| | π΄[S ] = ( ), Ξ(1 β p) || cos π || sin pπ 2 ( ) ) ( pπ p Ξ 1 β πΌ | πΎ | pπΎ cos πΌ p | | π΄[|S| ] = ( ), Ξ(1 β p) || cos π || cos pπ
for p β (β2, β1) βͺ (β1, πΌ).
for p β (β1, πΌ).
2
( ) πΌπ and Ξ denotes the Gamma function. From the above representations, where π = arctan π½ tan 2
moments with negative values of p are attainable. This results into the logarithmic moments approach that provides an easier way of estimating stable distribtuion parameters compared to the FLOM.
3.5.2. Logarithmic moments This approach is as a result of the challenges encountered when using the FLOM method which requires computing Gamma functions, the inversion of the sinc function and it only works for some p. The current method suggests computing derivatives with respect to the moment order p resulting in moments of the logarithms of the stable process. We illustrate in the following. Lemma 3.5β Let S denote a symmetric stable random variable and let p β β. Then Mn : = π΄[(log |S|)n ] = lim pβ0
n
d π΄[|S|p ], dpn
n = 1, 2, β¦ .
(25)
The moments follow readily for n = 1, 2, β¦. i.e. ( ) | π | 1 1 |. M1 = π΄[log |S|] = π0 1 β + log || | πΌ πΌ | cos π | ( M2 = π΄[(log |S| β π΄[log |S|])2 ] = π1
1 1 + 2 πΌ2
) β
(26) π2 . πΌ2
) ( 1 M3 = π΄[(log |S| β π΄[log |S|])3 ] = π3 1 β 3 . πΌ
(27)
(28)
where π = arctan(π½ tan πΌπβ2) and terms πk are given by π0 = β0.57721566, π1 = π 2 β6, π = 1.2020569 derived from the polygamma function πkβ1 =
d
k
dxk
log Ξ(x)|x=1 .
Proofβ 3.6 The proof is provided in KuruoΔlu (2001).
(29) β·
4. Parameter estimation of stable processes The four common methods for estimating parameters of stable processes include: quantiles method (see Fama & Roll, 1971; McCulloch, 1986, 1996), the logarithmic moments method (see KuruoΔlu, 2001), the empirical characteristics method (see Yang, 2012), and the ML method (see Nolan, 2001). We investigate their accuracy in the following.
4.1. The quantiles method The quantile method was pioneered by Fama and Roll (1971) but was much more appreciated through McCulloch (1986) after its extension to include asymmetric distributions and for cases where πΌ β [0.6, 2] unlike the former approach that restricts it to πΌ β₯ 1.
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Suppose sΜ is a given data sample then the estimates for πΌ and π½ are given by πΌΜ = Ξ1 (πΜ πΌ , πΜ π½ ) and Μ π½ = Ξ2 (πΜ πΌ , πΜ π½ ) where
πΜ πΌ =
sΜ 0.95 β sΜ 0.05 , sΜ 0.75 β sΜ 0.25
πΜ π½ =
sΜ 0.95 + sΜ 0.05 β 2sΜ 0.05 . sΜ 0.95 β sΜ 0.05
(30)
The notation sΜ q represents the qth quantile of sample sΜ and, πΌΜ and π½Μ are obtained by functions Ξ1 (πΜ πΌ , πΜ π½ ) and Ξ2 (πΜ πΌ , πΜ π½ ) given in Tables III and IV in McCulloch (1986) through linear interpolation. Consequently, the scale parameter is given by
πΜ =
sΜ 0.75 β sΜ 0.25 , Μ Ξ (πΌ, Μ π½)
(31)
3
Μ is given by Table V in McCulloch (1986). The consistent estimator π is then obtained Μ π½) where Ξ3 (πΌ, through interpolation. Finally the location parameter π is estimated through a new parameter defined by
{ π=
π + π½πΎ tan ππΌ ; 2 π;
πΌβ 1 πΌ = 1.
