Volatility Proxies for Discrete Time Models

Munich Personal RePEc Archive

Volatility Proxies for Discrete Time

Models

Robin G. de Vilder and Marcel P. Visser

Korteweg-de Vries Instute for Mathematics, University of

Amsterdam

14. September 2007

Download paper at IDEAS

MPRA Paper No. 4917, posted 14. September 2007 / 11:40

Page 2

Volatility proxies for discrete time models

Robin G. de Vilder

∗

Marcel P. Visser

†

September 14, 2007

Abstract

Discrete time volatility models typically employ a latent scale factor to represent

volatility. High frequency data may be used to construct proxies for these scale factors.

Examples are the intraday high-low range and the realized volatility. This paper devel-

ops a method for ranking and optimizing volatility proxies. It is possible to outperform

the quadratic variation as a proxy for the discrete time scale factor. For the S&P 500

index data over the years 1988-2006 this is achieved by a proxy which puts, among

other things, more weight on the highs than on the lows over intraday intervals.

JEL classification: C22,C52,C65.

Keywords: volatility proxy, realized volatility, quadratic variation, scale factor,

arch/garch/stochastic volatility, intraday seasonality.

∗

PSE (CNRS-EHESS-ENPC-ENS), Paris, France and Korteweg-de Vries Institute for Mathematics, Uni-

versity of Amsterdam, The Netherlands.

†

Corresponding author, Korteweg-de Vries Institute for Mathematics, University of Amsterdam. Plantage

Muidergracht 24, 1018 TV Amsterdam, The Netherlands. Tel. +31 20 5255861. Email: m.p.visser@uva.nl.

Page 3

1 Introduction

Volatility refers to the degree to which financial prices tend to fluctuate. It changes over

time and financial time series display periods of low and high volatility. A large and widely

used body of discrete time models for dealing with time varying volatility has a product

structure

= s

(1)

The observed financial return r

is modelled as the product of an iid innovation Z

and a

positive scale factor s

. The scale factor is generally called the volatility. Attention typically

centers on the volatility process (s

), specifications for which include Arch/Garch, stochastic

volatility, long memory, and Markov switching.

Since only the returns r

are observed, the values of the scale factors s

have to be

estimated. High frequency data may be used to construct proxies for the scale factor s

such as the intraday high-low range or the realized volatility. Proxies serve as measurements

of the scale factor. Good proxies are needed for efficient parameter estimation (see, e.g.,

Alizadeh, Brandt, and Diebold (2002)), they are essential for volatility forecast evaluation

(see, for instance, Andersen and Bollerslev (1998), and Hansen and Lunde (2006a)), and

good proxies will help to improve the economic understanding of the volatility process.

There is no straightforward way to tell, for any two proxies, which one is better, since

the scale factors s

are unobservable. Much of the current research focuses on efficiently

estimating the quadratic variation, which is (the limit of) the sums of squared intraday

returns, see, for instance, Barndorff-Nielsen and Shephard (2002), and Andersen, Bollerslev,

Diebold and Labys (2003). However, we shall see that the quadratic variation does not

necessarily lead to the most efficient proxy for the discrete time scale factor s

This paper proposes a way to rank and optimize proxies for the daily scale factor s

based on intraday high frequency data. We take three steps: extend the discrete time models

(1) to a generic continuous time model with intraday seasonality; fix a large class of volatility

proxies; and develop tools for ranking and optimizing those proxies.

We integrate high frequency data into the discrete time models (1), by replacing the iid

innovations (Z

) by iid stochastic processes (Ψ

) on the unit time interval:

(ϑ) = s

(ϑ)

ϑ ∈ [0, 1].

(2)

The continuous time models describe, for each day, the daily log return process R

from the

opening until closing, starting from the overnight return at the opening. The process Ψ

Page 4

may be any stochastic process, and is scaled by s

resulting in a bridge between the discrete

time, close-to-close returns r

in (1). These continuous time models are consistent with both

the persistence of volatility captured by discrete time volatility models, and with widely

observed empirical facts such as intraday seasonalities, leverage effects, and jumps. As in

the discrete time model, s

is a state of the market representing the level of the day-specific

price sensitivity. The actual fluctuations in the day n return process R

are the result of

the local fluctuations in the process Ψ

, scaled by the day-specific factor s

. The structure

in (2) is not a model of constant intraday volatility: depending on Ψ

it is possible to have

a hectic day (for instance, a large quadratic variation) when s

is low, and vice versa. For

discrete time models the term volatility tends to have a slightly different meaning than it

has for continuous time models. To avoid confusion we will, from now on, refer to s

as the

daily scale factor.

We proceed by formalizing the notion of a proxy in the spirit of Alizadeh, Brandt, and

Diebold (2002). The definition covers popular proxies such as the absolute return, the

intraday high-low range, and the realized volatility. For many applications it is helpful to

have proxies with small measurement variance. We develop some easy-to-implement tools

for ranking and improving proxies. In particular, proxies may be ranked by the variance of

their logarithm and they may be combined into a more efficient proxy by a technique similar

to the Markowitz (1952) optimization procedure. By the end of Section 3, we will have at

our disposal a largely model free methodology for ranking and improving proxies: optimality

of a proxy for s

does not depend on the particular discrete time model, as long as (1) holds.

Empirical analysis of intraday S&P 500 index futures market data from January 1988

to mid 2006 shows that the techniques provide a good proxy for the S&P 500 data. In

terms of the variance of the logarithm, the additional use of the high-low range over intra-

day intervals yields proxies that substantially outperform the realized volatility, which uses

only squared intraday returns. Martens and van Dijk (2007) arrive at a similar finding in

the context of estimating the quadratic variation. Moreover, our empirical results suggest

that the optimized proxy is more efficient than the (square root of) the quadratic variation.

Interestingly, the optimized proxy based on the highs, the lows, and the absolute returns

over ten-minute intervals, puts more weight on the highs than on the lows. From this point

of view, the upward price movements are more informative than the downward movements,

when proxying volatility.

The discrete time model class (1) provides a flexible way for dealing with time varying

volatility. These models are important tools for the practice of risk management and as-

set allocation, as well enable academic researchers to investigate fundamental questions in

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finance, such as the tradeoff between risk and return, see, for instance, Engle, Lilien, and

Robins (1987).

Many discrete time volatility models were developed when high frequency data were

not readily available, and were initially applied to daily, or lower frequency returns. The

continuous time models in (2) offer a natural way to introduce high frequency data into the

discrete time models and to construct proxies for the scale factors.

