Mancino_FourierForCovarianceEstimation

7/30/2019 Mancino_FourierForCovarianceEstimation

1/15Electronic copy available at: http://ssrn.com/abstract=1301031

Dynamic portfolio management: an application of Fourier

method for covariance estimation

Maria Elvira Mancino, Elena Rapiniand Simona Sanfelici

Abstract

The economic benefit of applying the Fourier covariance estimation methodology overother estimators in the presence of market microstructure noise is studied from the perspec-tive of an asset-allocation decision problem. We find that using Fourier methodology yieldsstatistically significant gains.

JEL: G11, C14, C22.Keywords: nonparametric covariance estimation, non-synchronicity, microstructure, Fourier

analysis, optimal portfolio choice.

1 Introduction

The recent availability of large high frequency financial data sets potentially provides a richsource of information about asset price dynamics. Specifically, nonparametric variance/covariancemeasures constructed by summing intra-daily return data (i.e. realized variances and covari-ances) have the potential to provide very accurate estimates of the underlying quadratic vari-

ation and covariation and, as a consequence, accurate estimation of betas for asset pricing,index autocorrelation, lead-lag patterns. These measures, however, have been shown to be sen-sitive to market microstructure noise inherent in the observed asset prices. Moreover, it is wellknown from [Epps, 1979] that the non-synchronicity of observed data leads to a bias towardszero in correlations among stocks as the sampling frequency increases. Motivated by these dif-ficulties, some modifications of realized covariance type estimators have been proposed in theliterature: [Martens, 2004], [Hayashi and Yoshida, 2005], [Voev and Lunde, 2007], [Large, 2007],[Barndorff-Nielsen and al., 2008].

A different methodology has been proposed in [Malliavin and Mancino, 2002], which isexplicitly conceived for the multivariate analysis. This method is based on Fourier analysisand does not rely on any data synchronization procedure but employs all the available data.

[Mancino and Sanfelici, 2008a] show that the univariate Fourier estimator is robust to marketmicrostructure effects. The analysis is extended to the multivariate case in [Mancino and Sanfelici, 2008b],where both the non-synchronicity issue and the effect of (dependent) microstructure noise aretaken into account.

Most of the works concerning the comparison of the efficiency of different variance/covariancesestimators consider only simple statistics such as bias and mean squared error (MSE). In this

DiMaD, University of Firenze, Italy, [email protected] Leasing Banca, Spa, Firenze, Italy, [email protected]. of Economics, University of Parma, Italy, [email protected]

1


2/15Electronic copy available at: http://ssrn.com/abstract=1301031

regard, among the most recent papers [Voev and Lunde, 2007] and [Griffin and Oomen, 2006]investigate the properties of three covariance estimators, namely realized covariance, realized co-variance plus lead- and lag-adjustments, and the covariance estimator by [Hayashi and Yoshida, 2005],when the price observations are subject to non-synchronicity and contaminated by (i.i.d.) mi-crostructure noise. They conclude that the ranking of the covariance estimators in terms ofefficiency depends crucially on the level of microstructure noise. [Gatheral and Oomen, 2007]compare twenty realized variance estimators using simulated data and find that the best vari-ance estimator is not always the one suggested by theory. The theoretical properties of theFourier estimator are studied by [Mancino and Sanfelici, 2008a, Mancino and Sanfelici, 2008b],who show that the Fourier estimator of covariance is not significantly affected by the microstruc-ture noise.

Nevertheless this approach to the comparison of covariance estimators does not have aneconomic basis and treats overestimates and underestimates of volatility of the same magnitudeas equally important. In this paper we consider the gains offered by the Fourier estimatorover other covariance measures from the perspective of an asset-allocation decision problem,following the approach of [Fleming et al., 2001], [Engle and Colacito, 2006], [Bandi et al., 2006]and [De Pooter at al., 2008] who study the impact of volatility timing versus unconditionalmean-variance efficient static asset allocation strategies and of selecting the appropriate samplingfrequency or choosing between different bias and variance reduction techniques for the realizedcovariance matrices. A preliminary result we prove here concerns the positive semi-definitenessof the estimated covariance matrix using Fourier methodology, when the Fejer kernel is used.This property has important consequences in the asset allocation framework. An investor isassumed to choose his/her portfolio to minimize variance subject to a required return constraints.Investors with different covariance forecasts will hold different portfolios. Correct covarianceinformation will allow the investor to achieve lower portfolio volatility. Therefore we study theforecasting power of the Fourier estimator and of other alternative realized variance measures inthe context of an important economic metric, namely the long-run utility of a conditional mean-

variance investor rebalancing his/her portfolio each period. We show that the Fourier estimatorcarefully extracts information from noisy high-frequency asset price data for the purpose ofrealized variance/covariance estimation and allows for non-negligible utility gains in portfoliomanagement.

