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Estimating Models Based on Markov JumpProcesses Given Fragmented Observation Series
Markus Hahn
Johann Radon Institute for Computational and Applied Mathematics (RICAM), LinzAustrian Academy of Sciences
Joint work with S. Fruhwirth-Schnatter (JKU Linz) and J. Sass (TU Kaiserslautern)
Linz, December 2, 2008
Work funded by FWF project P17947
Introduction
Problem
I Estimation from set of observed seriesI Independent series
I Data with breaks
I Each single series is (based) Markov processI Same generator for all series
I How to estimate common generator?
I How to cope with short observation series?
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Introduction
Outline
Introduction
Markov Jump Processes
Merged Markov Jump Processes
Inference for Merged Markov Jump Processes
Generalization: Markov Switching Models
Conclusion
3 / 37
Markov Jump Processes
Finite state Markov jump process (MJP)
I Y = (Yt)t∈[0,T ) is a continuous time Markov process
I Finite state space {1, . . . , d}
I Y is time homogeneous
I Jumps of Y are governed by rate matrix Q ∈ Rd×d
I Exponential rate of leaving state k:
λk = −Qkk =∑l 6=k
Qkl < ∞
i.e. average waiting time for leaving k is 1/λk
I Conditional transition probability:
P(Yt = l |Yt− = k,Yt 6= Yt−) = Qkl/λk
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Markov Jump Processes
Inference about rate matrix
I Ok occupation time of state k
I Nkl number of jumps from k to l
I Maximum likelihood estimation:
Qkl = Nkl/Ok
I Observing a path (Yt)t∈[0,T ), Ok and Nkl are sufficient forestimating Qkl
I Unbiased?
E(Qkl |Q) = ∞E(Qkl |Q,Ok > 0) = ∞
I Q is consistent, i.e.
limT→∞
P(|Qkl − Qkl | > ε) = 0
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Markov Jump Processes
Some Bayesian inference
I Using uninformative prior Qkl ∼ Ga(1, 0):
Qkl |Y ∼ Ga(Nkl + 1,Ok)
I Mean: (Nkl + 1)/Ok
I Variance: (Nkl + 1)/O2k , hence decreasing with 1/T
I Mode: Nkl/Ok
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Markov Jump Processes
Remarks on inference
I Q is well-known and looks nice
I In fact, estimation of Q is not so easy
I We need some observations for each transition k to l
I Basically, T needs to be as large as possible
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Merged Markov Jump Processes
Introduction
Markov Jump Processes
Merged Markov Jump Processes
Inference for Merged Markov Jump Processes
Generalization: Markov Switching Models
Conclusion
8 / 37
Merged Markov Jump Processes
Observing a number of series of MJPs
I M series of MJPs are observed
I All Y (m) characterized by same rate matrix Q
I Series may be independent or come from data with breaks
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
Figure: Processes Y (1), . . . , Y (M)
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Merged Markov Jump Processes
Observing merged MJPs
I Merged process of M single MJPs is observed
I Single MJPs characterized by same rate matrix Q
0 0.05 0.1 0.15 0.2 0.251
2
3
Figure: Merged process Y
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Merged Markov Jump Processes
Merged MJPs
I Given: Observation process Y = (Yt)t∈[0,T )
I Y is concatenation of MJPs Y (1), . . . ,Y (M)
I In detail, Y (m) = (Y(m)t )t∈[0,T ), T = M T , and
Yt = Y(1)t if 0 ≤ t < T ,
Yt = Y(2)t−T if T ≤ t < 2T ,
...
Yt = Y(m)t−mT if (m − 1)T ≤ t < mT
I All Y (m) characterized by same rate matrix Q
I NB: Y itself is not Markov!
