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304 V. Voev is 1.9%. From the daily data log monthly returns are constructed by using the opening price of the first trading day of the month and the closing price of the last day. These returns are then used to construct rolling window sample covariance matrices, used in the first two forecasting models. 4 Results In this section we present and discuss the results on the performance of the forecasting models described in Section 2. In order to asses the forecasting performance, we employ Diebold–Mariano tests for each of the variance and covariance series. Then we measure the deviation of the forecast as a matrix from its target by using again the Frobenius norm, which gives an overall idea of the comparative performance of the models. Of course, if the individual series are well forecast, so will be the matrix. As a target or ‘true’ covariance matrix, we choose the realized covariance matrix. First, we present some graphical results. Out of the total of 120 variance and covariance forecast series, Figure 1 plots nine representative cases, for the sample covariance and the RiskMetrics TM model, against the realized series. The name, which appears above each block in the figure, represents either a variance series (e.g. EK), or a covariance one (e.g. GE,AA). Both forecasts are quite close, and as can be seen, they cannot account properly for the variation in the series. As the tests show, however, the Riskmetrics TM fares better and is the best model among the sample based ones. It is already an acknowledged fact that financial returns have the property of volatility clustering. This feature is also clearly evident in the figure, where periods of low and high volatility can be easily distinguished, which suggests that variances and covariances tend to exhibit positive autocorrelation. Figure 2 shows the autocorrelation functions for the same nine series of realized (co)variances. The figure clearly shows that there is some positive serial dependence, which usually dies out quickly, suggesting stationarity of the series. Stationarity is also confirmed by running Augmented Dickey–Fuller (ADF) tests, which reject the presence of an unit root in all series at the 1% significance level. The observed dependence patterns suggest the idea of modelling the variance and covariance series as well as their shrunk versions as ARMA processes. This resulted in a few cases in which the matrix forecast was not positive definite (16 out of 176 for the original series and 8 out of 176 for the shrunk series). Thus the forecast in expression (24) seems to be reasonable and as we shall see later, compares well to the sample covariance based models. In a GARCH framework, the conditional variance equation includes not only lags of the variance, but also lags of squared innovations (shocks). When mean returns are themselves unpredictable (the usual approach is to model the mean equation as an ARMA process), the shock is simply the return. This fact led us to include lags of squared returns (for the variance series) and cross-products (for the covariance series) as in the ARMAX(p,q, 1) model in Eq. (23). This added flexibility, however, comes at the price of a drastic increase of the non-positive definite forecasts (108 and 96 out of 176, respectively). Thus the forecast in Eq. (24) comes quite close to the simple realized and shrunk realized covariance models in Sections 2.4 and 2.5, respectively. A solution to this issue is to decompose the matrices into their lower triangular Cholesky factors, forecast
Dynamic modelling of large-dimensional covariance matrices 305 GE.MO GE.AA EK 0.035 0.035 0.035 True Sample RM True Sample RM True Sample RM 0.025 0.025 0.025 0.015 0.015 0.015 0.005 0.005 0.005 100 120 140 160 180 200 220 240 260 280 −0.005 100 120 140 160 180 200 220 240 260 280 −0.005 100 120 140 160 180 200 220 240 260 280 −0.005 GE.CAT 0.035 GE.AA 0.035 GE.AXP 0.035 True Sample RM True Sample RM True Sample RM 0.025 0.025 0.025 0.015 0.015 0.015 Fig. 1 Comparison of the sample covariance based (Sample) and Riskmetrics TM (RM) forecast against the realized covariance (True). the Cholesky series, and then reconstruct the matrix. This leads to the forecasting formula in Eq. (25), which defines the drc − Chol and dsrc − Chol forecasting models for the simple realized and shrunk realized covariance case, respectively. A drawback of this approach is that the Cholesky series do not have an intuitive interpretation. They are simply used as a tool to constrain the forecasts to satisfy the complicated restrictions implied by the positive definiteness requirement. Another drawback is that the Cholesky decomposition involves nonlinear transformations of the original series. Thus, if one can adequately forecast the nonlinear transformation, this does not immediately mean that applying the inverse transformation 0.005 0.005 0.005 100 120 140 160 180 200 220 240 260 280 −0.005 100 120 140 160 180 200 220 240 260 280 −0.005 100 120 140 160 180 200 220 240 260 280 −0.005 GE 0.035 GE.EK 0.035 EK 0.035 True Sample RM True Sample RM True Sample RM 0.025 0.025 0.025 0.015 0.015 0.015 0.005 0.005 0.005 100 120 140 160 180 200 220 240 260 280 100 120 140 160 180 200 220 240 260 280 100 120 140 160 180 200 220 240 260 280
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High Frequency Financial Econometri
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Prof. Luc Bauwens CORE Voie du Roma
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vi Contents Intraday stock prices,
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2 L. Bauwens et al. component nicel
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4 L. Bauwens et al. but provides al
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Luc Bauwens . Dagfinn Rime . Genaro
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Exchange rate volatility and the mi
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32 K. Bien et al. Although economet
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34 K. Bien et al. 2.1 Copula functi
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36 K. Bien et al. and x k t ≡ (xk
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38 K. Bien et al. Fig. 3 Multivaria
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40 K. Bien et al. Bivariate model s
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42 K. Bien et al. deviation 0.0099,
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44 K. Bien et al. - Set ˆzt = Âxt
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46 K. Bien et al. % Frequency −0.