Inspired by [Fleming et al., 2001, Bandi et al., 2006], we construct daily variance/covarianceestimates using the Fourier method and the method proposed by [Hayashi and Yoshida, 2005],as well as estimates obtained by using conventional (in the existing literature) 1-, 5- and 10-minute intervals and MSE-based optimally sampled continuously-compounded returns for therealized measures. From each of these series, we derive one-day-ahead forecasts of the vari-ance/covariance matrix. A conditional mean-variance investor can use these forecasts to opti-mally rebalance his/her portfolio each period. We compare the investors long-run utility for

optimal portfolio weights constructed from each forecast. Our simulations show that the gainsyielded by the Fourier methodology are statistically significant and can be economically large,although the realized covariance with one lead-lag bias correction and suitable sampling fre-quency can be competitive. The analysis is conducted through Monte Carlo simulations, usingthe programming language Matlab.

The paper is organized as follows. In section 2 we describe the Fourier estimation methodol-ogy and we prove the positive semi-definiteness of the Fourier covariance matrix. In section 3 weexplain the asset allocation framework and metric to evaluate the economic benefit of differentcovariance forecasts. Section 4 presents several numerical experiments to value the gains offered

2


3/15

by Fourier estimator methodology in this context. Section 5 concludes.

2 Some properties of the Fourier estimator

The Fourier method for estimating co-volatilities was proposed in [Malliavin and Mancino, 2002]

having in mind the difficulties arising in the multivariate setting when applying the quadraticcovariation theorem to the true returns data, given the non-synchronicity of observed prices fordifferent assets. In fact the quadratic covariation formula is unfeasible when applied to estimatecross-volatilities, because it requires synchronous observations which are not available in realsituations. Being based on the integration of all data, the Fourier estimator does not need anyadjustment to fit non-synchronous data. We briefly recall the methodology.

Assume that p(t) = (p1(t), . . . , pk(t)) are Brownian semi-martingales satisfying the followingIto stochastic differential equations

dpj(t) =d

i=1ji (t) dW

i + bj(t) dt j = 1, . . . , k , (1)

where W = (W1, . . . , W d) are independent Brownian motions. The processes are observed ona fixed time window, which can be always reduced to [0, 2] by change of origin and rescaling,and and b

are adapted random processes satisfying hypothesis

(H) E[

20

(bi(t))2dt] < , E[

20

(ji (t))4dt] < i = 1, . . . , d, j = 1, . . . , k .

Moreover, we assume that the observed prices are affected by microstructure noise in theform

pi(t) = pi(t) + i(t) i = 1, . . . , k (2)

where the noise process is i.i.d. and the following assumptions hold:M1. p and are independent processes, moreover (t) and (s) are independent for s = t

and E[(t)] = 0 for any t.M2. E[i(t)j(t)] = ij < for any t, i, j = 1, . . . , k.From the representation (1) we define the volatility matrix, which in our hypothesis depends

upon time

ij(t) =d

r=1

ir(t)jr(t).

The Fourier method reconstructs ,(t) on [0, 2] using the Fourier transform of dp(t).The main result in [Malliavin and Mancino, 2005] relates the Fourier transform of , to the

Fourier transforms of the log-returns dp. More precisely the following result is proved: computethe Fourier transform of dpj for j = 1, . . . , k, defined for any integer z by

F(dpj)(z) =1

2

20

eiztdpj(t)

and consider the Fourier transform of the cross-volatility function defined for any integer z by

F(ij)(z) :=1

2

20

eizt ij(t)dt,

3


4/15

then the following convergence in probability holds

F(ij)(z) = limN

2

2N + 1

|s|N

F(dpi)(s)F(dpj)(z s).

As a particular case (by choosing z = 0) we can compute the integrated covariance, given

the log-returns of stocks, as the following limit in probability]0,2[

ij(t) dt = limN

(2)2

2N + 1

|s|N

F(dpi)(s)F(dpj)(s). (3)

From this convergence result, we can derive a suitable estimator for the integrated co-variance matrix. We assume that the price process for asset j (j = 1, . . . , k) is observed athigh-frequency intra-daily times {tjl , l = 1, . . . , nj}, which may be different on each daily tradingperiod normalized to length 2. Set

F(dpjnj )(s) :=

1

2

nj1l=1

exp(istjl )Ijl (p

j

),

where Ijl

(pj) := pj(tjl+1)pj(tjl ). The Fourier estimator of the integrated covariance

20

ij(t)dt

is then

ijN,ni,nj := (2)22N + 1 |s|N

F(dpini)(s)F(dpjnj

)(s) =

ni1u=1

nj1l=1

DN(tiu t

jl )Iiu(p

i)Ijl

(pj), (4)

where DN(x) =1

2N+1

sin[(N+ 12)x]

sin x2

is the rescaled Dirichlet kernel.

The construction of the estimator (4) can be modified by considering the Fejer summation,therefore in the sequel we will consider the variant obtained through the Fejer kernel

ijN,ni,nj := ni1u=1

nj1l=1

FN(tiu t

jl )Iiu(p

i)Ijl

(pj) (5)

where FN(x) =sinNxNx

2. This estimator has the advantage to preserve positivity of the covari-

ance matrix, as it is stated by the following

Proposition 2.1 The Fourier estimator

N is positive semi-definite.

Proof. Using Bochner theorem (see [Malliavin, 1995] pg 255) it suffices to prove that0

sin2 t

t2eitxdt 0 x R.

