Robustní monitorování stability v modelu CAPM

Transcription

1 Robustní monitorování stability v modelu CAPM Ondřej Chochola, Marie Hušková, Zuzana Prášková (MFF UK) Josef Steinebach (University of Cologne) ROBUST 2012, Němčičky,

2 Contents Introduction Robust sequential monitoring in CAPM Asymptotic results Numerical results

3 Introduction CAPM (Capital Assets Pricing Model) r tj r t,f = β j (r t,m r t,f ) + ε tj, j = 1,..., d, t = 1,... N r tj -return of an asset j at time t r t,m - return of the market portfolio at time t r t,f - riskless security r tj r t,f - access return (risk premium) β j - measure of risk of the asset j w.r.t. the market portfolio reparametrized model r tj = α j + β j r t,m + ε tj

4 Model with time varying betas: r tj = α tj + β tj r t,m + ε tj, j = 1,..., d, t = 1,..., N or more generally, r i = α i + β i r i,m + ε i, i = 1,..., r i, α i, β i, ε i are d-dimensional vectors

5 Model with time varying betas: r tj = α tj + β tj r t,m + ε tj, j = 1,..., d, t = 1,..., N or more generally, r i = α i + β i r i,m + ε i, i = 1,..., r i, α i, β i, ε i are d-dimensional vectors Sequential monitoring: We assume stable historical data of size m such that α 1 =... = α m = α 0, β 1 =... = β m = β 0, and with any new observation we want to decide whether the stability in parameters is violated or not.

6 Hypothesis testing problem: H 0 : β 1 =... = β 0 H 1 : β 0 = β 1 =... = β m+k β m+k +1 =... where k = k m is an unknown change point. The null hypothesis is rejected whenever for the first time Q(k, m)/q γ (k/m) c α where Q is a test statistic, q γ (t), t (0, ) is a boundary function and c α is an appropriately chosen critical value.

7 The stopping rule: { inf{1 k < mt + 1 : τ m (γ) = Q(k, m)/q γ (k/m) c α },, if Q(k, m)/q γ (k/m) < c α 1 k < mt + 1. ( closed-end procedure)

8 The stopping rule: { inf{1 k < mt + 1 : τ m (γ) = Q(k, m)/q γ (k/m) c α },, if Q(k, m)/q γ (k/m) < c α 1 k < mt + 1. ( closed-end procedure) The constant c = c α is chosen such that lim P( ) τ m < H 0 = α, m lim P( ) τ m < H 1 = 1, m

9 Sequential monitoring in linear models - Chu, Stinchcombe and White, Horváth, Hušková, Kokoszka, Steinebach, Aue, Hörmann, Horváth, Hušková, and Steinebach, 2011 for CAPM - Koubková 2006 (monitoring, L1-norm)

10 Robust procedures in linear models: - Wu (2007) robust estimators, dependent variables - Koenker, Portnoy, multivariate, independent - Bai et al. 1990, multivariate, independent - Chan, Lakonischok (1992), Genton and Ronchetti (2008) - empirical studies in CAPM

11 Sequential robust monitoring in CAPM r ij = α 0 j + β 0 j r im + (α 1 j + β 1 j r im)i {i > m + k } + ε ij, i = 1, 2,..., k - change point, αj 0, β0 j, α1 j, β1 j, j = 1,..., d unknown parameters r im = r i,m r M, r M = 1 m r i,m. m M estimators α jm, β jm of αj o, βo j, based on the training sample: min i=1 m ρ j (r ij a j b j r im ) i=1 ρ j are convex loss functions with the derivatives ψ j, j = 1,..., d M-residuals ψ( ε i ) = (ψ 1 ( ε i1 ),..., ψ d ( ε id )) T with ε ij = r ij α jm (ψ) r im βjm (ψ)

12 A test statistic based on first m + k observations Q(k, m) = ( m+k 1 m i=m+1 T ( m+k r im ψ( ε i )) Σ 1 1 m m i=m+1 ) r im ψ( ε i ), where the matrix Σ m is an estimator of the asymptotic variance { 1 lim var m } (r i,m Er i,m )ψ(ε i ) m m i=1 based on the first m observations

13 Typical score functions ψ(x) = ρ (x) ψ(x) = x, x R 1 (ρ(x) = x 2 ) - least squares estimators and L 2 residuals ψ(x) = sign x, x R 1 (ρ(x) = x ) - L 1 estimators and L 1 residuals Huber ψ(x) = { x for x R 1 and some K > 0 Ksign x x K x > K

14 Assumptions on score functions ψ j s 1 ψ j are monotone functions, 2 functions λ j (t) = ψ j (x t)df j (x), t R, satisfy λ j (0) = 0, λ j (0) > 0, λ j (t) exists in a neighborhood od 0 and is Lipschitz in neighborhood of 0 for t c o for some c o > 0. 3 ψj (t) 2+ df j (t) < for some > 0 and ψ j (x + t 2 ) ψ j (x + t 1 ) 2 df j (x) C 0 t 2 t 1 κ, for some 1 κ 2, c o > 0, C 0 > 0 F j distribution function of ε ij

15 Assumptions on regressors For any i Z, r i,m = h(ξ i, ξ i 1,...), where h is measurable, {ξ i } i is a sequence of i.i.d. random vectors and E r 0M 2+ < for some > 0. For all i Z, where ξ (L) i L, ξ(l) r (L) im L=1 r im r (L) im 2 < = h(ξ i, ξ i 1,... ξ i L+1, ξ (L) i L, ξ(l) i L 1,...), i L 1,... are i.i.d. with the same distribution as ξ i independent of {ξ i } i {r (L) (L) D i,m } is L dependent, r i,m = r im i Z.

