An Introduction to Robust Statistics with Economic and Financial Applications. Elvezio Ronchetti

Transcription

1 An Introduction to Robust Statistics with Economic and Financial Applications Elvezio Ronchetti Department of Econometrics University of Geneva Switzerland 1

2 Outline Introduction Sensitivity Curve and Influence Function Statistical Functionals Influence Function Breakdown Point M estimators Optimal Robust Estimators Robust Inference Leading Example (Finance) Messages 2

3 Introduction Robust statistics deals with deviations from ideal models and their dangers for corresponding inference procedures primary goal is the development of procedures which are still reliable and reasonably efficient under small deviations from the model, i.e. when the underlying distribution lies in a neighborhood of the assumed model Robust statistics is an extension of parametric statistics, taking into account that parametric models are at best only approximations to reality. 3

4 Main aims of robust procedures From a data-analytic point of view, robust statistical procedures will (i) find the structure best fitting the majority of the data; (ii) identify deviating points (outliers) and substructures for further treatment; (iii) in unbalanced situations : identify and give a warning about highly influential data points (leverage points). 4

5 In addition to the classical concept of efficiency, new concepts are introduced to describe the local stability of a statistical procedure (the influence function and derived quantities) its global reliability or safety (the breakdown point). 5

6 The ancient, vaguely defined problem of robustness has been partly formalized into mathematical theories which yield optimal robust procedures and which provide illumination and guidance for the user of statistical methods. 6

7 Some common misunderstandings Robust statistics replaces classical statistics. The normality assumption is guaranteed by the central limit theorem. If the errors are non-normal, I change the specification of the errors. I use classical procedures after removing outliers. Therefore I do not need any robust procedures. Robust statistics cannot be used when the errors are asymmetric. 7

8 Robustness its purpose is to safeguard against deviations from the assumptions. It makes unnecessary getting the stochastic part of the model right. Diagnostics Its purpose is to find and identify deviations from the assumptions. It helps to make the functional part of the model right. 8

9 Sensitivity Curve and Influence Function Sensitivity curve Observations z 1, z 2,... with underlying distribution (model) F. (function of the observa- Statistic T n tions) SC(z; z 1,..., z n 1, T n ) = n [T n (z 1,..., z n 1, z) T n 1 (z 1,..., z n 1 ) ] n IF (z; T, F ) 9

10 Influence function of the mean mean n (z 1,..., z n 1, z) mean n 1 (z 1,..., z n 1 ) 1 n 1 n (z z n 1 + z) n 1 1 (z z n 1 ) 1 n ( ) 1 n z 1 n 1 n 1 (z z n 1 ) 1 n z mean n 1 (z 1,..., z n 1 ) n z E F Z = IF (z; mean, F ) 10

11 Influence function of the least squares estimator Regression : y i = x T i β + e i Least Squares Est : ˆβ i = 1,..., n Q n 1 = 1 n 1 n 1 i=1 x i x T i Q, n. SC ( (x, y); (x 1, y 1 ),..., (x n 1, y n 1 ), LS ) = n n 1 Q 1 n 1 x(y xt ˆβ n 1 ) 1 1+ n 1 1 xt Q 1 n 1 x 1 Q 1 β 1 IF (x, y; LS, F ) = Q 1 x(y x T β) when n. 11

12 Can find IF (z; T, F ) for most est. Gross-error sensitivity : maximum (over z) of IF WANTED PROCEDURES WITH BOUNDED INFLUENCE FUNCTION Reward : ROBUSTNESS 12

13 Statistical Functionals Ex. Z F S : P R P, F S(F ) Characteristic of a distr. F T : F T (F ) = E F Z = zdf (z) = var F Z = F (a) = P F [Z < a] = F 1 ( 2 1 ) = F 1 ( 4 3 ) F 1 ( 1 4 )... Theil s index I : F I(F ) = E F [( ZµF ) µ F = E F Z ( )] log ZµF, 13

14 Maximum likelihood [ estimator ] T MLE : E θ F log g (Z T MLE (F )) Model g( θ), Z F T MLE : ML functional T MLE (G( θ)) = θ = 0 Level s functional {F θ θ Θ} H 0 : θ = θ 0 Test statistic T n α 0 = P Fθ0 [ Tn > k 1 α0 ] = α : F α(f ) = P F [ Tn > k 1 α0 ] α(f θ0 ) = α 0 14