As the Fourier transform of [ 12, 12

](t) is the functionsinxx ,

0

sin2 t

t2eitxdt = 2

0

[ 12, 12

](x t)[ 12, 12

](t)dt 0 x R.

4


5/15

For the sake of completeness we now recall the definition of the other estimators of covari-ance which will be considered in our analysis.

The realized covariance-type estimators are based on the choice of a synchronization pro-cedure, which gives the observations times {0 = 1 2 n 2} for both assets. Thequadratic covariation-realized covariance estimator is defined by

RCij :=n1u=1

u(pi)u(p

j),

where u(p) = p(u+1) p

(u). It is known that the realized covariance estimator is notconsistent under asynchronous trading, [Hayashi and Yoshida, 2005].

The realized covariance plus leads and lags estimator is defined by

RCLLij :=u

Lh=l

u+h(pi)u(p

j). (6)

The estimator (6) has good properties under microstructure noise contaminations of the prices,but it is still not consistent for asynchronous observations. This is due to the fact that allthe realized covariance type estimators need a data synchronization procedure, because of thedefinition of the quadratic covariation process. Nevertheless, the introduction of one lead andone lag appears to provide a correction for the downward bias by non-synchronous trading.

The [Hayashi and Yoshida, 2005] All-Overlapping (AO) estimator is

AOijn1,n2 :=l,u

Iil(pi)

Iju(pj)I

(IilIju=)

, (7)

where I(P) = 1 if proposition P is true and I(P) = 0 if proposition P is false. It is unbiased inthe absence of noise. From the practitioners point of view both this estimator and the Fourier

estimator are easy to implement as they do not rely on any choices of synchronization methodsand sampling schemes. However, in [Griffin and Oomen, 2006, Voev and Lunde, 2007] the AOestimator is proved to be not efficient in the presence of microstructure noise.

The theoretical properties of the Fourier estimator are studied by [Mancino and Sanfelici, 2008a,Mancino and Sanfelici, 2008b], who show that the bias of the covariance estimator is not affectedby the presence of i.i.d. noise. The Fourier estimator of covariance under microstructure noise isasymptotically unbiased, as in the case of the univariate Fourier estimator under microstructurenoise. Moreover, if the number of the Fourier coefficients N is conveniently chosen, the meansquared error of the Fourier estimator does not diverge and it is not significantly affected bythe microstructure noise. In contrast, the realized covariation and the AO estimator are notbiased by i.i.d. noise; nevertheless both realized covariation and AO estimator are inconsis-

tent under i.i.d. noise, because the MSE diverges as the number of observations increases, see[Mancino and Sanfelici, 2008b] for details on this point.

3 Forecasting and asset allocation

We use the methodology suggested by [Fleming et al., 2001] and [Bandi et al., 2006] to evaluatethe economic benefit of the Fourier estimator of integrated covariance in the context of an assetallocation strategy. Specifically, we compare the utility obtained by virtue of covariance forecasts

5


6/15

based on the Fourier estimator to the utility obtained through covariance forecasts constructedusing the more familiar realized covariance and other recently proposed estimators. In the fol-lowing, we adopt a notation which is common in the literature about portfolio management. Itwill not be difficult for the reader to match it with the one in the previous section.

Let Rf and Rt+1 be the risk-free return and the return vector on k risky assets over a day[t, t + 1], respectively. Define

t= E

t[R

t+1] and

t= E

t[(R

t+1

t)(R

t+1

t)] the conditional

expected value and the conditional covariance matrix of Rt+1. We consider a mean-varianceinvestor who solves the program

minwt

wttwt,

subject towtt + (1 w

t1k)R

f = p,

where wt is a k-vector of portfolio weights, p is a target expected return on the portfolio, and1k is a k 1 vector of ones. The solution to this program is

wt =(p R

f)1t (t Rf1k)

(t Rf

1k)

1

t (t Rf

1k)

. (8)

We estimate t using one-day-ahead forecasts Ct given a time series of daily covarianceestimates, obtained using the Fourier estimator, the realized covariance estimator, the realizedcovariance plus leads and lags estimator and the AO estimator. The out-of-sample forecast isbased on a univariate ARMA model.

Given sensible choices of Rf , p and t, each one-day-ahead forecast leads to the deter-mination of a daily portfolio weight wt. The time series of daily portfolio weights then leadsto daily portfolio returns. In order to concentrate ourselves on volatility approximation and toabstract from the issues that would be posed by expected stock return predictability, for alltimes t we set the components of the vector t = Et[Rt+1] equal to the sample means of the

returns on the risky assets over the forecasting horizon. Finally, we employ the investors long-run mean-variance utility as a metric to evaluate the economic benefit of alternative covarianceforecasts Ct, i.e.

