Online Appendix. A Simulation Details. A.1 Simulation Design. A.2 Practical Issues
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1 Online Appendix A Simulation Details A.1 Simulation Design The simulation design follows the approach in Barndorff-Nielsen et al. (211). Specifically, we simulate over the interval t [,1], which corresponds to 6.5 hours of second-by-second trading activity and a number N = 234 of intervals. The simulations are performed using 1 replications. The efficient price process is given by the following bivariate stochastic volatility model: dp j,t = µdt+ρσ j,t dw j,t + 1 ρ 2 σ j,t db t, p j, = σ j,t = exp(β +β 1 v t ), dv j,t = αv j,t dt+dw j,t, for j = 1,2, where dw 1, dw 2 and db are standard independent Brownian motions. In this setup, the second term of the price process is an idiosyncratic factor, while the third term is a common factor, whose strength is determined by the squared root of the correlation between p 1 and p 2, i.e. by 1 ρ 2. There is a perfect correlation (leverage) between the innovations of the idiosyncraticfactor and σ j, i.e. corr(ρσ j dw j,σ j ) = 1, while the leveragebetween the efficient price process and v j is ρ. The parametrization is also as in Barndorff-Nielsen et al. (211). We set [ µ =.3,β 1 =.125,α =.25,ρ =.3 and β = β1/2α, 2 which is a normalization ensuring that ] 1 E σ2 j,u du = 1. To initiate the process v j each day we use its stationary distribution, v j, N(,( 2a) 1 ). To generate the bivariate noise process, we start with an i.i.d. specification, N Noise 1: υ j N(,ωj) 2 with ωj 2 = ξ 2 N 1 σj,i 4. This specification means that the variance of the noise moves together with the spot volatility of the efficient price process. We assign two values to the noise-to-signal ratio.1 (low noise regime), and.1 (high noise regime). To pose an additional challenge we also consider two specifications, which allow for serial dependence (AR(1) and MA(1)) in the noise. Noise 2: υ j,t = φ j υ j,t 1 +υ j,t for φ 1 =.5 and φ 2 =.2. Noise 3: υ j,t = υ j,t +θ j υ j,t 1 for θ 1 =.7 and θ 2 =.4. To obtain different scenarios in terms of trading activity we generate random Poisson sampling times with constant intensities η 1 and η 2 for asset 1 and 2, respectively. This allows us to study the impact of non-synchronous trading and assess the two synchronization schemes, refresh time versus tick time. We consider 36 scenarios for eachnoisemodelbyvarying1/η 1 and 1/η 2 from1to6. Forexample, thepair(1/η 1,1/η 2 )=(1/,1/ )=(1,6) means that on average p 1 and p 2 is observed every 1 and 6 seconds, respectively. i=1 A.2 Practical Issues In our approach there are two parameters that need to be chosen: Q - the number of lags for the (cross) autocovariance function of the noise processes, and S - the number of subsamples. To choose S in the univariate 43
2 case we rely on results derived in Nolte & Voev (212), who provide the theoretical foundations 18 of how to choose S in the i.i.d. noise case. We denote this S as S and we use it as an indication of how to choose S for all noise models in both the univariate and multivariate estimation. The choice of Q should be data-driven as it depends on the serial dependence of the noise process. In practise we start with a base scenario, which sets S = S and Q = 5. Then, we perform a sensitivity analysis 19 to test whether our results can be further improved by selecting different values of S and Q. The values that minimize the RMSE in each case consist our optimal scenario. Specifically, the optimal scenario for the univariate estimation is S = S and Q = for the i.i.d. noise model and Q = 2 for the 2 dependant noise cases. The optimal scenario for the multivariate estimation is S = S and Q = across all noise models. Although the aforementioned sensitivity analysis is difficult to be implemented in a non-simulation based exercise, it provides insightful findings on how to choose S and Q in practise. On the one hand, a Q larger than might be beneficial when noise exhibits high dependence, especially for assets with very active trading activity. However, one might be conservative, as variance also increases with Q. On the other hand, a higher S might further improve the estimation precision when the trading frequency of the price processes is low. In our sensitivity analysis this occurs in the region of, > 5 mainly for the i.i.d. noise case. Values higher than S = 1.5S do not improve the estimation precision. A few remarks on the implementation of the competing estimators are required. The