15 Influence Function z 1,..., z n iid, z i F T n (z 1,..., z n ) T n (z 1,..., z n ) = T (F n ) T : functional on some subset of all distr. F n : empirical distribution (which assigns prob. 1 n to z 1,..., z n ). Influence Function of T at F : IF (z; T, F ) = lim ε 0 T ((1 ε)f +ε z ) T (F ) ε Hampel (1968), (1974), J. Am. Stat. Ass. z : distr. which puts mass 1 at any point z. Note : IF (z; T, F ) = ε T ((1 ε)f + ε z) ε=0 15

16 Properties IF describes the normalized influence on the estimate of an infinitesimal observation at z. IF is the Gâteaux derivative of T at F, or the integrand in the first term of the von Mises expansion T (G) = T (F ) + IF (z; T, F )d(g F )(z) + O( G F 2 ) Math. treatment (e.g.) : von Mises (1947), Ann. Math. Stat. Fernholz (1983), Springer Serfling (1980), Wiley 16

17 ε-neighborhood P ε (F ) of F : P ε (F ) = {G G = (1 ε)f + εh, H arbitrary } d(g, F ) = sup z G(z) F (z) = ε sup z H(z) F (z) ε. For G P ε (F ) : T (G) = T (F )+ε IF (z; T, F )dh(z)+o(ε 2 ) Bias curve: max bias over ε-neighborhood b(ε; T, F ) = sup T (G) T (F ) G P ε (F ) b(ε; T, F ) }{{} max bias over neighborh. ε γ (T, F ) }{{} gr err sens γ (T, F ) = sup z IF (z; T, F ) IF describes the robustness (stability) properties of T ( ) 17

18 For G = F n (empirical distr.) T n = T (F ) + 1 n n i=1 IF (z i ; T, F ) +... = n (T n T (F )) as N (0, V (T, F )) V (T, F ) = E F [IF (Z; T, F ) IF T (Z; T, F )] E F [IF (Z; T, F )] = 0 IF describes the efficiency properties of T ( ). 18

19 Connection to sensitivity curve SC(z; z 1,..., z n 1, T n ) = n [ T n (z 1,..., z n 1, z) T n 1 (z 1,..., z n 1 ) ] = T ( (1 1 n )F n 1+ 1 n z 1 n ) T (F n 1 ) 19

20 Connection to jackknife T (j) = T n 1 (z 1,..., z j 1, z j+1,..., z n ) Pseudo-values : j = 1, 2,... n T j = nt n (n 1)T (j) ] = T n + (n 1) [T n T (j) }{{} n 1 n SC(z j; z 1,..., z j 1, z j+1,... z n, T n ) n 1 n IF (z j; T, F ) Jackknife estimator : Tukey (1958) T = 1 n n T j j=1 T n + n 1 n j=1 IF (z j ; T, F ) (von Mises expansion; one-step est.) 20

21 The stability analysis by means of the influence function can be performed on any statistical functional e.g. (as)var F T n (as) level of a test = P F [T n > k α ]... 21

22 Breakdown Point The IF shows how an estimator reacts to a small proportion of outliers. Note that the sample mean cannot resist even one outlier! Other estimators can, because their IF is bounded. What is the maximum amount of perturbation they can resist? Breakdown Point Sample Z = (z 1,..., z n ) Statistic T n (Z) bias (m; T n, Z) = sup Z T n (Z ) T n (Z) Z : corrupted sample obtained by replacing any m of the original n data points by arbitrary values. Breakdown point of T n (at Z) : ε n (T n, Z) = min { m n bias (m; T n, Z) = } 22

23 Robustness notions as elementary calculus properties of a function of one argument, namely its continuity, differentiability, and vertical asymptote. The breakdown point tells us up to which distance the linear approximation provided by the influence function is likely to be of value. 23