U = Rp

2

1

m

mt=1

(Rpt+1 Rp)2,

where Rpt+1 = Rf + wt(Rt+1 R

f1k) is the return on the portfolio with estimated weights wt,Rp = 1m

mt=1 R

pt+1 is the sample mean of the portfolio returns across m n days, and is a

coefficient of risk-aversion.Following [Bandi et al., 2006], in order to avoid contaminations induced by noisy first mo-

ment estimation, we simply look at the variance component of U, namely

U =

2

1

m

mt=1

(Rpt+1 Rp)2, (9)

see [Engle and Colacito, 2006] for further justifications of this approach. The difference betweentwo utility estimations, say UA UB , can be interpreted as the fee that the investor would bewilling to pay to switch from covariance forecasts based on estimator A to covariance forecastsbased on estimator B. In other words, UA UB is the utility gain that can be obtained byinvesting in portfolio B, with the lowest variance for a given target return p.

6


7/15

4 Valuing the economic benefit by simulations

In the following sections we show several numerical experiments to assess the gains offered by theFourier estimator over other estimators in terms of in-sample and out-of-sample properties andfrom the perspective of an asset-allocation decision problem. In Section 4.1 our attention is fo-cused mainly on covariance estimation, since in this respect effects due to both non-synchronicity

and microstructure noise become effective. Nevertheless, the results in Sections 4.2, 4.3 and 4.4can be fully justified only by considering the properties of the different estimators for both thevariance and the covariance measures.

Following a large literature, we simulate discrete data from the continuous time bivariateHeston model

dp1(t) = (1 21(t)/2)dt + 1(t)dW1

dp2(t) = (2 22(t)/2)dt + 2(t)dW2

d21(t) = k1(1 21(t))dt + 11(t)dW3,

d22(t) = k2(2 22(t))dt + 22(t)dW4,

where corr(W1, W2) = 0.35, corr(W1, W3) = 0.5 and corr(W2, W4) = 0.55. The other pa-

rameters of the model are as in [Zhang et al., 2005]: 1 = 0.05, 2 = 0.055, k1 = 5, k2 = 5.5,1 = 0.05, 2 = 0.045, 1 = 0.5, 2 = 0.5. The volatility parameters satisfy the Fellers condition2k 2 which makes the zero boundary unattainable by the volatility process. Moreover, we as-sume that the additive logarithmic noises 1(t), 2(t) are i.i.d. Gaussian, contemporaneously cor-related and independent from p. The correlation is set to 0.5 and we assume (E[2])1/2 = 0.002,i.e. the standard deviation of the noise is 0.2% of the value of the asset price. From the simu-lated data, integrated covariance estimates can be compared to the value of the true covariancequantities.

We generate (through simple Euler Monte Carlo discretization) high frequency evenly sam-pled efficient and observed returns by simulating second-by-second return and variance pathsover a daily trading period of h = 6 hours, for a total of 21600 observation per day. In order to

simulate high frequency unevenly sampled data, we extract the observation times in such a waythat the durations between observations are drawn from an exponential distribution with means1 = 6 sec and 2 = 8 sec for the two assets respectively. Therefore, on each trading day the pro-cesses are observed at a different discrete unevenly spaced grid {0 = tl1 t

l2 t

lnl

2}for any l = 1, 2.

For the realized covariance type estimators, we generate equally-spaced continuously-compoundedreturns using the previous tick method. We consider 1, 5 and 10-min sampling intervals or opti-mally sampled realized covariances. [Bandi et al., 2006] provide an approximate formula for opti-mal sampling, which holds for uniform synchronous data. Given our general data setting, the op-timal sampling frequency can be obtained by direct minimization of the true mean squared error.In order to preserve the positive definiteness of the covariance matrices, we use a unique sampling

frequency for realized variances and covariances, given by the maximum among the three optimalfrequencies. For the Fourier and AO estimators, we employ all the available data set. In imple-menting the Fourier estimator 12N,n1,n2 , the smallest wavelength that can be evaluated in orderto avoid aliasing effects is twice the smallest distance between two consecutive prices (Nyquist

frequency). Nevertheless, as pointed out in the univariate case by [Mancino and Sanfelici, 2008a]and confirmed in the bivariate case in [Mancino and Sanfelici, 2008b], smaller values of N mayprovide better variance/covariance measures. More specifically, the optimal cutting frequenciesfor the various volatility measures can be obtained independently by minimizing the true MSE.

7


8/15

Although the positivity result of Proposition 2.1 is ensured only when the same N is used forall the entries of the covariance matrix, numerical experiments show that the use of differentcutting frequencies N for variances and covariances still preserves positive definiteness of thecovariance matrix, both in the sample and in the forecasting horizon.

4.1 Covariance estimation and forecastAs a first application, we perform an in-sample analysis in order to shed light on the propertiesof the different estimators in terms of different statistics of the covariance estimates, such asbias, MSE and others. More precisely, we consider the following relative error statistics

= E

C12

20

12(t)dt20

12(t)dt

, std =

V ar

C12

20

12(t)dt20

12(t)dt

1/2,

which can be interpreted as relative bias and standard deviation of an estimator C12 for thecovariance. The Fourier and RCopt estimators have been optimized by choosing the cuttingfrequency N of the Fourier expansion and the sampling interval on the basis of their MSE. Theresults are reported in Table 1. Within each table, entries are the values of, std, MSE and bias,using 750 Monte Carlo replications which roughly correspond to three years. Rows correspondto the different estimators. The sampling interval for the realized covariance-type estimatorsis indicated as a superscript. The optimal sampling frequency for RCopt is obtained by directminimization of the true MSE and corresponds to 2.33 min.

