WAVELETS AND ESTIMATION OF LONG MEMORY IN LOG VOLATILITY AND TIME SERIES PERTURBED BY NOISE

Percival and Walden (2002) present a wavelet methodology of the least squares estimation of the long memory parameter for fractionally differenced processes. We suggest that the general idea of using wavelets for estimating long memory could be used for the estimation of long memory in time series perturbed by noise. One prominent example thereof is the time series of log-Garman-Klass estimates of log volatility of fi nancial markets. The estimator of Percival and Walden (2002) is biased if the long memory time series is perturbed by noise. We propose a new estimator of the long memory parameter which combines (in its construction) the frequency-domain approach of Sun & Phillips (2003) and the approach of Percival & Walden (2002). We illustrate the properties of the proposed estimator via Monte Carlo simulations. The results show that the estimator may be useful for the estimation of the long memory in volatility.

The paper is organized as follows: an introduction to the maximal overlap discrete wavelet transform and the wavelet variance is given in Section 2. The logarithm of volatility and its estimators are defi ned and the long memory property of volatility is discussed in Section 3. The notion of long memory is introduced and the spectrum of log volatility is discussed in Section 4. Section 5 presents the methodology of estimation of the long memory parameter with the least squares method and wavelets of Percival & Walden (2002) and proposes a new estimator suitable for perturbed long memory processes.Monte Carlo simulations are used to assess the properties of this estimator.

MODWT and wavelet variance
In these paragraphs, we will give a brief introduction to the maximal overlap discrete wavelet transform (MODWT) and the wavelet variance, which will be used for the estimation of the long-memory parameter in the perturbed time series of the logarithm of volatility.
Our summary of the wavelet theory is based on Percival & Walden (2002) and Bašta (2010), which encompass further details and proofs of the statements given below.Further books may be recommended: e.g., Vidakovic (1999) gives a more technical and abstract introduction to wavelets.Gençay & Selçuk & Whitcher (2002) present a less technical introduction with applications in fi nance and economics.
2.1 Spectrum of the stochastic process and its properties Let {X t : t = …, -1, 0, 1, …} be a discrete-time second-order stationary (hereafter, stationary) stochastic process.Let {s X,τ : τ =…,-1, 0, 1, …} be the autocovariance sequence of this process.The spectrum of this process may be defi ned as , ( ) exp( 2 ), for , where i is the imaginary unit and f is a real-valued variable called frequency.Since spectrum is an even function of frequency and a periodic function of frequency with the period equal to unity, it is suffi cient to study and explore it only in the range of frequencies [0, 1/2].We may say informally that the spectrum S X (.) informs us about the "frequency content" of the stochastic process {X t } in the sense that 1/2 ,0 0 2 ( ) .
The contribution of the frequency interval (f, f + df), where 0 ≤ f ≤ f + df ≤ 1/2, to the variance s X,0 of the process {X t } is given as ( ) .
X S f df (3)

Linear filtering
Let the fi nite sequence {a t : t = 0, …, M -1} of real-valued (non-random) variables be called the linear fi lter of length M. For ease of notation, we will use the notation {a t } whenever the values that t can assume are clear from the text.Let the linear fi ltering of the discrete-time stationary stochastic process {X t : t = …,-1, 0, 1, …} with the linear fi lter {a t } be a discrete-time stationary stochastic process {Y t : t = …,-1, 0, 1,… } defi ned as 1 0 , for ..., 1,0,1,...
Let S X (.) and S Y (.) be the spectrum of the process {X t } and of the process {Y t } defi ned in the sense of Eq. 1.The effect of the linear fi ltering with the linear fi lter {a t } of the stochastic process {X t } is given by the characteristics of the linear fi lter {a t }.An important characteristics is the frequency response F a (.) of this fi lter defi ned as the Fourier transform of {a t }, i.e., 1 0 ( ) exp( 2), for .
The function |F a (.)| 2 , i.e., the square of the modulus of F a (.), is called the squared gain function of the fi lter {a t }.The squared gain function is an even and periodic function of frequency, with the period equal to unity.Therefore, further on it is suffi cient to explore it only in the frequency range [0, 1/2].The squared gain function informs us about frequencies that are passed and about frequencies that are attenuated by the effect of the linear fi ltering.More specifi cally, it can be shown that

