Model estimation through matrix equations in financial econometrics

Transcription

1 Model estimation through matrix equations in financial econometrics Federico Poloni 1 Joint work with Giacomo Sbrana 2 1 Technische Universität Berlin (A. Von Humboldt postdoctoral fellow) 2 Rouen Business School When Probability Meets Computation Varese, June 2012 F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

2 Last November, I received an message from a researcher in Econometrics looking for help with some matrix equations. We spent some time trying to understand each other s language... F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

3 Last November, I received an message from a researcher in Econometrics looking for help with some matrix equations. We spent some time trying to understand each other s language... ρ(m) QR a :,: O(n 2 log n) eigs! H t ARMA(1,1) P[X] E[y t ] Var Z! F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

4 Scalar GARCH A GARCH(1,1) model is a stochastic time series y t such that y t = h 1/2 t ɛ t, where ɛ t is IID noise with mean 0 and variance 1 h t depends linearly on y 2 t 1 and h t 1 : a + b 1 h t = c + ay 2 t 1 + bh t 1 Popular choice to model stock market volatility [Engle, 82], [Bollerslev, 86] Estimation problem Given a large number of observations, estimate the parameters c, a, b [Francq, Zakoïan, book 10] F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

5 Example Figure: Example GARCH(1,1) data F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

6 Multivariate GARCH Different possible generalizations. Simplest one: every entry of the variance H t may depend linearly on every entry of H t 1 and every (y t 1 ) i (y t 1 ) j Vectorization: a tool to express this y 1 y 1 1 y 1 y 2 y t yt T y x t = 1 y , H y 2 y 2 t = h t = y 2 y 3 5 y 3 y 3 6 Multivariate GARCH h t = c + Ax t 1 + Bh t 1 F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

7 Multivariate GARCH A multivariate GARCH(1,1) model is a time series y t of d 1 vectors such that y t = H 1/2 t ɛ t, where ɛ t is IID noise with mean 0 and variance I d H t is an affine linear function of H t 1 and y t 1 yt 1. T In formulas h t = c + Ax t 1 + Bh t 1 all the eigenvalues of A + B are in the unit circle Estimation problem Given a large number of observations, estimate the parameters c, A, B [Francq, Zakoïan, book 10] F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

8 Example Figure: Example GARCH(1,1) data F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

9 Quasi-Maximum Likelihood Most popular choice for estimating c, A, B: Quasi-Maximum Likelihood Given a guess ĉ, Â, ˆB, we can compute Ĥ t at each time and then the likelihood l t that a Gaussian N(0, H t ) generates the observed y t Remark the noise is not assumed Gaussian in general, so this is an approximation A good guess for the parameters is n max l t ĉ,â, ˆB t=1 (but it is not clear how to compute it) F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

10 Estimating a GARCH through ML Most popular choice for estimating c, A, B: Quasi-Maximum Likelihood Take likelihood function L(c, A, B) and feed it to a black-box optimizer optimizer proceeds blindly, no knowledge of function expression difficult optimization problem: nonconvex, high-dimensional the code is delicate: step-size control, numerical derivatives... slow, big and scary Our hope Finding a more manageable estimator F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

11 Estimating a GARCH through ML Most popular choice for estimating c, A, B: Quasi-Maximum Likelihood Take likelihood function L(c, A, B) and feed it to a black-box optimizer optimizer proceeds blindly, no knowledge of function expression difficult optimization problem: nonconvex, high-dimensional the code is delicate: step-size control, numerical derivatives... slow, big and scary Our hope Finding a more manageable estimator F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

12 Estimating c and A + B Key property y t y T t is almost H t, since H t = var[y t ] More precisely: ξ t := x t h t is an MDS, i.e., E [ξ t ] = 0 independently of everything that happens up to t 1 x t = ξ t + c + (A + B)x t 1 Bξ t 1 Yule-Walker type formulas If we compute autocorrelations M k = E M k+1 = (A + B)M k k 1 [ (x t E [x])(x t k E [x]) T ], Autocorrelations are easily estimated using sample autocorrelations ˆM k = 1 n x t+k xt T. From them we can get A + B and c. n t=1 F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