When we consider covariance estimates, the most important effect to deal with is the Eppseffect. The presence of other microstructure effects represents a minor aspect in this respect. Onthe contrary, it may in some sense even compensate the effects due to non-synchronicity, as wecan see from the smaller MSE of 1-minute realized covariance estimator with respect to 5-minuteestimator. We remark that the corresponding 1-minute estimator for variances is more affected

by the presence of noise, since it is not compensated by non-synchronicity. As any estimatorbased on interpolated prices, the realized covariance-type estimators suffer from the Epps effectwhen trading is non-synchronous, but the lead-lag correction reduces such an effect, at least interms of bias to the disadvantage of a slightly larger MSE. Note that the lead/lag correctioncontrasts the Epps effect, thus producing occasionally positive biases. On the other hand, thepresence of noise strongly affects the AO estimator. This is due to the Poisson trading schemewith correlated noise. In fact, the AO remains unbiased under independent noise wheneverthe probability of trades occurring at the same time is zero, which is not the case for Poissonarrivals. The Fourier estimator provides good covariance measures, both in terms of bias andMSE. Therefore, we can conclude that contrary to the AO estimator the Fourier covarianceestimator is not much affected by the presence of noise, so that it becomes a very interesting

alternative especially when microstructure effects are particularly relevant in the available data.

Before turning to asset allocation, we evaluate the forecasting power of the different esti-mators. In the tradition of [Mincer and Zarnowitz, 1969], we regress the real daily integratedcovariance over the forecasting period on one-step-ahead forecasts obtained by means of each co-variance measure. More precisely, following [Andersen and Bollerslev, 1998], we consider a largersample path of 1000 days and we split it into two parts: the first one containing 20% of totalestimates is used as a burn-in period to fit a univariate AR(1) model for the estimated covari-ance time series and then the fitted model is used to forecast integrated covariance on the next

8


9/15

Method std MSE bias

RC1min -0.07805275472999 0.22595032210904 0.00000085621669 -0.00032576060834RC5min -0.01130593672881 0.39847063473932 0.00000280603760 -0.00001894057172RC10min 0.01355081842814 0.54479459409222 0.00000571706029 0.00008501424105RCLL1min 0.01338414896851 0.32281801149180 0.00000173043743 0.00009896376793

RCLL

5min

-0.00869870282055 0.62833287597643 0.00000696972045 0.00003089840943RCLL10min -0.02385652069431 0.89431477251656 0.00001491460726 -0.00003947110061RCopt -0.02732364713452 0.29983219944312 0.00000150028381 -0.00008496777460AO 0.56243149971616 0.35295341500361 0.00000366031965 0.00176446312515Fourier -0.06518255776964 0.17089219158470 0.00000049500437 -0.00026715723226

Table 1: Relative and absolute error statistics for the in-sample covariance estimates for differentestimators.

day. The choice of the AR(1) model comes from [At-Sahalia and Mancini, 2007], who considerthe univariate Heston data generating process. The total number of out-of-sample forecasts m

is equal to 800. Each time a new forecast is performed, the corresponding actual covariancemeasure is moved from the forecasting horizon to the first sample and the AR(1) parametersare re-estimated in real time. For each time series of covariance forecasts, we project the realdaily integrated covariance on day [t, t + 1] on a constant and the corresponding one-step-aheadforecast Ct t+1

t12(s) ds = 0 + 1Ct + errort,

where t = 1, 2, . . . , m. Alternatively, we can regress simultaneously the real daily integratedcovariance over the forecasting period on various one-step-ahead forecasts obtained by means ofdifferent covariance measures. The regression now takes the form

t+1t

12(s) ds = 0 + CCt + CCt + errort,

where C and C are different covariance forecasts on day [t, t + 1]. The R2 from these regressionsprovides a direct assessment of the variability in the integrated covariance that is explainedby the particular estimates in the regressions. The R2 can therefore be interpreted as a simplegauge of the degree of predictability in the volatility process and hence of the potential economicsignificance of the volatility forecasts.

The results are reported in Tables 2, 3 and 4, using a Newey-West covariance matrix. Weremark that in this simulation the Fourier estimator is not optimal in the MSE sense, butwe set N1 = 155, N2 = 123, N = 271 from the previous experiment. When we consider a

single regressor, the R2

is the highest for the Fourier estimator while RCLL10min

explains lessthan five percent of the time series variability. Moreover, for the Fourier estimator we can notreject the hypothesis that 0 = 0 and 1 = 1 using the corresponding t tests. In contrast, wereject the hypothesis that 0 = 0 and 1 = 1 for all the other estimators except RC

5min andRCLL1min. When we include alternative forecasts besides Fourier estimator in the regression,the R2 improves very little relative to the R2 based solely on Fourier. Moreover, the coefficientestimates for FE are generally close to unity, while for the other estimators are near zero except5min for the realized covariance which differs significantly from zero at the 5% level. Therefore,we can conclude that the higher accuracy and lower variability of Fourier covariance estimates

