Economic Scenario Generation with Regime Switching Models

Transcription

1 Economic Scenario Generation with Regime Switching Models 2pm to 3pm Friday 22 May, ASB 115 Acknowledgement: Research funding from Taylor-Fry Research Grant and ARC Discovery Grant DP

2 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

3 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)

4 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; others 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

5 ERCH Model m series ERCH model is expressed in multivariate form as X t = M + ΘΨ t + ξ t, lnλ t = diag{ω 0 + ΩΦ t } Z t N(0, Σ z ) { E(Zt T 0, if t s, Z s ) = Σ z, if t = s. ξ t = Λ t Z t where M = E(X t ) is an m 1 column vector of unconditional series means, Θ is an m p conditional mean parameter matrix, Ψ t 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 = Ψ t E(Ψ t )

6 ERCH Model ξ t is an m 1 column vector of conditionally multivariate normal random errors or shocks to the series at time t, Λ t is an m m diagonal matrix of error standard deviations at time t, lnλ t is an m m diagonal matrix of the logarithms of the error standard deviations at time t, Z t 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

7 ERCH Model ω 0 is an m 1 column vector of parameters, Ω is an m q conditional volatility parameter matrix, Φ t is a q 1 column vector of lagged explanatory variable values at time t. Σ z 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 Z t, Harris (1994) estimated parameters based on quarterly Australian data.

8 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 There are 111 quarterly observations for each economic series or = 1221 data points in total.

9 Data Plots Variable Description G t the log return of GDP F t the log return of CPI R t the log return adjusted SPI (Share Price Index of ASX 200) Y t the log return of Dividend of the adjusted SPI T t the log return of 90-day Treasury notes yield B2 t the log return of 2-year Treasury Bond yield B10 t the log return of 10-year Treasury Bond yield AWE t the log return of average weekly earnings UR t the log return of unemployment rate RESH t the log return of residential house index in Sydney USB2 t the log return of US 2-year Treasury Bond yield

10 Plot in log return of GDP 0.04 lngdp log return % Quarter

11 Plot in log return of CPI logcpi log return % Quarter

12 Plot in log return of SPI and its Dvd lnspi lndvd log return % Quarter

13 Plot in log return of AUD interest rates log 90 day T note log 2 year T bond log 10 year T bond log return % Quarter

14 Plot in log return of AWE 0.05 log AWE log return % Quarter

15 Plot in log return of Unemployment Rate 0.25 log Unemployment Rate log return Quarter

16 Plot in log return of Residential property price index 0.1 log RESH log return Quarter

17 Plot in log return of 2year US interest rate 0.6 log US 2 year T bond log return Quarter

18 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)

19 Descriptive Statistics Table: Descriptive Statistics Economic Series Statistics Gt Ft Rt Dvdt Tt B2t B10t AWEt URt RESHt USB2t Mean Median Maximum Minimum Std.Dev Skewness Kurtosis Jarque-Bera P-value

20 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 p-value 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 Ft Rt Yt Tt B2t B10t AWEt URt RESHt USB2t

21 Univariate AR Model Table: AR Model Result Economic Series Statistics Gt Ft Rt Dvdt Tt B2t B10t AWEt URt RESHt USB2t intercept std error t-stat p-value AR(1) std error t-stat p-value R-square Adjusted R-square

22 Vector Autoregression Model (VAR) X ( ) t = AX ( ) t 1 + ε t, ε t = Λ t z t, z t = Ly t, (VAR) Log return of the quarterly value where X ( ) t = X t M X t = (G t, F t, R t, Y t, T t, B2 t, B10 t, AWE t, UR t, RESH t, USB2 t ) T M is an 11 1 column vector of the unconditional series mean, X t is an 11 1 column vector of the series values at time t, A is an conditional mean parameter matrix, ε t is an 11 1 column vector of conditionally multivariate normal random errors or shocks to the series at time t,

23 Vector Autoregression Model (VAR) Λ t is an column vector diagonal matrix of error standard deviations at time t given by σ σ Λ =......, σ 11 and z t 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. y t is an 11 1 column vector of independent standard normal errors or shocks to the series with correlation matrix I at time t.

24 Vector Autoregression Model (VAR) The VAR Model can be rewritten as X t = G t F t R t Y t T t B2 t B10 t AWE t UR t RESH t USB2 t = M + a 11 a a 111 a 21 a a a 111 a a 1111 [ G t 1 F t 1 R t 1 Y t 1 T t 1 B2 t 1 B10 t 1 AWE t 1 UR t 1 RESH t 1 USB2 t 1 M]+ε t,

25 Vector Autoregression Model (VAR) Contemporaneous correlations are modeled in the VAR system. We have { E(zt T 0, if t s, z s ) = Σ, if t = s. where is the contemporaneous covariance matrix of ε t. D is the contemporaneous correlation matrix of z t determined using Cholesky decomposition so that σ 2 1 ρ 12 σ 1 σ 2... ρ 111 σ 1 σ 11 ρ 12 σ 1 σ 2 σ ρ 211 σ 2 σ 11 Σ = = ΛDΛ ρ 111 σ 1 σ 11 ρ 211 σ 1 σ σ11 2

26 MLE Estimation for VAR Model Time series of k observations X t, X t+1,..., X t+k 1. The conditional expected values and variances are readily determined since ε t+k X t+k 1,..., X t N(0, Σ) E(X t+k+i X t+k+i 1, X t+k+i 2,..., X t+i ) = M + A(X t+k+i 1 M) Var(X t+k+i X t+k+i 1, X t+k+i 2,..., X t+i ) = Σ = ΛLL T Λ In general X t+k+i X t+k+i 1, X t+k+i 2,..., X t N(M +A(X t+k+i 1 M), Σ)