Wavelet and scaling filters
To introduce the maximal overlap discrete wavelet transform, hereafter abbreviated to MODWT, we fi rst need to defi ne a special set of linear fi lters.This special set may be created with the knowledge of just two elementary fi lters: the fi lter {h 1,t : t = 0, …, L 1 -1} and the fi lter {g 1,t : t = 0, … , L 1 -1}, both of lengths L 1 .The linear fi lters {h 1,t : t = 0, …, L 1 -1} and {g 1,t : t = 0, …, L 1 -1} are interconnected via the so-called quadrature mirror relationship, i.e., 1 1, 1, 1 1 ( 1) , for 0,..., 1 and fulfi l certain specifi c conditions, e.g., 1, 1, 2 0 0, for any nonzero integer An ideal fi lter with the nominal pass band [d, u], where d and u are real parameters satisfying 0 ≤ d ≤ u ≤ 1/2, is such a fi lter whose squared gain function is given as 2 , 1, for ( ) 0, for 0 and for 1 / 2 It can be shown that {h For a given pair of fi lters {h 1,t } and {g 1,t }, two sets of fi lters may be created via the so-called pyramid algorithm.This algorithm will not be specifi ed here because it is technical and will be of no use for the purpose of this text (details on the algorithm can be found, e.g., in Percival & Walden, 2002;Gençay & Selçuk & Whitcher, 2002;Vidakovic, 1999;and Bašta, 2010).Only the properties of the fi lters from both sets will be discussed in this text.
Similarly, let us call the fi lters from the second set the scaling fi lters and let us denote them as {g 1,t : t = 0, ... , L 1 -1}, {g 2,t : t = 0, ... , L 2 -1}, {g 3,t : t = 0, ... , L 3 -1}, etc., where the fi lter {g 1,t } is the fi lter already defi ned in the previous paragraphs and where L 1 , L 2 , L 3 are the lengths of these fi lters.Again, the set of the scaling fi lters is infi nite and generally the jth fi lter of this set is denoted as {g j,t : t = 0, …, L j -1} and has the same length as fi lter {h j,t }, i.e., L j .
There are different families of sets of wavelet and scaling fi lters, for example the Haar family, the D(4) family, the LA(8) family, etc. (for details see, e.g., Percival & Walden, 2002).

MODWT wavelet coefficients and wavelet variance
Let {X t : t = …, -1, 0, 1, …} be a discrete-time stationary stochastic process with the variance σ X 2 ≡ s X,0 and the spectrum S X (.).The linear fi ltering of {X t } with the wavelet fi lter {h j,t : t = 0, …, L j -1} (for j = 1,2, …) results in a discrete-time stationary stochastic process defi ned as 1 , , , 0 , for ..., 1, 0,1, ..., called the process of the jth level MODWT wavelet coeffi cients.jth level MODWT wavelet coeffi cients posses certain intriguing properties: they are stationary and it can be shown that their unconditional mean is zero and their unconditional variance -called the jth level wavelet variance -is defi ned as It can be shown Moreover, let S Wj (.) be the spectrum of jth level wavelet coeffi cients.It holds that (see Eq. 6 where |F hj (f)| 2 is the squared gain function of the fi lter {h j,t }.Therefore (see Eq. 2), The approximation ≈ in Eq. 15 is meant in the sense that {h j,t } is approximately an ideal fi lter with the nominal pass band [2 -(j + 1) , 2 -j ] (see Section 2.3) and thus, |F hj (f)| 2 may be approximately replaced with |F i d,u (f)| 2 with d = 2 -(j + 1) and u = 2 -j .Eq. 13 and Eq. 15 tell us that ν X,j 2 is associated with the contribution of the frequency band [2 -(j + 1) , 2 -j ] to the variance of the stochastic process {X t }.

Estimation of wavelet variance
Let us be given a portion X 0 , X 1 , …, X N -1 of the stationary process {X t : t = …, -1, 0, 1, ..} and let us defi ne {W X,j,t : where mod denotes the operation modulo.The modulo operation affects the construction of W X,j,t for 0 ≤ t < L j -1, whereas the construction of W X,j,t for L j -1 ≤ t ≤ N -1 would remain unaffected if the modulo operation was left out in the defi nition of Eq. 16.
Therefore, we say that W X, j,t for 0 ≤ t < L j -1 are affected by the circularity assumptions, whereas W X, j,t for L j -1 ≤ t ≤ N -1 are not.
An unbiased estimator of the wavelet variance ν X,j 2 of Eq. 12 for N -L j ≥ 0 is defi ned as where 1, i.e., only W X, j,t not affected by the circularity assumption are involved in the construction of the estimator.

