4 Case (B): Time-Varying Scale Process
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replaced by αˆ p,n since we already know that
∗
:=
σˆ 2q,n
√
n(αˆ p,n − α ) = O p (1). Therefore,
n2q/αˆ p,n−1 n
∑ |∆i X|2q → p σ2q∗ .
µˆ (2q, 0) i=1
(18)
Once again, let us remind that µˆ (2q, 0) can be easily computed in view of (2) with
σ = 1. By the same token, we could deduce that (still under 4q < α , of course)
∗
σˆ 4q,n
:=
n4q/αˆ p,n−1 n−1
∑ |∆i X|2q|∆i+1X|2q → p σ4q∗ .
µˆ (2q, 2q) i=1
∗
∗
, σˆ 4q,n
) can serve as a desired consistent estimator.
After all, V (ρˆ n , αˆ p,n , σˆ 2q,n
Now we are in a position to complete our main objectives (A) and (B).
3.3 Case (A): Geometric Skewed Stable L´evy Process
When σt ≡ σ > 0, our model reduces to the geometric skewed stable L´evy
process. In this case we can perform a full-joint interval estimation
concerning
√
the dominating (three-dimensional) parameter (ρ , α , σ ) at rate n.
We keep using the framework of the last subsection. It directly follows from
(15) that
ˆ
−
ρ
ρ
n
√
n αˆ p,n − α →d N3 (0,V (ρ , α , σ )),
(19)
(σˆ p,n ) p − σ p
where V (ρ , α , σ ) explicitly depends on the three-dimensional parameter
(ρ , α , σ ); recall that p = 2q < α /2. Applying the delta method to (19) in orp
ˆ
ˆ
der
√ to convert (σ p,n ) to σ p,n in (19), we readily get the asymptotic normality of
n(ρˆ n − ρ , αˆ p,n − α , σˆ p,n − σ ); we omit the details. Our first objective (A) is thus
achieved.
In summary, we may proceed with the choice q = 1/4 (so p = 1/2) as follows.
1. Compute the estimate ρˆ n of ρ by (7).
2. Using the ρˆ n , find the root αˆ 1/2,n of (16).
3. Using (ρˆ n , αˆ 1/2,n ) thus obtained, an estimate of σ is provided by, e.g. (recall
(18)),
2
n1/(2αˆ p,n)−1 n
σˆ 1/2,n :=
|∆i X| .
∑
µˆ (1/2, 0) i=1
3.4 Case (B): Time-Varying Scale Process
Now we turn to the case (B). Again by means of the argument give in Section
3.2, it remains to construct an estimator of σα∗ = 01 σsα ds. The point here is that,
different from the case (A), a direct use of (15) is not sufficient to deduce the
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distributional result concerning estimating σα∗ because the dependence of (r, r )
on α is not allowed there. In order to utilize Mn (r) with r depending on α , we
need some additional arguments.
Extracting the second row of (12), we have
√
∗
n{Mn (r) − µ (r)σr∗+ } →d N1 0, B(r, r)σ2r
.
(20)
+
In view of the condition maxl≤m rl < α /2, we need (at least) a tripower variation
for setting r+ = α . For simplicity, we set m = 3 and
r = r(α ) =
α α α
, ,
.
3 3 3
With this choice, we are going to provide an estimator of σα∗ with specifying its
rate of convergence and limiting distribution.
Let Mn∗ (α ) := Mn (α /3, α /3, α /3). In this case the normalizing factor is
r
/
+
n α −1 ≡ 1, so that
Mn∗ (α ) =
n−2 3
∑ ∏ |∆i+l−1X|α /3,
i=1 l=1
which is computable as soon as we have an estimate of α . We have already obtained the estimator αˆ p,n , hence want to use Mn∗ (αˆ p,n ). For this, we have to look
at the asymptotic behavior of the gap
√
√
n{Mn∗ (r(α )) − µ (r(α ))σα∗ } − n{Mn∗ (αˆ p,n ) − µ (r(αˆ p,n))σα∗ },
namely, the effect of “plugging in αˆ p,n ”.
