### Table 5. Volatility Forecasts

### Table 3: Out-of-Sample MSE Comparisons of MIDAS Models with Daily Regressors - DJ Index Each entry in the table corresponds to the ratios MSEMIDAS=MSEABDL; where MSEABDL is the out-of-sample MSE for the benchmark ARFI(5,d) model in equation (1.10) and MSEMIDAS is the out-of-sample MSE from the MIDAS model (equations (1.1) and (1.2)) using lagged daily predictors, shown in the corresponding column and discussed in section 1.1. We obtain the out-of-sample forecasts by estimating the models with data from April 1, 1993 to March 31, 2001 and then use the estimates from these regressions (shown in Table 2 above) to forecast future realized volatilities. The regressions are run on a weekly, bi-weekly, tri-weekly, and monthly data sampling schemes with non-overlapping ~ Q(Hm) t+H;t; for H = 5 days (1wk), H = 10 (2wks), 15 (3wks) and 20 (4wks) days using two measures of volatility Qt+H;t and log (Qt+H;t). For the log speci cation, models with log r2 t;t 1 and log jrt;t 1j regressors are identical, hence the vertical bar to avoid duplication.

in Abstract

2003

"... In PAGE 18: ... However, the ultimate test of forecasting models is whether they maintain their performance out-of-sample. The out-of-sample MSEs of the MIDAS models relative to the benchmark are displayed in Table3 which features a few interesting results. First, the realized power ~ P (m) t k;t k 1 is the dominant out-of-sample predictor of future realized volatility.... In PAGE 19: ... To avoid reporting many results, we simply focus on the out-of-sample performance. Table 4 is analogous to Table3 for the DJ index. The results obtained with the DJ index remain true for the individual stocks.... In PAGE 41: ...This table displays similar results as in Table3 , but for six individual stocks rather than the DJ Index. More speci cally, for each of the six stocks { Disney Co (DIS), General Electric Co (GE), J.... ..."

### Table 6: Relative Pricing Errors (%) of Alternative Models with Implied Volatility Risk Premium for SV Models and Implied Volatility for Constant Volatility Models Moneyness Days-to-Expiration

"... In PAGE 26: ... This finding is also consistent with the conjecture in Lamoureux amp; Lastrapes (1993) and explains why the implied volatility is an inefficient forecast of the underlying volatility.16 Table6 reports the relative pricing errors (%) for alternative models in terms of option prices. In 16 As suggested by the referee, these results could potentially explain the findings in Lamoureux amp; Lastrapes (1993) if we extract the implied volatility from various models and regress daily realized volatility over the life of the option.... ..."

### Table 1 Unbiasedness tests of S amp;P 100 volatility proxies

1998

"... In PAGE 12: ... Hansen and Singleton 1982 . The unbiasedness regression results are reported in Table1 . Comparing the 2 R -statistics across regressions, the call option implied volatility has slightly more explanatory power than the put option implied volatility, but each of these measures exhibits a much stronger relation with realized volatility than does the historical volatility.... In PAGE 13: ...X 2 where d s 0,1 . The CS -statistics reported in Table1 indicate strong unbiased- 0 ness rejections for each volatility forecast. For the historical volatility, this finding is not unexpected.... ..."

Cited by 12

### Table 2 Efficiency tests of S amp;P 100 volatility proxies

1998

"... In PAGE 15: ... Specifically, any forecast error that is entirely realized prior to t should be orthogonal to the time t forecast error. Table2 provides the regression results and OI2-statistics for two monthly lagged forecast errors, m and m . ml represents the forecast error using the tyKty2Kt a and b estimates provided on the first line of the table for each forecast.... ..."

Cited by 12

### Table III Volatility Measures from Survey Data The table presents summary statistics by survey date. Realized one-year S amp;P 500 return (%) is realized one-year S amp;P 500 return from the survey date. Average forecast error (%) is defined as Realized one-year S amp;P 500 return (%) minus Average one-year S amp;P 500 expected return (%). % below confidence range is the percentage of respondents for whose the realized one-year S amp;P 500 return is below the lower confidence bound. % in confidence range is the percentage of respondents for whose the realized one-year S amp;P 500 return is within the confidence bounds. % above confidence range is the percentage of respondents for whose the realized one-year S amp;P 500 return is above the upper confidence bound. VIX is the Chicago Board of Options Exchange (CBOE) volatility index which reflects the average of imputed volatility across traded options on the S amp;P 500 futures index. Realized volatility is the annual volatility of the S amp;P 500 one year from the survey date measured with daily returns. Past qtr S amp;P 500 return is the S amp;P 500 returns in the quarter preceding the survey date.

