between realized and forecasted volatility, as well as encompassing regressions, and Taylor and Xu (1997) use measures of relative forecast errors. Because there is no consensus in the forecasting literature about the best method, we report several of the most popular performance measures. The first method is to calculate the root mean squared error (RMSE). If σˆ t is the forecasted volatility for period t, the RMSE for T forecasts is. T 1 (|Rt | − σˆ t )2 . (21) RMSE =

(9) T t=1 The second method is to calculate the logarithmic loss function employed in Pagan and Schwert (1990), LL =. 5. T 1 ¯ − ln(σˆ t ))2 . (ln |Rt − R| T t=1. (22). Alternatively, realized volatility can be measured by the squared return with forecasts given by conditional variances instead of by conditional standard deviations. The absolute return is preferred here because it reduces the effect of outliers. When all results are reproduced using squared returns and variances, the results and conclusions are similar to those reported here..

(10) Seasonal Adjustment Methods. 291. This alternative emphasizes the importance of accurate predictions when the realized volatility is close to zero, and it reduces the effect of outliers. The third method is to simply calculate the correlation between realized and forecasted volatility. The fourth method uses the regression |Rt | = α + β σˆ t + εt. (23). and reports the goodness of fit measured by the squared correlation, R 2 . Finally, we use the encompassing regression in which we include more than one forecast in the previous regression and check whether one forecast dominates the forecast from another method.. IV. Results Intraday Seasonal Pattern For two of the four methods we first filter (deseasonalize) the returns. The seasonal pattern is estimated using the mean of the squared returns (equations (9) and (10)) or the FFF (equations (14) and (15)). The seasonal is estimated for both ¯ 2 ) and Rt2 . The seasonals are estimated for the first nine months of ln((Rt − R) 1996. The averages taken over all thirty-nine weeks provide the resulting seasonals in Figures Ia and Ib for the DEM/USD and in Figures Ic and Id for the YEN/USD. A difference between using squared returns and their logarithms is that outliers have less effect in the latter method. Also, the FFF tries to find a smooth fit of the R 2 seasonal. If the R 2 seasonal is driven by outliers explaining the spikes on many occasions, we expect it to do well in-sample (because the effect of the. Figure Ia. DEM/USD Volatility Seasonals Based on Squared Returns..

(11) 292. The Journal of Financial Research. Figure Ib. DEM/USD Volatility Seasonals Based on the Log of Squared Returns.. Figure Ic. YEN/USD Volatility Seasonals Based on Squared Returns.. Figure Id. YEN/USD Volatility Seasonals Based on the Log of Squared Returns..

(12) Seasonal Adjustment Methods. 293. outliers will be reduced) but worse out-of-sample (because large returns are not repeated in the same thirty-minute intervals). The seasonals for the YEN/USD seem to be less pronounced than the seasonals for the DEM/USD, partly reflected by many more spikes for the YEN/USD, and the average level of the seasonal is higher for the YEN/USD. This is probably because our sample period shows a relatively volatile year for the YEN/USD. In February 1996 the Bank of Japan intervened several times to support the dollar against the yen, and for much of the remainder of 1996 there was strong speculation of pending Japanese interest rate rises. In-Sample Fit Although we cannot compare directly the in-sample fit for all the models, we can obtain some idea of their adequacy by estimating them for the first nine months. The seasonals presented in Figure I are employed for this purpose. The results are presented in Table 1. The parameter estimates for α and β are significantly different from zero at the 5% level for all six models. The inclusion of the FFF in the GARCH specification (the P-GARCH model) leads to a highly significant improvement over the use of a standard GARCH(1,1) model. The R 2 and FFF methods estimate GARCH(1,1) parameters from the deseasonalized returns, R˜ d ,n = Rd ,n /sd ,n . This explains the substantial variation among the reported log-likelihoods. We adjust the log-likelihood by recalculating the loglikelihood after multiplying the mean-adjusted returns by the seasonal sd ,n and the estimated conditional variances by the squared seasonal. The result is that the R 2 method (with seasonal based on the log of squared returns) has the highest log-likelihood. We must be cautious about drawing any conclusions from these likelihoods, because the models are generally nonnested. The R 2 method is likely to be favored in this comparison because the spikes in the seasonal reflect some large outliers that will now be fit well. This advantage disappears when forecasting the volatility out-of-sample, as we see later. The use of the log of the squared returns (ln R 2 ) instead of squared returns (R 2 ) to calculate the seasonals seems to improve the in-sample fit. The P-GARCH model is at a disadvantage compared with the FFF method that uses a different FFF for each day of the week, as well as news dummies. To check whether P-GARCH is more efficient than the two-step method based on the FFF, we extend equation (19) for the DEM/USD with a FFF for each day of the week and the news dummies. The log-likelihood increases to 14,028, which is better than the two-step method regardless of whether the FFF is based on the squared returns or on the log of squared returns. Using an exponential version of the FFF in equation (19) with sn2 replaced by exp(sn2 ) gives a log-likelihood of 14,035. Repeating this exercise for equation (20) gives log-likelihoods of 14,039 and 14,045, respectively. Hence, P-GARCH is more efficient in-sample, and the.