There have been several ways in the literature to deal with high frequency data in discrete

time volatility models. The first one is to apply the model to a five-minute grid, or even

study the continuous time limit and to infer the implications for the daily or weekly sampling

frequency, see, for example, Drost and Nijman (1993), Drost and Werker (1996), and Meddahi

and Renault (2004). In empirical applications this approach leads to serious time aggregation

problems: the parameters estimated using very short intervals, typically turn out to be

internally inconsistent with those obtained from daily or lower frequencies. These problems

have often led to the impression that the discrete time model structure (1) is fundamentally

flawed and cannot be made consistent with the data. Andersen and Bollerslev (1997) argue,

on the other hand, that intraday seasonality causes a daily barrier below which stationary

models should not be applied. Their daily barrier agrees with our choice for the business day

as a unit of time in the models (1) and (2). They propose a model for five-minute returns,

which explicitly takes into account intraday periodicities on top of the daily volatility s

Alizadeh, Brandt, and Diebold (2002) assume an intraday Brownian motion, and use the

daily high-low range for quasi maximum likelihood estimation of a discrete time stochastic

volatility model. They obtain improved estimators due to the high-low range being superior

to absolute or squared daily returns as a proxy for the volatility s

The remainder of the paper is organized as follows. Section 2 provides a continuous time

extension of the discrete time models. Section 3 introduces proxies, and develops tools for

ranking and optimizing them. Section 4 constructs a good proxy for the S&P 500 data.

In Section 5 we conclude. Appendices A, B, and C contain a description of the data, the

empirical technique of prescaling and a number of proofs, respectively.

2 The Continuous Time Models

In the classical Black Scholes world the log return process is a continuous time Brownian

motion σB

with constant volatility σ > 0. This leads to discrete time returns

= σZ

(3)

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where the Z

are iid standard Gaussian random variables. The Black Scholes model has

been extended to deal with time varying volatility. In the continuous time extensions the

term volatility refers to either the instantaneous diffusion coefficient, or the (square root of)

the quadratic variation over a given time period. The discrete time generalizations replace

the constant σ in (3) by a sequence (s

) of strictly positive random variables:

= s

(1)

The innovations (Z

) are iid, and the innovation Z

is independent of s

. The term volatil-

ity now refers to the scale factor s

, which generally differs from the continuous time concept

of daily volatility.

In order to study intraday based proxies for the scale factor s

, we integrate high frequency

data into the discrete time models by replacing the iid innovations (Z

) by iid stochastic

processes (Ψ

) :

(ϑ) = s

(ϑ),

ϑ ∈ [0, 1].

(2)

The day n return process R

is the log return process with respect to the previous day’s

close. The return process R

is the product of the daily scale factor s

and a stochastic

process Ψ

. The process Ψ

is a cadlag process

representing the intraday price pattern.

The time of day ϑ is in [0, 1]. The processes Ψ

are iid. The sequence of scale factors (s

)

may be any strictly positive stochastic process, as long as the scale factor s

is independent

of all (Ψ

) for k = n up to infinity. In particular, the scale factor s

and the process Ψ

are

independent, for each day n. The continuous time model structure (2) will be referred to as

the scaling hypothesis. A model that satisfies the scaling hypothesis leads to the well-known

product structure (1) for the daily close-to-close returns,

= R

(1) = s

(1).

The continuous time models have the following interpretation. The discrete time scale

factor s

describes the state of the market on day n, and represents the level of overall price

sensitivity. The factor s

may be determined by the process over the past days, (s

n−k

, Ψ

n−k

k ≥ 1, for instance like a Garch model. It may also include an external source of randomness.

The actual fluctuations in the process Ψ

determine the pattern of the intraday return

For identification, the innovations are typically assumed mean zero, unit variance. Such identification

assumptions will not be necessary in this paper.

The sample paths are right continuous and have left limits.

Page 7

process, such as up or down days, quiet or hectic days. The process Ψ

offers a flexible way

of taking into account intraday seasonality. It may also have, for instance, leverage effects,

jumps, stochastic spot volatility.

As an example, consider the case that Ψ is a diffusion with stochastic instantaneous

volatility v(ϑ),

dΨ(ϑ) = v(ϑ)dB(ϑ),

which implies for the n-th day return process R

(ϑ) = s

(ϑ) dB

(ϑ),

ϑ ∈ [0, 1].

(4)

So, under the scaling hypothesis, in the often used diffusive model for the log price process

p(t),

dp(t) = σ(t)dB(t),

the instantaneous volatility σ(t) allows a decomposition into a daily scale factor s

and an

independent intraday stochastic volatility component v

(ϑ), modelling intraday effects such

as seasonalities. The intraday volatility processes (v

) are iid and independent of events up

to the close of the n-1

day

Let us provide the remaining formal details for the continuous time process R

. The

trading hours are normalized to cover the interval [0, 1]. Time advances only during trading

hours. Let Ψ be a cadlag process on the closed interval [0, 1], left continuous in 1. The

processes (Ψ

) are independent copies of the process Ψ. In order to describe the dependence

structure, it is convenient to introduce the discrete time model filtration (G

). The σ-field

includes the history of (s

, Ψ

) extended with s

n+1

. So, G

= σ{(Ψ

)

i≤n

, (s

)

i≤n+1

}. The

σ-field G

represents the model information at the start of day n+1. The model information

includes the observable day n return process, as R

= s

. The process Ψ

n+1

is independent

of G

, and the processes (Ψ

) are identically distributed.

Note that Ψ(1) is not standardized

. More generally, processes that satisfy the scaling

Andersen and Bollerslev (1997) propose to model the seasonal intraday volatility in five-minute returns

n,ϑ

on day n as the product of a daily scale factor s

and deterministic intraday components (v

) :

n,ϑ

= s

· v

· ε

n,ϑ

, ϑ

= i/K, i = 1, . . ., K, where ε

n,ϑ

is an iid mean zero, unit variance error. Their

model is a special case of the continuous time model (4), discretized to five-minute returns.

If Ψ(1) is not standardized, then the discrete time returns satisfy r

= aσ

+ σ

, where the ε

are

iid(0,1). So the continuous time model (2) also embeds a simple version of the ARCH-M model introduced

by Engle, Lilien, and Robins (1987).

Page 8

hypothesis do not have a unique representation. Consider a representation (s

, Ψ

) for (R

)

that satisfies the scaling hypothesis. Set s

′

= s

/2 and Ψ

′

= 2Ψ

. Then

′

≡ R

(5)

As we will see later, identification of s

and Ψ

is not necessary for the study of proxies.