(32)
Moreover, π is estimated by (33)
Μ Μ 5 (πΌ, Μ π½), πΜ = sΜ 0.5 + πΞ
Μ is obtained from Table VII (McCulloch, 1986) by linear interpolation. The location Μ π½) where Ξ5 (πΌ, parameter is estimated consistently by πΜ = πΜ + π½ΜπΜ tan
π πΌΜ . 2
(34)
4.2. Empirical characteristic function method Suppose a set of observable data {s1 , s2 , β¦ , sN } follows a stable distribution. Then we can approximate the characteristic function of this data by applying a basic Monte Carlo approach based on the law of large numbers i.e.
Μ Ξ¦(u) = π΄[eiusj ] β Ξ¦(u) =
N 1 β iusj e . N j=1
(35)
We can express the characteristic function (12) in terms of the cosine and sine function from basic trigonometric principles, i.e. πΌ
Ξ¦(u) = eβ|πu| (cos π + i sin π),
(36)
where
π = πu β |πu|πΌ π½ sign(u)π(u, πΌ) { tan ππΌ , πΌβ 1 2 π(u, πΌ) = 2 log |u| , πΌ=1 π As a result, we observe that
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(37)
|Ξ¦(u)| = eβ|πu| . πΌ
The estimated characteristic function relates to the model parameters by
Μ )| = π πΌ |u |πΌ ; for k = 1, 2, u > 0, πΌ β 1. log |Ξ¦(u k k k
(38)
Solving this system leads to the estimation representation formulas for the stability and variance parameters: Μ log |Ξ¦(u )|
1 log log |Ξ¦(u Μ )|
2 . | u1 | log || u || | 2| Μ Μ log |u1 | log(β log |Ξ¦(u 2 )|) β log |u2 | log(β log |Ξ¦(u1 )|) log πΜ = . | |u log || u1 || | 2|
πΌΜ =
The real and imaginary parts of the characteristic function (36) provide estimates for π½Μ and πΜ:
arctan
Im(Ξ¦(u)) = πu β |πu|πΌ π½ sign(u)π(u, πΌ). Re(Ξ¦(u))
(39)
Suppose Ξ₯(u): = arctan(ImΞ¦(u)βReΞ¦(u)) and choose another set of positive numbers uk , k = 3, 4 together with πΌΜ and πΜ then the estimates of the location and skewness parameters are given respectively by
πΜ =
π½Μ =
u4πΌΜ Ξ₯(u3 ) β u3πΌΜ Ξ₯(u4 ) u3 u4πΌΜ β u3 u4πΌΜ
u4 Ξ₯(u3 ) β u3 Ξ₯(u4 ) πΜ tan π2πΌΜ (u4 u3πΌΜ β u3 u4πΌΜ ) πΌΜ
(40)
.
.
(41)
Notice, it can be deduced from Equation (36) that
log(β log(|Ξ¦(u)|2 )) = log(2π πΌ ) + πΌ log(u). This provides an alternative way to envision the regression estimation method:
yk = m + πΌxk + πk ;
k = 1, 2, β¦ , M;
Μ )| ), m = log(2π ), x = log(u ) and π is an error term. The stability where yk = log(β log |Ξ¦(u k k k k πk parameter πΌ and the scale parameter π can be estimated by selecting uk = , k = 1, 2, β¦ , M; of 25 real data (see Koutrouvelis, 1980, Table I). The estimates πΌΜ and πΜ are then used to estimate π½ and π using the following relation 2
zl = π l + π l ,
πΌ
l = 1, 2, β¦ , Q.
Μ and πl is some random error. The prowhere zl = Ξ₯n (ul ) + πkn (ul ), πl = πΜl u β |πΜl u| π½ sign(u)π(u, πΌ) πl posed real data set for Q (see Koutrouvelis, 1980, Table II) is ul = , l = 1, 2, β¦ , Q. πΌΜ
50
4.3. Logarithmic moments method This approach follows the theory discussed in Section 3.5.2. The key innovation with this method is that there is no need of computing Gamma functions and the sinc function as in the FLOM. Secondly, techniques of parameter estimation for symmetric stable random variables (i.e. π½ = 0) can be Page 12 of 28
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applied to skewed stable random variables (i.e. π½ β 0) and, techniques of parameter estimation for centered stable random variables (i.e. π = 0) to non-centered ones (i.e. π β 0) through centro-symmetrization. However, this comes at a cost of losing almost half of the sample data. Therefore to obtain better estimates one has to use large sample data sets.