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48 K. Bien et al. References Amilon
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50 1 Introduction A. Escribano and
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52 A. Escribano and R. Pascual Jang
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54 A. Escribano and R. Pascual the
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56 The generating processes of mark
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58 with AtðLÞ ¼ 0 B @ 1 ð ÞAab
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60 A. Escribano and R. Pascual cros
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62 Hasbrouck (1991). The system is
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64 unitary seller-initiated shock (
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66 6.1 Estimation of the baseline m
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Table 2 The base-line VEC model for
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70 Table 3 Simulation of the base-l
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72 revert towards narrow levels. As
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Table 5 Impulse-response functions
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76 We also show that NYSE buyer-ini
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78 As 0 < a m < 1; L ð Þ ¼ 1 a m
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80 NYSE 2000 Stocks AOL America Onl
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82 A. Escribano and R. Pascual Madh
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84 1 Introduction S. Frey, J. Gramm
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86 develops the empirical methodolo
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Table 1 Sample descriptives Company
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90 comparability across stocks, we
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92 subtracting the deviations from
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94 market order distribution that c
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96 Table 2 First stage GMM results
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Table 4 First stage GMM results bas
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100 S. Frey, J. Grammig quotes on e
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102 S. Frey, J. Grammig approximate
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104 To obtain the estimates in the
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106 Hence, using the liquidity stat
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108 In the main text we discuss the
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Pierre Giot . Joachim Grammig How l
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How large is liquidity risk in an a
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134 A. D. Hall, N. Hautsch In this
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136 includes order book variables,
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138 An important determinant of liq
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140 The innovation term ei is compu
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Table 1 Order book characteristics
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144 data covering the normal tradin
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146 Table 3 Descriptive statistics
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Table 4 Fully specified ACI models
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Table 4 (continued) BHP NAB NCP TLS
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Table 5 ACI models without dynamics
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Table 6 ACI models without covariat
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160 specification which includes or
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162 5.2.5 The impact of the bid-ask
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164 contrast to those of Pascual an
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Roman Liesenfeld . Ingmar Nolte . W
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Modelling financial transaction pri
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200 W. B. Omrane, H. V. Oppens anal
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202 W. B. Omrane, H. V. Oppens of s
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204 The extrema detection method ba
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206 price 0.8565 0.8575 0.8585 0.85
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208 We distinguish three possible c
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210 originates. The most active tra
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212 Table 2 Predictability of the c
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214 6 Conclusion Using 5-min euro/d
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216 where σk is the standard devia
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218 If we meet a particular case su
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220 C.5. Triple bottom (TB) TB is c
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222 References W. B. Omrane, H. V.
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Juan M. Rodríguez-Poo · David Ver
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Semiparametric estimation for finan
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Page 270 and 271: 264 A. S. Tay, C. Ting More interes
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Page 274 and 275: 268 Acknowledgements Tay gratefully
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Page 278 and 279: 272 contrast, price responses to po
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Page 302 and 303: 296 V. Voev 2.1 A sample covariance
Page 304 and 305: 298 V. Voev Using the equicorrelate
Page 306 and 307: 300 V. Voev where �kl(t) is the (
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