(13) 294. The Journal of Financial Research. TABLE 1. In-Sample Fit, January–September 1996. R2 2. FFF ln R. 2. R. 2. ln R 2. Raw. R. 0.001 (0.001) 0.001 (0.000) 0.290 (0.028) 0.604 (0.032) 3.716 (0.153) 13,745 13,745. 0.020 (0.010) 0.011 (0.024) 0.038 (0.037) 0.953 (0.058) 4.813 (0.262) −12,631 14,024. 0.047 (0.014) 1.068 (0.337) 0.157 (0.031) 0.672 (0.082) 4.117 (0.187) −20,476 14,074. 0.019 (0.009) 0.083 (0.124) 0.104 (0.088) 0.826 (0.196) 4.232 (0.204) −12,590 13,903. 0.043 (0.018) 0.983 (0.405) 0.151 (0.038) 0.692 (0.101) 4.193 (0.194) −20,531 14,019. 0.001 (0.001) 0.001 (0.000) 0.229 (0.025) 0.654 (0.037) 3.780 (0.155) 11,286 11,286. 0.013 (0.006) 0.053 (0.029) 0.104 (0.031) 0.852 (0.056) 4.810 (0.235) −12,537 11,472. 0.036 (0.029) 0.586 (0.235) 0.143 (0.034) 0.771 (0.067) 4.028 (0.170) −20,407 11,513. 0.014 (0.131) 0.082 (0.012) 0.131 (0.007) 0.803 (0.013) 4.130 (0.002) −12,300 11,388. 0.034 (0.018) 0.578 (0.269) 0.140 (0.038) 0.776 (0.077) 4.052 (0.176) −20,458 11,462. P-GARCH Eq. (19). Panel A. DEM/USD µ ω α β ν Log-likelihood Adj. log-likelihood. 0.010 (0.006) — 0.175 (0.044) 0.605 (0.130) 4.226 (0.216) −7,558 13,994. Panel B. YEN/USD µ ω α β ν Log-likelihood Adj. log-likelihood. 0.010 (0.005) — 0.157 (0.010) 0.715 (0.023) 4.019 (0.001) 11,424 11,424. Note: Generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) models are estimated for the first nine months of 1996 (9,360 observations). The first column (Raw) is for the original thirty-minute returns. The second column (R 2 ) is for returns divided by the seasonal based on averages of squared returns (R 2 ) or averages of the log of squared returns (ln R 2 ). The third column (FFF) is for returns divided by the seasonal based on the flexible Fourier form for squared returns (R 2 ) or the log of squared returns (ln R 2 ). The fourth column (P-GARCH) is for the periodic GARCH model defined in equations (18) and (19). The log-likelihood is based on the data used to estimate the GARCH model. Hence, for the R 2 and FFF methods, which use the deseasonalized returns, the adjusted log-likelihood is also reported, where the residuals are multiplied by the seasonal and the estimated conditional variances are multiplied by the square of the seasonal. µ is the conditional mean; ω, α, and β are conditional variance parameters; and ν is the degrees of freedom of the conditional (student-t) distribution. Heteroskedasticity-consistent standard errors are shown in parentheses.. exponential form is an improvement, although the effect is much smaller. Repeating the calculations for the YEN/USD gives a highest log-likelihood of 11,455 for equation (12) using exp(sn2 ), still short of the 11,462 for the two-step method based on the FFF for the log of squared returns. However, as we show next, out-of-sample it is better to use a more parsimonious model like the one provided in equation (19)..