Log volatility
In this section, we introduce the notion of volatility and its two estimators.For a given trading day t (i.e., from the open of the day t to the close of the day t), let us assume that the logarithm of the stock price is a Brownian motion.Further, let Close t be the closing stock price for the trading day t and Open t be the opening stock price for the trading day t.Let the return for the trading day t be defi ned as log log( ) log( ).
Furthermore, let V t be the variance of R t , hereafter called volatility, i.e., 2 ( ) and let log volatility H t be defi ned as the logarithm of V t , i.e., log( ) Further on, we will assume that log volatility is constant for a given trading day but can differ from one trading day to another.

Log-squared-return estimator
With respect to the defi nitions and assumptions of the previous section, we may write ) and (see, e.g., Breidt et al., 1998) The variance of this estimator is var(A t |H t ) = π 2 /2.We will call A t the log-squared-return estimator.

Log-Garman-Klass estimator
If the information on the opening, closing, high and low prices for the trading day t is available, it is possible to obtain another unbiased estimator with a smaller variance than that of the log-squared-return estimator.Let us defi ne (see Garman & Klass, 1980

Stochastic processes
So far we have analyzed the estimation of the log volatility for a single trading day t based on the information for this day.Now, let us generalize our approach to a sequence of trading days indexed by …, -1, 0, 1, ….Each of these trading days is characterized by H t , log(U t 2 ) and Z t .Again, we stress that log volatility is assumed to be constant during a given trading day but can differ from one trading day to another.Specifi cally, let {H t : t = …, -1, 0, 1, …}, {log(U t 2 ): t = …, -1, 0, 1, …} and {Z t : t = …, -1, 0, 1, …} be discrete-time stochastic processes.From the assumptions of the previous sections it can be shown that the process {H t } is independent of {log(U t 2 )} and also of {Z t }.Moreover, {log(U t 2 )} and {Z t } are white noise processes.In practice realizations of {H t } and {log(U t 2 )} are not observable.For each trading day t = …, -1, 0, 1, … either the log-squared-return estimator A t or the log-Garman-Klass P t estimator may be constructed to estimate log volatility H t of the day.We may write 2 1 log( ) , ..., 1, 0,1, ...

Long memory and volatility
In this section, we introduce long memory processes and one example of these processes -the fractionally differenced process -and argue that log volatility processes are long memory ones.We discuss their spectra.
where B is the lag operator defi ned as BX t ≡ X t -1 , {E t : t = …, -1, 0, 1, …} is a Gaussian white noise process with the variance σ E 2 , and δ is a real-valued parameter satisfying 0 < δ < 0.5.Hereafter, the parameter δ will be called the long memory parameter.It can be shown that the spectrum S X (.) of the fractionally differenced process is given as
The spectrum S H (.) of {H t } is thus given as (see Eq. 28; we assume the frequency range The logarithm of Eq. 30 results in 2 2 log ( ) log(4sin ( )) log( ).
For "f close to zero" we can use the approximation 2 log ( ) 2 log(2 ) log( ).
Thus, we can write Following the idea of Sun & Phillips (2003) we can write Taking the logarithm of both sides results in For "close to zero" and f "close to zero" we can write 2 2 2 2 log ( ) 2 log(2 ) log( ) (2 ) .

Long memory parameter estimation
In this section, we build up on the methodology by Percival & Walden (2002), who present the least squares estimation of the long memory parameter with the use of wavelets in unperturbed long memory time series.We enhance this methodology so that it enables us to estimate the long memory parameter in the case of long memory processes perturbed with independent white noise (which is the case of, e.g., the log-Garman-Klass estimates of volatility of fi nancial series).