4.3.1. Centro-symmetrization of stable random data sets Let Sk be a sequence of n independent stable random variables distributed according to
Sk βΌ S(πΌ, π½, π, π). Then the distribution of a weighted sum of the above sequence with weights ak can be estimated using their characteristic function: n β β β a n n k β β οΏ½ οΏ½ οΏ½ k=1 ak πβ, οΏ½ak οΏ½πΌ π, π½, ak Sk βΌ SβπΌ, n Z= β β β k=1 k=1 οΏ½ak οΏ½πΌ k=1 β β β β k=1 n
(42)
where the pth power of a number x is defined by
x
= sign(x)|x|p . As a result, it is easy to obtain sequences of independent stable random variables with zero π, zero π½ as well as both zero π and zero π½ for πΌ β 1. This yields the centred, deskewed, and symmetrized sequences:
SkC
= S3k + S3kβ1 β 2S3kβ2
( [ ] ) 2 β 2πΌ πΌ βΌ S πΌ, π½, [2 + 2 ]π, 0 , 2 + 2πΌ
(43)
SkD = S3k + S3kβ1 β 21βπΌ S3kβ2 βΌ S(πΌ, 0, 4π, [2 β 21βπΌ ]π),
(44)
SkS = S2k β S2kβ1 βΌ S(πΌ, 0, 2π, 0).
(45)
4.3.2. Parameter estimation Suppose Sk is a data set assumed to be drawned from S(πΌ, π½, π, π). Then the exponent parameter πΌ is estimated by setting π = 0 in (27), and the log moment M2 is estimated from the obverted data (45). That is,
( πΌΜ =
M2 1 β π1 2
)β1β2 (46)
.
The estimated πΌΜ is used to estimate π using (26) where M1 is estimated from the obverted data (44). That is,
Μ = |π|
(( π
1
2
) )1β2 β M2 πΌΜ 2 + π1 .
(47)
From the definition of π, |π½0 | can be estimated by
tan πΜ π½Μ0 = . Μ tan πΌπ 2
(48)
Μ of the original Centering (see (43)) requires |π½Μ0 | to be multiplied by (2 + 2 )β(2 β 2 ) to obtain |π½| data where the sign of π½ is determined by πΌ
πΌ
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K = sign(|Smax β S md | β |Smin β S md |),
Μ such that π½Μ = K|π½|.
where Smax, S md, Smin is the maximum, median and minimum of the original data. Next we estimate the scale parameter πΜ0 using (26) where M1 is estimated from the obverted data (43). That is
Μ exp((M β π )πΌΜ + π ). πΜ0 = | cos π| 1 0 0
(49)
Again centering (see (43)) gives the parameter estimate πΜ of the original data by πΜ = πΜ0 (2 β 2
1βπΌ β1
) .
Finally, the location parameter π is estimated by
πΜ = πΜ 0 (2 β 21βπΌ )β1 .
(50)
where π0 is the median or mean of the obverted data ().