By means of Taylor’s formula
ax = ay + (loga)y (x − y) + (loga)2
1
0
(1 − u)ay+u(x−y)du(x − y)2
applied to the function x → ax (x, y, a > 0), we get
√
α α α
n Mn∗ (αˆ p,n ) − µ
σα∗
, ,
3 3 3
=
√
+
n Mn∗ (α ) − µ
n−2
α α α
1√
α /3
σα∗ +
n(αˆ p,n − α ) ∑ xi log xi
, ,
3 3 3
3
i=1
1√
n(αˆ p,n − α )
3
2
1 n−2
√ ∑ (log xi )2
n i=1
1
0
{α +u(αˆ p,n−α )}/3
(1 − u)xi
du,
(21)
where we wrote xi = ∏3l=1 |∆i+l−1 X|. We look at the right-hand side of (21)
termwise. Let yi := ∏3l=1 |n1/α ∆i+l−1 X|.
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• The first term is O p (1), as is evident from (20).
• Concerning the second term, we have
n−2
α /3
∑ xi
log xi =
i=1
1 n−2 α /3
3
1 n−2 α /3
yi log y j − (log n) ∑ yi
∑
n i=1
α
n i=1
1
α α α
3
µ
σα∗ + O p √
, ,
α
3 3 3
n
3
α α α
, ,
.
= O p (1) − (logn) µ
α
3 3 3
= O p (1) − (logn)
√
• Write the third term as { n(αˆ p,n − α )/3}2 Tn , and let us show that Tn =
o p (1). Fix any ε > 0 and ε0 ∈ (0, α /2) in the sequel. Then
P[|Tn | > ε ] ≤ P[|αˆ p,n − α | > ε0 ] + P |Tn | > ε , |αˆ p,n − α | ≤ ε0 =: pn + pn .
Clearly pn → 0 by the
inf
u∈[0,1]
√
n-consistency of αˆ p,n . As for pn , we first note that
1
ε0
{α + u(αˆ p,n − α )} ≥ 1 − > 0
α
α
on the event {|αˆ p,n − α | ≤ ε0 }. We estimate pn as follows:
pn = P |αˆ p,n − α | ≤ ε0 ,
1 n−2
√ ∑ (log xi )2
n i=1
1
0
≤ P |αˆ p,n − α | ≤ ε0 , nε0 /α −1/2
≤ P nε0 /α −1/2
≤
{α +u(αˆ p,n −α )}/3 −{α +u(αˆ p,n −α )}/α
(1 − u)yi
1 n−2
∑ (log xi )2
n i=1
n
1
0
du > ε
{α +u(αˆ p,n −α )}/3
(1 − u)yi
du > ε
1 n−2
∑ {(log n)2 + (log yi )2 }(1 + yi )(α +ε0)/3 > Cε
n i=1
1 ε0 /α −1/2
n
(log n)2 → 0
Cε
for some constant C > 0. Here we used Markov’s inequality in the last
inequality; note that (α + ε0 )/3 < α /2, hence the moment does exist.
Piecing together these three items and (21), we arrive at the asymptotic relation:
√
√
1
n
α α α
1
α α α
Mn∗ (αˆ p,n ) − µ
, ,
, ,
.
σα∗ = − µ
σα∗ n(αˆ p,n − α ) + O p
log n
3 3 3
α
3 3 3
log n
(22)
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Now, recalling (2) we note that the quantity µ (α /3, α /3, α /3) is a continuously
differentiable
function of (ρ , α ). Write µ¯ (ρ , α ) = µ (α /3, α /3, α /3). In view of
√
the n-consistency of (ρˆ n , αˆ p,n ) and the delta method, we obtain
1
µ¯ (ρ , α ) = µ¯ (ρˆ n , αˆ p,n ) + O p √ .