13 Moving average Now we use the moving average vector j t := x t (A + B)x t 1 c = ξ t Bξ t 1 (the second form is obtained using the formulas for h t ) We get, with Φ := A + B and Σ = Var(ξ t ) [ ] Γ 0 = E j t jt T = M 0 M 1 Φ T ΦM1 T + ΦM 0 Φ T = Σ + BΣB T [ ] Γ 1 = E j t+1 jt T = M 1 ΦM 0 = BΣ F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

14 Matrix equations [ ] Γ 0 = E j t jt T = M 0 M 1 Φ T ΦM1 T + ΦM 0 Φ T = Σ + BΣB T [ ] Γ 1 = E j t+1 jt T = M 1 ΦM 0 = BΣ We can eliminate either Σ or B, getting Γ T 1 Γ 0 B T + Γ 1 (B T ) 2 = 0 (P) Γ 0 = Σ + Γ 1 Σ 1 Γ T 1. (P) is a palindromic matrix equation [Mackey et al 05, Gohberg et al book 09] (R) is a baby-riccati nonlinear matrix equation (NME) [Engwerda et al 93, Meini 02, + others] (R) F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

15 A closed-form estimator Γ1 T Γ 0 B T + Γ 1 (B T ) 2 = 0 (P) We can solve (P) via linearization [... ] or cyclic reduction/doubling [... ] The complete procedure Compute a few of the first sample autocorrelations ˆM k Estimate  + B and ĉ using Yule-Walker results (how?) Estimate ˆΓ 0 and ˆΓ 1, autocorrelations of j t = x t ( + B)x t 1 ĉ Solve the matrix equation (P) to get ˆB Suggested by [Linton, Kristensen 06] for the univariate GARCH, where (P) is simply a quadratic equation We can now generalize it to the more interesting multivariate case F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

16 Solving matrix equations (when it is possible) Γ T 1 Γ 0 B T + Γ 1 (B T ) 2 = 0 (P) Γ 0 = Σ + Γ 1 Σ 1 Γ T 1. Solvability theory, putting together mostly known results for (R): If P(λ) = Γ T 1 λ 1 + Γ 0 + Γ 1 λ 0 on the unit circle, a unique stable B (ρ(b) 1) exists Otherwise, no pair (B, Σ) with Σ 0 exists If our model holds, B exists (but often eigenvalues close to the unit circle). Problem Γ 0, Γ 1 come from the data and must be estimated. If large errors, (P) and (R) are perturbed to unsolvable equations (R) F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

17 Example Work in progress (with T. Brüll, C. Schröder): find small perturbations to Γ 0, Γ 1 which move the lines up. F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

18 So, does it really work? Yes, but still not as accurate as MLE. Weak point: the convergence ˆM k M k is not that fast However, much faster than MLE. Suggested uses: use it as a starting value for optimization in MLE or use an iterative feasible least-squares procedure to refine it Almost finished: asymptotic consistency/normality properties of LS F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

19 Idea of feasible GLS Method to refine an estimate ĉ, Â, ˆB x t = h t + (error); h t = f (c, A, B, x t 1, h t 1 ) linear in c, A, B Problems: min f (c, A, B, xt 1, h t 1 ) x t 2 c,a,b We do not know h t 1 in the formula Replace it by ĥ t 1 Variance of the error varies (it is h t ), least squares methods start from the assumption that it is fixed Rescale it with ĥt We run a few iterations of this method and check which one has the largest likelihood F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

20 Simulated (Monte-Carlo) results Only some preliminary results , 1000 or 5000 observations ρ(a + B) up to 0.9 Speedup in QML around 30% for sufficiently hard experiments In some cases, convergence problems in QML with our starting values we are trying to understand this Closed form estimator iterations of FGLS: accuracy on par with QML, much faster, simple to code. F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

21 Conclusions New application for matrix equations and structured linear algebra Correct estimations are crucial to assess risk in economic models F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

22 Conclusions New application for matrix equations and structured linear algebra Correct estimations are crucial to assess risk in economic models We help practitioners make the same errors, but much faster! Thanks for your attention And a happy retirement to Guy! F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19

23 Conclusions New application for matrix equations and structured linear algebra Correct estimations are crucial to assess risk in economic models We help practitioners make the same errors, but much faster! Thanks for your attention And a happy retirement to Guy!??? Profit! F. Poloni (TU Berlin) Matrix eqns in econometrics Varese / 19