3 Proxies

The daily scale factor s

in the continuous time model (2) is difficult to estimate, even if we

observe the full sample path of the asset price. This stands in contrast to the situation of

obtaining a perfect estimate for the diffusion coefficient using the full sample path, as Merton

(1980) discusses. We will work with (ex post) statistics of the sample path to estimate s

Let us introduce the concept of volatility proxy in the spirit of Alizadeh, Brandt, and Diebold

(2002).

Consider volatility proxies that are positive functionals H(R

) of the day n return process

. If the functional H is positively homogeneous,

H(αR

) = αH(R

α ∈ [0, ∞),

then a proxy H(R

) is linear in s

H(R

) = s

H(Ψ

Applying logarithms leads to a measurement equation: the log of a proxy is the sum of two

independent terms, the log of the scale factor and a measurement error U

= log(H(Ψ

)) :

log(H(R

)) = log(s

) + U

(6)

The measurement errors U

form an iid sequence. Naturally the quality of the proxy is

related to the bias and the variance of the measurement error U

. Rewriting equation (6),

log(H(R

)) = log(s

) + EU

+ (U

− EU

makes clear that for a given functional H the measurement error introduces a constant bias,

= µ. In applications where the proxy is used as a variable in a regression, the bias will

be corrected by the regression parameters. It is also possible to rescale the proxy, replacing

H by aH, in order to obtain a bias-corrected version. So, the key determinant of the quality

Page 9

is the measurement variance λ

= var(U

(7)

An optimal proxy functional H

∗

will satisfy

var(log(H

∗

(Ψ))) = inf

var(log(H(Ψ))).

For a proxy functional H, the measurement error U

only depends on the process Ψ

. This

means that the optimality of a proxy functional is independent of the particular discrete

time model for the scale factors (s

3.1 Definition and Basic Properties

Let us now provide the formal details for volatility proxies. Recall from Section 2 that the

process Ψ is cadlag on [0, 1]. Let D[0, 1] denote the Skorohod space of cadlag functions on

[0, 1], which are left continuous in 1. Endow D[0, 1] with the Skorohod topology. The space

D[0, 1] is a separable, complete metric space (see Billingsley (1999)). The space C[0, 1] of

continuous functions on the unit interval is a linear subspace of D[0, 1].

From now on we assume the scaling hypothesis for the day n return process: R

(ϑ) =

(ϑ). A proxy is the result of applying a certain estimator, the proxy functional H, to

the day n return process R

. We restrict attention to positively homogeneous functionals in

order to ensure that the decomposition in (6) holds.

Definition 3.1. Let H be a measurable, positively homogeneous functional D → [0, ∞), on

a linear subspace D of D[0, 1]. Assume Ψ ∈ D a.s., and H(Ψ) > 0 a.s. Then H is a proxy

functional. The random variable H(R

) is a proxy.

Remark 1. Proxy functionals are nonlinear: H(Ψ) + H(−Ψ) > 0 a.s., but H(Ψ − Ψ) =

H(0) = 0; proxy functionals need not be symmetric: H(Ψ) = H(−Ψ), in general.

Example 3.1.1. Here are some examples of proxies: the absolute daily return; the absolute

overnight return; the high-low range; realized volatility, defined for any grid as the square

root of the sum of the squared returns over the grid; maximal absolute two-minute return;

the square root of the daily increment in quadratic variation. In these cases, if Ψ(1) has a

density, the almost sure positivity of H(Ψ) is ensured.

Combining different proxies through a positively homogeneous function, generates a new

proxy. Write x >> 0 if min

> 0.

Page 10

Proposition 3.2. Let H

(i)

, i = 1, . . ., d, be proxy functionals on D. Let G : [0, ∞)

→ [0, ∞)

be a measurable, positively homogeneous function. Moreover, assume G(x) > 0 for x >> 0.

Then the functional H : f ↦→ G(H

(1)

(f), . . ., H

(d)

(f)) is a proxy functional on D.

Proof. Since every H

(i)

(Ψ) > 0 a.s. we have (H

(1)

(Ψ), . . ., H

(d)

(Ψ)) >> 0 a.s., so H(Ψ) > 0

a.s. Homogeneity holds, since

H(αf) = G(H

(1)

(αf), . . ., H

(d)

(αf)) = G(αH

(1)

(f), . . ., αH

(d)

(f)) = αH(f).

□

As a consequence, given two proxy functionals H

(1)

and H

(2)

, it is possible to obtain

a new proxy functional, for example, by scaling one of them with a positive number, by

adding them, or by taking the maximum or the minimum. One may also take a geometric

combination:

H(R

) ≡ (H

(1)

))

(2)

))

, w

∈ R, w

+ w

= 1.

(8)

Scaling a proxy with a positive number a > 0 does not change the measurement variance

var(log(aH(Ψ))) = var(log(H(Ψ))) = λ

(9)

Assume 0 < var(log(s

)) < ∞. It then follows, using the decomposition (6), that

corr(log(H(R

)), log(s

)) = (1 +

var(log(s

))

)

−1/2

(10)

This means that proxies with smaller measurement variance λ

have larger correlation with

log(s

). The ideal situation of zero measurement variance gives perfect correlation. The

proxy is then a perfect proxy:

H(R

) = cs

for a certain constant c > 0. Such perfect proxies do exist, in special cases. For example,

if Ψ is the standard Brownian motion on [0, 1]. The quadratic variation of Ψ then equals

one, and s

is the square root of the quadratic variation: √QV

= s

. If a proxy has zero

measurement variance, then one knows the value of s

, without knowing the model for the

sequence (s

). The following example shows that the square root of the quadratic variation

is not necessarily the most efficient proxy for s

Page 11

Example 3.1.2. Consider the case that Ψ is a diffusion as in (4):

dΨ(ϑ) = v(ϑ)dB(ϑ),

where B denotes standard Brownian motion. Let the instantaneous volatility process v(ϑ) be

deterministic at the opening and stochastic for the rest of the day. More specifically, suppose

v(ϑ) equals 1 before time of day ϑ

= 1/2, and v(ϑ) equals either c

or c

after ϑ

, both with

probability 1/2. The square root of the truncated quadratic variation over [0, 1/2] equals s

times a constant, and as such is perfect. The square root of the quadratic variation of R

the product of s

and a random variable with positive variance

Estimating the daily scale factor s

is a different exercise from estimating (the square

root of) the quadratic variation. The quadratic variation is a measure for the price fluctu-

ations that actually occurred during the whole trading day. As an estimator of the daily

scale factor s

, the square root of the quadratic variation is merely one out of many possible

candidates.