n
Substituting (23) in (22) we end up with
√
Mn∗ (αˆ p,n )
1 √
1
n
,
− σα∗ = − σα∗ n(αˆ p,n − α ) + O p
log n µ¯ (ρˆ n , αˆ p,n )
α
log n
(23)
(24)
which implies that
σˆ α∗ ,n :=
Mn∗ (αˆ p,n )
µ¯ (ρˆ n , αˆ p,n )
(25)
√
serves as ( n/ log n)-consistent estimator of σα∗ . Its asymptotic distribution is the
centered normal scale mixture with limiting variance being
∗
v(ρ , α , σα∗ , σ p∗ , σ2p
) :=
σα∗
α
2
∗
V22 (ρ , α , σ p∗ , σ2p
),
where V22 denotes the (2, 2)th entry of V ; recall that p is a parameter-free con∗ ) can be constant (see Section 3.2). A consistent estimator of v(ρ , α , σα∗ , σ p∗ , σ2p
structed by plugging in the estimators of its arguments.
(24) indicates an asymptotic linear dependence of
√ The stochastic expansion
√
n(αˆ p,n − α ) and ( n/ log n)(σˆ α∗ ,n − σα∗ ). Of course, this occurs even for constant√σ , if we try to estimate (α , σ α ) instead of (α , σ ). The point is that, plugging
in a n-consistent estimator
of
√
√α into the index r of the MPV Mn (r) slows down
estimation of σα∗ from n to n/(log n). It is beyond the scope of this article to
explore a better alternative estimator of σα∗ .
4. Simulation Experiments
Based on the discussion above, let us briefly observe finite-sample performance of our estimators. For simplicity, we here focus on nonrandom σ .
4.1 Case (A)
First, let σ is a positive constant, so that X is the geometric skewed stable
L´evy process and the parameter to be estimated is (ρ , α , σ ).
As a simulation design, we set α = 1.3, 1.5, 1.7, and 1.9 with common β =
−0.5 and σ = 1; hence (α , ρ ) = (1.2, 0.7638), (1.5, 0.5984), (1.7, 0.5467), and
(1.9, 0.5132). The sample size are taken as n = 500, 1000, 2000, and 5000. In
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all cases, the tuning parameter q is set to be 1/4, and 1000 independent sample
paths of X are generated. Empirical means and empirical s.d.’s are given with the
1000 independent estimates obtained. The results are reported in Table 4.1. We
see that estimation of (ρ , α ) is, despite of its simplicity, quite reliable. On the
other hand, estimation variance of σ is relatively large compared with those of ρ
and α . Nevertheless, it is clear that the bias is small. Moreover, as α gets close
to 2, the performance of σˆ n becomes better, while that of (ρˆ n , αˆ p,n ) is seemingly
unchanged.
In the unreported simulation results, we have observed that a change of q
within its admissible region does not lead to a drastic change unless it is too small
(see Remark 3.3).
Table 1. Estimation results for the true parameters (ρ , α , σ ) = (0.7638,1.2,1), (0.5984,1.5,1),
(0.5467,1.7,1), and (0.5132,1.9,1) with the geometric stable L´evy processes. In each case, the
empirical mean and standard deviation (in parenthesis) are given.
α = 1.2
n
500
1000
2000
5000
α = 1.5
n
500
1000
2000
5000
α = 1.7
n
500
1000
2000
5000
α = 1.9
n
500
1000
2000
5000
ρ
0.7627
0.7634
0.7645
0.7636
(0.0186)
(0.0137)
(0.0096)
(0.0061)
1.2026
1.2031
1.2031
1.2023
α
(0.0790)
(0.0575)
(0.0437)
(0.0313)
(0.0222)
(0.0162)
(0.0106)
(0.0073)
1.4929
1.5010
1.4986
1.4983
α
(0.1030)
(0.0757)
(0.0564)
(0.0364)
1.6810
1.6830
1.6930
1.6977
α
(0.1103)
(0.0823)
(0.0625)
(0.0375)
1.8553
1.8767
1.8870
1.8971
α
(0.1026)
(0.0808)
(0.0579)
(0.0401)
ρ
0.5988
0.5981
0.5986
0.5984
ρ
0.5476
0.5474
0.5472
0.5466
(0.0219)
(0.0158)
(0.0113)
(0.0070)
ρ
0.5129
0.5133
0.5131
0.5128
(0.0224)
(0.0164)
(0.0109)
(0.0073)
σ
1.1021
1.0450
1.0253
1.0123
1.0751
1.0289
1.0284
1.0169
(0.8717)
(0.4643)
(0.5102)
(0.2854)
σ
(0.4066)
(0.2549)
(0.2355)
(0.1516)
σ
1.0633
1.0567
1.0308
1.0126
(0.2359)
(0.1948)
(0.1611)
(0.1022)
σ
1.0821
1.0535
1.0330
1.0097
(0.1767)
(0.1568)
(0.1111)
(0.0809)
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4.2 Case (B)
Next we observe a case of time-varying but nonrandom scale. We set
σtα =
3
2
cos(2π t) +
,
5
2
(26)
0.6
0.4
0.2
Varying scale
0.8
1.0
so that σα∗ = 0.6.