multivariate Parzen kernel is implemented in the spirit of Barndorff-Nielsen et al. (211). Specifically, we use a refresh time synchronization scheme, with Parzen weights, and with bandwidth H = 3.51n 3/5 mean j=1,...,d (φ 2 j /RV j,1/9) 2/5.The nominator inside the parenthesis approximates the spot variance and is obtained by averaging over subsampled realized variances computed at 6-second grid and dividing by twice the number of returns. The denominator is the realized variance estimator based on 2 min subsampling returns, an approximation for the squared root of the integrated quarticity. To calculate the flat-top realized kernel we follow the recommendations of Varneskov (216). We use the refresh time sampling scheme, with a Tukey-Hanning type kernel function and tuning parameters c = H 1/2 and b = 2. The parameter c controls the flat-top region of the weighting function. Finally, we adopt all Varneskov (216) suggestions for the calculation of the bandwidth. ( 1 ln(ln(n)) ln(n) ), and α = 18 Briefly, optimal S is given by S = α N β 1, where N is the number of intraday prices, β = ω 4 (π 2 4(γ 2+2γ 1)). Intuitively, α is a noise-to-signal ratio and resembles to a similar quantity which determines IQ the length of the bandwidth in the realized kernel approach. 19 We do not report these result to save space. However, they are available upon request. 44
3 B Performance Evaluation Metrics In this section, we describe performance evaluation measures we used in out-of-sample asset allocation in Section 5.2 to Section 5.4. We consider five performance measures, i.e. ex post portfolio volatility (VOL), Sharpe ratio (SR), turnover ratio (TO), portolio concentration (CO), and short positions (SP) commonly used in the literature (DeMiguel et al. 213, Hautsch et al. 215, Bollerslev et al. 216). First, we construct the portfolio volatility measure. VOL 2 t+1 = w t RCOV t+1w t (15) where w t is the optimal GMV portfolio weight we obtained at time t, RCOV t,t+1 is five minutes sub-sampled realized covariance matrix at time t + 1, the annualized mean of the square root of VOL 2 t+1 is the ex post portfolio volatility measure (VOL). The lower the portfolio volatility, the better the performance. Second, we compute Sharpe ratio. SR = µ p σ p (16) where µ p and σ p are annualized unconditional mean and volatility of the obtained GMV portfolio. 2 The higher the Sharpe ratio, the better the portfolio performance. Next, we report portfolio turnover ratio. TO = N wt i wi t 1 i=1 1+rt 1 i 1+r p t 1 (17) where w t and w t 1 are optimal GMV portfolio weights at time t and t 1, rt 1 i and rp t 1 are indiviudal stock and portfolio returns at time t 1. Portfolio turnover is closely linked to transaction costs, and hence an optimal portfolio should avoid extreme high turnover in practice. Another portfolio performance measure is portfolio concentration. N CO = ( (wt i )2 ) 1/2 (18) i=1 This measure quantifies the concentrations of weights among different stocks. In practice, portfolios should avoid taking extreme positions in particular assets. The special case is the equal weight portfolio, which provides a lower bound for this measure. Finally, we also consider short selling positions. SP = N wt1 i w i t < (19) i=1 Also, taking too much short selling positions are costly and risky, and hence practical implementation should control for short selling positions. For equal weighted portfolio, the short selling position is zero by construction. 2 Here we closely follow DeMiguel et al. (213) and use raw returns to compute Sharpe ratios, our main results remain valid when excess returns are used. 45
4 C Simulation Tables and Graphs Table B1: Noise Model 1 (i.i.d.) - Covariance (V12) - BIAS Panel A. OLS V12 (base scenario) Panel B. OLS V12 (optimal Q and S) Panel C. HY V Panel D. HYS V Panel E. MK V Panel F. FTRK V
5 Table B2: Noise Model 1 (i.i.d.) - Covariance (V12) - RMSE Panel A. OLS V12 (base scenario) Panel B. OLS V12 (optimal Q and S) Panel C. HY V Panel D. HYS V Panel E. MK V Panel F. FTRK V
6 Table B3: Noise Model 1 (i.i.d.) - Variance of Process 1 (V1) - BIAS Panel A. OLS V1 (base scenario) Panel B. OLS V1 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
7 Table B4: Noise Model 1 (i.i.d.) - Variance of Process 1 (V1) - RMSE Panel A. OLS V1 (base scenario) Panel B. OLS V1 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
8 Table B5: Noise Model 1 (i.i.d.) - Variance of Process 2 (V2) - BIAS Panel A. OLS V2 (base scenario) Panel B. OLS V2 (optimal Q and S Panel C. UK V Panel D. MK V Panel E. FTRK V