Percival & Walden estimator of the long memory parameter
Percival & Walden (2002) present a general methodology that allows the estimation of the long memory parameter δ in unperturbed fractionally differenced processes.Let us assume that log volatility {H t : t = …, -1, 0, 1, …} is a stationary fractionally differenced process with the long memory parameter δ.Let S H (.) be the spectrum of {H t }, ν H,j 2 the jth level wavelet variance of {H t } and n H,j the estimate of the jth level wavelet variance.
Let us further assume that the spectrum S H (.) is approximately constant and equal to C j in the frequency range 2 -(j + 1) < f < 2 -j for j = 1, 2, .... From Eq. 15 it follows that 2 , 2 .
Again, for j such that "2 -(j + 1/2) is close to zero" we can write 2 where 2 -(j + 1/2) is the centre of the frequency interval [2 -(j + 1) , 2 -j ] on the logarithmic scale.Furthermore, we get Now, let us assume (at least for a while) that realizations of {H t } be observable, which is not the case in reality.Then, let us calculate the estimates n H,j 2 of the wavelet variances ν H,j 2 in the sense of Eq. 17 for all integers j satisfying α ≤ j ≤ β, where α and β are positive integers chosen in such a way so that:  2 -(a + 1/2) is "close to zero".This condition ensures that 2 -(j + 1/2) is "close to zero" for every integer j satisfying j ≥ α.  N -L β ≥ 0, where L β is the length of the wavelet fi lter {h β,t }.This ensures that the variance of the jth level wavelet coeffi cients can be estimated for all integers j satisfying 1 ≤ j ≤ β, since M j ≥ 1 for 1 ≤ j ≤ β (see also Eq. 17).
For further on, let us assume that the integer j satisfi es α ≤ j ≤ β.Further, we can use the approximation (see, e.g., Percival & Walden, p. 313 where χ 2 [η j ] is a chi-square random variable with η j degrees of freedom, where max{ / 2 ,1}, 1 For ease of notation we use the equality sign (=) instead of the approximation sign (≈) having in mind that the equality is only approximate. 2 Again, we use the equality sign (=) instead of the approximation sign (≈) having in mind that the equality is only approximate.3 d stands for equality in distribution where M j is defi ned in Eq. 18.Thus 2 2 2 , ,
We can write (see Bartlett & Kendall, 1946;Percival & Walden, 2002) 2 where ψ(.) and ψ ′ (.) is the digamma and trigamma function.Thus, for we can write where is a random variable with a zero mean and variance equal to ψ ′(η j /2).Moreover, e j * is approximately normally distributed (for η j > 10) and e j * and e i * (for integer j and integer i satisfying α ≤ j ≤ β, α ≤ i ≤ β and i ≠ j) are approximately uncorrelated (see Percival & Walden, 2002).Afterwards, an estimate of δ -let us call it the PW estimate -can be obtained from the estimate of the slope of the linear regression model of Eq. 50.Moreover, the estimate of log(σ E 2 ) can be obtained from the estimate of the intercept in the linear regression model of Eq. 50.The parameters of the regression model can be estimated by the weighted least squares with the weight at the level j given as 1/ψ ′(η j /2).
However, one major diffi culty occurs.During the construction of the PW estimator we have assumed that the realizations of {H t } be observable so that n H,j 2 could be calculated in the sense of Eq. 17.However, this is not the case as realizations of {H t } are not observable since {H t } is buried in noise.We will tackle this problem in the next section.

An estimator for perturbed time series -Approach A
First let us give some notations.Let S P (.) and ν P, j 2 (for j = 1, 2, ...) be the spectrum of {P t } and the jth level wavelet variance of {P t }.Furthermore, let n P, j 2 be the estimate of ν P, j 2 calculated in the sense of Eq. 17 from a given realization of a portion of {P t }.
The Approach A assumes that n P, j 2 equals n H, j

2
. Afterwards, the estimate of δ may be easily obtained based on the model of Eq. 50.Thus, said in a word, in Approach A we estimate the long memory parameter with the approach of Percival & Walden as if the time series was not perturbed by noise.

An estimator for perturbed time series -Approach B
In contrast to the Approach A, the Approach B takes into account a fi rst-order correction for the presence of noise.More specifi cally, the Approach B assumes that is "close to zero" and f is "close to zero".Then (see Eq. 38) Afterwards, we follow ideas and steps similar to those used in the construction of the PW estimator above (see Section 5.1).At this moment, the positive integers α and β have to be chosen in such a way that   and δ are not known in advance, the upper bound of

  
has to be estimated. N -L β ≥ 0, where L β is the length of the wavelet fi lter {h β, t }.
Then, for integer j satisfying α ≤ j ≤ β we can write where and where χ 2 [η j ] is a chi-square random variable with η j degrees of freedom where η j is defi ned in Eq. 44.The error term e j in Eq. 54 has a zero mean and variance equal to ψ ′(η j /2).Moreover, e j and e i (for integer j and integer i satisfying α ≤ j ≤ β, α ≤ i ≤ β and i ≠ j) are approximately uncorrelated.The parameter δ can be estimated within the non-linear regression model of Eq. 54 via the weighted least squares estimator with the weight at the jth level given as 1/ψ ′(η j /2).