2006

### Table 2. Correlation Matrices of Realized DM/$ and Yen/$ Volatilities

2001

"... In PAGE 13: ... Such questions are difficult to answer using conventional volatility models, but they are relatively easy to address using our realized volatilities and correlations. The sample correlations in the first panel of Table2 , along with the lstddt-lstdyt scatterplot in the top panel of Figure 2, clearly indicate a strong positive association between the two exchange rate volatilities. Thus, not only do the two exchange rates tend to move together, as indicated by the positive means for covt and corrt , but so too do their volatilities.... In PAGE 13: ... This suggests factor structure, as in Diebold and Nerlove (1989) and Bollerslev and Engle (1993). The correlations in the first panel of Table2 and the corrt-lstddt and corrt-lstdyt scatterplots in the second and third panels of Figure 2 also indicate positive association between correlation and volatility. Whereas some nonlinearity may be operative in the corrt-lstddt relationship, with a flattened response for both very... In PAGE 17: ... Next, turning to the multivariate unconditional distributions, we display in the lower panels of Table2 the correlation matrices of all volatility measures for h = 5, 10, 15, and 20. Although the correlation between the different measures drops slightly under temporal aggregation, the positive association between the volatilities, so apparent at the one-day return horizon, is largely preserved under temporal aggregation.... ..."

Cited by 30

### Table 2 Correlation Matrices of Realized DM/$ and Yen/$ Volatilities

"... In PAGE 12: ... Although such questions are difficult to answer using conventional volatility models, they are relatively easy to address using our realized volatilities and correlations. The sample correlations in the first panel of Table2 , along with the lstdd t -lstdy t scatterplot in the top panel of Figure 2, indicate a strong positive association between the two exchange rate volatilities. Thus, not only do the two exchange rates tend to move together, as indicated by the positive means for cov t and corr t , but their volatilities are also closely linked.... In PAGE 12: ... This provides empirical justification for the use of multivariate volatility models with a factor structure, as in Diebold and Nerlove (1989) and Bollerslev and Engle (1993). The correlation figures in Table2 along with the corr t -lstdd t scatterplot in the second panel of Figure 2 also indicate a positive association between correlation and volatility. To quantify further this volatility effect in correlation, we show in the top panel of Figure 3 kernel density estimates of corr t when both lstdd t and lstdy t are less than -0.... In PAGE 15: ... In contrast to previously, however, the unconditional variances of lstdd t,h and lstdy t,h now decrease with h , but again at a rate linked to the fractional integration parameter, as we document below. Next, turning to the multivariate unconditional distributions, we display in the lower panels of Table2 the correlation matrices of all volatility measures for h = 5 , 10 , 15 , and 20 . While the correlation between the different measures of volatility drops slightly under temporal aggregation, the strong positive association between the volatilities so apparent at the one-day return horizon is largely preserved under temporal aggregation.... ..."

### Table 3. Dynamic Dependence Measures for Realized DM/$ and Yen/$ Volatilities

2001

"... In PAGE 14: ... It is therefore striking that the time series plot for corrt shows equally pronounced persistence, with readily identifiable periods of high and low correlation. The visual impression of strong persistence in the volatility measures is confirmed by the highly significant Ljung-Box tests reported in the first panel of Table3 .... In PAGE 16: ...Hurvich, Deo and Brodsky (1998). The estimates of d are given in the first panel of Table3 . The estimates are highly statistically significant for all eight volatility series, and all are fairly close to the typical value of 0.... In PAGE 17: ...2 The Conditional Distribution: Dynamic Dependence, Fractional Integration and Scaling Andersen, Bollerslev and Lange (1999) show that, given the estimates obtained at the daily level, the integrated volatility should, in theory, remain strongly serially correlated and highly predictable, even at the monthly level. The Ljung-Box statistics for the realized volatilities in the lower panels of Table3 provide strong empirical backing. Even at the monthly level, or h = 20, with only 122 observations, all of the test statistics are highly significant.... In PAGE 17: ...ntegration is invariant to the sampling frequency; see, e.g., Beran (1994). This strong prediction is borne out by the estimates for d for the different levels of temporal aggregation, reported in the lower panels of Table3 . All of the estimates are within two asymptotic standard errors of the average estimate of 0.... In PAGE 18: ...ottom panels are 1.780 and 1.728, respectively, corresponding to values of d of 0.390 and 0.364. Because a non-linear function of a sum is not the sum of the non-linear function, it is not clear whether lstddt,h and lstdyt,h will follow similar scaling laws. The estimates of d reported in Table3 suggest that they should. The corresponding plots for the logarithm of the h-day logarithmic standard deviations log(Var(lstddt,h)) and log(Var( lstdyt,h )) against log(h), for h = 1, 2, .... ..."

Cited by 30