9 Table B6: Noise Model 1 (i.i.d.) - Variance of Process 2 (V2) - RMSE Panel A. OLS V2 (base scenario) Panel B. OLS V2 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
10 Table B7: Noise Model 2 (AR(1)) - Covariance (V12) - BIAS Panel A. OLS V12 (base scenario) Panel B. OLS V12 (optimal Q and S) Panel C. HY V Panel D. HYS V Panel E. MK V Panel F. FTRK V
11 Table B8: Noise Model 2 (AR(1)) - Covariance (V12) - RMSE Panel A. OLS V12 (base scenario) Panel B. OLS V12 (optimal Q and S) Panel C. HY V Panel D. HYS V Panel E. MK V Panel F. FTRK V
12 Table B9: Noise Model 2 (AR(1)) - Variance of Process 1 (V1) - BIAS Panel A. OLS V1 (base scenario) Panel B. OLS V1 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
13 Table B1: Noise Model 2 (AR(1)) - Variance of Process 1 (V1) - RMSE Panel A. OLS V1 (base scenario) Panel B. OLS V1 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
14 Table B11: Noise Model 2 (AR(1)) - Variance of Process 2 (V2) - BIAS Panel A. OLS V2 (base scenario) Panel B. OLS V2 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
15 Table B12: Noise Model 2 (AR(1)) - Variance of Process 2 (V2) - RMSE Panel A. OLS V2 (base scenario) Panel B. OLS V2 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
16 Table B13: Noise Model 3 (MA(1)) - Covariance (V12) - BIAS Panel A. OLS V12 (base scenario) Panel B. OLS V12 (optimal Q and S) Panel C. HY V Panel D. HYS V Panel E. MK V Panel F. FTRK V
17 Table B14: Noise Model 3 (MA(1)) - Covariance (V12) - RMSE Panel A. OLS V12 (base scenario) Panel B. OLS V12 (optimal Q and S) Panel C. HY V Panel D. HYS V Panel E. MK V Panel F. FTRK V
18 Table B15: Noise Model 3 (MA(1)) - Variance of Process 1 (V1) - BIAS Panel A. OLS V1 (base scenario) Panel B. OLS V1 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
19 Table B16: Noise Model 3 (MA(1)) - Variance of Process 1 (V1) - RMSE Panel A. OLS V1 (base scenario) Panel B. OLS V1 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
20 Table B17: Noise Model 3 (MA(1)) - Variance of Process 2 (V2) - BIAS Panel A. OLS V2 (base scenario) Panel B. OLS V2 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
21 Table B18: Noise Model 3 (MA(1)) - Variance of Process 2 (V2) - RMSE Panel A. OLS V2 (base scenario) Panel B. OLS V2 (optimal Q and S) Panel C. UK V Panel D. MK V Panel E. FTRK V
22 (h) OLS V (i) HY V12 (j) HYS V (k) MK V12 (l) FTRK V12 Graph B1: Noise Model 1 (i.i.d.) - Covariance (V12) - RMSE. 64
23 (a) OLS V (b) HY V12 (c) HYS V (d) MK V12 (e) FTRK V12 Graph B2: Noise Model 1 (i.i.d.) - Covariance (V12) - BIAS. 65
24 (a) OLS V1 (b) UK V (c) MK V1 (d) FTRK V1 Graph B3: Noise Model 1 (i.i.d.) - Variance of Process 1 (V1) - RMSE. 66
25 (a) OLS V1 (b) UK V (c) MK V1 (d) FTRK V1 Graph B4: Noise Model 1 (i.i.d.) - Variance of Process 1 (V1) - BIAS. 67
26 (a) OLS V2 (b) UK V (c) MK V2 (d) FTRK V2 Graph B5: Noise Model 1 (i.i.d.) - Variance of Process 2 (V2) - RMSE. 68
27 (a) OLS V2 (b) UK V (c) MK V2 (d) FTRK V2 Graph B6: Noise Model 1 (i.i.d.) - Variance of Process 2 (V2) - BIAS. 69
28 (a) OLS V (b) HY V12 (c) HYS V (d) MK V12 (e) FTRK V12 Graph B7: Noise Model 2 (AR(1)) - Covariance (V12) - RMSE. 7
29 (a) OLS V (b) HY V12 (c) HYS V (d) MK V12 (e) FTRK V12 Graph B8: Noise Model 2 (AR(1)) - Covariance (V12) - BIAS. 71
30 (a) OLS V1 (b) UK V (c) RK V1 (d) FTRK V1 Graph B9: Noise Model 2 (AR(1)) - Variance of Process 1 (V1) - RMSE. 72
31 (a) OLS V1 (b) UK V (c) MK V1 (d) FTRK V1 Graph B1: Noise Model 2 (AR(1)) - Variance of Process 1 (V1) - BIAS. 73
32 (a) OLS V2 (b) UK V (c) MK V2 (d) FTRK V2 Graph B11: Noise Model 2 (AR(1)) - Variance of Process 2 (V2) - RMSE. 74
33 (a) OLS V2 (b) UK V (c) MK V2 (d) FTRK V2 Graph B12: Noise Model 2 (AR(1)) - Variance of Process 2 (V2) - BIAS. 75
34 (a) OLS (b) HY V12 (c) HYS V (d) MK V12 (e) FTRK V12 Graph B13: Noise Model 3 (MA(1)) - Covariance (V12) - RMSE. 76
35 (a) OLS V (b) HY V12 (c) HYS V (d) MK V12 (e) FTRK V12 Graph B14: Noise Model 3 (MA(1)) - Covariance (V12) - BIAS. 77
36 (a) OLS V1 (b) UK V (c) MK V1 (d) FTRK V1 Graph B15: Noise Model 3 (MA(1)) - Variance of Process 1 (V1) - RMSE. 78
37 (a) OLS V1 (b) UK V (c) MK V1 (d) FTRK V1 Graph B16: Noise Model 3 (MA(1)) - Variance of Process 1 (V1) - BIAS. 79
38 (a) OLS V2 (b) UK V (c) MK V2 (d) FTRK V2 Graph B17: Noise Model 3 (MA(1)) - Variance of Process 2 (V12) - RMSE. 8
39 (a) OLS V2 (b) UK V (c) MK V2 (d) FTRK V2 Graph B18: Noise Model 3 (MA(1)) - Variance of Process 2 (V12) - BIAS. 81
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