(14) Seasonal Adjustment Methods. 295. Out-of-Sample Forecasting Performance Using the estimated seasonals in Figure I and assuming they can be used for the last three months of 1996, we now forecast for each method. The GARCH(1,1) model is repeatedly estimated using data from the 60 days (2,880 observations) preceding the forecasting period for the DEM/USD and 195 days for the YEN/USD.6 The parameters are reestimated every twenty-four hours. This results in 3,120 forecasts. In the comparisons, we remove the last three hours on Fridays and the first two hours on Mondays, as well as intervals containing prescheduled news and the two subsequent intervals following the news announcements, leaving us with 2,963 forecasts. The latter deletions are made to make the competition fair for the “raw” and P-GARCH methods, which do not use news dummies. It is also beyond the scope of this study to forecast volatility around news announcements. The forecasts are compared using the measures discussed in section III. The results for the univariate measures are provided in Table 2. First, note that modeling the seasonal pattern in volatility improves outof-sample forecasting. For nearly all univariate measures, the model using the (original) raw returns is the worst of the six models under consideration; the other five models take into account the intraday seasonal. The only exceptions are for the R 2 method for the YEN/USD. Nevertheless, the forecasts using GARCH(1,1) for the raw returns still have a correlation of 0.245 with the realized volatility measure (the absolute returns) for the DEM/USD and 0.237 for the YEN/USD. ¯ 2 ) improves out-of-sample foreSecond, the seasonal based on ln((Rt − R) casting compared with the seasonal based on Rt2 . For both the R 2 and the FFF methods we find a higher correlation with realized volatility, as well as a lower RMSE and a higher coefficient of determination. For example, for the FFF the correlation is 0.294 versus 0.268 for the DEM/USD, whereas it is 0.269 versus 0.264 for the YEN/USD. Third, it seems worthwhile to estimate the FFF rather than using the simpler R 2 method. If we compare, for example, the results using the seasonal based on ¯ 2 ), we find for the DEM/USD that the correlation for the FFF method ln((Rt − R) is 0.294 versus 0.273 for the R 2 method. For the YEN/USD, the correlation for the FFF method is 0.269 versus 0.251 for the R 2 method. Also, the RMSE is lowest for the FFF method. A logical explanation is that the smooth fit of the FFF more realistically describes the true underlying seasonal pattern. The R 2 method is more affected by outliers (resulting in the spiky seasonals in Figures Ia to Id), which are not likely to happen again in the same intraday intervals in other weeks. Finally, the P-GARCH model using the specification in equation (19) ranks first for both the DEM/USD and the YEN/USD, using simply one FFF for the entire. 6 Several robustness checks revealed that the GARCH parameters changed more rapidly for the DEM/USD over the out-of-sample period than for the YEN/USD. For the YEN/USD all models performed better using 195 days preceding the forecasting period instead of 60 days..

