Change point tests in functional factor models with application to yield curves

Transcription

1 Change point tests in functional factor models with application to yield curves Department of Statistics, Colorado State University Joint work with Lajos Horváth University of Utah Gabriel Young Columbia University Patrick Bardsley University of Texas

2 Scalar change point model Null hypothesis: X i = µ i +u i, 1 i N, Eu i = 0. H 0 : µ 1 = µ 2 =... = µ N. H A : the means change in some way: abruptly, gradually, epidemically... It is typically assumed that the errors u i are homoskedastic, the series u i is stationary. There has been limited work for nonstationary u i : Cavaliere, Taylor, Dalla, Giraitis, Phillips...

3 CUSUM approach CUSUM process: [Nt] Z N (t) = N 1/2 X l [Nt] N l=1 N l=1 X l. Functionals of this process are used as tests statistics. If the u i are iid, Z N typically leads to optimal tests, it can be connected to the likelihood method. There has been recent work for the u i which are dependent and and exhibit change/points or irregular trends in variability. Change points or trends in the variability of the u i can confound change points in the mean of the X i.

4 Assumptions on the u i u i = u i,n = a(i/n)e i [0,1] t a(t) has regular variation. [Nt] N 1/2 (no mixing needed). Set Then l=1 e l D[0,1] σw(t). t b(t) = σ 2 a 2 (u)du. [Nt] N 1/2 l=1 0 u l D[0,1] W(b(t)). Main difficulty: the function b( ) is unknown.

5 Approach to testing Under H 0, Z N (t) D[0,1] Γ(t), where Γ(t) = W(b(t)) tw(b(1)). C(t,s) = E[Γ(t)Γ(s)] = b(t s) tb(s) sb(t)+tsb(1). b(t) is estimated by the LRV of X i,i [Nt]. λ i eigenvalues of C(, ) ZN 2 (t)dt D Γ 2 (t)dt. 0 D = λ i ξi 2. i=1 Weighted functionals, e.g. Anderson Darling, are possible.

6 Yield curves i - day (or month) t j - time to maturity (1 month, 3 months,..., 30 years) Fractions of bonds with continuous maturities are traded. Our theory can be in discrete or continuous time t. X i (t j ) - yield of a bond bought on day i and held for time t j. For the theory, we can use, the yield curves, X i (t),t [0,T].

7 Yield curves US yield curves over five business days around the Lehman Brothers bankruptcy filing on Sep

8 Yield curves Level, range, shape (mean structure) US yield curves over N = 100 business days; the central time point corresponds to the Lehman Brothers collapse.

9 Factor models X i (t j ) = K β i,k f k (t j )+ε i (t j ). k=1 The f k are deterministic functions, treated as known in standard finance methodology The K time series {β i,k, = 1,2,3,...} are modeled as time series (modern dynamic models). If β i,k β k, static model.

10 Nelson Siegel factors Nelson Siegel factors f 1 (t,λ),f 2 (t,λ),f 3 (t,λ).

11 Functional vs. multivariate modeling {X i (t j ), i = 1,2,3,...} is a multivariate time series of dimension J = 10. Why not model it as a VAR process? The yield curves have a specific shape whose components have economic interpretation. Factors lead to nontrivial and practically relevant dimension reduction and improved estimation and prediction.

12 Change point test in a functional factor model K X i (t) = β i,k f k (t)+ε i (t), 1 i N. k=1 (Continuous time) β i,k = µ i,k +b i,k, Eb i,k = 0. µ i = [µ i,1,µ i,2,...,µ i,k ] H 0 : µ 1 = µ 2 =... = µ N.

13 Change point vs. break point Recall: H 0 : mean structure does not change Error functions: K k=1 b i,kf k (t)+ε i (t). The distribution of the error functions can change at known points: 1 = i 0 < i 1 < i 2 <... < i M < i M+1 = N. (Similar to prior information in Bayesian inference) On each segment, error functions are realizations of potentially different weakly dependent processes in L 2. Compare: unknown change points in mean structure known break points in error structure

14 Determining break points Dates of substantial central bank intervention, Dates of events of economic impact, Exploratory analysis of the variability of the yield curves. After the application of the test, some change points may be close to break points.

