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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion Economic Scenario Generation with Regime Switching Models Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar 2pm to 3pm Friday 22 May, ASB

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Economic Scenario Generation with Regime Switching Models

Michael Sherris and Boqi Zhang

UNSW Actuarial Research Seminar

2pm to 3pm Friday 22 May, ASB 115

Acknowledgement: Research funding from Taylor-Fry Research Grant and ARC Discovery Grant DP0663090

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Presentation Overview

Introduction, Background and ERCH Model Data, Descriptive Statistics and Other Tests Univariate AR Model and Vector Autoregression Model (VAR) Univariate Regime Switching RSAR(1) Model Multivariate Regime Switching RSVAR(1,2) Model Models Simulation Comparison and Conclusion

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Economic Scenario Generators

Economics Scenario Generators increasingly used - life, non-life, superannuation; solvency, DFA, investment strategy ERCH model developed in Australia for life insurance solvency VAR models in economics and econometrics Regime switching models for univariate series - used by SoA for solvency, product guarantees for equity returns Multivariate regime switching model using the VAR model structure less well developed - issues with multivariate models (parsimony, data)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

ESG Models

Early models - cascade structure,Box-Jenkins transfer function - Wilkie (1986, 1995) Development of commercial models - consultants: Towers Perrin CAP:Link; in-house DFA models; model specialists Barrie Hibbert;

  • thers Algorithmics etc

Hamilton (1989, 1990) - regime switching Harris (1994) developed the Exponential Regressive Conditional Heteroscedasticity (ERCH) model Hardy (2001) - SoA solvency and products with guarantees

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

ERCH Model

m series ERCH model is expressed in multivariate form as Xt = M + ΘΨ∗

t + ξt,

ξt = ΛtZt lnΛt = diag{ω0 + ΩΦt} Zt ∼ N(0, Σz) E(Z T

t Zs) =

  • 0,

if t = s, Σz, if t = s. where M = E(Xt) is an m ∗ 1 column vector of unconditional series means, Θ is an m ∗ p conditional mean parameter matrix, Ψt is a p ∗ 1 column vector of lagged explanatory variable values at time t, with the superscript asterix referring to unconditional mean adjustment so that Ψ∗

t = Ψt − E(Ψt)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

ERCH Model

ξt is an m ∗ 1 column vector of conditionally multivariate normal random errors or shocks to the series at time t, Λt is an m ∗ m diagonal matrix of error standard deviations at time t, lnΛt is an m ∗ m diagonal matrix of the logarithms of the error standard deviations at time t, Zt is an m ∗ 1 column vector of multivariate standard normal standardized error or shocks to the series at time t, diag{...} is a diagonal matrix whose i-th non-zero element is equal to the i-th element of its vector arguments

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

ERCH Model

ω0 is an m ∗ 1 column vector of parameters, Ω is an m ∗ q conditional volatility parameter matrix, Φt is a q ∗ 1 column vector of lagged explanatory variable values at time t. Σz is an m ∗ m contemporaneous correlation matrix, the i, jth element of which is equal to the contemporaneous correlation between the ith and jth components of the Zt, Harris (1994) estimated parameters based on quarterly Australian data.

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Data

Data used for fitting the models is from Reserve Bank of Australia (RBA), the Australia Bureau of Statistics (ABS) and Residex for their residential house index series. Quarterly data for all the following 11 series are taken from these sources and has been modelled in the form of difference of log value. The sample period is from the first quarter of 1979 to the third quarter of 2006. There are 111 quarterly observations for each economic series or 111 ∗ 11 = 1221 data points in total.

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Data Plots

Variable Description Gt the log return of GDP Ft the log return of CPI Rt the log return adjusted SPI (Share Price Index of ASX 200) Yt the log return of Dividend of the adjusted SPI Tt the log return of 90-day Treasury notes yield B2t the log return of 2-year Treasury Bond yield B10t the log return of 10-year Treasury Bond yield AWEt the log return of average weekly earnings URt the log return of unemployment rate RESHt the log return of residential house index in Sydney USB2t the log return of US 2-year Treasury Bond yield

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of GDP

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96100 104 108 111 −0.03 −0.02 −0.01 0.01 0.02 0.03 0.04 Quarter log return % lnGDP Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of CPI