0.0
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0.0
0.2
0.4
0.6
0.8
1.0
Time
Figure 3. The plot of the function t → σtα given by (26).
With the same choices of (ρ , α ), q, and n as in the previous case, we obtain
the result in Tables 4.2; the estimator of σα∗ here is based on (25). There we can
observe a quite similar tendency as in the previous case.
5. Concluding Remarks
We have studied some statistical aspects in the calibration problem of a geometric skewed stable asset price models. Estimation of stable asset price models with possibly time-varying scale can be done easily by means of the simple empirical-sign statistics and MPVs. Especially, we could estimate integrated
scale, which is a natural quantity as in the integrate variance in the framework of
Brownian semimartingales, with multistep estimating procedure: we estimate ρ ,
α , and σ (or σα∗ ) one by one in this order. Our simulation results say that finitesample performance of our estimators are unexpectedly good despite of their simplicity, except for a relatively bigger variance in estimating σ (or σα∗ ).
We close with mentioning some possible future issues.
• Throughout we supposed the independence between the scale process σ
and the driving skewed stable L´evy process Z. This may be disappointing
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Table 2.
Estimation results for the true parameters (ρ , α ) = (0.7638,1.2), (0.5984,1.5),
(0.5467,1.7), and (0.5132,1.9) with σα∗ = 0.6 in common under (26). In each case, the empirical mean and standard deviation (in parenthesis) are given.
α = 1.2
n
500
1000
2000
5000
α = 1.5
n
500
1000
2000
5000
α = 1.7
n
500
1000
2000
5000
α = 1.9
n
500
1000
2000
5000
ρ
0.7632
0.7636
0.7638
0.7641
1.1951
1.2042
1.2044
1.2025
α
(0.0794)
(0.0619)
(0.0472)
(0.0305)
σα∗
0.6730 (0.3857)
0.6274 (0.3094)
0.6105 (0.2323)
0.6029 (0.1521)
(0.0220)
(0.0159)
(0.0111)
(0.0069)
1.4877
1.4908
1.4960
1.4990
α
(0.1023)
(0.0733)
(0.0573)
(0.0376)
σα∗
0.6697 (0.3031)
0.6551 (0.2488)
0.6349 (0.2033)
0.6151 (0.1414)
(0.0216)
(0.0160)
(0.0113)
(0.0071)
1.6727
1.6801
1.6931
1.6988
α
(0.1038)
(0.0820)
(0.0600)
(0.0393)
0.6832
0.6714
0.6318
0.6116
1.8440
1.8703
1.8851
1.8956
α
(0.1039)
(0.0823)
(0.0588)
(0.0411)
σα∗
0.7196 (0.2233)
0.6762 (0.1897)
0.6412 (0.1349)
0.6168 (0.0998)
(0.0179)
(0.0139)
(0.0098)
(0.0059)
ρ
0.5978
0.5981
0.5985
0.5987
ρ
0.5460
0.5465
0.5468
0.5465
ρ
0.5130
0.5131
0.5138
0.5135
(0.0229)
(0.0159)
(0.0114)
(0.0068)
σα∗
(0.2465)
(0.2280)
(0.1607)
(0.1135)
as it excludes accommodating the leverage effect, however, the simple constructions of our estimators (especially, ρˆ n ) break down if they are allowed
to be dependent. We may be able to deal with correlated σ and Z if we have
an extension of the power-variation results obtained in Corcuera et al. [7]
to the MPV version. To the best of author’s knowledge, such an extension
does not seem to have been explicitly mentioned as yet.