15 Detection through projections onto factors Vectors of projections: z i = [ X i,f 1,..., X i,f K ] Introduce the deterministic matrix C = [ f k,f j, 1 k,j K]. and random vectors b i = [b i,1,b i,2,...,b i,k ], ε i = [ ε i,f 1, ε i,f 2,..., ε i,f K ]. Then z i = Cµ i +γ i, γ i = Cb i +ε i, Change in the vectors µ i is equivalent to a change in the z i at the same change points.

16 Detection through projections onto factors CUSUM process: [Nx] α N (x) = N 1/2 z i [Nx] N i=1 N i=1 z i, 0 x 1. Even under H 0, the distribution of the z i can change at break points. { γ (m) i } - model for the vector errors γ i over the interval (i m,i m+1 ]. Long run covariance matrices: V m = l= cov(γ (m) i,γ (m) i+l ).

17 Detection through projections onto factors Limit distribution of the CUSUM process: α N G 0, in D K ([0,1]), G 0 (x) = G(x) xg(1). {G(x), x [0,1]} is a mean zero R K valued Gaussian process with covariances E[G(x)G (y)] = m (θ j θ j 1 )V j +(x θ m )V m+1, θ m x θ m+1,y x j=1 The covariances of the process G 0 can be computed explicitly (a long formula). Cramér von Mises functional 1 0 α N (x) 2 dx 1 0 G 0 (x) 2 dx. We must simulate the distribution of the right hand side.

18 Detection through projections onto factors 1. Eigenvalue method (Eigen): Estimate R(x,y) = E[G 0 (x)g 0 (y) ]. Use equality in distribution: 1 0 G 0 (x) 2 dx j=1 λ j Z 2 j. The Z j are independent standard normal; 1 0 R(x,y)φ j (y)dy = λ j φ j (x).

19 Detection through projections onto factors 2. Direct simulation method (Sim): Generate replications of the process G 0. Enough to generate G(x) = m ( ) x (θ j θ j 1 ) 1/2 G j (1)+(θ m+1 θ m ) 1/2 θm G m+1, θ m+1 θ m j=1 E[G j (x)g j (y) ] = min(x,y)v j, G j (x) = L j W(x), L j L j = V j, W = [W 1,W 2,...,W K ], K independent standard Wiener processes. x (θ m,θ m+1 ].

20 Nonparametric functional approach Motivation: the factor model can be rewritten as where τ i (t) = X i (t) = τ i (t)+η i (t), 1 i N, K µ i,k f k (t), η i (t) = k=1 K b ik f k (t)+ε i (t). k=1 The first line is a valid model without the second line. We can test H 0 : τ 1 = τ 2 =... = τ N. A version of the Eigen method can be derived, It has to be custom coded, one needs to solve U 0 (x,y;t,s)ϕ j (y,s)dyds = λ j ϕ j (x,t). The 4D kernel is expressed in terms suitably defined LRV kernels.

21 Application to yield curves Case (1)

22 Application to yield curves Case (2)

23 Application to yield curves Case (3)

24 Application to yield curves Sampling Period Method Break Point P value ProjSim yes 1.5% (1) ProjEigen yes 1.7% 03/20/ /19/2009 NFEigen yes 0.1% ProjSim no 87.9% ProjEigen no 85.2% NFEigen no 26.2% ProjSim yes 0.1% (2) ProjEigen yes 0.0% 06/30/ /29/2006 NFEigen yes 0.2% ProjSim no 56.7% ProjEigen no 50.5% NFEigen no 57.2% ProjSim yes 68.1% (3) ProjEigen yes 66.9% 02/16/ /14/2013 NFEigen yes 55.8% ProjSim no 80.4% ProjEigen no 77.7% NFEigen no 76.3%

25 Conclusions Conclusions based on similar data examples and simulations: 1 If a possible break point in the error structure is not taken into account in any of the testing procedures, an existing change point in the mean structure may not to be detected. 2 Break points can be misplaced, ±50 days is fine. 3 The tests are generally well calibrated if N = 500. If N = 250, all tests have a tendency to overreject by some 2 percent. 4 If the intelligible factors (Lengwiller and Lenz (2010) are used, the empirical size of projsim test improves at the 5% nominal level, but the size of the ProjEigen test deteriorates. 5 We recommend ProjSim with intelligible factors and NFEigen. The paper contains complete asymptotic theory and details of numerical implementation.