Let us return to the issue of identification. The following proposition states that different

representations (s

, Ψ

) for the scaling hypothesis result in the same ordering for proxy

functionals. So, for our purposes identification plays no role.

Proposition 3.3. Suppose H

(1)

and H

(2)

are proxy functionals. Moreover, assume (s

, Ψ

)

and (s

′

, Ψ

′

) both satisfy the scaling hypothesis for R

. If H

(1)

is better than H

(2)

for Ψ, then

(1)

is also better than H

(2)

for Ψ

′

Proof. See appendix C.

□

3.2 Existence of Optimal Proxies

Recall that an optimal proxy functional H

∗

satisfies

var(log(H

∗

(Ψ))) = inf

var(log(H(Ψ))).

Theorem 3.4. If there exists a proxy functional with finite measurement variance, then

there exists an optimal proxy functional.

Proof. See appendix C.

□

The next proposition states that optimal proxies are scaled versions of one another, except

possibly on a set of measure zero.

Page 12

Proposition 3.5. Suppose H

(1)

and H

(2)

are two optimal proxy functionals. Then there

exists a constant a > 0, such that H

(1)

(Ψ)

a.s.

= aH

(2)

(Ψ).

Proof. See appendix C.

□

3.3 Empirically Ranking Proxies

This section provides a practical way to compare proxies. By definition, a proxy functional

(1)

is more efficient than H

(2)

if it has smaller measurement variance:

(λ

(1)

)

≤ (λ

(2)

)

Comparison by measurement variances is infeasible in an empirical situation: the measure-

ment variances cannot be estimated, since the processes (Ψ

) are not observed. However,

taking variances on both sides of the decomposition (6) gives

var(log(H

(i)

))) = (λ

(i)

)

+ var(log(s

)).

(11)

There is no covariance term by the independence of s

and Ψ

. Equation (11) shows that the

variances of the proxies all have the common term var(log(s

)). It follows that if the variance

of the log proxy is smaller, then the measurement variance must be smaller. So, proxies may

be ranked by the variances of the log proxies. Let us summarize this in a proposition.

Proposition 3.6. Let H

(1)

and H

(2)

be two proxy functionals. Assume U

(i)

= log(H

(i)

(Ψ)),

i = 1, 2, and log(s

) have finite variances. Assume R

satisfies the scaling hypothesis. Then

var(log(H

(1)

))) − var(log(H

(2)

))) = (λ

(1)

)

− (λ

(2)

)

The autocorrelation function is an additional indicator of the quality of a proxy, see Figure

2 below. Let ρ

(j) denote j-th order autocorrelation. If the process (s

, Ψ

) is stationary,

then, using (6),

log(H(R

))

(j) = a

· ρ

log(s

)

(j) + e

(j)

(12)

≈ a

· ρ

log(s

)

(j).

where a

∈ (0, 1], see Appendix C. The proportion factor a

is close to unity if the measure-

ment variance λ

is small. In practice, the correction term e

(j) vanishes for j → ∞, and

the autocorrelations ρ

log(s

)

(j) are large and decay slowly. Under suitable assumptions the

Page 13

ratio of the autocorrelation functions for different proxies converges,

log(H

(1)

))

(j)

log(H

(2)

))

(j) →

var(log(s

)) + (λ

(2)

)

var(log(s

)) + (λ

(1)

)

j → ∞.

(13)

This means that better proxies have larger autocorrelations. See Appendix C for details.

3.4 Improving Proxies

This section provides a way to combine given proxy functionals H

(1)

, . . ., H

(d)

into a more

efficient proxy functional. Consider the geometric combination of these functionals,

(w)

) =

∏

i=1

(i)

))

+ . . . + w

= 1, w

∈ R.

(14)

Here, the column vector w is the d-dimensional coefficient vector. The restriction ∑ w

= 1

is needed to obtain a proxy functional, but the coefficients are not restricted to the interval

[0, 1]. Let Λ denote the covariance matrix of the measurement errors U

(i)

= log(H

(i)

(Ψ)):

Λ = cov([U

(1)

, . . ., U

(d)

]

′

(15)

The measurement variance λ

of the geometric combination U

(w)

in (14) is λ

= w

′

Λw and,

as for the minimal variance portfolio in Markowitz portfolio theory, λ

is minimal for

∗

−1

′

−1

ι = (1, . . ., 1)

′

(16)

with optimal variance λ

∗

′

−1

This solution is empirically infeasible: it is impossible to estimate the variance matrix Λ,

since the measurement errors U

(i)

are not observed. Let Λ

p,n

denote the covariance matrix

of the log proxies:

p,n

= cov([log(H

(1)

)) . . .log(H

(d)

))]

′

(17)

The covariance matrix Λ

p,n

is the covariance matrix Λ with a common noise term var(log(s

))

added to each element:

p,n

= Λ + var(log(s

)) ιι

′

(18)

Page 14

Assuming stationarity for (s

), the covariance matrix Λ

= Λ

p,n

may be estimated from the

data. Formula (19) in Theorem 3.7 below shows how to obtain the optimal coefficients w

∗

in (16) from the covariance matrix Λ

p,n

Theorem 3.7. Let R

satisfy the scaling hypothesis. Assume var(log(H

(i)

(Ψ))) < ∞ for

i = 1, . . ., d. Let the covariance matrices Λ and Λ

p,n

be defined by (15) and (17), respectively.

The optimal coefficient vector w

∗

in (16) does not depend on the form of the process (s

)

and may be expressed as

∗

−1

p,n

′

−1

p,n

(19)

Let λ

∗

′

−1

. The variance of the logarithm of the optimal geometric proxy is

var(log(H

∗

)

))) = λ

∗

+ var(log(s

)).

Proof. The optimal coefficient w

∗

does not depend on (s

): it follows from equation (18)

that

arg min

′

p,n

w = arg min

′

Λw + var(log(s

))) = arg min

′

Λw.

(20)

Define the Lagrangian w

′

p,n

w+µ (1−w

′

ι). Differentiating the Lagrangian with respect

to w yields 2Λ

p,n

w −µι = 0, hence w = 1/2 Λ

−1

p,n

µι. By ι

′

w = 1, this yields µ = 2/ι

′

−1

p,n

ι and

w = Λ

−1

p,n

ι/ι

′

−1

p,n

ι. Since w

′

p,n

w is convex in w and there is a unique solution to the first

order condition, it is the optimum.

Use (20) to obtain the equalities w

∗

= Λ

−1

p,n

ι/ι

′

−1

p,n

ι = Λ

−1

ι/ι

′

−1

ι, which imply

var(log(H

∗

)

))) = λ

∗

+ var(log(s

)).