24 M-estimators z 1,..., z n iid Huber(1964) Parametric model {F θ θ Θ} n M-estimator T n : ψ(z i, T n ) = 0 i=1 M- estimators generalize M LE (for which ψ(z, θ) = score = θ log f θ(z)) To any asymptotically normal estimator, there exists an asymptotically equivalent M-estimator. Properties : IF (z; ψ, F ) = M(ψ, F ) 1 ψ(z, T (F )) n(tn T (F )) D N (0, V (ψ, F )) V (ψ, F ) = M(ψ, F ) 1 Q(ψ, F )M(ψ, F ) T M(ψ, F ) = E F [ θ ψ(z, T (F ))] Q(ψ, F ) = E F [ψ(z, T (F )) ψ(z, T (F )) T ] 24

25 Optimal Robust Estimators z 1,..., z n Parametric model {F θ θ Θ} score s(z, θ) = θ log f θ(z) where ψ A,a c ˆθ c : n i=1 ψ A,a c (z i, ˆθ c ) = 0 (z, θ) = h c {(A[s(z, } θ) a]) h c (z) = z min 1, Huber function A, a : c z E Fθ [ψc A,a (Z, θ)] = 0 E Fθ [ψc A,a (Z, θ)s(z, θ) T ] = I Then ˆθ c is optimal B-robust in the sense that it minimizes trace {V (ψ, F θ )}, subject to a given upper bound c on the IF : sup z IF (z, ψ, F θ c. Hampel(1968) Hampel, Ronchetti, Rousseeuw, Stahel (1986) 25

26 26

27 ψ A,a c can be interpreted as a truncated version of s. Because of the truncation, s must be shifted by a in order to satisfy Fisher consistency at the model. Moreover, the second condition ensures that c is an upper bound on the gross-error sensitivity. Optimal robust estimator defined by : ψc A,a (z, θ) = h c (A[s(z, θ) a]) = A[s(z, θ) a] w c (z, θ) s(z, θ) = θ log f θ(z) w c (z, θ) = min { 1, c A[s(z,θ) a] } 27

28 Choice of the norm : Euclidean norm c dim(θ)/e θ s Self-standardized norm IF = (IF T V 1 IF ) 1/2 ( 2nd equation defining A is different) c dim (θ) 28

29 Sketch of the proof Class of M-est. {ψ} minimize trace {V (ψ, F θ )} subject to IF (z; ψ, F θ ) c z V (ψ, F θ ) = M(ψ, F θ ) 1 Q(ψ, F θ )M(ψ, F θ ) T IF (z; ψ, F θ ) = M(ψ, F θ ) 1 ψ(z, θ) M(ψ, F θ ) = θ [ψ(z, θ)]df θ (z) = ψ(z, θ)s(z, θ) T df θ (z) ψ is determined up to the multipl. with a matrix w.l.g. M(ψ, F θ ) = I We have to solve the following problem : minimize trace ψ(z, θ)ψ(z, θ) T df θ (z) ψ(z, θ) c z subject to M(ψ, F θ ) = I 29

30 A, a defined by the given implicit equations. Then {ψ A[s a]} {ψ A[s a]} T = ψψ T + A [s a][s a] T A T } {{ } fixed A [s a]ψ T ψ[s a] T A T } {{ } } {{ } I I minimize trace ψψ T equivalent to min trace {ψ A[s a]}{ψ A[s a]} T minimize ψ A[s a] 2 subject to ψ c ψ = ψ A,a c = h c (A[s a]) 30

31 Robust Inference robustness of validity The level of the test should be stable under small, arbitrary departures from the distribution under the null hypothesis. robustness of efficiency The test should still have a good power under small, arbitrary departures from the distribution under specified alternatives. Heritier & Ronchetti (1994), J. Am. Stat. Ass. 31

32 z 1,..., z n Parametric model {F θ θ Θ R p } Notation : θ = [ θ(1) θ (2) ] }p q }q A (ij) i, j = 1, 2 corresp. partition of p p matrix Test : H 0 : θ (2) = 0. Classical tests (as. equivalent): Wald, score (Rao), likelihood ratio Robust tests will be based on robust M- estimators T n of θ. 32

33 Classes of tests (i) Wald type test statistic W 2 n = (T n) T (2) [ V (22) (ψ, F θ )] 1 (Tn ) (2) (ii) Score (Rao) type test statistic Z n = 1 n T 0 n n i=1 Rn 2 = ZT n C 1 Z n ψ(z i, Tn 0) (2) is the M-est. in the reduced model n ψ(z i, Tn 0 ) (1) = 0, Tn(2) 0 = 0 i=1 C = M (22 1) V (22) M(22 1) T = as cov (Z n ) M (22 1) = M (22) M (21) M 1 (11) M (12) 33