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104 108 111 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Quarter log return % logCPI Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of SPI and its Dvd

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 111 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 Quarter log return % lnSPI lnDvd Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of AUD interest rates

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108 111 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 Quarter log return % log 90−day T−note log 2−year T−bond log 10−year T−bond

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of AWE

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108 111 −0.02 −0.01 0.01 0.02 0.03 0.04 0.05 Quarter log return % log AWE

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of Unemployment Rate

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108 111 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25 Quarter log return log Unemployment Rate

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of Residential property price index

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108 111 −0.04 −0.02 0.02 0.04 0.06 0.08 0.1 Quarter log return log RESH Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Plot in log return of 2year US interest rate

4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100104108 111 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 Quarter log return log US 2−year T−bond Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Data Summary

Tables in slides to follow show: Most series have (positive or negative) skewness and also kurtosis (except for bond interest rates) All series (except perhaps for inflation) are stationary (unit root tests) No strong evidence of multi-collinearity (correlations) Economic series show autoregression but financial series do not (AR univariate)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Descriptive Statistics

Table: Descriptive Statistics

Economic Series Statistics Gt Ft Rt Dvdt Tt B2t B10t AWEt URt RESHt USB2t Mean 0.00802 0.01211 0.023517 0.021328

  • 0.003099
  • 0.00356
  • 0.004226

0.013588

  • 0.002691

0.021894

  • 0.006413

Median 0.008546 0.009062 0.034403 0.028435

  • 0.008043

0.012838

  • 0.011285

0.016246

  • 0.00722

Maximum 0.035147 0.041359 0.242154 0.151386 0.437565 0.237217 0.193786 0.044282 0.231661 0.09916 0.557806 Minimum

  • 0.023841
  • 0.004591
  • 0.533806
  • 0.146122
  • 0.25272
  • 0.280762
  • 0.164303
  • 0.016745
  • 0.09251
  • 0.026259
  • 0.533253

Std.Dev 0.008982 0.009486 0.093412 0.048093 0.114911 0.099263 0.072923 0.010604 0.048324 0.026495 0.148109 Skewness

  • 0.308195

0.581466

  • 2.037098
  • 0.389754

0.536843

  • 0.252947

0.140172 0.488395 1.497614 0.773178

  • 0.017305

Kurtosis 4.368876 2.821585 13.635 4.190385 4.268755 2.967431 2.814097 3.912792 7.180143 3.25018 5.167795 Jarque-Bera 10.42363 6.402128 599.8735 9.363992 12.77674 1.188573 0.523333 8.266297 122.308 11.34885 21.73997 P-value 0.005452 0.040719 0.009261 0.001681 0.551956 0.769768 0.016032 0.003433 0.000019

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Unit Root Test and Correlation

Table: Unit Root Test

Economic Series Augmented Dickey-Fuller Test Gt Ft Rt Dvdt Tt B2t B10t AWEt URt RESHt USB2t t-stat

  • 9.576011
  • 2.466313
  • 11.54542
  • 4.96337
  • 9.410926
  • 10.58258
  • 9.633236
  • 7.054811
  • 4.010385
  • 4.31123
  • 4.455761

p-value 0.1266 0.0001 0.002 0.0007 0.0004 Hypothesis result reject not reject reject reject reject reject reject reject reject reject reject unit root result no yes no no no no no no no no no stationarity result yes no yes yes yes yes yes yes yes yes yes