4.4. Maximum likelihood method The ML method is the most favored parameter estimation method in economic and financial applications. The method relies on the density function which in the case of stable distributions poses a closed form representation problem. In this case we propose a numerical estimation of the density function. For a vector s = (s1 , s2 , β¦ , sn ) of independent identically distributed random variables assumed to follow a stable distribtion, the ML estimate of the parameter vector Ξ = (πΌ, π½, π, π) is obtained by maximizing the log-likelihood function given by
LΞ (s) =
n β
Μ ; Ξ), ln h(s i
(51)
i=1
Μ Ξ) denotes a numerically estimated stable probability density function. It is shown for where h(s; instance in Mittnik, Rachev, Doganoglu, and Chenyao (1999) that the best algorithms to compute the ML is by using Fast Fourier Transforms (FFT) or by direct integration method as in Nolan (2001). The ML algorithms require carefully chosen initial input parameters which in our case can be obtained for example, through the quantiles method described above. The FFT is faster for large data sets and the direct integral approach is suitable for smaller data sets since it can be evaluated at any arbitrary point. In the following section, we analyze commodities and apply the empirical characteristic functions method to estimate the stable distribution parameters. It is important to mention the restrictions on the parameters under which the different estimation methods operate.
4.5. Error analysis In this section we simulate datasets from the stable family of distributions based on the theory in Chambers et al. (1976) and Weron and Weron (1995). Then use the above four methods to retrieve the stable parameters from the simulated data. Our focus is on the πΌ and π½ but the arguments extend to the other two parameters.
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Table 1. Estimation methods and their parameter restrictions Estimation method Quantile Logarithm moments Maximum likelihood Empirical characteristic function
Parameter restrictions πΌ β₯ 0.1
π½=π=0 πΌ β₯ 0.4
πΌ β₯ 0.1
First, it is important to mention that all the four methods perform poorly close to the boundaries i.e. πΌ β 0, πΌ β 2 and π½ β Β±1. Moreover, literature shows that the methods operate efficiently under the parameter restrictions in Table 1. In addition, the MLE seems the most preferred and used estimation method. However, we observe in our analysis that this method fails for particular parameter ranges and it is not robust. For instance in estimating 0.1 < πΌ < 1.0 with respect to π½, the MLE fails to converge and returns huge unrealistic errors. This is why we do not include it in Figure 2(a). Similarly, for π½ = 0.4 estimation with respect to πΌ, the logarithm moments method returns either negative or very large π½ values which is expected according to the constraints in Table 1. We omit its graph in Figure 2(b). Meanwhile, we Figure 2. Method comparison for πΌ = 0.4 and π½ = 0.4 estimation.
(a)
(b)
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notice that in both cases, the quantile and ECF methods work well with the latter providing relatively the best estimates. The graphs in Figure 3 show the error associated with estimating 1.0 < πΌ < 2.0 for different π½ values. Note that all the four methods work well and we still notice the ECF being relatively the most accurate and robust method. Recall that for πΌ β 1 and πΌ β 2 the estimation methods perform poorly. An example is Figure 3(a) (for πΌ = 1.4) which was the closest for which the ML would converge but for higher πΌ > 1.4 values but far less than 2.0 (see for instance, Figure 3(a) for πΌ = 1.7) the methods performed relatively better except for the logarithm moments methods.
Figure 3. Method comparison for πΌ = 1.4 and πΌ = 1.7 estimation.
(a)
(b)
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The graphs in Figure 4 illustrate convergence of the quantile, ECF and the MLE in estimating
πΌ = 1.4 and πΌ = 1.7. We simulated 50,000 points and divided it into 100 sets starting with a 500-sized set and increasing it by 500 to 50,000. The logarithm moments method performed extremely poorly and incomparable to the above three methods. It is not included in Figure 4(a) and (b). The ECF is seen to perform better than the quantile and ML methods with a relatively better convergence rate. Similary Figure 5 shows the convergence rates for the quantile, ECF and ML estimation methods. The ECF still provides a better precision in both cases i.e. Figure 5(a) and (b).
Figure 4. π½ estimation for differing data set sizes and πΌ values.
(a)
(b)
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Figure 5. πΌ estimation for differing data set sizes for π½ = 0.4 values.
(a)
(b)
In summary the empirical characteristic function method outperforms all the three other methods discussed in this paper in the following way: (1)βIt is robust and can consistently estimate a wide range of πΌ and π½ parameters. (2)βIt provides a better precision compared to the quantile, logarithm moments and MLE methods for a wide range of πΌ and π½ parameters. (3)βIt has a better convergence rate. Therefore the quantile, logarithm moments or the ML methods can be used to provide initial parameters for the ECF method. Similarly, the latter can be used to provide initial parameters for better future estimators. The following section is devoted to extracting stable parameters from log-returns commodity futures data using the ECF method.