34 (iii) Likelihood ratio type test statistic S 2 n = 2 n n i=1 [ ] ρ(z i, T n ) ρ(z i, Tn 0 ) ρ(z, 0) = 0, θ ρ(z, θ) = ψ(z, θ) T n, Tn 0 M-est. in full and reduced model resp. 34

35 The three test statistics can be written as functionals of the empirical distribution function. Functionals as quadratic forms U(F ) T U(F ) : U W (F ) = V (ψ, F θ ) 1/2 (22) T (F ) (2) W ald U R (F ) = C(ψ, F θ ) 1/2 Z(F ) Score When M is symmetric positive definite, the likelihood ratio test statistic is as. equivalent to the quadratic form defined by U LR (F ) = M(ψ, F θ ) 1/2 (22.1) T (F ) (2) T (F ) (resp. Z(F )) are the functionals associated with T n (resp. Z n ). 35

36 Asymptotic distribution Under the sequence of alternatives H 1,n : θ (2) = n nw 2 n, nr2 n are asymptotically χ2 q (δ), δ = T V (ψ, F θ0 ) 1 (22). nsn 2 q is asymptotically (λ 1/2 i=1 i N i + µ i ) 2, where N 1,..., N q are iid N(0, 1). 36

37 Robustness H 0 : θ = θ 0, θ 0(2) = 0, θ 0(1) unspecified Contamination : F ε,n = (1 ε n )F θ0 + ε n G Statistical functional U n = U(F (n) ), where F (n) is the empir. distr. function, such that U(F θ0 ) = 0, IF (z; U, F θ0 ) bounded and n(un U(F ε,n )) D N (0, I q ). α(f ) level of the test based on quadratic form nu T n U n when the underlying distr. is F and α(f θ0 ) = α 0 the nominal level. 37

38 Then : lim n α(f ε,n) = α 0 + ε 2 µ + 0(ε 2 ) IF (z; U, F θ0 )dg(z) 2 where µ = δ H q(η 1 α0 ; δ) δ=0, H q ( ; δ) is the cumulative distr. fct. of χ 2 q (δ), and η 1 α0 the 1 α 0 quantile of a central χ 2 q. Similar result for the power. To have a stable level in a neighborhood around the hypothesis, bound the influence function of the functional U(F )! 38

39 Special case : G = z (point mass at z) U = U W, U = U R lim n α(f ε,n) = α 0 +ε 2 µ IF (2) T V 1 (22) IF (2) +0(ε 2 ) }{{} self-stand. infl. where IF (2) = IF (z; T (2), F θ0 ) To obtain robust Wald and score type tests, bound the self-standardized influence function of the est. T (2). Optimality theory. 39

40 General references (books) Huber, P.J.(1981) Robust Statistics, Wiley (paperback 2004) Hampel,F.R., Ronchetti,E.M., Rousseeuw,P.J., Stahel, W.A. (1986) Robust Statistics: The Approach Based on Influence Functions, Wiley (paperback 2005) Maronna R. A., Martin, R.D., Yohai, V. J. (2006) Robust Statistics: Theory, and Methods, Wiley Dell Aquila,R., Ronchetti, E. M. (2007) Robust Statistics and Econometrics with Economic and Financial Applications, Wiley, to appear 40

41 Leading Example (Finance) 41

42 Stylized Facts Volatility clustering Variance of daily price changes varies over time (heteroscedasticity) Model conditional distribution at time t given the information F t 1 up to time t 1 Low signal-to-noise ratio Robustness issue Financial markets are complex; therefore financial models are at best only approximate descriptions of the underlying structure 42

43 Robustness in Finance Two main research fields on robustness: Modelling preferences for robustness and robust decision processes of agents that take into account some forms of model misspecification in their decisions Developing robust statistics for the econometric analysis of financial time series using models that are possibly misspecified 43

44 Wanted Robust procedures that take into account model misspecifications both when determining optimal policies in financial models when estimating the parameter inputs for a financial model 44