Table: Correlation Matrix

Gt Ft Rt Yt Tt B2t B10t AWEt URt RESHt USB2t Gt 1

  • 0.1399

0.0815 0.1924 0.0605 0.0987 0.0985 0.0573

  • 0.4081

0.2131 0.1738 Ft

  • 0.1399

1 0.0879 0.2485 0.0396 0.0827 0.0926 0.4873 0.1417 0.0427

  • 0.1459

Rt 0.0815 0.0879 1 0.1787

  • 0.1491
  • 0.1100
  • 0.1252
  • 0.0571
  • 0.0760
  • 0.0042

0.0382 Yt 0.1924 0.2485 0.1787 1 0.1816 0.1999 0.1121 0.2107

  • 0.2730

0.1624

  • 0.0765

Tt 0.0605 0.0396

  • 0.1491

0.1816 1 0.6659 0.5107 0.0724

  • 0.2350

0.1347 0.1504 B2t 0.0987 0.0827

  • 0.1100

0.1999 0.6659 1 0.8410 0.0836

  • 0.2104

0.1196 0.4305 B10t 0.0985 0.0926

  • 0.1252

0.1121 0.5107 0.8410 1 0.1073

  • 0.1237

0.1124 0.4540 AWEt 0.0573 0.4873

  • 0.0571

0.2107 0.0724 0.0836 0.1073 1 0.0478 0.1215 0.0155 URt

  • 0.4081

0.1417

  • 0.0760
  • 0.2730
  • 0.2350
  • 0.2104
  • 0.1237

0.0478 1

  • 0.2294
  • 0.2258

RESHt 0.2131 0.0427

  • 0.0042

0.1624 0.1347 0.1196 0.1124 0.1215

  • 0.2294

1 0.1070 USB2t 0.1738

  • 0.1459

0.0382

  • 0.0765

0.1504 0.4305 0.4540 0.0155

  • 0.2258

0.1070 1

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Univariate AR Model

Table: AR Model Result

Economic Series Statistics Gt Ft Rt Dvdt Tt B2t B10t AWEt URt RESHt USB2t intercept 0.007732 0.011946 0.022941 0.021265

  • 0.003537
  • 0.003927
  • 0.0049

0.013197

  • 0.002816

0.021062

  • 0.006487

std error 0.000937 0.001872 0.008091 0.005356 0.012202 0.009379 0.007557 0.0015 0.007238 0.004576 0.013217 t-stat 8.249534 6.381619 2.835495 3.970247

  • 0.289873
  • 0.418704
  • 0.648361

8.797614

  • 0.389085

4.602459

  • 0.4908

p-value 0.0055 0.0001 0.7725 0.6763 0.5181 0.698 0.6246 AR(1) 0.124081 0.616753

  • 0.102243

0.145303 0.098984

  • 0.017421

0.077562 0.388275 0.415415 0.529426

  • 0.075214

std error 0.09147 0.075648 0.09547 0.095369 0.095741 0.096141 0.095756 0.08671 0.087736 0.081631 0.096033 t-stat 1.356523 8.1529

  • 1.070946

1.523589 1.033869

  • 0.181199

0.810003 4.477843 4.734819 6.485606

  • 0.783214

p-value 0.1778 0.2866 0.1305 0.3035 0.8566 0.4197 0.4352 R-square 0.016753 0.380982 0.010508 0.021041 0.0098 0.000304 0.006038 0.156587 0.171897 0.280303 0.005648 Adjusted R-square 0.007649 0.37525 0.001346 0.011977 0.000632

  • 0.008953
  • 0.003165

0.148777 0.164229 0.273639

  • 0.003559

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Vector Autoregression Model (VAR)

X (∗)

t

= AX (∗)

t−1 + εt,

εt = Λtzt, zt = Lyt, (VAR) Log return of the quarterly value where X (∗)

t

= Xt − M Xt = (Gt, Ft, Rt, Yt, Tt, B2t, B10t, AWEt, URt, RESHt, USB2t)T M is an 11 ∗ 1 column vector of the unconditional series mean, Xt is an 11 ∗ 1 column vector of the series values at time t, A is an 11 ∗ 11 conditional mean parameter matrix, εt is an 11 ∗ 1 column vector of conditionally multivariate normal random errors or shocks to the series at time t,

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Vector Autoregression Model (VAR)

Λt is an 11 ∗ 11 column vector diagonal matrix of error standard deviations at time t given by Λ =        σ1 . . . σ2 . . . . . . . . . ... . . . . . . σ11        , and zt is an 11 ∗ 1 column vector of multivariate independent standard normal errors or shocks to the series with correlation matrix D (defined shortly after) at time t. yt is an 11 ∗ 1 column vector of independent standard normal errors or shocks to the series with correlation matrix I at time t.