9


10/15

Method R2 F p 0 1Fourier 0.210362 212.589597 0.000000 0.000251 0.996739

(0.000251) (0.068361)RC1min 0.186222 182.611691 0.000000 0.000004 1.091056

(0.000288) (0.080739)RC2min 0.159487 151.420599 0.000000 0.000646 0.872224

(0.000265) (0.070882)RC5min 0.158765 150.605610 0.000000 0.000653 0.839401

(0.000265) (0.068399)RC10min 0.107694 96.311707 0.000000 0.000654 0.842651

(0.000328) (0.085863)RCLL1min 0.155834 147.311660 0.000000 0.000824 0.781413

(0.000254) (0.064382)RCLL5min 0.067863 58.096974 0.000000 0.000705 0.811731

(0.000413) (0.106497)RCLL10min 0.041545 34.590271 0.000000 0.002408 0.364288

(0.000248) (0.061940)AO 0.186796 183.303989 0.000000 -0.001162 0.888731

(0.000373) (0.065642)

Table 2: Regression of real integrated covariance on each covariance forecast over the forecastinghorizon. Standard deviations are listed in parenthesis.

translate into superior forecasts of future covariances and this explains the superior performanceof the Fourier forecasts.

4.2 Dynamic portfolio choice and economic gains

In this section, we consider the benefit of using the Fourier estimator from the perspective ofthe asset-allocation problem of Section 3.

Given any time series of daily variance/covariance estimates, we split our sample of 750days into two parts: the first one containing 30% of total estimates is used as a burn-inperiod, while the second one is saved for out-of-sample purposes. The out-of-sample forecast isbased on univariate ARMA models. More precisely, following [At-Sahalia and Mancini, 2007],the estimated series of 225 in-sample covariance matrices is used to fit univariate AR(1) models

Method R2 F p 0 FE 1min 5min 10minRC 0.218 55.438 0.000 0.000064 0.762475 0.052218 0.322030 -0.098934

Std (0.000338) (0.162943) (0.199279) (0.143406) (0.145376)T-statistics (0.188921) (4.679382) (0.262034) (2.245592) (-0.680538)

RCLL 0.212 53.448 0.000 0.000319 0.895311 0.147186 -0.052503 -0.017447Std (0.000405) (0.119065) (0.118386) (0.177799) (0.089995)T-statistics (0.788629) (7.519509) (1.243270) (-0.295294) (-0.193873)

Table 3: Regression of real integrated covariance on Fourier (FE) and RC/RCLL -type estimatorsover the forecasting horizon. The first panel refers to the realized covariance estimator with 1,5, 10-min sampling, the second one to its lead/lag bias correction.

10


11/15

Method R2 F p 0 FE 2min AOOthers 0.2110 7 0.962 0 .000 -0.000025 0.852102 0.030607 0.121691Std (0.000434) (0.194413) (0.136529) (0.170537)T-statistics (-0.056499) (4.382940) (0.224183) (0.713576)

Table 4: Regression of real integrated covariance on Fourier (FE), RC2min and AO estimators

over the forecasting horizon.

for each variance/covariance estimates separately. The total number of out-of-sample forecastsm for each series is equal to 525. Each time a new forecast is performed, the correspondingactual variance/covariance measure is moved from the forecasting horizon to the first sampleand the AR(1) parameters are re-estimated in real time. Given sensible choices of Rf, p and t,each one-day-ahead variance/covariance forecast leads to the determination of a daily portfolioweight wt. The time series of daily portfolio weights then leads to daily portfolio returns andutility estimation.

We implement the criterion in (9) by setting Rf equal to 0.03 (converted to daily values by

dividing by 250) and considering three targets p, namely 0.09, 0.12, 0.15. In order to concentrateon volatility timing and abstract from issues related to expected stock return predictability, forall times t we set the components of the vector t = Et[Rt+1] equal to the sample means of thereturns on the risky assets over the forecasting horizon. For all times t, the conditional covariancematrix is computed as an out-of-sample forecast based on the different variance/covarianceestimates.

We interpret the difference UCUFourier between the average utility computed on the basisof the Fourier estimator and that based on alternative estimators C, as the fee that the investorwould be willing to pay to switch from covariance forecasts based on estimator C to covarianceforecasts based on the Fourier estimator. Table 5 contains the results for three levels of risk-aversion and three target expected returns. Due to the presence of microstructure noise effects

which spoils the sum of squared high-frequency intra-day returns, besides the All-overlappingestimator we consider the AO + RCLL1min estimator which is based on the AO estimator forthe covariances and on the 1-minute RCLL estimator for the variances. When the target is0.09, the investor would pay between 0.52% (when = 2) of his portfolio return and 2.59%(when = 10) per year to use the Fourier estimator versus the RC1min estimator. When thetarget is 0.12, the investor would pay between 0.92% (when = 2) of his portfolio returnand 4.60% (when = 10). Finally, when the target is 0.15, the investor would pay between1.44% (when = 2) of his portfolio return and 7.19% (when = 10). The same investorwould pay marginally less to abandon RC5min, according to the better in-sample propertiesof this estimator for the whole covariance matrix which translate into more precise forecasts.The remaining part of the table can be read similarly. Strikingly, the utility gain of the Fourier

estimator over the AO is very large, but this is due to the presence of microstructure effects. Evenwhen considering the AO + RCLL1min estimator, the Fourier estimator is superior, while onlya very small utility loss is encountered when considering the RCLL1min estimator. Notice thatthe optimally sampled realized covariance estimator cannot achieve the same performance. Inparticular, this evidence partially contradicts the conclusions of [De Pooter at al., 2008] aboutthe greater effects obtainable by a careful choice of the sampling interval rather than by biascorrection procedures.