(15) 296. The Journal of Financial Research. TABLE 2. Out-of-Sample Forecast Performance, October–December 1996. R2. FFF ln R 2. P-GARCH Eq. (19). 0.268 0.046 5.249 1.691 0.072. 0.294 0.045 5.201 1.642 0.086. 0.298 0.044 5.168 1.636 0.089. 4.14 0.31 4.74 3.59. 4.15 0.25 4.66 3.47. 4.38 0.26 4.95 3.91. 4.47 0.30 5.24 3.92. 0.227 0.060 5.873 1.760 0.052. 0.251 0.058 5.812 1.715 0.063. 0.264 0.057 5.751 1.712 0.069. 0.269 0.057 5.750 1.697 0.072. 0.270 0.056 5.736 1.686 0.073. 4.33 0.27 4.82 3.91. 3.90 0.11 4.06 3.67. 4.02 0.07 4.19 3.81. 4.00 0.07 4.09 3.83. 3.95 0.08 4.05 3.77. 2. ln R. 2. Raw. R. 0.245 0.046 5.290 1.741 0.060. 0.267 0.047 5.302 1.696 0.071. 0.273 0.047 5.308 1.687 0.074. 3.80 0.12 4.14 3.60. 4.43 0.37 5.25 3.83. 0.237 0.057 5.799 1.736 0.056 3.70 0.07 3.81 3.56. R. 2. Panel A. DEM/USD Correlation with realized (|Rt |) Mean forecast (realized: 0.048) RMSE (×10−2 ) LL Regression adjR 2 Degrees of freedom Mean Std. dev. Max. Min. Panel B. YEN/USD Correlation with realized (|Rt |) Mean forecast (realized: 0.0575) RMSE (×10−2 ) LL Regression adjR 2 Degrees of freedom Mean Std. dev. Max. Min.. Note: The 2,963 forecasts for the last three months of 1996 are compared with the realized volatility measured by the absolute return. The models are reestimated every twenty-four hours using data from the 60 days (DEM/USD) or 195 days (YEN/USD) preceding the forecasting period. The first column (Raw) is for the generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model for the original thirty-minute returns. The second column (R 2 ) is for the GARCH(1,1) model for returns divided by the seasonal based on averages of squared returns (R 2 ) or averages of the log of squared returns (lnR 2 ). The third column (FFF) is for returns divided by the seasonal based on the flexible Fourier form for squared returns (R 2 ) or the log of squared returns (lnR 2 ). The fourth column (P-GARCH) is for the periodic GARCH model defined in equations (18) and (19). “Regression adjR 2 ” relates to the adjusted coefficient of determination in the regression |Rt | = α + β σˆ t + εt , where σˆ t is the volatility forecast for one of the models. The last part of the table shows statistics for the estimated degrees of freedom of the scaled t-distribution.. week (in contrast to different FFFs for every day of the week) and an adjustment to the constant in the GARCH specification. P-GARCH outperforms the raw method because the seasonal is included. However, it also outperforms the R 2 method and the FFF method. P-GARCH ranks first and the FFF method, with seasonal based ¯ 2 ), ranks second for both currencies and for each of the perforon ln((Rt − R) mance measures given by the correlation, coefficient of determination, RMSE, and LL. The results for the encompassing regressions are provided in Table 3. They support the conclusions based on correlation, coefficient of determination, RMSE,.