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Vector Autoregression Model (VAR)

The VAR Model can be rewritten as

Xt =                  Gt Ft Rt Yt Tt B2t B10t AWEt URt RESHt USB2t                  = M +        a11 a12 . . . a111 a21 a22 . . . a211 . . . . . . ... . . . a111 a112 . . . a1111        [                  Gt−1 Ft−1 Rt−1 Yt−1 Tt−1 B2t−1 B10t−1 AWEt−1 URt−1 RESHt−1 USB2t−1                  −M]+εt,

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Vector Autoregression Model (VAR)

Contemporaneous correlations are modeled in the VAR system. We have E(zT

t zs) =

  • 0,

if t = s, Σ, if t = s. where is the contemporaneous covariance matrix of εt. D is the contemporaneous correlation matrix of zt determined using Cholesky decomposition so that Σ =        σ2

1

ρ12σ1σ2 . . . ρ111σ1σ11 ρ12σ1σ2 σ2

2

. . . ρ211σ2σ11 . . . . . . ... . . . ρ111σ1σ11 ρ211σ1σ11 . . . σ2

11

       = ΛDΛ

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

MLE Estimation for VAR Model

Time series of k observations Xt, Xt+1, . . . , Xt+k−1. The conditional expected values and variances are readily determined since εt+k|Xt+k−1, . . . , Xt ∼ N(0, Σ) E(Xt+k+i|Xt+k+i−1, Xt+k+i−2, . . . , Xt+i) = M + A(Xt+k+i−1 − M) Var(Xt+k+i|Xt+k+i−1, Xt+k+i−2, . . . , Xt+i) = Σ = ΛLLTΛ In general Xt+k+i|Xt+k+i−1, Xt+k+i−2, . . . , Xt ∼ N(M +A(Xt+k+i−1−M), Σ)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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MLE Estimation for VAR Model

The conditional probability density is f (Xt+k+i|Xt+k+i−1, Xt+k+i−2, . . . , Xt+i) = 1 (2π)m/2|Σ|1/2 exp

  • −1

2(Xt+k+i − M − A(Xt+k+i−1 − M))TΣ−1 (Xt+k+i − M − A(Xt+k+i−1 − M)) and the conditional log-likelihood is ln f (Xt+k+i|Xt+k+i−1, Xt+k+i−2, . . . , Xt+i) = −m 2 ln(2π) − 1 2 ln |Σ| − 1 2(Xt+k+i − M − A(Xt+k+i−1 − M))T Σ−1(Xt+k+i − M − A(Xt+k+i−1 − M))

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

MLE Estimation for VAR Model

ln L =

T

  • i=1

[− m 2 ln(2π) − 1 2 ln |Σ| − 1 2 (Xt+k+i − M − A(Xt+k+i−1 − M))T Σ−1(Xt+k+i − M − A(Xt+k+i−1 − M))] = − mT 2 ln(2π) − 1 2

T

  • i=1

ln |Σ| − 1 2

T

  • i=1

(Xt+k+i − M − A(Xt+k+i−1 − M))T Σ−1(Xt+k+i − M − A(Xt+k+i−1 − M))

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

VAR Model Parameters

Table: VAR Model Estimation

Economic Series Statistics Gt Ft Rt Dvdt lnTt lnB2t lnB10t AWEt URt RESHt lnUSB2t m 0.00785 0.01184 0.02288 0.02123

  • 0.00338
  • 0.00378
  • 0.00482

0.01318

  • 0.00264

0.02104

  • 0.0064

sigma 0.00772 0.00733 0.09317 0.0481 0.10704 0.09848 0.07266 0.00881 0.03839 0.02236 0.1447

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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VAR Model Estimation

A=                 −0.20177 −0.05544 0.64447 0.03078 −0.63259 −0.10968 −0.17315 0.40705 0.02075 −0.0122 0.16029 −1.29615 1.6804 −0.31954 −0.25909 0.54016 −0.57478                 L=                 1 1 0.10996 0.08358 0.99042 0.13441 0.25303 0.14342 0.94728

  • 0.06413
  • 0.07169
  • 0.12778

0.16853 0.97263 0.03159 0.07227

  • 0.11866

0.19948 0.63493 0.73265 0.05763 0.02906

  • 0.14227

0.12266 0.48512 0.66616 0.5305 0.16123 0.2381

  • 0.12883

0.12898 0.00585 0.03123 0.02694 0.93933

  • 0.26078

0.02598

  • 0.1094
  • 0.16909
  • 0.15937
  • 0.06609

0.00838

  • 0.01267

0.92777 0.14953

  • 0.18736

0.01009 0.05079 0.00553 0.02888 0.03413 0.01936

  • 0.0357

0.96756 0.11254

  • 0.16582

0.0567

  • 0.11929

0.10464 0.50585 0.13367 0.03905

  • 0.08386

0.11189 0.79789                

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Univariate Regime Switching RSAR(1) Model Structure