□

Remark 2. In empirical applications one uses estimates of the variances. In order to reduce

estimation error, we shall use the technique of prescaling, see Appendix B.

Assuming stationarity for the process (s

, Ψ

), the covariance matrix Λ

is consistently

estimated by the sample covariance matrix of the log of the proxies, thereby providing

coefficientsw that are consistent for w

∗

. However, this estimator for w

∗

may remain consistent

while allowing, for example, for structural breaks in the scale factors (s

). See the consistency

condition (22) in Appendix C for more details.

Page 15

4 A good proxy for the S&P 500

This section applies the techniques of Section 3 to the S&P 500 futures tick data. Appendix

A describes the data.

4.1 Microstructure Noise Barrier

On small time scales financial prices are subject to market microstructure effects, such as

the bid-ask bounce, price discreteness, and asynchronous trading, see, for instance, Zhang,

Mykland, and Aıt-Sahalia (2005), Oomen (2006), and Hansen and Lunde (2006b). These

effects may invalidate the model assumptions. Microstructure effects may be avoided by

sampling at sufficiently wide intervals.

In this paper the measure of comparison is the variance of the logarithm. The standard

realized volatility RV and the high-low based realized volatility RV HL (see Table 1) depend

on the sampling interval ∆ϑ. Figure 1 shows the graph of ∆ϑ → ̂

var(log(H

∆ϑ

))), for

∆ϑ ranging from zero to sixty minutes. These curves suggest that a qualitative change of

behaviour occurs for ∆ϑ ≈ five minutes for realized volatility, and ∆ϑ ≈ eight minutes for

realized high-low. From now on realized volatility is based on five-minute sampling intervals

or larger. For realized high-low our minimal sampling interval will be ten minutes.

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

Interval length (min.)

var (log(RV))

(a) Realized volatility

0.14

0.145

0.15

0.155

0.16

0.165

0.17

0.175

0.18

0.185

0.19

Interval length (min.)

var (log(RVHL))

(b) Realized high-low

Figure 1:

Plots of the sample variance of the log of a proxy with ∆ϑ ranging from zero to 60 minutes (zero

is tick per tick). (a) Realized volatility. (b) Realized high-low.

4.2 Ranking Proxies

Table 1 compares twelve simple proxies constructed from the data. We emphasize that these

proxies are a set of twelve out of many possible proxies. They are of no special importance

Page 16

themselves.

full

1st

2nd

3rd

4th

name

RV5

0.064

0.070 0.070 0.073 0.042

RV10

0.080

0.085 0.093 0.090 0.052

RV15

0.089

0.096 0.105 0.093 0.061

RV20

0.100

0.110 0.117 0.103 0.071

RV30

0.117

0.133 0.134 0.113 0.087

abs-r

0.611

0.683 0.550 0.635 0.568

0.161

0.179 0.176 0.160 0.130

maxar2

0.118

0.134 0.124 0.118 0.088

RAV5

0.058

0.060 0.065 0.066 0.040

RAV10

0.072

0.072 0.085 0.082 0.049

RVHL10

0.053

0.057 0.061 0.061 0.034

RAVHL10 0.047

0.048 0.055 0.054 0.031

Table 1:

Performance of twelve proxies. The full sample is split into four subsamples. Prescaling by

EWMA(0.7) predictor for RV5. The following proxies are included.

RV5: root of sum of squared 5 min

returns; RV10: root of sum of squared 10 min returns; RV15: root of sum of squared 15 min returns; RV20:

root of sum of squared 20 min returns; RV30: root of sum of squared 30 min returns; abs-r: absolute close-

to-close return; hl: high-low of the intraday return process; maxar2: maximum of the absolute 2-minute

returns; RAV5: sum of absolute 5-minute intraday returns; RAV10: sum of absolute 10-minute intraday

returns; RVHL10: root of sum of 10-minute squared high-lows; RAVHL10: sum of 10-minute high-lows;

full

1st

2nd

3rd

4th

name

RV5-up

0.066

0.070 0.070 0.073 0.051

RV5-down

0.094

0.103 0.101 0.104 0.068

RV10-up

0.091

0.094 0.101 0.095 0.074

RV10-down

0.133

0.138 0.146 0.149 0.100

RAV5-up

0.064

0.066 0.070 0.067 0.051

RAV5-down

0.097

0.099 0.101 0.110 0.077

RAV10-up

0.093

0.093 0.102 0.097 0.081

RAV10-down

0.147

0.145 0.155 0.168 0.120

RAV10HIGH 0.053

0.055 0.061 0.054 0.041

RAV10LOW

0.081

0.082 0.086 0.090 0.064

Table 2:

Performance of upward/downward decomposed proxies. The table contains proxies from Table

1 splitted according to upward and downward price movements. For example, RV5-up is root of sum of

squared 5 min positive returns, RAV10HIGH is the sum of 10-minute highs, and RAV10LOW is the sum of

10-minute absolute lows.

For each proxy a measure of comparison is given for five samples: first the full sample

(days 2 to 4575) and then for four subsamples spanning the full sample (2:1144, 1145:2287,

2288:3431, 3432:4575). The measure of comparison is PV (prescaled variance), which is

the variance of the logarithm of a proxy after prescaling, PV = var(log(H(R

)/p

)); see

Section 3.3 and Appendix B. The first observation cannot be prescaled and is left out of

the variance computations. Smaller variances correspond to more efficient proxies. For

the prescaling sequence (p

) we take an exponentially weighted moving average predictor

Page 17

of five-minute realized volatility with smoothing parameter β = 0.7, yielding a prescaling

sequence p

= 0.7 p

n−1

+ 0.3 RV 5

n−1

. We have set the smoothing parameter so that the

sample variance of the logarithm of prescaled five-minute realized volatility is minimal. The

prescaled variance ranks proxies, but its value is not a measure for the quality of a proxy.

The first column of Table 1 shows that the quality of the realized volatility RV improves if

one increases the sampling frequency from 30 minutes to 5 minutes. The prescaled variance is

maximal for the absolute close-to-close returns, confirming the fact that absolute or squared

daily returns are poor proxies

. Note that the maximal absolute two-minute return is better

than high minus low, which tends to use returns based on much longer time spans. Overall,

we find that sums of absolute values lead to more efficient proxies than sums of squared

values. This observation relates to a finding of Barndorff-Nielsen and Shephard (2003), whose

simulations indicate that absolute power variation, based on the sum of absolute returns,

has better finite sample behaviour than realized quadratic variation. The best performing

proxy in Table 1 is RAV HL10, the sum of the ten-minute high-low ranges. The remaining

columns of Table 1 show that the ranking of the different proxies in the various subsamples

is the same as in the full sample, with one exception in the second subsample for RV 30 and

maxar2.