45 Estimation and Inference for Single Factor Models Data set of Chan, Karolyi, Longstaff, Sanders (1992): One-month yields based on the average of bid and ask prices for Treasury bills normalized to reflect a standard month of 30.4 days. They are monthly observations covering the period from June 1964 to December 1989, for a total of 307 observations. Ahn and Gao (1999): McCulloch and Kwon (1993) dataset over the period from December 1946 to February for comparability with the Ahn and Gao (1999) study. 45

46 0.16 US interest rates and implied weights 0.14 US interest rate Jan65 Jan70 Jan75 Jan80 Jan Weights Jan65 Jan70 Jan75 Jan80 Jan85 US Short Term Rates 46

47 US rates: first differences r(t) r(t 1) t 47

48 Single Factor Short Rate Models Basic Setting dr t = µ(r t )dt + σ(r t )dw t, where r t is the short rate at time t and (W t ) t 0 is a standard Brownian motion in R 48

49 Chan, Karolyi, Longstaff, Sanders (1992) (CKLS), linear drift term µ(r t ) = α + βr t σ(r t ) = σr γ t Ahn and Gao (1999), quadratic drift term µ(r t ) = α + α 1 r t + α 2 rt 2 σ(r t ) = σr 3/2 t Ait Sahalia (1996) µ(r t ) = α + α 1 r t + α 2 rt 2 + α 3rt 1 σ(r t ) = β 0 + β 1 r t + β 2 r β 3 t 49

50 Alternative Models of the Short Rate M odel α β σ γ restrictions M erton 0 0 (0 attainable) V asicek 0 β < 0 (0 attainable) Cox Ing. Ross 1 2 β < 0 and 2α σ 2 Dothan Geom. B. Mot. 0 1 β < 0, (0 attainable) Brenn. Schw. 1 β < 0 and α > 0 V ariab. Rate (0 attainable) Const. El. V ar. 0 β < 0, (0 attainable) Natural restrictions have to be imposed on the parameter values to ensure that the drift is meanreverting at high interest rate values (infinity not attainable) and zero is unattainable; see Aït-Sahalia (1996). 50

51 GMM Estimation of CKLS Models Crude discretization r t r t 1 = α + βr t 1 + ɛ t where E(ɛ t ) = 0 and E(ɛ 2 t ) = σ2 r 2γ t 1 Orthogonality conditions used in CKLS: E(ɛ t ) = 0 E(ɛ t r t 1 ) = 0 E(η t ) = 0 E(η t r t 1 ) = 0 where η t = ɛ 2 t σ2 r 2γ t 1 51

52 GMM Estimators X := (X n ) n N stationary ergodic sequence defined on an underlying probability space (Ω, F, IP). Parametric model P := {P θ, θ Θ}, θ Θ R p. True parameter vector: θ 0. 52

53 Method of moments: 1 n 1 n n X i = E θ X 1 = g 1 (θ) i=1 n i=1 Xi 2 = E θ X1 2 = g 2(θ)... Equivalently: n [X i g 1 (θ)] = 0 i=1 n [Xi 2 g 2(θ)] = 0 i=1... i.e. E θ h(x 1 ; θ) = 0 where h(x 1 ; θ) = (h 1 (X 1 ; θ), h 2 (X 1 ; θ),... ) Orthogonality conditions 53

54 GMM: Estimate indirectly some function a : P A := a(p) R k of parameters of interest by introducing a function h : R N A R H enforcing a set of orthogonality conditions E θ0 h(x 1 ; a(p θ0 )) = 0, (1) on the structure of the underlying model. 54

55 W := (W n ) n N sequence of weighting symmetric positive definite matrices converging a.s to W 0, the inverse of the covariance matrix of h(x 1, a(p θ0 )), Generalized method of moments estimator (GMME) associated with W: (ã(p θn )) n N solution to (Hansen, 1982) min a A 1 n (= min a A n i=1 h (X i ; a)w n 1 n n i=1 h(x i ; a) E θ n h (X 1 ; a)w n E θn h(x 1 ; a)) where P θn := 1 n δxi is the empirical distribution of X 1,.., X n and δ x denotes the point mass distribution at x R N ). 55

56 Under appropriate regularity conditions the GMME exists, is strongly consistent and asymptotically normally distributed with asymptotic covariance matrix h (X 1 ; a(p θ0 )) Σ θ0 (W 0 ) = [E θ0 W 0 a h(x 1 ; a(p θ0 )) E θ0 a ] 1. 56