Process follows Yt = µ1 + α1(Yt−1 − µ1) + σ1εt, εt ∼ N(0, 1) (1) Yt = µ2 + α2(Yt−1 − µ2) + σ2εt, εt ∼ N(0, 1) (2)

  • r concisely

Yt|ρt = µρt+αρt(Yt−1−µρt)+σρtεt, εt are iid ∼ N(0, 1), ρt = 1, 2 hence Yt|ρt ∼ N(µρt + αρt(Yt−1 − µρt), σ2

ρt), ρt = 1, 2

The transition matrix P denotes the probabilities of moving between regimes, given by pij = Pr[ρt+1 = j|ρt = i], i = 1, 2, j = 1, 2. (3)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Algorithm for Maximum Likelihood Estimation

Two-regime AR(1) model has 8 parameters to estimate, Θ = µ1, µ2, α1, α2, σ1, σ2, p12, p21. Likelihood for the observations y = (y1, y2, . . . , yn) is L(Θ) = f (y1|Θ)f (y2|Θ, y1)f (y3|Θ, y1, y2) · · · f (yn|Θ, y1, . . . , yn−1) where f is the conditional pdf for y. Contribution to the log-likelihood of the t-th observation is logf (yt|yt−1, yt−2, . . . , y1, Θ). Determined recursively by calculating for each t:

f (ρt, ρt−1, yt|yt−1, . . . , y1, Θ) = p(ρt−1|yt−1, . . . , y1, Θ)∗p(ρt|ρt−1, Θ)∗f (yt|ρt, yt−1, Θ)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Algorithm for Maximum Likelihood Estimation

p(ρt|ρt−1, Θ) is the transition probability between the regimes f (yt|ρt, yt−1, Θ) = φ((yt − µρt)/σρt) where φ is the standard normal probability density function Probability function p(ρt−1|yt−1, yt−2, . . . , y1, Θ) is found from recursion with

f (ρt−1, ρt−2 = 1, yt−1|yt−2, yt−3, . . . , y1, Θ) + f (ρt−1, ρt−2 = 2, yt−1|yt−2, yt−3, . . . , y1, Θ) f (yt−1|yt−2, yt−3, . . . , y1, Θ)

f (yt|yt−1, yt−2, . . . , y1, Θ) is the sum over the four possible values for ρt = 1, 2 and ρt−1 = 1, 2.

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

RSAR(1) Model Fitting

Numerical routine reproduces Hardy’s NAAJ paper results Most series benefit from Regime switching - captures skewness and kurtosis Improvement in modeling marginal series Model is effectively a mixture of two normal distributions (different means and volatilities)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

RSAR(1) Model Estimation

Table: Maximum Likelihood Estimates of the Univariate Regime Switching Model

para mu1 mu2 a1 a2 s1 s2 p12 p21 Loglikehood Gt 1.2609 0.0082 1.0005

  • 0.2499

0.0065 0.0074 0.7191 0.5163 372.548 Ft 0.0110 0.0186 0.7847

  • 0.2373

0.0042 0.0127 0.0901 0.4568 406.217 Rt 0.0329

  • 0.0934

0.0247

  • 1.2985

0.0693 0.0049 0.0188 0.5063 135.330 Yt 0.0332 0.0118 0.5452

  • 0.1163

0.0245 0.0605 0.1027 0.1323 192.670 lnTt

  • 0.0108

0.0113 0.4564

  • 0.1053

0.0720 0.1604 0.0095 0.0154 99.836 lnB2t 0.0092

  • 0.0644

0.1000

  • 0.6367

0.0871 0.0984 0.0411 0.2153 101.922 lnB10t 0.0120

  • 0.0676

0.1353

  • 0.2728

0.0663 0.0402 0.1584 0.8617 133.799 AWEt 0.0271 0.0103 0.7018 0.0588 0.0117 0.0076 0.0491 0.0098 366.079 URt