(16) Seasonal Adjustment Methods. 297. TABLE 3. Encompassing Regressions, Coefficients for Explanatory Variables. Panel A. DEM/USD R Row 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16. Raw. R2. 2. ln R 2. Panel B. YEN/USD R2. FFF R2. ln R 2 P-GARCH. 0.305 0.411 (4.8) (7.7) 0.327 0.414 (5.7) (8.9) 0.272 0.487 (4.0) (7.4) 0.124 0.640 (1.7) (9.4) −0.027 (−0.32) 0.278 0.350 (3.8) (5.1) 0.302 0.380 (3.9) (4.2) 0.116 0.618 (1.5) (7.1) 0.209 (3.5) 0.340 0.353 (5.6) (4.8) 0.160 0.567 (2.2) (6.5) 0.232 (4.1) 0.085 0.657 (0.86) (6.8) 0.191 (2.5) 0.346 (3.7) −0.031 0.158 0.187 (−0.37) (2.1) (1.5). Raw. R2. ln R 2. FFF R2. ln R 2 P-GARCH. 0.502 0.329 (6.5) (5.3) 0.379 0.441 (4.5) (6.4) 0.139 0.723 (1.2) (6.6) 0.211 0.640 (2.2) (7.5) −0.090 (−0.69) 0.200 0.504 (2.6) (6.5) 0.118 0.724 (1.5) (7.7) 0.154 0.661 (2.2) (8.4) 0.181 (2.7) 0.291 0.559 (3.7) (5.9) 0.129 0.663 (1.2) (5.6) 0.233 (2.8) 0.380 0.482 (3.2) (4.4) 0.358 (3.0) 0.395 (3.3) −0.053 0.103 0.296 (−0.41) (0.97) (1.9). 0.959 (9.7). 0.707 (8.3). 0.670 (7.9). 0.740 (7.8) 0.549 (4.7) 0.577 (4.2). 0.977 (7.4). 0.732 (8.7). 0.657 (6.4). 0.566 (4.5) 0.494 (3.6) 0.537 (2.9). Note: Coefficients and t-values (in parentheses) for encompassing regressions. Using, for example, two models, we estimate the regression (1). |Rt | = α + β1 σˆ t. (2). + β2 σˆ t. + εt (i). using ordinary least squares where |Rt | is the absolute realized thirty-minute return, and σˆ t is the volatility forecast for model i. The rows of the table then report the regression coefficients β1 and β2 and their t-values. In total there are 2,963 forecasts for each model for the last three months of 1996. All models are estimated every twenty-four hours using data from the 60 days (DEM/USD) or 195 days (YEN/USD) preceding the forecasting period. “Raw” indicates the original thirty-minute returns are used in a generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. “R 2 ” indicates the original returns are first divided by the seasonal based on averages of squared returns (R 2 ) or the log of squared returns (lnR 2 ) and then the GARCH(1,1) model is estimated for the resulting deseasonalized returns. “FFF” indicates the original returns are divided by the seasonal based on the flexible Fourier form for squared returns (R 2 ) or the log of squared returns (lnR 2 ). “P-GARCH” is the periodic GARCH model defined in equations (18) and (19)..

(17) 298. The Journal of Financial Research. and LL. First, the model ignoring the seasonal performs the worst. Including forecasts from P-GARCH makes the raw forecasts insignificant (see row 5 for both Panels A and B in Table 3), which is also the case using forecasts for the DEM/USD from the FFF method based on the “log” seasonal (see row 4 in Panel A) and for the YEN/USD from the FFF method based on the squared returns (see row 3 in Panel B). In the other cases, however, the raw forecasts have some incremental information over the forecasts from the other methods. Second, the forecasts based ¯ 2 ) are better than the forecasts based on Rt2 . For the FFF, the latter on ln((Rt − R) are insignificant when both are included in one regression for the DEM/USD (row 13 in Panel A of Table 3). Third, the FFF method improves over the R 2 method, although the latter provides incremental information for the DEM/USD (but not for the YEN/USD). V. Conclusions In this article we investigate whether modeling the intraday seasonal volatility pattern in financial markets improves out-of-sample volatility forecasting. Using a GARCH(1,1) model for the original returns as the benchmark, we show that modeling the seasonal improves forecasting performance. We also seek the best method of modeling the intraday seasonal volatility pattern for the DEM/USD and the YEN/USD exchange rate returns, measured by out-of-sample forecasting performance. First, we assess whether the seasonal pattern should be determined using squared returns or using the log of the squared returns. We recommend the latter as the best method. Second, we compare two methods that first determine the seasonal and then analyze the deseasonalized return series. The FFF improves over the mean of the squared returns for each intraday interval. Third, we compare these two-step methods with the P-GARCH model of Bollerslev and Ghysels (1996). The P-GARCH model could be seen as a simultaneous optimization of the seasonal pattern and the dynamic volatility pattern. We find that the P-GARCH model provides the best forecasts out-ofsample. All of these conclusions apply to both currencies. There is a strong case for giving preference to the two-step method based on the FFF method rather than the P-GARCH method. First, it is computationally more efficient. Second, the forecasting performance of the P-GARCH model is only marginally better than the performance of the two-step FFF. Third, P-GARCH potentially suffers from the restriction that the amount of historical data used for the seasonal is the same as for the GARCH parameters. The results provide important insights into the methods of modeling the intraday volatility seasonal pattern. They show that it is worthwhile to do so when forecasting intraday volatility, and they show the most promising way to do so. Our results help solve the problem of selecting from the wide variety of approaches available for modeling seasonal volatility..

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