5. Commodity data The data sets used here are obtained from Quandl Financial and Economic Data website. The sets differ in sizes and include settled prices of Corn Futures Continuous Contract C#1 from 1959-07-01
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to 2017-02-10; Crude Oil Futures Continuous Contract C#1 from 1983-03-30 to 2017-02-10; Gasoline Futures Continuous Contract C#1 from 2005-10-03 to 2017-02-10; Gold Futures Continuous Contract C#1 from 1974-12-31 to 2017-02-10; Natural Gas Futures Continuous Contract C#1 from 1990-0403 to 2017-02-10; Platinum Futures Continuous Contract C#1 from 1969-01-02 to 2017-02-10; Silver Futures Continuous Contract C#1 from 1963-06-13 to 2017-02-10; Soybeans Futures Continuous Contract C#1 from 1959-07-01 to 2017-02-10; Wheat Futures Continuous Contract C#1 from 1959-07-01 to 2017-02-10. To avoid multi-distributional effects, we work with log-returns of the data sets.
5.1. The t-location-scale distribution The t-location-scale distribution is most suited for modeling data distributions with heavier tails, more prone to outliers than the Gaussian distribution. The distribution uses the following parameters Parameter
Description
Support
π
Location parameter
ββ < π < β
π
Scale parameter
π>0
πΌβ
Shape parameter
πΌβ > 0
The probability density function (pdf) of the t-location-scale distribution is given by οΏ½
οΏ½ οΏ½2 β οΏ½ β οΏ½ β β β xβπ πΌ + Ξ πΌ 2+1 β β π h(x) = β οΏ½ β οΏ½β β β πΌ β πΌ β π πΌ πΞ 2 β β β
οΏ½ β πΌ +1 2
,
where Ξ(β
) denotes the gamma function. The mean of the t-location-scale distribution is given by π β and it is defined for πΌ > 1 and undefined otherwise. The variance is given by
π ar = π 2
πΌβ . πΌ β2 β
The t-location-scale distribution approaches the Gaussian distribution as πΌ approaches infinity and β smaller values of πΌ yield heavier tails. This distribution does not take skewness into consideration and its three parameters are usually estimated using the ML estimation method. β
Using algorithms by Sheppard (2012) on our log-returns commodity futures data we obtained Βfittings in Figures 6β8. According to the πΌ values, the log-returns data exhibit some tails. To determine the nature of the details one would require to run some QQ plots but this can also be observed directly from the Figures 6β8. β
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Figure 6. Energy: The data exhibits high peaks and skinny tails.
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Figure 7. Grains: The data exhibits high peaks and skinny tails.
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Figure 8. Metals: The data exhibits high peaks and skinny tails.
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Table 2. t-location-scale distribution parameters extracted from the log-returns data t-location scale parameters Energy
π
πΆ
Gasoline
0.000905848
0.0193048
4.45922
Natural gas
0.000117668
0.0181944
2.50848
Crude oil Grains
Corn Soy beans Wheat
Metals
β
π
Gold Platinum Silver
0.000313708
0.00912801
1.75246
5.52294e β 05
0.00439924
3.03782
0.000400308
0.00535733
2.43218
1.63112e β 05
0.0113397
3.41863
0.000905944
0.0108491
2.70553
0.00044219
0.0110761
3.13631
β1.72459e β 05
0.000682631
0.512196
It is important to mention that QQ plots do not straight away provide conclusive evidence about the nature of the tails. More tests would still need to be made. For instance under the t-location scale it is not obvious to observe any skewness in the data. We however, view this effect when we fit the data to stable distribution (see Table 2) as discussed in the following section.