57 In our case: X = (r t 1, r t ) t 0 Orthogonality function h: h(x, y; α, β, σ, γ) = y x α βx (y x α βx)x (y x α βx) 2 σ 2 x 2γ ((y x α βx) 2 σ 2 x 2γ )x. 57

58 Classical GMM Estimates of Alternative Models for the Short-Term Interest Rate The parameters are estimated by the classical GMM induced by the original orthogonality function h t-statistics are in parentheses. The values of Hansen s statistics (ξ for short) are reported with p-values in parentheses. 58

59 Classical GMM Model α β σ γ ξ Unres (1.85) (-1.55) (1.56) (5.95) Merton (1.44) (14.54) (0.034) Vasicek (0.33) (-0.04) (14.33) (0.009) CIR (0.67) (-0.36) (15.28) (0.027) Dothan (15.94) (0.133) GBM (1.50) (16.07) (0.206) BS (1.24) (-0.92) (16.18) (0.137) VR (15.66) (0.098) CEV (1.53) (1.18) (3.59) (0.084) Classical GMM Estimates 59

60 Classical Analysis Models that allow for values of γ 1 are not rejected using Hansen s statistic while models where γ [0, 1) are. The estimates γ in the corresponding models are strongly significant. 60

61 How stable is this analysis? How reliable are these conclusions? 61

62 Sensitivity analysis of the p value of Hansen s statistic Change one observation: ɛ=1/306=0.3%. Vary the value corresponding to March 1980 (observed value: ) from to by steps of size Variability of the short-term rate changes around March 1980 is very high: a change from a 15% to a 9.5% interest rate level just before March 1980 the magnitude of the sensitivity analysis seems to be realistic with respect to the structure of the short-rate observations over this particular period. 62

63 Sensitivity Analysis Merton, Vasicek, Cox et al., Dothan 63

64 Classical Hansen s test Very steep p values curves E.g. Merton and Cox et al. models: a change of the value of the observation of basis points (100 basis points = 0.01) is sufficient for obtaining p-values not rejecting the model specification at a 5% significance value, as opposed to the results for the uncontaminated model. This analysis shows that it is very difficult to distinguish by means of the classical Hansen s test the specification properties of the models, even when only a single observation is changed in the data. 64

65 Alternative Analysis: Robust Statistics 65

66 Local stability properties of the GMME a: IF (x; a, P θ0 ) = Σ θ0 (W 0 )E θ0 h (X 1 ;a(p θ0 )) a W 0 h(x; a(p θ0 )). The IF of a GMME is proportional to the orthogonality function of the model and is bounded if and only if the function inducing the orthogonality conditions of the model is bounded. Examples of GMME with unbounded orthogonality conditions: linear and nonlinear LS instrumental variables estimators 66

67 In our model, the orthogonality function h is unbounded in x and y The classical GMM estimator and tests corresponding to this orthogonality function are therefore not robust. 67

68 Function h for Vasiceck, Cox Ing. Ross, Brenn. Schw., Variab. Rate model 68

69 Whereas the original moment conditions h are usually dictated by economic theory, the truncated version h A,τ c takes into account the realistic case that only the majority of the data can reasonably fit the original moment conditions. The weights w c (A[h(x; a) τ]) assigned to each observation x can be used to detect outlying points. The bound imposed on the self-standardized influence of any GMME cannot be chosen arbitrarily small. Indeed, c H. No further model assumptions are needed in order to do this construction. 69

70 Robust Inference with GMM Estimators Hansen s test: For the case r := H k > 0 the GMM specification test is a test of the overidentifying restrictions implied by the null hypothesis (1). It is constructed with a functional ξ G (P θn ) := U G (P θn )U G (P θn ), where U G (P θn ) := W 1 20 E θ n h(x 1 ; ã(p θn )). Can also consider GMM versions of the classical ML-tests (Wald, score, likelihood-ratio type) to test a null hypothesis {a A g(a) = 0} defined using a smooth function g : A R r. 70

71 Theorem: Let ã be a GMME induced by an orthogonality function h and denote by α : dom(α) R the level functional of the test. Let further (P ɛ,n,g ) n N be a sequence of ɛ, n, G contaminations of the underlying null distribution P θ0, each of them belonging to a corresponding neighborhood { U ɛ,n (P θ0 ) := P ɛ,n,g := (1 ɛ )P θ0 + ɛ } G. n n 71