  • 0.0216

0.0530 0.0157 0.2040 0.0268 0.0542 0.0715 0.2093 203.177 RESHt 0.0065 0.0398 0.1960 0.3351 0.0144 0.0250 0.0610 0.0711 270.355 lnUSB2t

  • 0.0059
  • 0.0111

0.1804

  • 0.2213

0.0998 0.2186 0.0468 0.1152 64.284 Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Historical Log Returns of CPI and UR with corresponding RSAR(1) regime estimation

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Mixed Density of GDP,CPI,SPI and DVD

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Mixed Density of AU interest rates and AWE

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Mixed Density of UR, RESH and US 2y rate

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Multivariate Regime Switching RSVAR(1) Model Structure

Two data series selected to be modelled with regime switching in the VAR model (parsimony) Regimes are assumed to be global regimes for all series. Univariate regime switching results identify CPI and Unemployment Rate, two important economic indicators, as best candidates for the switching series (means and variances and AR) Correlation matrix not regime switching

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

RSVAR(1) Model Estimation

Table: Estimated Mean and Volatility Parameters of the Multivariate Regime Switching Model

Series Gt Ft 1 Rt Yt lnTt lnB2t lnB10t AWEt URt 1 RESHt lnUSB2t Ft 2 URt 2 mean 0.00378 0.01881 0.02184 0.02177

  • 0.02692
  • 0.0111
  • 0.00571

0.02317 0.02288 0.01391

  • 0.0477

0.04634 0.04526 sigma 0.00751 0.01081 0.0925 0.04788 0.10422 0.09965 0.07204 0.00846 0.04641 0.02066 0.13966 0.00398 0.02528

L=                   1 0.02387 0.99972 0.08306

  • 0.02737

0.99617 0.16012 0.18561 0.17396 0.95376

  • 0.13734
  • 0.14476
  • 0.16061

0.19787 0.94617 0.02487 0.05371

  • 0.10272

0.19781 0.67305 0.70272 0.01635

  • 0.01516
  • 0.13913

0.13726 0.46671 0.69235 0.51395 0.2538 0.2692

  • 0.09235

0.16482 0.00491 0.04241 0.10561 0.90331

  • 0.1384

0.04017

  • 0.08138
  • 0.25695
  • 0.061
  • 0.18983

0.18441

  • 0.04239

0.91161 0.13757

  • 0.21472

0.07146 0.10996 0.09275 0.00454 0.12459 0.10552

  • 0.08558

0.9355 0.09813

  • 0.20181

0.08151

  • 0.14948

0.23098 0.46873 0.19727 0.01504

  • 0.21123
  • 0.01517

0.75073                  

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

RSVAR(1) Model Estimation

A1=                

  • 0.22922
  • 0.08375

0.58189 0.05714

  • 1.19903
  • 0.19411
  • 0.04232

0.27439 0.00402 0.00138 0.6713

  • 3.05078

1.6523

  • 0.42851
  • 0.41759
  • 0.00076

0.62592                 A2=                

  • 0.07537
  • 0.04368

0.97334

  • 0.00094
  • 0.33954
  • 0.02988
  • 0.2837

0.38136 0.03892

  • 0.02187
  • 0.09428
  • 0.3024

1.24057

  • 0.26621
  • 0.28814

0.77569

  • 1.3903

               

p12 = 0.441277 p21 = 0.250059 maxloglikehood = 2458.05

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Historical Data with estimated RSVAR(1,2) global regime

Ft

  • 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 Regime preference Ft

URt

  • 0.15
  • 0.1
  • 0.05

0.05 0.1 0.15 0.2 0.25 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 Regime preference urt

Gt and AWEt

  • 0.03
  • 0.02
  • 0.01

0.01 0.02 0.03 0.04 0.05 0.06 0.07 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 6 3 6 6 6 9 7 2 7 5 7 8 8 1 8 4 8 7 9 9 3 9 6 9 9 1 2 1 5 1 8 Gt Regime preference awet