• Assuming that σ is indeed time-varying and possibly random, estimation
of “spot” scales σt is an open problem. Needless to say, this is much more
difficult and delicate to deal with than the integrated scale. We know several results for Brownian-semimartingale cases (see, among others, Fan and
Wang [8] and Malliavin and Mancino [12]), however, yet no general result
for the case of pure-jump Z.
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• Finally, it might be interesting to derive an option-pricing formula for the
case of time-varying scale, which seems more realistic than the mere geometric skewed stable L´evy processes.
References
1. Applebaum, D. (2004), L´evy Processes and Stochastic Calculus. Cambridge University Press, Cambridge.
2. Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J. and Shephard, N. (2006), Limit
theorems for bipower variation in financial econometrics. Econometric Theory 22,
677–719.
3. Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J., Podolskij, M. and Shephard,
N. (2006), A central limit theorem for realised power and bipower variations of continuous semimartingales. From Stochastic Calculus to Mathematical Finance, 33–68,
Springer, Berlin.
4. Barndorff-Nielsen, O. E. and Shephard, N. (2005), Power variation and time
change.Teor. Veroyatn. Primen. 50, 115–130; translation in Theory Probab. Appl. 50
(2006), 1–15.
5. Bertoin, J. (1996), L´evy Processes. Cambridge University Press.
6. Borak, S., Hăardle, W. and Weron, R. (2005), Stable distributions. Statistical tools for
finance and insurance, 21–44, Springer.
7. Corcuera, J. M., Nualart, D. and Woerner, J. H. C. (2007), A functional central limit
theorem for the realized power variation of integrated stable processes. Stoch. Anal.
Appl. 25, 169–186.
8. Fan, J. and Wang, Y. (2008), Spot volatility estimation for high-frequency data. Stat.
Interface 1, 279–288.
9. Fujiwara, T. and Miyahara, Y. (2003), The minimal entropy martingale measures for
geometric L´evy processes. Finance Stoch. 7, 509–531.
10. Kallsen, J. and Shiryaev, A. N. (2001), Time change representation of stochastic integrals. Teor. Veroyatnost. i Primenen. 46, 579–585; translation in Theory Probab. Appl.
46 (2003), 522–528.
11. Kuruo˘glu, E. E. (2001), Density parameter estimation of skewed α -stable distributions. IEEE Trans. Signal Process. 49, no. 10, 2192–2201.
12. Malliavin, P. and Mancino, M. E. (2009), A Fourier transform method for nonparametric estimation of multivariate volatility. Ann. Statist. 37, 1983–2010.
13. Masuda, H. (2009), Joint estimation of discretely observed stable L´evy processes with
symmetric L´evy density. J. Japan Statist. Soc. 39, 1–27.
14. Masuda, H. (2009), Estimation of second-characteristic matrix based on realized multipower variations. (Japanese) Proc. Inst. Statist. Math. 57, 17–38.
15. Miyahara, Y. and Moriwaki, N. (2009), Option pricing based on geometric stable processes and minimal entropy martingale measures. In “Recent Advances in Financial
Engineering”, World Sci. Publ., 119–133.
16. Sato, K. (1999), L´evy Processes and Infinitely Divisible Distributions. Cambridge University Press.
17. van der Vaart, A. W. (1998), Asymptotic Statistics. Cambridge University Press, Cambridge.
May 3, 2010
15:41
Proceedings Trim Size: 9in x 6in
007
202
18. Woerner, J. H. C. (2003), Purely discontinuous L´evy processes and power variation:
inference for integrated volatility and the scale parameter. 2003-MF-08 Working Paper
Series in Mathematical Finance, University of Oxford.
19. Woerner, J. H. C. (2004), Estimating the skewness in discretely observed L´evy processes. Econometric Theory 20, 927–942.
20. Woerner, J. H. C. (2007), Inference in L´evy-type stochastic volatility models. Adv. in
Appl. Probab. 39, 531–549.