Table 2 provides the prescaled variances for the upward and downward components of a

number of proxies. For instance, the five-minute realized volatility is decomposed according

to upward and downward price movements:

RV5 =

∑

i=1

n,ϑ

∑

i=1

n,ϑ

>0}

∑

i=1

n,ϑ

<0}

= RV5-up + RV5-down.

Note that RV5 (PV=0.064) is better than RV5-up (PV=0.066) and RV5-down (PV=0.094).

The upward proxies are consistently more efficient than their downward counterparts. This

difference suggests that, when proxying the scale factor s

, one should put more weight on

the positive returns.

We use the absolute returns larger than 0.001, or 10 basis points, in order to avoid taking the log of

zero. This leaves 4079 daily returns.

Page 18

4.3 Optimized Proxy

Let us now combine the proxies in Tables 1 and 2 into a more efficient one using the technique

of Section 3.4. The five-minute realized volatility RV5 has PV=0.064. Let us improve

upon this value. First, by using the high-low range over intraday intervals, RVHL10 has

PV = 0.053, see Table 1. It is even better to use absolute values: RAVHL10 has PV = 0.047.

Now use the theory of Section 3.4 to combine the high-low ranges in RAVHL10 with the

absolute returns in RAV10: inserting the covariance matrix

p,n

of the log of these two

prescaled proxies into formula (19), yields the proxy

H(R

) = (RAV HL10

)

1.82

(RAV 10

)

−0.82

(PV = 0.041).

Decomposing RAVHL10 into its upward and downward components we obtain the proxy

(w)

) = (RAV 10HIGH

)

1.04

(RAV 10LOW

)

0.72

(RAV 10

)

−0.76

(PV = 0.038).(21)

Of course, one may also apply the optimal coefficient formula (19) to all twenty-one proxies

in Tables 1 and 2 at once

. The full combination yields a proxy with PV = 0.037, which

only marginally outperforms the proxy obtained in (21). We prefer to work with the simpler

proxy in (21) and we will refer to it as the optimized proxy H

(w)

The optimized proxy easily outperforms all proxies in Tables 1 and 2. If one extrapolates

the full sample prescaled variances of the realized volatilities of Table 1 to a time interval

of length zero (corresponding to the limiting case of the quadratic variation), one obtains a

value between 0.050 and 0.060. The value PV = 0.038 for the optimized proxy is well below

these values, suggesting that it is a more efficient proxy for the daily scale factor than the

square root of the quadratic variation.

Observe that the coefficient for RAV10 in the optimized proxy is negative (w

= −0.76).

In geometrical terms this negative coefficient may be explained as follows. The log proxies

are vectors in an affine space. The proxies are highly related, since they all approximate

the same daily scale factor s

. The optimal proxy is not in the convex hull of the proxies in

Tables 1 and 2. The original proxies do not completely reflect the direction of the optimal

proxy. The coefficients outside [0, 1] correct the direction.

Table 3 investigates the stability of the optimized proxy H

(w)

). Similarly to Table 1

it reports performance measures for the full sample and for four subsamples. The first row

reports the performance of H

(w)

) in the different subsamples; comparison with Tables

1 and 2 shows that H

(w)

) outperforms all those proxies in every subsample. The proxy

(w,i)

) is constructed using the coefficients that are optimal for the i-th subsample. In

We exclude the absolute close-to-close return in performing this calculation.

Page 19

full

1st

2nd

3rd

4th

name

(w)

0.038

0.039 0.043 0.043 0.028

(w,1)

0.038

0.039 0.043 0.043 0.028

(w,2)

0.039

0.039 0.043 0.043 0.029

(w,3)

0.039

0.040 0.044 0.042 0.029

(w,4)

0.039

0.040 0.045 0.044 0.027

Table 3:

Geometric proxies, optimized for different sub-

samples: performance and stability.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Lag (days)

Figure 2:

Autocorrelations of log proxies.

Lags 1 to 50 days. From bottom to top:

RV30, RV15, RV10, RV5, H

(w)

the first subsample the performance of the globally optimized H

(w)

) (PV=0.039) is not

substantially improved by H

(w,1)

) (PV=0.039). A similar statement holds for the other

subsamples. Moreover, proxies based on a particular subsample are close to optimality in

all other subsamples. For instance, the proxy optimized for the first subsample (the years

1988–1992) is nearly optimal for the years 2002–2006. We conclude that the optimality of

(w)

) is stable.

Figure 2 shows the autocorrelations of log(H

(w)

)) and the log of four different realized

volatilities, RV30, RV15, RV10, and RV5. The autocorrelations of the different realized

volatilities increase as the sampling interval decreases. The autocorrelations for the optimized

proxy are substantially larger than those for five-minute realized volatility. Hence, formula

(13) provides an additional indication that H

(w)

is the best.

Table 4 explores the quality of the optimized proxy in a heuristic way. It gives the

coefficient of determination, R

, of a linear regression of the logarithm of a proxy on the

logarithm of another proxy lagged one day:

log(H

(j)

)) = α + β log(H

(i)

n−1

)) + ε

Large R

’s in a particular column mean that the proxy in that column is largely predictable,

suggesting that it is a good proxy for daily volatility. The R

’s attain their maximum at the

optimized proxy, in the most right column.

Finally, Figure 3 shows the time series graphs of four different proxies. The proxies were

standardized to have mean one, by dividing them by their mean. From top to bottom the

The optimized proxy is constructed as an optimized proxy for s

, not as an optimal predictor for s

n+1

Even so, the R

’s attained in the row H

(w)

(−1) are large.

Page 20

RV30 RV20 RV15 RV10 RV5

(w)

RV30(-1)

0.35

0.39

0.42

0.46

0.50

0.58

RV20(-1)

0.38

0.42

0.45

0.49

0.54

0.61

RV15(-1)

0.39

0.44

0.47

0.50

0.55

0.63

RV10(-1)

0.41

0.45

0.48

0.52

0.57

0.66

RV5(-1)

0.43

0.48

0.51

0.55

0.60

0.69

(w)

(−1)

0.44

0.48

0.51

0.54

0.60

0.71

Table 4:

of the regression log(H

(j)

)) = α + β log(H

(i)

n−1

)) + ε

, for i, j = 1, . . ., 6, and n =

2, . . ., 4575.

curves become ’less erratic’, suggesting a decrease in the measurement errors U

. Each step

shows a marked improvement.