72 The following two statements then hold: The bias of the level functional α under P ɛ,n,δx contaminations is bounded if and only if h is bounded. If h is bounded, the bias of α is uniformly bounded by the inequality: lim n where α(p ɛ,n,g) α 0 ɛ 2 µ sup x ( h(x, a(p θ0 )) W0 ) 2 + o(ɛ 2 ), µ = β H r(η 1 α0 ; β) β=0 = (1 α 0) H r+2(η 1 α0 ; 0), H r ( ; β) is the cumulative distribution function of a noncentral X 2 (r; β) distribution with r degrees of freedom and noncentrality parameter β 0, and η 1 α0 is the 1 α 0 quantile of a X 2 (r; 0). 72

73 Maximal asymptotic bias of the level of tests that are derived from the robust GMME can be bounded by lim n α(p ɛ,n,g) α 0 µ (ɛc) 2 + o(ɛ 2 ). Can choose c in dependence of both the maximal amount of contamination (ɛ) expected by the researcher given some prior information on the nature of the data and the maximal bias for the level (maxbias), he (she) is going to accept from his (her) research strategy, i.e. c = 1 ɛ maxbias/µ. For α 0 = 5%, this can be approximated by c 3 ɛ r0.3 (maxbias) 1/2. For instance, for ɛ = 5% and maxbias = 0.5%, we obtain values of c which vary from 4 for r = 1 to 6 for r = 5. Similar result for the power. 73

74 Robust Analysis (c = 6) Robust GMM specifications tests reject practically all constrained models at a 5% significance level. Some observations are identified as potentially influential (e.g. March 1980). 74

75 This analysis shows that it is very difficult to distinguish by means of the classical Hansen s test the specification properties of the models, even when only a single observation is changed in the data. A robust Hansen s test should be used if one is interested in obtaining decisions that are not primarly determined by a few observations in the sample. Models with γ 1 cannot be motivated, when using robust model selection strategies. 75

76 Influential Observations Some observations are identified as potentially influential. A clustering of influential observations is visible in the period. Other influential observations are found in the (the first OPEC crisis) and after the stock market crash in The period is well-known to coincide with a temporary change in the monetary policy of the Federal Reserve. The influential observations coincide with the typical regimes found in regime switching studies, e.g. Gray(1996) who identified similar regimes at exactly those points where the infuential observations are found. 76

77 Extensions of the CKLS Models Misspecification of the CKLS models Need sophisticated multi-factor models or more complex single factor models? Regime-switching models (e.g. as proposed in Cai (1994), Gray (1996), Ang and Bekaert (2000b)) Models allowing for nonlinearities in the drift and diffusion term (e.g. Ait Sahalia (1996), Stanton (1997), Jiang (1998) and Ahn and Gao (1999)) Models adding GARCH and similar features (e.g. Brenner, Harjes and Kroner (1996), Koedijk, Nissen, Schotman and Wolff (1997) and Ball and Torous (1999)) 77

78 We re-analyze the Ahn and Gao (1999) model which adds a quadratic drift term. Dataset spanning the period Dec February Model α 1 α 2 α 3 σ ξ d.f Cl. AG ( 1.08) (1.73) ( 1.80) (19.93) (0.103) Rob. AG ( 0.60) (1.50) ( 1.50) (26.17) (0.000) 78

79 The classical Hansen test does not reject the AG model, while the robust test does. The classical Hansen test is still highly sensitive with respect to a contamination of a single point. There are still influential observations clustering in the period. Conclusion It seems that adding a nonlinearity to the drift, does still not suffice to explain the period. Similar conlusions are found for models allowing parameter shifts (e.g. Bliss and Smith(1999)). 79

80 Details can be found in : Ronchetti, E. and Trojani, F. (2001) Robust Inference with GMM Estimators Journal of Econometrics, 101, Dell Aquila, R., Ronchetti, E., and Trojani, F. (2003) Robust GMM Analysis of Models for the Short Rate Process Journal of Empirical Finance, 10,

81 Messages There exist robust statistical procedures which complement classical estimators and tests for general parametric models. Whenever you can do a likelihood analysis, you can do a robust analysis. 81