21. Zolotarev, V. M. (1986), One-Dimensional Stable Distributions. American Mathematical Society, Providence, RI. [Russian original 1983]
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A Note on a Statistical Hypothesis Testing for
Removing Noise by the Random Matrix Theory
and Its Application to Co-Volatility Matrices
Takayuki Morimoto1,∗ and Kanta Tachibana2
1
School of Science and Technology, Kwansei Gakuin University,
2-1 Gakuen, Sanda-shi, Hyogo 669-1337, Japan.
2
Faculty of Informatics, Kogakuin University, 1-24-2 Nishi-shinjuku,
Shinjuku-ku, Tokyo 163-8677, Japan.
Email: morimot@kwansei.ac.jp and kanta@cc.kogakuin.ac.jp
It is well known that the bias called market microstructure noise will arise,
when estimating realized co-volatility matrix which is calculated as a sum
of cross products of intraday high-frequency returns. An existing conventional technique for removing such a market microstructure noise is
to perform eigenvalue decomposition of the sum of cross products matrix and to identify the elements corresponding to the decomposed values
which are smaller than the maximum eigenvalue of the random matrix as
noises. Although the maximum eigenvalue of a random matrix follows
asymptotically Tracy-Widom distribution, the existing technique does not
take this asymptotic nature into consideration, but only the convergence
value is used for it. Therefore, it cannot evaluate quantitatively such a
risk that regards accidentally essential volatility as a noise. In this paper,
we propose a statistical hypothesis test for removing noise in co-volatility
matrix based on the nature in which the maximum eigenvalue of a random
matrix follows Tracy-Widom distribution asymptotically.
Keywords: Realized volatility, market microstructure noise, random matrix theory.
∗
Corresponding author.
203
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1. Introduction
In recent years, we can easily obtain “high frequency data in finance”, so we
may estimate and forecast (co)-volatility more correctly than before using by Realized Volatility (RV) which is a series of the sum of intraday squared log return
and Realized Co-volatility (RC) which is a series of the sum of cross-product of
two log returns, see [2] or [1]. However, it is well known that when forecasting
volatility, RV and RC are contaminated by large biases, so called micro structural
noise which is progressively increased as sampling frequency becomes higher,
see [7]. Thus, the research considers a statistical method of removing such a noise
in RV and RC by using random matrix theory. Doing eigenvalue decomposition
of cross product matrix, we consider that noises in a covolatility matrix are elements corresponding to eigenvalues smaller than the maximum eigenvalue of the
random matrix. It is known that the maximum eigenvalue of a random matrix
will follow Tracy-Widom distribution asymptotically. However, existing methods haven’t taken into consideration a distribution of the maximum eigenvalue of
a random matrix, but have used only the maximum eigenvalue itself, for example, see [9]. Therefore, they cannot evaluate quantitatively a risk of considering
accidentally that essential volatility is a noise.
Therefore, we propose a statistical hypothesis test for removing noise in covolatility matrix based on the nature in which the maximum eigenvalue of a random matrix follows Tracy-Widom distribution asymptotically.
This paper is organized as follows. Section 2 describes theoretical background
of this study and gives brief explanation of random matrix theory and our proposal.
Section 3 investigates empirical analysis. Section 4 concludes.
2. Theoretical Background
In this section, we will introduce theoretical properties of random matrix with
some simulation results.
2.1 Random matrix
Random matrix is a matrix which has random variables as its elements. First,
[16] and [17] developped a eigenvalue distribution of N × N real symmetric matrix A = (ai j ) with elements {ai j } ∼ i.i.d.(0, 1/N). Following [16] and [17], we
introduce N × N real symmetric random matrix A = (ai j ) with elements {ai j |i ≤ j}
which independently follows a distribution with a mean 0 and a variance 1/N. If
eigenvalues of A are λ1 , . . . , λN and an empirical eigenvalue distribution of A is
defined by
N
1
ρA (λ) =
δ(λ − λi ),
N i=1
then
√ 2
4−λ
2π
lim ρA (λ) =
0
N→∞
(|λ| ≤ 2)
,
(otherwise)