5 Conclusions

This paper provides a theoretical basis for ranking and optimizing volatility proxies, based

on intraday data. The theory is founded on a natural class of continuous time extensions of

discrete time, daily volatility models, taking into account intraday seasonality. Good proxies

are needed for parameter estimation and forecast evaluation of discrete time volatility models.

In this paper a volatility proxy is the result of applying a positively homogeneous func-

tional to the intraday return process. By definition, a good proxy has small measurement

variance. We show that optimal proxies exist and provide easy-to-implement tools for rank-

ing and improving proxies. The approach is to a large extent model free: an optimal proxy

for the scaling factor s

is an optimal proxy under all possible discrete time models of the

form r

= s

, where the Z

are iid innovations, and the innovation Z

is independent of

the scale factor s

For the S&P 500 data a combination of the highs, the lows, and the absolute returns over

ten-minute intervals yields a good proxy. One should put more weight on the highs than on

the lows, when proxying volatility. The empirical results indicate that the optimized proxy,

although it uses only a finite sampling grid, is more efficient for the scale factor s

than (the

square root of) the quadratic variation, which is based on continuous sampling.

This paper has addressed the problem of ranking and optimizing proxies for today’s scale

factor s

. We see opportunities to use proxies for the specification of the daily volatility

process (s

), and accordingly use proxies to forecast future volatility. We aim to pursue

these ideas in future research.

Page 21

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

(a) Absolute close-to-close returns

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

(b) Intraday high-low range

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

1988

1990

1992

1994

1996

1998

2000

2002

2004

2006

(d) Optimized proxy from formula (21)

Figure 3: Time series of four standardized proxies, H(R

H(R

Page 22

A Data

Our data set is the U.S. Standard & Poor’s 500 stock index future, traded on the Chicago

Mercantile Exchange (CME), for the period 1st of January, 1988 until May 31st, 2006. The

data were obtained from Nexa Technologies Inc. (www.tickdata.com). The futures trade

from 8:30 A.M. until 15:15 P.M. Central Standard Time. Each record in the set contains

a timestamp (with one second precision) and a transaction price. The tick size is $0.05 for

the first part of the data and $0.10 from 1997-11-01. The data set consists of 4655 trading

days. We removed sixty four days for which the closing hour was 12:15 P.M. (early closing

hours occur on days before a holiday). Sixteen more days were removed, either because of

too late first ticks, too early last ticks, or a suspiciously long intraday no-tick period. These

removals leave us with a data set of 4575 days with nearly 14 million price ticks, on average

more than 3 thousand price ticks per day, or 7.5 price ticks per minute.

There are four expiration months: March, June, September, and December. We use the

most actively-traded contract: we roll to a next expiration when the tick volume for the next

expiration is larger than for the current expiration.

An advantage of using future data rather than the S&P 500 cash index is the absence

of non-synchronous trading effects which cause positive autocorrelation between successive

observations, see Dacorogna et al. (2001). As in the cash index there are bid-ask effects in

the future prices which induce negative autocorrelation between successive observations. We

deal with these effects by taking large enough time intervals, see Section 4.1. Since we study

a very liquid asset the error term due to microstructures is relatively small.

B Prescaling

The methods of comparing proxies in Proposition 3.6 and of improving proxies in Theorem

3.7 are formulated in terms of population variances and covariances. In practical situations,

one has to work with the sample counterparts of these quantities, which introduces sampling

error. To reduce the sampling error caused by the scale factors (s

), we propose the technique

of prescaling. The idea is to stabilize the sequence (s

), by scaling it by a predictable sequence

of random variables (p

). Let F

= σ(R

, i ≤ n) denote the observable information up until

day n.

Definition B.1. A prescaling sequence (p

) is an (F

n−1

) adapted sequence of strictly positive

random variables.

Page 23

The prescaling factors p

will be used to define adjusted scale factors

˜s

= s

Proposition B.2. Assume the processes (R

) satisfy the scaling hypothesis. Prescale the

scale factors (s

) to obtain the sequence (˜s

) above. The corresponding processes (

), where

= ˜s

, satisfy the scaling hypothesis.

Proof. The variables (˜s

) are positive. Both p

n+1

and s

n+1

are G

-measurable, hence so is

˜s

n+1

. Therefore

satisfies the scaling hypothesis.

□

As a result one may define proxies for ˜s

. These are prescaled proxies:

) = H(R

)/p

The proxy H(

) for ˜s

has the same measurement error as the proxy H(R

) for s

= log(H(

)) − log(˜s

)

= log(H(R

)) − log(p

) − (log(s

) − log(p

))

= U

So, ranking and optimizing proxies before and after prescaling are equivalent in terms of

population statistics; therefore one may replace s

by ˜s

, and H(R

) by H(

) in Proposition

3.6 and Theorem 3.7. As a consequence, the population value of the noise term var(log(s

))

in equations (11) and (18) changes into

var(log(˜s

)) = var(log(s

)).

A good predictor p

of s

results in a small term var(log(˜s

)).

C Proofs

Proof of Proposition 3.3. By assumption s

′

= s

. Independence of s

and Ψ

implies

var(log(s

′

)) + var(log(H

(1)

(Ψ

′

))) = var(log(s

)) + var(log(H

(1)

(Ψ

)))

≤ var(log(s

)) + var(log(H

(2)

(Ψ

)))

= var(log(s

′

)) + var(log(H

(2)

(Ψ

′

))).

Hence var(log(H

(1)

(Ψ

′

))) ≤ var(log(H

(2)

(Ψ

′

))).

□

Page 24

Proof of Theorem 3.4. We have to show that there exists a measurable, positively homoge-

neous functional H

∗

: D → [0, ∞), with H

∗

(Ψ) > 0 a.s., and var(log(H

∗

(Ψ))) ≤ var(log(H(Ψ)))

for all proxy functionals H.

For a proxy functional H, write U = log(H). Define λ

= var(log(H(Ψ))). Let U denote

the space of all log proxy functionals with λ

< ∞. The space U is not empty, by assump-

tion. If EU(Ψ) = a = 0, then H

′

= e

−a

H is an equally good proxy functional for which

Elog(H

′

(Ψ)) = 0. Therefore we may restrict attention to the subspace U

of U of centered

functionals. The space U

is affine: if U

, U

∈ U

, and w ∈ R, then wU

+ (1 − w)U

∈ U

since (H

(1)

)

(2)

)

(1−w)

is a proxy functional, see equation (8).