11


12/15

Method p = 0.09 p = 0.12 p = 0.15 2 7 10 2 7 10 2 7 10

RC1min 0.52 1.81 2.59 0.92 3.22 4.60 1.44 5.03 7.19RC5min 0.28 0.97 1.38 0.49 1.72 2.46 0.77 2.70 3.85RC10min 0.62 2.18 3.12 1.11 3.88 5.55 1.73 6.07 8.67

RCLL

1min

-0.30 -1.07 -1.52 -0.54 -1.90 -2.71 -0.85 -2.97 -4.24RCLL5min 1.50 5.26 7.52 2.68 9.37 13.38 4.18 14.64 20.91RCLL10min 1.76 6.14 8.78 3.12 10.93 15.61 4.88 17.08 24.40RCopt 0.004 0.01 0.02 0.007 0.02 0.04 0.01 0.04 0.06AO 1.60 5.60 7.99 2.84 9.96 14.22 4.45 15.56 22.23AO + RCLL1min 0.38 1.33 1.89 0.67 2.36 3.37 1.05 3.69 5.26

Table 5: Annualized fees UC UFourier that a mean-variance investor would be willing to payto switch from C to Fourier estimates.

4.3 The statistical significance of the economic gains

One way to assess the statistical significance of the economic gains resulting from Table 5 is toperform the following joint statistical test. For any target p and any estimator, one can define

alternative covariance forecasts Ct and portfolio returns Rp(C)t+1 . Define

aCt+1 = (Rp(Fourier)t+1 R

p(Fourier))2 (Rp(C)t+1 R

p(C))2.

Assessing the statistical significance of the economic gains of the Fourier estimate over alternative

forecasts can be conducted by testing whether the mean of aCt+1 is larger than (or equal to) zeroagainst the alternative that the mean is smaller than zero.

Following [Bandi et al., 2006], for any target return d = 0.09, 0.12, 0.15, we define the vector

Adt+1 =

aC1t+1, aC2t+1, a

C3t+1

,

where the triple of estimators (C1, C2, C3) is given by (RC1min, RC5min, RC10min), (RCLL1min,

RCLL5min,RCLL10min) and (RCopt,AO,AO+RCLL1min) respectively. We write the regressionmodel

Adt+1 = d13 + t+1,

where d is a scalar parameter. We perform the one-sided test H0 : d 0 against HA :

d < 0.The parameter d is estimated by GMM using a Bartlett HAC covariance matrix. The t-statisticsof all the tests imply rejection of the null and hence statistical significance of the economic gains

at 5% level.

4.4 A small-sample Monte Carlo experiment

Another way to asses the superiority of the Fourier estimator over the others is based on the

work of [Diebold and Mariano, 1995] and consists in examining each aCt time series separately ina Monte Carlo experiment. By regressing on a constant, the null hypothesis is simply a test that

the mean ofaCt is zero. Therefore, a negative number is evidence in favor of better performance ofthe Fourier estimator over C. A similar approach is used also by [Engle and Colacito, 2006]. We

12


13/15

RC1min RC5min RC10min

86 69 68

RCLL1min RCLL5min RCLL10min

54 76 75

RC2min AO AO + RCLL1min

76 99 86

Table 6: Number of times (out of 100 Monte Carlo trials) that the mean of aCt has a signifi-cant negative value, i.e. the asset allocation based on the Fourier estimator has a statisticallysignificant benefit over the others.

explore the significance of the proposed methods on a sample of 1000 days, with m = 800 out-of-sample forecasts, and simulate a total of 100 samples. We allocate assets according to (8) andrun the one-sided Diebold-Mariano test in each Monte Carlo trial. In Table 6 we list the timesthat the asset allocation based on the Fourier estimator has a statistically significant benefit overthe others at a 5% significance level. We remark that in this automatic Monte Carlo experiment,

the Fourier and RC estimators have not been optimized with respect to MSE. On the contrary,we arbitrarily fix the sampling period for RCopt at 2 min and N1 = 155, N2 = 123, N = 271 forthe Fourier estimator for the two variances and the covariance respectively. The table reveals asuperiority of the Fourier procedure over all the other estimators, with a percentage of successbetween 54% and 99%.

5 Conclusions

We have analyzed the gains offered by the Fourier estimator from the perspective of an asset-allocation decision problem. The comparison is extended to realized covariance-type estimators,to lead-lag bias corrections and to the All-Overlapping estimator.