Rt and Yt

  • 0.6
  • 0.5
  • 0.4
  • 0.3
  • 0.2
  • 0.1

0.1 0.2 0.3 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 6 3 6 6 6 9 7 2 7 5 7 8 8 1 8 4 8 7 9 9 3 9 6 9 9 1 2 1 5 1 8 Regime preference Rt Yt

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Historical Data with estimated RSVAR(1,2) global regime (continue)

RESHt

  • 0.04
  • 0.02

0.02 0.04 0.06 0.08 0.1 0.12 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 6 3 6 6 6 9 7 2 7 5 7 8 8 1 8 4 8 7 9 9 3 9 6 9 9 1 2 1 5 1 8 Regime preference resht

lnTt lnB2t lnB10t lnUSB2t

  • 0.7
  • 0.5
  • 0.3
  • 0.1

0.1 0.3 0.5 0.7 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 Regime preference lnTt lnB2t lnB10t lnusb2t

Global Regime Probability

0.2 0.4 0.6 0.8 1 1.2 1 4 7 1 1 3 1 6 1 9 2 2 2 5 2 8 3 1 3 4 3 7 4 4 3 4 6 4 9 5 2 5 5 5 8 6 1 6 4 6 7 7 7 3 7 6 7 9 8 2 8 5 8 8 9 1 9 4 9 7 1 1 3 1 6 1 9 regime 1 regime 2

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Simulation

Unconditional transition probabilities - maximum likelihood estimates for full data series Conditional transition probabilities - used to determine historical regimes - maximum likelihood for regime probabilities conditional

  • n history

Simulation of multi-variate series Comparison of models with historical data over 10 year horizon

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Key features

Tables in slides to follow show: VAR model provides generally a good fit Regime switching for the univariate series gives good fits for series with kurtosis Regime switching VAR model with constant correlation similar to VAR but an improvement Over the 10 year horizon models provide a reasonable quantification of the distributions based on historical data

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Simulation Comparison for GDP and CPI

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Simulation Comparison for SPI and DVD

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Simulation Comparison for 90day Tnote and 2year Tbond

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Simulation Comparison for 10year Tbond and AWE

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Simulation Comparison for UR, RESH and US 2y rate

URt Distribution Comparison

50 100 150 200 250 300 350

  • .

2 5

  • .

2 6

  • .

1 6 2

  • .

1 1 8

  • .

7 4

  • .

3 . 1 4 . 5 8 . 1 2 . 1 4 6 . 1 9 . 2 3 4 Bin Frequency URt_univariate_RSAR1 URt_VAR1 URt_Historical URt_RSVAR(1,2)

RESHt Distribution Comparison

50 100 150 200 250

  • .

1

  • .

8

  • .

6

  • .

4

  • .

2 . 2 . 4 . 6 . 8 . 1 . 1 2 Bin Frequency RESHt_univariate_RSAR1 RESHt_VAR1 RESHt_Historical RESHt_RSVAR(1,2)

LnUSB2t Distribution Comparison

50 100 150 200 250 300 350

  • 0.8
  • 0.62
  • 0.44
  • 0.26
  • 0.08

0.1 0.28 0.46 0.64 0.82 1 Bin Frequency LnUSB2t_univariate_RSAR1 LnUSB2t_VAR1 LnUSB2t_Historical LnUSB2t_RSVAR(1,2)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Issues and research developments

Modeling marginal series and dependence separately with regime switching in dependence All consider example with small number of series. Actuaries interested in jointly modeling a large number of financial and economic series Many fit marginal series parameters and use these in the dependence estimation (inference function for marginals) Marginal series with heavier tailed distribution such as t along with a regime switching canonical vine copula (Chollette, Heinen, Valdesogo, 2009) - 4 and 5 series Regime switching correlation matrix (Pelletier, 2006)

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models

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Introduction Data Analysis VAR Regime Switching Multivariate RS Simulation Conclusion

Conclusions and Summary

Regime switching models for univariate Australia data series fitted

  • capture non-normal distributions of series

VAR model fitted for multivariate Australian series - provides econometric relationships between series and lagged values and multivariate (dependent) error structure Fitted common regime switching model to multivariate series for Australian data Simulation using conditional regime probabilities Further research: regime switching for marginals along with copula for dependence.

Michael Sherris and Boqi Zhang UNSW Actuarial Research Seminar Economic Scenario Generation with Regime Switching Models