I Assumption of equal length is for notational convenience only
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Merged Markov Jump Processes
Merged MJPs – Example
M = 5, d = 3, T = 0.05, T = 0.25, Q =
(−50 30 2020 −40 2030 40 −70
)
0 0.05 0.1 0.15 0.2 0.251
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
Figure: Merged process Y and single processes Y (1), . . . , Y (M)
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Inference for Merged Markov Jump Processes
Introduction
Markov Jump Processes
Merged Markov Jump Processes
Inference for Merged Markov Jump Processes
Generalization: Markov Switching Models
Conclusion
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Inference for Merged Markov Jump Processes
Splitting merged process
I Split Y into Y (1), . . . ,Y (M)
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
0 0.051
2
3
I Pooling:
Qkl =1
M
M∑m=1
Q(m)kl
I Problem: MLE does not exist if some O(m)k = 0;
short occupation times lead to unstable results
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Inference for Merged Markov Jump Processes
Estimating directly from merged process
I Consider Y
0 0.05 0.1 0.15 0.2 0.251
2
3
I Number of jumps and occupation time:
Nkl =M∑
m=1
N(m)kl + N+
kl , where N+kl =
M−1∑m=1
I{Y
(m)T−=k, Y
(m+1)0 =l
},
Ok =M∑
m=1
O(m)k
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Inference for Merged Markov Jump Processes
Estimating directly from merged process – first try
I First attempt: Qkl = Nkl/Ok
I Problem: Bias caused from “artificial” jumps N+kl /Ok
I If N+kl is observed explicitly, things are easy
I We assume N+kl cannot be observed
I Location of splitting points unknown
I Process not directly observed
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Inference for Merged Markov Jump Processes
Estimating directly from merged process – bias
I Assume Y(m)0 ∼ π, where π stationary distribution
I Extra jumps N+ do not affect distribution of occupation times andstationary distribution
I Hence, π = π and π = π(Q) is “unbiased” estimate for π
I Joining two independent stationary processes generates a jump fromk to l with probability πkπl :
P(N+
kl |π)
= Bin(M − 1, πkπl)
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Inference for Merged Markov Jump Processes
Estimating directly from merged process – bias (2)
I P(N+
kl |π)
= Bin(M − 1, πkπl) is justified if
I Y (m) are independent series
I Data with breaks:Length of break τ such that ρτ (Y ) is close to zero, where
ρt(Y ) =
Pdk=1 πk(Xkk(t) − πk)Pd
k=1 πk(1 − πk)
and
Xkl(t) = P(Yt = l |Y0 = k) = exp(Q t)kl
I.e. Xkk(τ) should be close to πk
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Inference for Merged Markov Jump Processes
Estimating directly from merged process – bias (3)
I Note: If π is given, N+kl and Ok are independent
I Recall: P(N+
kl |π)
= Bin(M − 1, πkπl)
I As
T
Ok
P−−−−→T→∞
1
πk,
we have
N+kl
Ok
≈ (M − 1)πkπl
πkT= πl
M − 1
T
I As π is unbiased estimate for π, these quantities can be estimatedknowing Q
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Inference for Merged Markov Jump Processes
Estimating directly from merged process – correction
I 2-step construction for corrected estimate:
1) Qkl = Nkl/Ok
2) Qkl = Qkl − (M − 1) πl/T
I “Merging”
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Inference for Merged Markov Jump Processes
Comparison: Splitting vs. Merging
I Variance for Splitting for Ga(1, 0) prior:
Var
(M−1
M∑m=1
Q(m)kl
∣∣∣Y (1), . . . ,Y (M)
)=
1
M2
M∑m=1
N(m)kl + 1(O
(m)k
)2I Variance for Merging for Ga(1, 0) prior:
Var(Qkl | Y
)=
∑Mm=1 N
(m)kl + 1(∑M
m=1 O(m)k
)2I For rather short single observation times T we expect Merging to
give more reliable results
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Inference for Merged Markov Jump Processes
Numerical example
I M = 100, d = 3, T = 0.25, T = 25, Q =
(−100 60 4040 −70 3040 60 −100
)
I On average, about 22 jumps in each single process Y (m)
I Simulate merged data and apply Splitting and Merging
I Repeat 100 000 times and consider sampling distributions of Qkl
and Qkl
I π = (0.29 0.46 0.25) and Q = Q −
(−2.8 1.8 1.01.1 −2.1 1.01.1 1.8 −2.9
)
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Inference for Merged Markov Jump Processes
Numerical example – results
−140 −120 −100 −800
0.05
0.1
50 70 900
0.05
0.1
0.15
30 40 50 600
0.1
0.2
30 40 500
0.1
0.2
−90 −80 −70 −600
0.1
0.2
20 30 400
0.1
0.2
30 40 50 600
0.1
0.2
50 70 900
0.05
0.1
0.15
−140 −120 −100 −800
0.05
0.1
Green circles: true Qkl , blue: Merging Qkl , red: Splitting Qkl
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Inference for Merged Markov Jump Processes
Numerical example – remarks
I Qkl outperforms Qkl wrt. both location and dispersion
I Qkl is skewed towards over-estimation:If some O
(m)k is very small, Q
(m)kl heavily over-estimates Qkl and
also Qkl is too high
I This is the more severe, the smaller T is
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Inference for Merged Markov Jump Processes
Numerical example (2)
I M = 62, d = 3, T = 1/M, T = 1, Q =
(−800 450 350100 −200 100300 400 −700
)
I On average, about 6 jumps in each single process Y (m)
I Probability that one state is visited never or only for a very shorttime is high!