27 MLE Estimation for VAR Model The conditional probability density is f (X t+k+i X t+k+i 1, X t+k+i 2,..., X t+i ) ( 1 = (2π) m/2 exp 1 Σ 1/2 2 (X t+k+i M A(X t+k+i 1 M)) T Σ 1 (X t+k+i M A(X t+k+i 1 M)) and the conditional log-likelihood is ln f (X t+k+i X t+k+i 1, X t+k+i 2,..., X t+i ) = m 2 ln(2π) 1 2 ln Σ 1 2 (X t+k+i M A(X t+k+i 1 M)) T Σ 1 (X t+k+i M A(X t+k+i 1 M))

28 MLE Estimation for VAR Model ln L = T [ m 2 ln(2π) 1 ln Σ 2 i=1 1 2 (X t+k+i M A(X t+k+i 1 M)) T Σ 1 (X t+k+i M A(X t+k+i 1 M))] = mt 2 ln(2π) 1 T ln Σ 2 i=1 1 T (X t+k+i M A(X t+k+i 1 M)) T Σ 1 (X t+k+i M A(X t+k+i 1 M)) 2 i=1

29 VAR Model Parameters Table: VAR Model Estimation Economic Series Statistics Gt Ft Rt Dvdt lntt lnb2t lnb10t AWEt URt RESHt lnusb2t m sigma

30 VAR Model Estimation A= L=

31 Univariate Regime Switching RSAR(1) Model Structure Process follows or concisely Y t = µ 1 + α 1 (Y t 1 µ 1 ) + σ 1 ε t, ε t N(0, 1) (1) Y t = µ 2 + α 2 (Y t 1 µ 2 ) + σ 2 ε t, ε t N(0, 1) (2) Y t ρ t = µ ρt +α ρt (Y t 1 µ ρt )+σ ρt ε t, ε t are iid N(0, 1), ρ t = 1, 2 hence Y t ρ t N(µ ρt + α ρt (Y t 1 µ ρt ), σ 2 ρ t ), ρ t = 1, 2 The transition matrix P denotes the probabilities of moving between regimes, given by p ij = Pr[ρ t+1 = j ρ t = i], i = 1, 2, j = 1, 2. (3)

32 Algorithm for Maximum Likelihood Estimation Two-regime AR(1) model has 8 parameters to estimate, Θ = µ 1, µ 2, α 1, α 2, σ 1, σ 2, p 12, p 21. Likelihood for the observations y = (y 1, y 2,..., y n ) is L(Θ) = f (y 1 Θ)f (y 2 Θ, y 1 )f (y 3 Θ, y 1, y 2 ) f (y n Θ, y 1,..., y n 1 ) where f is the conditional pdf for y. Contribution to the log-likelihood of the t-th observation is logf (y t y t 1, y t 2,..., y 1, Θ). Determined recursively by calculating for each t: f (ρ t, ρ t 1, y t y t 1,..., y 1, Θ) = p(ρ t 1 y t 1,..., y 1, Θ) p(ρ t ρ t 1, Θ) f (y t ρ t, y t 1, Θ)

33 Algorithm for Maximum Likelihood Estimation p(ρ t ρ t 1, Θ) is the transition probability between the regimes f (y t ρ t, y t 1, Θ) = φ((y t µ ρt )/σ ρt ) where φ is the standard normal probability density function Probability function p(ρ t 1 y t 1, y t 2,..., y 1, Θ) is found from recursion with f (ρ t 1, ρ t 2 = 1, y t 1 y t 2, y t 3,..., y 1, Θ) + f (ρ t 1, ρ t 2 = 2, y t 1 y t 2, y t 3,..., y 1, Θ) f (y t 1 y t 2, y t 3,..., y 1, Θ) f (y t y t 1, y t 2,..., y 1, Θ) is the sum over the four possible values for ρ t = 1, 2 and ρ t 1 = 1, 2.

34 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)

35 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 Ft Rt Yt lntt lnb2t lnb10t AWEt URt RESHt lnusb2t

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

37 Mixed Density of GDP,CPI,SPI and DVD

38 Mixed Density of AU interest rates and AWE

39 Mixed Density of UR, RESH and US 2y rate

40 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

41 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 sigma L=

42 RSVAR(1) Model Estimation A1= A2= p 12 = p 21 = maxloglikehood =

43 Regime preference Ft Regime preference urt Gt Regime preference awet Regime preference Rt Yt Historical Data with estimated RSVAR(1,2) global regime Ft URt Gt and AWEt Rt and Yt

44 Regime preference resht Regime preference lntt lnb2t lnb10t lnusb2t regime 1 regime Historical Data with estimated RSVAR(1,2) global regime (continue) RESHt lntt lnb2t lnb10t lnusb2t Global Regime Probability

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

46 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

47 Simulation Comparison for GDP and CPI

48 Simulation Comparison for SPI and DVD

49 Simulation Comparison for 90day Tnote and 2year Tbond

50 Simulation Comparison for 10year Tbond and AWE

51 Simulation Comparison for UR, RESH and US 2y rate URt Distribution Comparison Frequency Bin URt_univariate_RSAR1 URt_VAR1 URt_Historical URt_RSVAR(1,2) RESHt Distribution Comparison Frequency Bin RESHt_univariate_RSAR1 RESHt_VAR1 RESHt_Historical RESHt_RSVAR(1,2) LnUSB2t Distribution Comparison Frequency LnUSB2t_univariate_RSAR1 Bin LnUSB2t_Historical LnUSB2t_VAR1 LnUSB2t_RSVAR(1,2)

52 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)

53 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.