5.2. Stable distribution fitting On the other hand, by assuming stable distribution for our log-returns commodity futures data, we employed the ECF method and obtained the stable parameters in Table 3. Log-returns of commodity futures are not only high peaked but they also have left and right skinny tails with extreme outliers as observed from the QQ-plots for energy commodities (i.e. Crude oil, Natural gas and Gasoline) in Figure 9, the grains commodities in Figure 10 and the precious metals in Figure 11. Table 3 shows stable distribution parameters extracted from the log-returns data using the empirical characteristic function parameter estimation method. We notice that the data exhibit a bit of skewness which is not reflected in the t-location-scale distribution fitting. Table 3. Stable distribution parameters extracted from the log-returns data Stable distribution parameters Energy
Grains
Metals
πΆ
π·
π
π
Gasoline
1.7504
β0.3806
0.0152
β0.0005
Natural gas
1.5329
0.0371
0.015
0.0005
Crude oil
1.2322
β0.1526
0.0075
β0.0022
Corn
1.651
0.2117
0.0036
0.0004
Soy beans
1.4665
β0.0968
0.0043
0.0001
Wheat
1.638
0.0929
0.0091
0.0003
Gold
1.5007
β0.1324
0.0088
0.0001
Platinum
1.5943
β0.1339
0.0089
β0.0001
Silver
0.4461
0.0176
0.0011
β0.0001
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Figure 9. Energy: In all, the left and right tails are skinny.
(a)
(b)
(c)
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Figure 10. Grains: In all, the left and right tails are skinny.
(a)
(b)
(c)
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Figure 11. Metals: In all, the left and right tails are skinny.
(a)
(b)
(c)
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6. Conclusion First we showed that the ECF provides the best precision in estimating a wide range of πΌ and π½ parameters, it is robust and provides better convergence compared to the quantile, ML, and the logarithm moments. Secondly, we have illustrated that in general, the distribution of the commodity futures log-returns data is closest to a t-location-scale distribution due to its high peaks, skinny tails and extreme outliers. Moreover, by using the ECF estimation method we realize some minor skewness effects not captured in the t-location-scale fitting. We recommend the ECF as a suitable approach for estimating parameters of any skewed financial market data and could be used to obtain initial input parameters for future and better estimation techniques. Funding This work was supported by funds from the National Research Foundation of South Africa (NRF), the African Institute for Mathematical Sciences (AIMS) and the African Collaboration for Quantitative Finance and Risk Research (ACQuFRR) which is the research section of the African Institute of Financial Markets and Risk Management (AIFMRM), which delivers postgraduate education and training in financial markets, risk management and quantitative finance at the University of Cape Town in South Africa. Author details M. Kateregga1 E-mail: [email protected] S. Mataramvura1 E-mail: [email protected] ORCID ID: http://orcid.org/0000-0002-8073-2070 D. Taylor1 E-mail: [email protected] 1 Actuarial Science, University of Cape Town, Rondebosch, Cape Town 7700, South Africa. Citation information Cite this article as: Parameter estimation for stable distributions with application to commodity futures logreturns, M. Kateregga, S. Mataramvura & D. Taylor, Cogent Economics & FinanceΒ (2017), 5: 1318813. Notes 1. Note that characteristic functions always exist. 2. These are easily obtained from in-built functions in MATLAB Cover image Source: Original image from Parameter estimation for stable distributions with application to commodity futures log-returns by M. Kateregga, S. Mataramvura and D. Taylor. References Ament, S., & OβNeal, M. (2016). Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics, arXiv: 1607.04247v1. Applebaum, D. (2004). LΓ©vy processes and stochastic calculus. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. Bachelier, L. (1900). ThΓ©orie de la spΓ©culation. Annales scientifiques de lβΓ.N.S. 3e sΓ©rie, 17, 21β86. Chambers, J. M., Mallows, C. L., & Stuck, B. W. (1976). A method for simulating stable random variables. Journal of the American Statistical Association, 71, 340β344. Dumouchel, W. H. (1971). Stable distributions in statistical inference. The Journal of the American Statistical Association, 78, 469β477.
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