Recomendaciones

of multiple correlation coefficients, or R2_{, for single regressor equations projecting fu-}

ture integrated (realized) volatility onto the corresponding (conditional expectation) forecasts (and a constant). In addition, we derive approximately optimal sampling frequency in terms of MSE of the prediction. This analysis parallels that of Ander- sen, Bollerslev and Meddahi (2004), in particular, they show that for n-period ahead forecasts, R2(RVdt+nh,nh,RVdt,h) = CovRVdt+nh,nh,RVdt,h 2 VarRVdt+nh,nh VarRVdt,h (3.9) 93

From the above equation we note that the relative R2_{-performance of different RV}

t,h measures only depends on the variances of RVdt,h, since CovRVdt+nh,nh,RVdt,h

= Cov (IVt+nh,nh, IVt,h) and Var

d

RVt+nh,nh

is fixed. As a result, whenever VarRVdA_t,h≥

VarRVdB_t,h then R2_(RVd

t+nh,nh,RVd A

t,h) ≤ R2(RVdt+nh,nh,RVd B

t,h). The above setting also applies to multiple regressor case, which will be relevant for the empirical analysis reported in the next sections, namely:

R2_dRV

t+nh,nh;RVdt,h,RVdt−h,h, . . . ,RVdt−lh,h

(3.10)

To proceed, we simplify the notation, as in Andersen, Bollerslev and Meddahi (2004), namely for a covariance-stationary random variable (yτ, zt) and an integer l, we let C(yτ, zt, l) denote the (l+ 1) vector defined by:

C(yτ, zt, l) = (Cov (yτ, zt),Cov (yτ, zt−1), . . . ,Cov (yτ, zt−l))0. (3.11)

Moreover, let M(zt, l) denote the (l+ 1)×(l+ 1) matrix whose (i, j)’th component is given by

M(zt, l)[i, j] = Cov (zt, zt+i−j) (3.12)

We can express then R2 _{for a regression of RV}

t+nh,nh onto a constant and the set of regressors (RVt,h, RVt−h,h, . . . , RVt−lh,h), l ≥0, denoted R2(RVt+nh,nh, RVt,h, l) as:

R2(RVt+nh,nh, RVt,h, l) =

C(.)0[M(RVt,h, l)]−1C(.) Var (RVt+nh,nh)

(3.13)

whereC(.) =C(RVt+nh,nh, RVt,h, l). The following result is shown in Appendix A.2:

Theorem 3.3.2 For multiple regressions and realized volatility estimators A, B, and C yieldingR2_{’s appearing in (3.13), and wheneverCov}d_RVi

t+a,a,RVd j t−δ,b = Cov (IVt+a,a, IVt−δ,b), 94

∀δ≥0 and i, j ={A, B, C} we have that:

R2RVdC_t+nh,nh,dRVA_t,h, l_≥R2RVdC_t+nh,nh,RVdB_t,h, l (3.14)

iff VarRVdA_t,h≤VarRVdB_t,h. (3.15)

With the above result we are able to proceed with the comparison of the impact of various volatility measures on forecasting performance. To apply Theorem 3.3.2 to our framework, we have to assume that the covariances between daily estimators are equal to the covariances between daily integrated volatilities. This assumption consists of two parts. The first part states that the overnight change in prices can be neglected. The implication of the second part is that all daily estimators that we consider contain the daily integrated volatility. In reality, however, the second assumption holds only asymptotically. The exact covariances (assuming daily integrated volatility can be consistently estimated without overnight returns) are shown in (A.38) and summarized in the consecutive table. All numerical results are obtained using these formulas.

Consider two groups of estimators: group A = _{RV, RVAC1} and group B =

{RV , RVT S}. We expect that group B estimators should have smaller variance both in noise-free and noisy environments. In the noise-free environment the variances are smaller given that variance of the discretization noise (appearing in (A.34)) averaged over subsamples is smaller than the variance of the discretization noiseMVar Zj/M+h,h

(appearing in (A.31)). As sampling and subsampling frequencies go to infinity, all co- variances converge to the integrated volatility covariance.

When microstructure noise is present, as the sampling frequency becomes large, the variances of group A estimators diverge to infinity. As a result, the R2_{s of the}

regressions converge to 0. Group B estimators feature a different pattern. These two estimators have asymptotically zero variance, and the two scales correction only

removes the bias from the estimator averaging over subsamples. However, if the bias is constant over time, it does not affect the predicting power of the regression but only changes the intercept. Thus, the performance of group B estimators should be approximately the same, with averaging over subsamples estimator performing better as the “sparse” grid M approaches M. Finally, in real-data applications, the variance of market microstructure noise is time-varying, which makes the two scales estimator preferable.

To appraise the performance of the realized volatility estimators, we compare the population predictive power of the estimators using three models suggested by An- dersen, Bollerslev and Meddahi (2004). The details of the models and the numerical values of the model parameters of interest such asλi and ai are available in Appendix A.2. Table 3.2 presents the sample results for models M1 – M3 (A.39 – A.41), for different frequencies, number of lags, zero and nonzero microstructure noise. Mar- ket microstructure noise is assumed i.i.d. with κ = 3, σ2

η = 0.03. The full version of the table is available online in Ghysels and Sinko (2006b). We consider three values for lags: 1, 15, 50; six values of the variance of market microstructure noise

σ2 = 0,0.005,0.01,0.015,0.02,0.025,0.03, and six values for the microstructure noise kurtosis κ= 1.5,2,2.5,3,3.5,4. The main findings are the following:

In the noise-free environment the averaging over subsampling estimator performs the best across all estimators. The two scales estimator produces slightly worse results. These two estimators outperform the plain realized volatility estimator and RVAC1 es-

timator. Moreover, the plain realized volatility estimator performs better thanRVAC1.

