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300 V. Voev where �kl(t) is the (k, l) element of the �(t) process. With this notation, we will now interpret the ‘true’ covariance matrix as � t+1 �t+1 = t �(u)du. (18) Thus the covariance matrix at time t + 1 is the increment of the integrated covariance matrix of the continuous local martingale from time t to time t + 1. The realized covariance as defined in expression (1) consistently estimates �t+1 as given in Eq. (18). Furthermore, Barndorff-Nielsen and Shephard (2004) show that under a set of regularity conditions the realized covariation matrix follows asymptotically, as M →∞, the normal law with N × N matrix of means � t+1 t �(u)du. The asymptotic covariance of � � √ M � RC t+1 − � t+1 t �(u)du is �t+1, aN 2 × N 2 array with elements �� t+1 � �t+1 = {�kk ′(u)�ll ′(u) + �kl ′(u)�lk ′(u)} du t k,k ′ ,l,l ′ =1,...,N Of course, this matrix is singular due to the equality of the covariances in the integrated covariance matrix. This can easily be avoided by considering only its unique lower triangular elements, but for our purposes it will be more convenient to work with the full matrix. The result above is not useful for inference, since the matrix �t+1 is not known. Barndorff-Nielsen and Shephard (2004) show that a consistent, positive semi-definite estimator is given by a random N 2 ×N 2 matrix: Ht+1 = M� j=1 xj,t+1x ′ j,t+1 − 1 2 M−1 � j=1 � xj,t+1x ′ j+1,t+1 + xj+1,t+1x ′ � j,t+1 , (19) � where xj,t+1 = vec r j r t+ M ′ t+ j � M and the vec operator stacks the columns of a matrix into a vector. It holds that MHt+1 p → �t+1 with M →∞. With the knowledge of this matrix, we can combine the asymptotic results for the realized covariance, with the result in Eq. (7) to compute the estimates for πt, ρt and νt. For the equicorrelated matrix F we have that4 � fij =¯r σ (RC) ii σ (RC) jj , where ¯r is the average value of all pairwise correlations, implied by the realized covariance matrix, and σ (RC) is the (i,j) element of the realized covariance matrix. Thus �, ij the population equicorrelated matrix, has a typical element φij =¯̺ √ σiiσjj , where σij is the (i,j) of the true covariance matrix � and ¯̺ is the average correlation 4 In the following exposition, the time index is suppressed for notational convenience. .
Dynamic modelling of large-dimensional covariance matrices 301 implied by it. Substituting σ (RC) ij for sij in Eq. (7) and multiplying by M gives for the optimal shrinkage intensity: Mα ∗ = �Ni=1 � � �√Mσ � � Nj=1 (RC) √Mfij, Var ij − Cov √ Mσ (RC) �� ij �Ni=1 � � � Nj=1 Var fij − σ (RC) � ij + (φij − σij) 2 � . (20) Note that this equation resembles expression (8). The only difference is the scaling by √ M instead of √ T , which is due to the √ M convergence. In this case πt, the first summand in the numerator, is simply the sum of all diagonal elements of �t. By using the definition of the equicorrelated matrix, it can be shown that the second term, ρt, can be written as (suppressing the index t) ρ = N� �√ � (RC) AVar Mσ ii i=1 + N� N� i=1 j=1,j�=i � √M � ACov ¯r σ (RC) ii σ (RC) jj , √ Mσ (RC) � ij . (21) Applying the delta method the second term can be expressed as 5 ¯r 2 ⎛� � ⎜ � � ⎝ σ (RC) jj σ (RC) ii � � � + � σ (RC) ii σ (RC) jj �√ (RC) ACov Mσ ii , √ Mσ (RC) � ij �√ (RC) ACov Mσ jj , √ Mσ (RC) � ij ⎞ ⎠ . From this expression we see that ρ also involves summing properly scaled terms of the � matrix. In the denominator of Eq. (20), the first term is of order O(1/M), and the second one is consistently estimated by ˆν = �N � � Nj=1 i=1 fij − σ (RC) �2 ij . Since we have a consistent estimator for �, we can now also estimate π and ρ. In particular, we have ˆπ = ˆρ = N� N� i=1 j=1 N� i=1 hij,ij hii,ii + ¯r 2 5 cf. Ledoit and Wolf (2004). N� � N� � � � i=1 j=1 (RC) σ jj σ (RC) hii,ij + ii � � � � (RC) σ ii σ (RC) hjj,ij, jj
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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 262 and 263: 256 simply aggregated. Even after a
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Page 266 and 267: 260 together with the restrictions
Page 268 and 269: Table 3 Estimated probabilities of
Page 270 and 271: 264 A. S. Tay, C. Ting More interes
Page 272 and 273: Table 4 Estimated probabilities of
Page 274 and 275: 268 Acknowledgements Tay gratefully
Page 276 and 277: 270 This paper explores the effect
Page 278 and 279: 272 contrast, price responses to po
Page 280 and 281: 274 Fig. 1 Continuous line is the T
Page 282 and 283: 276 the β’s can form a convex sh
Page 284 and 285: 278 fixing some intervals around th
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Page 288 and 289: 282 30 25 20 15 10 5 CC UNEMW ISM U
Page 290 and 291: Appendix Table A1 Consumer confiden
Page 292 and 293: Table A3 Non-farm payrolls • ✓
Page 294 and 295: Table A5 Weekly unemployment claims
Page 296 and 297: 290 Table A8 Retail sales • ✓ 1
Page 298 and 299: 292 Table A12 GDP, BI, TB and PI Re
Page 300 and 301: 294 V. Voev with the problem of how
Page 302 and 303: 296 V. Voev 2.1 A sample covariance
Page 304 and 305: 298 V. Voev Using the equicorrelate
Page 308 and 309: 302 V. Voev where hkl,k ′ l ′ i
Page 310 and 311: 304 V. Voev is 1.9%. From the daily
Page 312 and 313: 306 V. Voev ACF ACF ACF 0.5 0.5 0.5
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Page 316 and 317: 310 V. Voev 5 Conclusion Table 2 Ro
Page 318: 312 V. Voev Engle R (1982) Autoregr
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