I Simulate merged data and apply Splitting and Merging
I Repeat 100 000 times and consider sampling distributions of Qkl
and Qkl
I π = (0.15 0.68 0.17) and Q = Q −
(−51.8 41.4 10.49.1 −19.6 10.59.1 41.4 −50.5
)
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Inference for Merged Markov Jump Processes
Numerical example (2) – results
−3,000 −2,000 −1,000 00
0.003
0.006
0 500 1000 15000
0.004
0.008
0 500 1000 15000
0.005
0.01
0 100 2000
0.02
0.04
−400 −200 00
0.01
0.02
0.03
0 100 2000
0.02
0.04
0 500 1,000 1,5000
0.005
0.01
0 500 1000 15000
0.005
0.01
−3,000 −2,000 −1,000 00
0.004
0.008
Green circles: true Qkl , blue: Merging Qkl , red: Splitting Qkl
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Generalization: Markov Switching Models
Introduction
Markov Jump Processes
Merged Markov Jump Processes
Inference for Merged Markov Jump Processes
Generalization: Markov Switching Models
Conclusion
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Generalization: Markov Switching Models
A continuous-time Markov switching model
I Observation process R = (Rt)t∈[0,T ] (e.g. stock returns) withdynamics
dRt = µt dt + σt dWt ,
Rt =
∫ t
0
µs ds +
∫ t
0
σs dWs
I W standard Brownian motion
I Drift and volatility jump between d levels:µt = µ(Yt), σt = σ(Yt)
I State process Y is a MJP with state space {1, . . . , d}, Y ⊥ W
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Generalization: Markov Switching Models
Example
0 0.2 0.4 0.6 0.8 11
2
3State process
0 0.2 0.4 0.6 0.8 1−2
02
Drift process
0 0.2 0.4 0.6 0.8 10.1
0.15
0.2
Volatility process
0 0.2 0.4 0.6 0.8 1−0.03
0
0.03 Daily stock returns
0 0.2 0.4 0.6 0.8 10.8
1
1.2Price process
∆t = 1250 , µ = (3, 0,−2), σ = (0.20, 0.12, 0.15), Q =
(−70 40 3020 −40 2030 50 −80
)29 / 37
Generalization: Markov Switching Models
MSMs in Finance
I Short rate models: Elliott/Hunter/Jamieson (2001)
I Investment problems: Zhang (2001), Guo (2005)
I Risk measures for derivatives: Elliott/Siu/Chan (2008)
I Portfolio optimization: Honda (2003), Zhou/Yin (2003),Sass/Haussmann (2004), Bauerle/Rieder (2005), . . .
I Option pricing: Guo (2001), Buffington/Elliott (2002),Chan/Elliott/Siu (2005), Liu/Zhang/Yin (2006),Yao/Zhang/Zhou (2006), . . .
I . . .
