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216 where σk is the standard deviations for the observations that occur within the window k. However, the optimal bandwidth for Silverman (1986) involves a fitted function which is too smooth. In other words this optimal bandwidth places too much weight on prices far away from any given time t, inducing too much averaging and discarding valuable information in local price movements. Like Lo et al. (2000), through trial and error, we found that an acceptable solution to this problem is to use a bandwidth equal to 20% of hopt,k : B. Extrema detection methods h ¼ 0:2 hopt;k: (18) Technical details for both extrema detection methods and projection procedure are presented below: B.1. M1 M1 is the extrema detection method using the close prices. After smoothing the data by estimating the Nadaraya–Watson kernel function, ^mhðXPj;k Þ, we compute maxima and minima respectively noted by max ^mhðXP j;k Þ and min ^mhðXP j;k Þ : max ^mh XPj;k ¼ ^mh XPj;k min ^mh XPj;k ¼ ^mh XPj;k S ^m0 h S ^m0 h XPj;k ¼þ1; S ^m0 h XPj;k ¼ 1; S ^m0 h XPjþ1;k ¼ 1 XPjþ1;k ¼þ1 ; where S(X) is the sign function, equal to +1 (−1) when the sign of X is positive (negative), and ^m 0 h XPj;k is the first derivative of the kernel function ^mh XPj;k .By construction we obtain alternate extrema. We denote respectively by tM ^mh XPj;k and tm ^mh XPj;k the moments correspondent to detected extrema such that: tM ^mh XPj;k ¼ j j 2 max n o n ^mhðXP Þ j;k o (19) : (20) tm ^mh XPj;k ¼ j j 2 min ^mhðXP j;k Þ After recording the moments of the detected extrema we realize an orthogonal projection of selected extrema, from the smoothing curve, to the original one. We deduce the corresponding extrema to construct the series involving both maxima, and minima, minPj;k such that: maxPj;k W. B. Omrane, H. V. Oppens maxPj;k ¼ max P tM ð ^mhðXP j;k ÞÞ 1;k; P tM ð ^mhðXP j;k ÞÞ;k; P tM ð ^mhðXP j;k ÞÞþ1;k minPj;k ¼ min P tmð ^mhðXP j;k ÞÞ 1;k; P tmð ^mhðXP j;k ÞÞ;k; P tmð ^mhðXP j;k ÞÞþ1;k : For each window k we get alternate maxima and minima. This is assured by the bandwidth h which provide at least two time intervals between two consecutive
The performance analysis of chart patterns 217 extrema. The final step consists to scan the extrema sequence to identify an eventual chart pattern. If the same sequence of extremum was observed in more than one window, only the first sequence is retained for the recognition study to avoid the duplication of results. B.2. M2 M2 is the extrema detection method built on high and low prices. According to this method, maxima and minima have to be detected onto separate curves. Maxima on high prices curve and minima on the low one. Let Ht and Lt (t =1,...,T), be respectively the series for the high and the how prices, and k a window containing l regularly spaced time intervals such that: Hj;k fHtjktkþl1g (21) Lj;k fLtjktkþl1g; (22) j =1,...,l and k =1,...,T−l+1. We smooth these series through the kernel estimator detailed in Appendix A to obtain ^mh XHj;k and ^mh XLj;k . We detect maxima on the former series and minima on the latter one in order to construct two separate extrema series max ^mhðXH j;k Þ and min ^mhðXL j;k Þ such that: max ^mhðXH j;k Þ ¼ ^mh XHj;k S ^m 0 h min ^mhðXL j;k Þ ¼ ^mh XLj;k S ^m 0 h XHj;k ¼þ1; S ^m0 h XLj;k ¼ 1; S ^m0 h XHjþ1;k ¼ 1 XLjþ1;k ¼þ1 ; where S(x) is the sign function defined in the previous Section. We record the moments for such maxima and minima, denoted respectively by tM ^m XHj;k h and tm ^m XLj;k h and we project them on the original high and how curves to deduce the original extrema series maxHj;k and minLj;k , such that: maxHj;k ¼ max H tM ð ^mhðXH j;k ÞÞ 1;k; H tM ð ^mhðXH j;k ÞÞ;k; H tM ð ^mhðXH j;k ÞÞþ1;k minLj;k ¼ min L tmð ^mhðXL j;k ÞÞ 1;k; L tmð ^mhðXH j;k ÞÞ;k; L tmð ^mhðXH j;k ÞÞþ1;k : However, this method does not guarantee alternate occurrences of maxima and minima. It is easy to observe, in the same window k, the occurrence of two consecutive minima on the low series before observing a maximum on high series. To resolve this problem we start by recording the moments for the selected maxima on high curve, tM(Hj,k), and minima in low curve, tm(Lj,k). Then we select, for window k the first extremum from these two series, E1,k , and its relative moment, , such that: tE1;k tE1;k ¼ min t tM Hj;k ; tm Hj;k (23) E1;k ¼ maxH j;k [ minLj;k j ¼ tE1;k : (24)
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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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Page 175 and 176: Modelling financial transaction pri
Page 203: Modelling financial transaction pri
Page 206 and 207: 200 W. B. Omrane, H. V. Oppens anal
Page 208 and 209: 202 W. B. Omrane, H. V. Oppens of s
Page 210 and 211: 204 The extrema detection method ba
Page 212 and 213: 206 price 0.8565 0.8575 0.8585 0.85
Page 214 and 215: 208 We distinguish three possible c
Page 216 and 217: 210 originates. The most active tra
Page 218 and 219: 212 Table 2 Predictability of the c
Page 220 and 221: 214 6 Conclusion Using 5-min euro/d
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Page 226 and 227: 220 C.5. Triple bottom (TB) TB is c
Page 228 and 229: 222 References W. B. Omrane, H. V.
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Page 262 and 263: 256 simply aggregated. Even after a
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Table 4 Estimated probabilities of
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268 Acknowledgements Tay gratefully
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270 This paper explores the effect
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272 contrast, price responses to po
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274 Fig. 1 Continuous line is the T
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276 the β’s can form a convex sh
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278 fixing some intervals around th
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280 . That is for a given absolute
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282 30 25 20 15 10 5 CC UNEMW ISM U
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Appendix Table A1 Consumer confiden
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Table A3 Non-farm payrolls • ✓
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Table A5 Weekly unemployment claims
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290 Table A8 Retail sales • ✓ 1
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292 Table A12 GDP, BI, TB and PI Re
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294 V. Voev with the problem of how
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296 V. Voev 2.1 A sample covariance
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298 V. Voev Using the equicorrelate
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300 V. Voev where �kl(t) is the (
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302 V. Voev where hkl,k ′ l ′ i
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304 V. Voev is 1.9%. From the daily
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306 V. Voev ACF ACF ACF 0.5 0.5 0.5
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308 V. Voev autocorrelated. Indeed,
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310 V. Voev 5 Conclusion Table 2 Ro
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312 V. Voev Engle R (1982) Autoregr
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