Define λ

inf

= inf

H:log(H)∈U

{λ

}. Consider the space L

(D, B), of equivalence classes [U] of

log proxy functionals U, with inner product < [U

(1)

], [U

(2)

] >= E(U

(1)

(Ψ)U

(2)

(Ψ)). Here, B

denotes the Borel sigma-field for D. Notice that U

is a subset of L

and that λ coincides

with the L

-norm ||.|| on U

. Let U

, U

, . . . ∈ U

be a sequence for which ||U

|| → λ

inf

. Then

], [U

], . . . is a Cauchy sequence in L

: apply the parallelogram law to obtain

0 ≤ ||U

− U

≤ −4||

+ U

+ 2||U

Since U

is affine, (U

+ U

)/2 ∈ U

, hence ||

≥ λ

inf

. Therefore ||U

− U

≤

−4λ

inf

+ 2λ

→ 0 for m, n → ∞.

By completeness of L

the sequence [U

], [U

], . . . converges to an element [U

] in L

and by

continuity of the norm λ

= λ

inf

. Pick a functional U

∈ U

from [U

]. Let us use U

construct a functional H

∗

that satisfies the conditions stated at the start of the proof. For

every L

convergent sequence there exists a subsequence that converges almost surely. Let

= log(H

)

(f)) → U

(f) on a set C almost everywhere in D. Define on the convergence

set C: H

∗

(f) = limH

)

(f). For {αf : f ∈ C, αf /∈ C, α ∈ [0, ∞)}, define H

∗

(αf) =

αH

∗

(f). For remaining f ∈ D define H

∗

(f) ≡ 0. The functional H

∗

assigns a single value

to each f ∈ D : consider f

, f

∈ C, α

, α

> 0, and f = α

= α

. Then H

∗

(α

) ≡

∗

) = α

∗

(α

/α

) By homogeneity of H

∗

on C this equals α

∗

) ≡ H

∗

(α

Being the result of a limit, the functional H

∗

is measurable. Positive homogeneity follows

by construction. Moreover, H

∗

(Ψ) > 0 almost surely, since U

(Ψ)

a.s.

= log(H

∗

(Ψ)) and

var(U

(Ψ)) = λ

< ∞. Finally, var(log(H

∗

(Ψ))) = λ

inf

≤ λ

for all H.

□

Lemma C.1. If H

∗

is an optimal proxy functional, and H is a proxy functional, then

cov(log(H

∗

(Ψ)), log(H(Ψ))) = (λ

∗

)

Page 25

Proof of Lemma C.1. Consider the proxy functional H(f) ≡

∗

(f))

(H(f))

1−w

, with

measurement variance λ

= w

(λ

∗

)

+ 2w(1 − w) cov

(log(H

∗

(Ψ)), log(H(Ψ))) + (1 − w)

Since H

∗

is optimal, ∂λ

/∂w |

w=1

= 0. Hence cov(log(H

∗

(Ψ)), log(H(Ψ))) = (λ

∗

)

□

Proof of Proposition 3.5. Both proxy functionals have measurement variance (λ

∗

)

. Let H

denote the centered proxy: H

= exp(−Elog(H(Ψ)) ) H, with Elog(H

(Ψ)) = 0. Consider

the covariance of the centered log proxies: cov(log(H

(1)

(Ψ)), log(H

(2)

(Ψ))). By Lemma C.1

this covariance equals (λ

∗

)

. By Cauchy-Schwarz this equality holds if and only if H

(1)

(Ψ)

a.s.

(2)

(Ψ). In other words, if and only if H

(1)

(Ψ)

a.s.

= aH

(2)

(Ψ), for certain a > 0.

□

Autocorrelation formulas (12) and (13). Correlation calculations yield for a

and e

(j) :

var(log(s

))

var(log(s

)) + λ

, and e

(j) = corr(log(s

), U

−j

) ·

√var(log(s

)) λ

var(log(s

)) + λ

The proportion factor a

∈ (0, 1] is large when the measurement variance λ

is small. The

correction term e

(j) depends on the impact of U

−j

= log(H(Ψ

−j

)) on log(s

). The factor

to the right hand side of the multiplication dot is in [0, 1]. The ratio of two autocorrelation

functions equals

log(H

(1)

))

(j)

log(H

(2)

))

(j)

(1)

· ρ

log(s

)

(j) + e

(1)

(j)

(2)

· ρ

log(s

)

(j) + e

(2)

(j)

Assuming that the impact of U

−j

is small compared to the autocorrelation in log(s

) :

corr(U

(i)

−j

, log(s

))/corr(log(s

−j

), log(s

)) → 0

j → ∞,

gives, upon dividing both numerator and denominator by ρ

log(s

)

(j) (assuming ρ

log(s

)

(j) = 0

for all j > 0),

log(H

(1)

))

(j)

log(H

(2)

))

(j) →

(1)

(2)

var(log(s

)) + (λ

(2)

)

var(log(s

)) + (λ

(1)

)

j → ∞.

(13)

□

Consistently estimating the optimal coefficients. First some notation. Let (X

)

n∈1...N

be a

series of vectors. Let ̂

var(X

) and ̂

cov(X

) denote the standard empirical variance and covari-

ance matrices of the series (X

), respectively. Let H(R

) be shorthand for the d-dimensional

column vector of proxies H

(i)

), and let U

denote the accompanying measurement errors.

Let log(H(R

)) denote the element wise logarithms. So, log(H(R

)) = log(s

) · ι + U

Page 26

The standard formula for the sample variance of the sum of random vectors gives:

var(log((H(R

)) = ̂

var(log(s

))ιι

′

+ ̂

var(log(U

)) + 2 · ̂

cov(U

, log(s

) · ι).

The estimatorw is given byw = arg min

′

var(log(H(R

)))w. As in the proof of Theorem

3.7, the variance of log(s

) drops out:

w = arg min

′

(

var(log(U

)) + 2 · ̂

cov(U

, log(s

) · ι)

) w.

Ifw is consistent for w

∗

, then asymptotically it should solve arg min

′

Λw. The term

var(log(U

)) converges to Λ for increasing sample sizes, since the measurement error vectors

are iid. So, the consistency ofw comes down to the consistency condition that the sample

covariance converges to zero in probability:

cov(U

, log(s

) · ι)

→ 0,

N → ∞.

(22)

In addition to existence of second moments and the independence of U

and log(s

), the

stationarity for (s

, Ψ

) is a sufficient, but not necessary, condition which ensures that the

consistency condition (22) holds.

□

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