An iterative approach
We propose a way of estimating δ within the model of Eq. 54 which avoids using non-linear least squares.Since several approximations have already been used in the construction of the model of Eq. 54, this model is unlikely to be an exact description of reality.Therefore, it might be suffi cient -instead of using non-linear regressionto apply a procedure consisting of the following steps: Let us obtain the starting estimate d 0 of the long memory parameter δ as well as the starting estimate s E,0 of the parameter σ E .More specifi cally, d 0 is related to the weighted least squares estimate of the slope in the linear regression model of Eq. 50, whereas s E,0 can be obtained as a function of the weighted least squares estimate of the intercept of the model of Eq. 50.
Consequently, the corrected estimates d 1 and s E,1 of the long memory parameter δ and of the parameter σ E are obtained from the following linear regression model estimated by ordinary weighted least squares with the weight at the level j given as 1/ψ ′(η j /2).
Consequently, the fi nal estimate d of δ is obtained from estimated by ordinary weighted least squares with the weight at the level j given as 1/ψ ′(η j /2).

Monte Carlo simulations
We illustrate the properties of the Approach A (i.e., the approach of Percival & Walden applied to the time series as if the time series were not perturbed by noise) and the Approach B (which takes into account a fi rst-order correction for the presence of noise) via a Monte Carlo simulation.Several realizations of fractionally differenced processes of Eq. 29 (with different values of δ and σ E 2 ) perturbed with a Gaussian white noise with variance σ Z 2 = 0.26 (see Section 3.2) were generated.The following settings of the simulation were chosen:  Length N: 512. Long memory parameter δ: 0.1, 0.2, 0.3, 0.4, 0.45, 0.49.
where σ E 2 is the variance of process {E t } of Eq. 29 and σ Z 2 = 0.26; see Section 3.2): 0.1, 0.5, 1, 1.5, 2, 2.5, 3, 5, 10.  Wavelet family: Haar.On the one hand, Haar wavelet fi lters are the shortest ones.Therefore, only a few wavelet coeffi cients are affected by the circularity assumption.On the other hand, Haar wavelet fi lters might be quite poor approximations to ideal band-pass fi lters for the respective frequency bands.
Our choice of the Haar wavelet family is due to its simplicity.
 For each possible combination of {δ, σ E 2 /σ Z 2 } 500 realizations of length N of the process of Eq. 29 were generated perturbed with a Gaussian white noise with a variance σ Z 2 = 0.26. α = 2, β = 9. 4 This choice of α and β corresponds to the case where it is assumed that the time series can be modelled well as a realization of a perturbed long memory process (with parameters δ, σ E 2 and σ Z

2
) in the low frequency region as well as in the high frequency region. log(n P,j 2 ) of this perturbed series was calculated for α ≤ j ≤ β.The long memory parameter was estimated with Approach A and B.
In Fig. 1, we present the bias of the estimation of δ for both the approaches (Approach A and Approach B).Clearly, the Approach A suffers from a downward bias in the estimation of δ.This bias is especially large for small values of σ E 2 /σ Z 2 .For example, for δ ≥ 0.4 and σ E 2 /σ Z 2 ≤ 1.5, the absolute bias is larger than 0.1.For a given δ the bias of the Approach A tends approximately to zero with increasing σ E 2 /σ Z 2 .On the other hand, the Approach B seems to be superior to the Approach A in terms of bias.Even for 0.1 ≤ σ E 2 / σ Z 2 ≤ 1.5, the absolute bias is bounded by about 0.06 for δ = 0.1, 0.5, 1, 1.5, 2, 2.5, 3, 5, 10.For σ E 2 /σ Z 2 ≥ 2 the absolute bias is not larger than 0.02 for δ = 0.1, 0.5, 1, 1.5, 2, 2.5, 3, 5, 10.
We may thus conclude that estimating the long memory parameter following the Approach A (i.e., the Percival & Walden's approach as if the time series were not perturbed by noise) may lead to a large bias in the estimation of this parameter.This bias is substantially reduced if the Approach B is applied (i.e., a fi rst-order correction for the presence of noise is applied).The reduction of the bias is not complete in the Approach B due to several approximations in the model building.
In Fig. 2, we present the root mean squared error (RMSE) of the estimation of δ for the Approaches A and B. Both the approaches generally lead to a higher RMSE for smaller values of σ E 2 /σ Z

2
. We can see that the Approach B is superior to the Approach A in terms of RMSE for δ ≥ 0.2 for any considered value of σ E 2 /σ Z 2 .This result (together with the results of the bias study above) suggests that the application of the Approach B (i.e., the fi rst-order correction for the presence of noise) is desirable if the long memory parameter is close to 0.5.In the case of the long memory parameter being close to 0, the Approach A (i.e., no correction for the presence of noise) might be well acceptable.
In this paper, we have not explored the characteristics of both the approaches for other settings (i.e., for other choices of N, δ, the ratio σ E 2 /σ Z 2 , the wavelet family, α and β).In spite of this fact, it can be largely expected that the Approach B (fi rst-order correction for the presence of noise) might be superior to the Approach A (no correction for presence of noise) under many circumstances.