We show that the Fourier estimator carefully extracts information from noisy high-frequencyasset price data and allows for non-negligible utility gains in portfolio management. Specifically,our simulations show that the gains yielded by the Fourier methodology are statistically signif-icant and can be economically large, while only the realized covariance with one lead-lag biascorrection and suitable sampling frequency can be competitive.

Analyzing the in-sample and out-of-sample properties of different covariance measures,we find that the Fourier estimator provides more precise variance/covariance estimates whichtranslate into more precise forecasts.

References

[At-Sahalia and Mancini, 2007] At-Sahalia, Y. and Mancini, L. (2007) Out of sample forecastsof quadratic variations. Working paper.

[Andersen and Bollerslev, 1998] Andersen, T. and Bollerslev, T. (1998) Answering the skeptics:yes, standard volatility models do provide accurate forecasts. International Economic Review,39/4, 885905.

[Bandi et al., 2006] Bandi, F.M., Russel, J.R. and Y. Zhu (2006) Using high-frequency data indynamic portfolio choice. Econometric Reviews, forthcoming .

13


14/15

[Barndorff-Nielsen and al., 2008] Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A. and Shep-hard, N. (2008) Multivariate Realised kernels: consistent positive semi-definite estimators ofthe covariation of equity prices with noise and non-synchronous trading. Working paper.

[De Pooter at al., 2008] De Pooter, M., Martens, M. and Dick van Dijk. (2008) Predicting thedaily covariance matrix for S&P100 stocks using intraday data: but which frequency to use?

Econometric Reviews, 27, 199229.

[Diebold and Mariano, 1995] Diebold, F.X. and Mariano, R.S. (1995) Comparing predictiveaccuracy. Journal of Business & Economic Statistics, 13, 253263.

[Engle and Colacito, 2006] Engle, R. and Colacito, R. (2006) Testing and valuing dynamiccorrelations for asset allocation. Journal of Business & Economic Statistics, 24(2), 238253.

[Epps, 1979] Epps, T. (1979) Comovements in stock prices in the very short run. Journal ofthe American Statistical Association, 74, 291298.

[Fleming et al., 2001] Flaming, J., Kirby, C. and Ostdiek, B. (2001) The economic value of

volatility timing. The Journal of Finance, LVI, 1, 329352.

[Gatheral and Oomen, 2007] Gatheral, J. and Oomen, R.C.A. (2007) Zero-intelligence realizedvariance estimation. Working Paper.

[Griffin and Oomen, 2006] Griffin, J.E. and Oomen, R.C.A. (2006) Covariance Measurement inthe Presence of Non-Synchronous Trading and Market Microstructure Noise. Working Paper.

[Harris et al., 1995] Harris, F.H deB., McInish, T.H., Shoesmith, G.L. and Wood, R.A. (1995)Cointegration, error correction, and price discovery on informationally linked security markets.

Journal of Financial and Quantitative Analysis, 30, 563579.

[Hayashi and Yoshida, 2005] Hayashi, T. and Yoshida, N. (2005) On covariance estimation ofnonsynchronously observed diffusion processes. Bernoulli, 11(2), 359379.

[Large, 2007] Large, J. (2007) Accounting for the Epps effect: Realized covariation, cointegrationand common factors . Working paper.

[Malliavin, 1995] Malliavin, P. (1995). Integration and Probability, Springer Verlag.

[Malliavin and Mancino, 2002] Malliavin, P. and Mancino, M.E. (2002). Fourier series methodfor measurement of multivariate volatilities. Finance and Stochastics, 4, 4961.

[Malliavin and Mancino, 2005] Malliavin, P. and Mancino, M.E. (2005). A Fourier transformmethod for nonparametric estimation of multivariate volatility. Annals of Statistics, forth-

coming.

[Mancino and Sanfelici, 2008a] Mancino, M.E. and Sanfelici, S. (2008) Robustness of FourierEstimator of Integrated Volatility in the Presence of Microstructure Noise. ComputationalStatistics and Data Analysis, 52(6), 29662989.

[Mancino and Sanfelici, 2008b] Mancino, M.E. and Sanfelici, S. (2008) Estimating covariancevia Fourier method in the presence of asynchronous trading and microstructure noise. Sub-mitted to J. Financial Econometrics.

14


15/15

[Martens, 2004] Martens, M. (2004). Estimating Unbiased and Precise Realized Covariances.

EFA 2004 Maastricht Meetings Paper No. 4299.

[Mincer and Zarnowitz, 1969] Mincer, J. and Zarnowitz, V. (1969). The evaluation of economicforecasts. In: Mincer J. (Ed.) Economic forecasts and expectations. National Bureau of Eco-nomic Research, New York.

[Voev and Lunde, 2007] Voev, V. and Lunde, A. (2007) Integrated Covariance Estimation UsingHigh-Frequency Data in the Presence of Noise. Journal of Financial Econometrics, 5.

[Zhang et al., 2005] Zhang, L., Mykland, P. and At-Sahalia, Y. (2005). A tale of two time scales:determining integrated volatility with noisy high frequency data. Journal of the AmericanStatistical Association, 100, 13941411.

15

Mancino_FourierForCovarianceEstimation

Documents

Transcript of Mancino_FourierForCovarianceEstimation