16 Assumptions on errors For any i Z, ε i = g(ζ i, ζ i 1,...), where g is measurable, {ζ i } i is sequence of i.i.d. random vectors For all i Z, L=1 L=1 ψ(ε i ) ψ(ε (L) i ) 2 < sup ψ(ε i a) ψ(ε (L) i a) 2 < a a 0 for some a 0 > 0, where ζ (L) i L, ζ(l) ε (L) i = g(ζ i, ζ i 1,... ζ i L+1, ζ (L) i L, ζ(l) i L 1,...) i L 1,... are i.i.d., independent of {ζ i} i, with the same distribution as ζ i.

17 Asymptotic results Model under the null hypothesis Theorem. Let the above assumptions be satisfied and Σ m Σ = o P (1) as m. Then under the null hypothesis as m max 1 k mt Q(k, m) q 2 γ(k/m) D sup 0<t<T /(T +1) d j=1 W 2 j (t) t 2γ, where {W j (t), t (0, 1)}, j = 1,..., d are independent Brownian motions and q γ (t) = (1 + t) (t/(t + 1)) γ, t (0, ), γ [0, 1/2)

18 Critical values c α satisfies ( P sup 0<t<T /(T +1) d j=1 W 2 j (t) t 2γ c α ) = α. The explicit form of the limit distribution is unknown Model under local alternatives: r ij = α 0 j + β 0 j r im + (α 1 j + β 1 j r im )δ m I {i > m + k } + ε ij δ m 0 and k < Tm + 1 Theorem (consistency): When δ m 0, δ m m 1/2, βj 1 at least one j and k = ms, 0 < s < T, as m 0 for Q(k, m) max 1 k mt, in probability. q γ (k/m)

19 Estimator of asymptotic variance matrix Σ Σ = i= Bartlett -type estimator E[(r 0M Er 0M )(r im Er im )ψ(ε 0 )ψ(ε i ) T ] Σ m = ω q (k) Γ k k q 1 m k m j=1 Γ k = r jm r j+k,m ψ( ε j )ψ( ε j+k ) T k 0 Γ T k k < 0 ω q (x) = ( 1 x ) I { x q} q q(m) as m, q(m)/m 0

20 Numerical results Simulations Critical values computed from simulated limit distribution Empirical quantiles of sample values of test statistic by Monte Carlo method Empirical quantiles of sample values of test statistic generated by pair bootstrap from historical period Empirical level of test Empirical power

21 L 2 Huber L 1 m \γ r im AR(1) ε i VAR(1) r im AR(1), ε i VAR(1) Table: Empirical sizes for 5% level, T = 10, dependent observations.

22 L 2 Huber L 1 m \γ N t t Table: Empirical sizes for 5% level, T = 10, different error distribution.

23 L 2 Huber L 1 m \γ N t ND ND t ND ND ND Table: Medians of detection delays, k = 10.

24 L 2 Huber L 1 m \γ Table: Empirical power of the test (in %) for t 1 errors

25 Application to real data Data: not traded indices computed from sector data of global world economy, serve as benchmarks for investors: World Consumer Disretionary (3) World Consumer Staples (4) World Energy (5) World Financials (6) World Health Care (7) Information Technology (9) Telecommunication Services (11) Market portfolio: MSCI World Daily Index (NDDUWI) Risk-free asset: S&P 3M US Treasury Bill Training period: (m=500) Monitoring period:

26 450 global world indices,

27 L 2 L 1 Huber ˆα ˆβ ˆα ˆβ ˆα ˆβ Table: Estimators of α, β, historical period : 4-Consumer Staples, 6-Financial, 7-Health Care

28 Index: 6, 7 Type: 3 Test statistic I 2007 I 2008 Financials (6), Health Care (7): red - γ = 0, blue - γ = 0.25, black - γ = 0.45 solid line - asymptotic critical values, dashed - closed end

29 Index: 4, 6 Type: 3 Test statistic I 2007 I 2008 Consumer Staples (4), Financials (6)

30 Index: 4, 7 Type: 3 Test statistic VII 2007 IX 2007 XI 2007 I 2008 III 2008 V 2008 Consumer Staples (4), Health Care (7)

31 Consumer Staples, Health Care and Financials Consumer Staples and Financials Test statistic Test statistic XII 2006 III 2007 VI 2007 IX 2007 XII 2007 XII 2006 III 2007 VI 2007 IX 2007 XII 2007 Health Care and Financials Consumer Staples and Health Care Test statistic Test statistic XII 2006 III 2007 VI 2007 IX 2007 XII 2007 VII 2007 XI 2007 III 2008

32 References 1 Aue, A., Hörmann, S., Horváth, L., Hušková, M. and Steinebach, J. (2011). Sequential stability testing for stability of high frequency portfolio betas. To appear in Econometric Theory. 2 Berkes, I., Hörmann, S., Schauer, J. (2011). Split invariance principles for stationary processes. Ann. Probab. 39, Hörmann, S. and Kokoszka P.(2010). Weakly dependent functional data. Ann. Statist. 38, Jurečková, J., Sen, P.K. (1996). Robust Statistical inference. Asymptotics and Interrelations. Wiley, New York. 5 Wu, W.B. (2007). M-estimation of linear models with dependent errors. Ann. Statist. 35,

33 Thank you for your attention