Conclusions
Wavelets are an intriguing tool of data analysis.They may be utilized for the estimation of the long memory parameter in long memory processes: Percival & Walden (2002) summarize a wavelet methodology of the least squares estimation of the long memory parameter for fractionally differenced processes.Building up on their work, we have suggested an approach based on wavelets that may be used in the estimation of the long memory parameter in time series perturbed by noise such as log-Garman-Klass estimates of log volatility of fi nancial markets.The proposed approach makes a fi rst-order correction for the presence of noise.Our Monte Carlo simulations suggest that this correction leads to superior estimation when compared to a case where no correction for the perturbation is taken into account.In future research, it might be possible to exploit the fact that wavelets are capable of depicting timevarying properties of time series and our approach might be extended to the estimation of the time-varying long memory parameter in time series perturbed by noise.

WAVELETS AND ESTIMATION OF LONG MEMORY IN LOG VOLATILITY AND TIME SERIES PERTURBED BY NOISE
Abstract: Percival and Walden (2002) present a wavelet methodology of the least squares estimation of the long memory parameter for fractionally differenced processes.We suggest that the general idea of using wavelets for estimating long memory could be used for the estimation of long memory in time series perturbed by noise.One prominent example thereof is the time series of log-Garman-Klass estimates of log volatility of fi nancial markets.The estimator of Percival and Walden (2002) is biased if the long memory time series is perturbed by noise.We propose a new estimator of the long memory parameter which combines (in its construction) the frequency-domain approach of Sun & Phillips (2003) and the approach of Percival & Walden (2002).We illustrate the properties of the proposed estimator via Monte Carlo simulations.The results show that the estimator may be useful for the estimation of the long memory in volatility.

4
For small values of σ E 2 /σ Z 2 , this choice of α and β does not satisfy the condition of being close to zero.Despite of this, it might be instructive to see what the properties of the proposed estimators are in this case (i.e., for α = 2 and β = 9 and for small values of σ E

Figure 1
Figure 1 The bias (on the y-axis) of the estimation of the long memory parameter δ in fractionally differenced processes (see the text for further specifi cations).Points connected with the solid line correspond to the Approach B, whereas points connected with the dashed line correspond to the Approach A. The ratio σ E 2 /σ Z 2 is plotted on the x-axis.Different subplots correspond to different values of the long memory parameter δ.The top row corresponds to (from left to right) δ = 0.1, δ = 0.2 and δ = 0.3.The bottom row corresponds to (from left to right) δ = 0.4, d = 0.45 and δ = 0.49.

Figure 2
Figure 2 The decadic logarithm (on the y-axis) of the root mean squared error of the estimation of the long memory parameter δ in fractionally differenced processes (see the text for further specifi cations).The points connected with the solid line correspond to the Approach B, whereas the points connected with the dashed line correspond to the Approach A. The ratio σ E 2 /σ Z 2 is plotted on the x-axis.Different subplots correspond to different values of the long memory parameter δ.The top row corresponds to (from left to right) δ = 0.1, δ = 0.2 and δ = 0.3.The bottom row corresponds to (from left to right) δ = 0.4, δ = 0.45 and δ = 0.49.
Molnár, 2011) ) -log(Open t ), B t ≡ log(Min t ) -log(Open t ), C t ≡ log(Close t ) -log(Open t ), where Max t , Min t , Close t , Open t are the high, low, closing and opening stock prices for the day t.It can be shown that Gaussian distribution with (see for exampleMolnár, 2011)γ ≡ E(Z 0,t ) = -0.13,var(Z 0,t ) = 0.26 and is independent of H t .Let us defi ne P t ≡ P 0,t -γ, Z t ≡ Z 0,t -γ.Then, σ Z 2 ≡ var(Z t ) = 0.26 and P t = H t + Z t is an unbiased estimator of H t in the sense that E(P t |H t ) = H t .The variance of this estimator is var(P t |H t ) = 0.26.We will call P t the log-Garman-Klass estimator.