Infeasible linear regressions with integrated variance on the right hand side also perform better compared to the feasible ones. This finding goes along the same line as Theorem 3.3.2. Lower variance of the estimator implies higher R2 _{assuming all other theorem}

assumptions hold. As sampling frequency increases, the variance of the discretization

noise decreases andR2 _{of all estimators converge to the the R}2 _{of the infeasible regres-}

sion on integrated volatility (R2

IV). For all models fifteen lags are sufficient and further increase in the number of lags does not provide increase in R2_{s. The results are not}

surprising given the fact that the models have small number of independent parameters and exponential decay of the lag coefficients.

The results change in an environment with microstructure noise. Namely, they depend not only on microstructure noise variance, but also on its kurtosis. Forκ≤ 2, the plain realized volatility estimator (RV) performs better compared toRVAC1 and the

averaging over subsamples realized estimatorRV produces better results thanRVT S for all frequencies. Moreover, in absolute terms, when microstructure noise varianceσ2 _≥

0.015, all R2_{s monotonically decrease with the increase in the subsampling frequency}

for a given set of model parameters.

The aforementioned finding can be explained by the fact that, for this range of microstructure noise variances, the optimal sampling in terms of R2 _{becomes lower}

than five minutes. The main factor that impacts the performance ofR2 _{is the variance}

of the realized volatility estimators. The biases are captured by the constant terms of the regressions. Figure 3.1 and 3.2 show the optimal sampling frequencies that maximize R2 _{of the estimators, conditional on specific values for noise variance for}

κ= 1.5 andκ= 3. For large values of market microstructure noise variance increasing the number of lags is helpful. For some extreme cases the difference between theR2 _for

a regression with 15 lags and a regression with 50 lags is 10%. The explanatory power of the class A estimators monotonically decreases for all values of microstructure noise variance considered.

For κ > 2 the situation changes. First, for higher frequencies, the plain realized volatility R2

pl is smaller than R2AC1. Second, R 2

T S becomes larger than R2av for some range of near-optimal frequencies around the maximum of R2_{. This result supports}

the findings of A¨ıt-Sahalia, Mykland and Zhang (2005b) who show that the two scales estimator performs better with the optimal frequency compared to the “second best” averaging over subsamples estimator. However, for the non-optimal frequencies the averaging over subsamples estimator outperforms the two scales one.

The Meddahi (2001) model also allows us to derive an approximately optimal fre- quency. However, analytical solution even in this case can be obtained only for the simplest cases of RVAC1 and “plain” RV estimators. For the other two cases the so-

lution can be found as a root of third and fourth power polynomials. For these two it is more convenient to express the solution in terms of φ ≡ M /M, φ ∈ (0,1), with larger M corresponding to higher frequency. As a result, φ _' 0 corresponds to the case of the lowest possible frequency, φ _' 1 corresponds to the case of the highest frequency. In our analysis only M changes. There are two major differences between the optimal sampling we derive and the optimal sampling derived in Bandi and Russel (2005). First, for prediction purposes, the bias of the estimator does not matter as long as it is constant over time. As a result, using Theorem 3.3.2, the optimal sampling fre- quency is the one that minimizes the variance of the estimators. Second, we are using a stochastic volatility model while Bandi and Russel (2005) assume that the variance is deterministic function of time. The following proposition summarizes our results:

Proposition 3.3.1 Let Assumption 3.3.1 and the conditions of Theorem 3.3.2 hold,

Mh _' 1, m _' M/M. Also, lets define Q = Pp_i=0a2

i Then the approximate optimal

sampling frequencies for the RV, RVAC1, RVT S and RV estimators are: MRV ' s Q 2κσ4 η , MRVAC1 ' s 3Q 4σ4 η M_RV 'arg minn8a0φσ2η+ 4φ2Mκση4−2φση4(κ−1) +Q(1−φ){2M(2−φ)−(1/φ+ 1)} 3M2_φ MRVT S 'arg min Q_{2M(2₋φ)₋(1/φ+ 1)_} 3M2_φ(1−φ) − 2φσ 4 η(κ−1) 1−φ + 8φa0ση2+ 8φ2Mση4 (1−φ)2 (3.16)

Proof: see Appendix A.42.

We note from the above proposition that MRV/MRVAC₁ '

p

2/(3κ).Although the result is based on approximations, it matches with exact computations. In particular, consider the findings reported in Figures 3.1 and 3.2 where we computed the R2 _{as a}

function of sampling frequency, second and fourth moments of market microstructure noise and mean and variance of realized volatility directly for three models M1 – M3 (A.39 – A.41) using an exact formula for the variance. The optimal sampling frequency forRVAC1 estimator is always greater than the optimal sampling frequency for “plain”

realized volatility estimator RV, and the difference increases as kurtosis κ of market microstructure noise increases.

We showed in Theorem 3.3.2 that the optimum in terms of MSE of prediction is equivalent to the optimum in terms of minimum of estimator’s variance. Comparing optimal frequencies in terms of MSE of prediction and optimal frequency in terms of

MSE of estimatorwe can conclude that they will coincide whenever the estimators are unbiased. Under Assumption 3.3.1, RVAC1 and RVT S are unbiased. In this case the

minimum in terms of MSE of estimator coincides with the minimum in terms of variance of the estimator which is equivalent to the minimum in terms of MSE of prediction. For

all other cases, when the bias is non-zero and is linearly increasing with the sampling frequency, the optimal sampling frequency in terms of MSE of prediction will be higher than the optimal sampling frequency in terms of MSE of estimator, as the increase in the bias introduces an additional penalty as the sampling frequency increases. Also, note that for RV and RV the bias is the same and equal to 2Mσ2

η.