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Generalization: Markov Switching Models
Estimation from discretely observed data
I Return process R is observable at times t = i ∆t,
Vi = ∆Ri =
∫ i ∆t
(i−1) ∆t
µs ds +
∫ i ∆t
(i−1) ∆t
σs dWs , i = 1, . . . ,H
I Vi daily stock returns
I State process Y is independent of W and not observable (hidden)
I Wanted: µ(k), σ(k), Q
I Problems:I ∆t given and fixedI Noise high compared to signalI High-frequency switching of states, i.e. λk ∆t highI Number of observations low (say, less than 5000)
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Generalization: Markov Switching Models
Data with breaks / merged processes
I Often, we encounter data with breaks
I Weekends for daily data, nights for intra-day data, . . .
I Observable discrete time process:V = (Vi )i=1,...,H is concatenation of V (1), . . . ,V (M)
V (m) = (V(m)i )i=1,...,H and H = M H
I Hidden continuous-time state process:Y is concatenation of Y (1), . . . ,Y (M)
Y (m) = (Y(m)t )t∈[0,T ) and T = M T
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Generalization: Markov Switching Models
Estimation from data with breaks
I Proceed similarly as for MJPs
I Use merged data V for estimation
I Employ arbitrary method to obtain estimates µ, σ, and Q
I µ and σ are not affected by merging, hence µ = µ, σ = σ
I Correction for Q: As described for MJPsI Point estimates: Correct Q to obtain Q
I Simulation based: Correct each sample Qj to obtain samples Qj
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Generalization: Markov Switching Models
Numerical example
I T = 10, H = 10 000, ∆t = 1/1000
I d = 2, Q =(−60 60
40 −40
), µ = (2 − 1), σ = (0.10 0.05),
i.e. µ∆t = (0.002 − 0.001), σ√
∆t = (0.0032 0.0016)
I M, T , H varying s.t. T = M T and H = M H
I M = 1 corresponds to one coherent seriesM = 400 corresponds to 400 series with H = 25 observations each
I Simulate merged data
I Perform method of moments-type estimation for merged data
I Repeat 1 000 times
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Generalization: Markov Switching Models
Numerical example – results
µ(1) µ(2) σ(1) σ(2) λ1 λ2 λ1 λ2
true 2.00 -1.00 0.100 0.050 60.0 40.0 60.0 40.0
M = 1, 2.02 -0.97 0.100 0.052 58.6 37.5 58.6 37.5τ = 10 0.11 0.05 0.002 0.002 9.8 6.5 9.8 6.5
M = 50, 2.02 -0.97 0.100 0.052 61.5 39.5 58.6 37.7τ = 0.2 0.12 0.05 0.001 0.003 10.7 6.3 10.7 6.7
M = 100, 2.03 -0.97 0.100 0.052 64.7 41.2 58.6 37.3τ = 0.1 0.11 0.05 0.001 0.003 11.6 6.4 10.5 6.8
M = 200, 2.03 -0.96 0.100 0.052 71.2 45.1 59.0 37.4τ = 0.05 0.12 0.06 0.001 0.003 15.6 8.2 10.7 6.7
M = 400, 2.06 -0.96 0.100 0.053 85.3 52.9 60.7 37.6τ = 0.025 0.13 0.06 0.002 0.003 28.4 15.1 12.6 8.0
Table: Results for MSM (T = 10, H = 10 000): mean (top), RMSE (bottom)
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Generalization: Markov Switching Models
Numerical example – remarks
I Estimates of µ(k), σ(k) are not affected by merging
I Quality of corrected estimates Qkl is (nearly) independent of M
I Ignoring breaks can lead to considerable bias in estimate for Q
I Method of moments-type estimation requires a lot of observations,Splitting for M > 2 not applicable;ML or Bayesian methods could be applied for M ≤ 10, but arecomputationally much more costly
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Conclusion
Conclusion
I Estimation for set of short (independent) observation series ordata containing (long) breaks
I Applicable to processes based on MJPs
I First, estimate parameters for merged seriesSecond, correct rates for bias afterwards
I Post-processing correction
I Works with arbitrary estimation approach for coherent series
I Single series need not be of same length /splitting times need not be known –only number of breaking points required
I Works similarly for discrete time processes
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