Quantiles, Expectiles and Splines

Transcription

1 Quantiles, Expectiles and Splines Andrew Harvey University of Cambridge December 2007 Harvey (University of Cambridge) QES December / 40

2 Introduction The movements in a time series may be described by time-varying quantiles. These may be estimated non-parametrically by tting a simple moving average or a more elaborate kernel. An alternative approach is to formulate a partial model, the role of which is to focus attention on some particular feature - here a quantile - so as to provide a (usually nonlinear) weighting of the observations that will extract that feature by taking account of the dynamic properties of the series. Time-varying quantiles can be tted to a sequence of observations by setting up a state space model and iteratively applying a suitably modi ed signal extraction algorithm. Satisfy the de ning property of xed quantiles in having the appropriate number of observations above and below. Harvey (University of Cambridge) QES December / 40

3 GM Q(05) Q(95) Q(25) Q(75) Figure: Quantiles tted to GM returns Harvey (University of Cambridge) QES December / 40

4 Expectiles are similar to quantiles except that they are de ned by tail expectations; see Newey and Powell (1987). Time-varying expectiles can be estimated by a state space algorithm. This is similar to the algorithm used for quantiles, but estimation is more straightforward and much quicker. Harvey (University of Cambridge) QES December / 40

5 Quantiles and expectiles Let ξ(τ) - or, when there is no risk of confusion, ξ - denote the τ th quantile. The probability that an observation is less than ξ(τ) is τ, where 0 < τ < 1. Given a set of T observations, y t, t = 1,.., T, the sample quantile, eξ(τ), can be obtained as the solution to minimizing S τ = T ρ τ (y t t=1 ξ) = (τ 1)(y t ξ) + τ(y t ξ) (1) y t <ξ y t ξ with respect to ξ, where ρ τ (.) is the check function, de ned for quantiles as ρ τ (y t ξ) = (τ I (y t ξ < 0)) (y t ξ) (2) and I (.) is the indicator function. Harvey (University of Cambridge) QES December / 40

6 Expectiles, denoted µ(ω), 0 < ω < 1, are similar to quantiles but they are determined by tail expectations rather than tail probabilities. For a given value of ω, the sample expectile, eµ(ω), is obtained by minimizing the asymmetric least squares function, S ω = ρ ω (y t µ(ω)) = jω I (y t µ(ω) < 0)j (y t µ(ω)) 2, (3) with respect to µ(ω). Di erentiating S ω and dividing by minus two gives T jω I (y t µ(ω) < 0)j (y t µ (ω)) (4) t=1 The sample expectile, eµ(ω), is the value of µ(ω) that makes (4) equal to zero. Setting ω = 0.5 gives the mean, that is eµ(0.5) = y. For other ω it is necessary to iterate. Harvey (University of Cambridge) QES December / 40

7 Signal extraction A framework for estimating time-varying quantiles, ξ t (τ), can be set up by assuming that they are generated by stochastic processes and are connected to the observations through a measurement equation y t = ξ t (τ) + ε t (τ), t = 1,..., T, (5) where Pr(ε t (τ) < 0) = τ with 0 < τ < 1. The disturbances, ε t (τ), are assumed to be serially independent and independent of ξ t (τ). The problem is then one of signal extraction. The assumption that the quantile or expectile follows a stochastic process can be regarded as a device for inducing local weighting of the observations. One possibility is a random walk, ξ t (τ) = ξ t 1 (τ) + η t (τ), η t (τ) v IID(0, σ 2 η(τ) ), (6) Harvey (University of Cambridge) QES December / 40

8 A smoother quantile can be extracted by a local linear trend ξ t = ξ t 1 + β t 1 + η t (7) β t = β t 1 + ζ t where β t is the slope and ζ t is IID(0, σ 2 ζ ). It is well known that in a Gaussian model setting Var(η t ) = σ 2 ζ /3 and Cov(η t, ζ t ) = σ2 ζ /2 results in the smoothed estimates being a cubic spline. The model for expectiles is set up in a similar way with (5) replaced by y t = µ t (ω) + ε t (ω). Harvey (University of Cambridge) QES December / 40

9 Derivation and properties The proposed solution minimizes t ρ τ (y t ξ t ) subject to a set of constraints imposed by the time series model for the quantile. Suppose this model is a rst-order autoregression,. The criterion function is then J = T ρ τ (y t ξ t ) t=1 ω 1 (1 φ 2 )(ξ 1 ξ ) 2 2 σ 2 η T 1 η 2 2 t t=2 σ 2. (8) η where ω is a scaling constant. Given the observations, the estimated time-varying quantiles, eξ 1,.., eξ T, are the values that maximise J. Harvey (University of Cambridge) QES December / 40

10 In a classical signal extraction framework, y t = µ t + ε t, t = 1,..., T (9) where µ t is a Gaussian stochastic process and ε t is NID(0, σ 2 ). If J is rede ned with µ t in place of ξ t and ρ τ (y t ξ t )/ω replaced by (y t µ t ) 2 /2σ 2, it can be interpreted as the logarithm of the joint density of the observations and µ t 0 s. Di erentiating with respect to µ t, t = 1,..., T, setting to zero and solving gives the modes, eµ t, t = 1,..., T, of the conditional distributions of the µ t 0 s. For a multivariate Gaussian distribution these are the conditional expectations, which by de nition are the smoothed (minimum mean square error) estimators. Harvey (University of Cambridge) QES December / 40

11 Returning to the quantiles and di erentiating J with respect to ξ t gives J = 1 ξ t ω IQ (y t ξ t ) + φξ t 1 (1 + φ 2 )ξ t + φξ t+1 + (1 φ) 2 ξ σ 2, η (10) for t = 2,..., T 1, (modi ed at the endpoints), where IQ (y t ξ t ) is de ned as in (24). For t = 2,.., T 1, setting J/ ξ t to zero gives an equation that is satis ed by the estimated quantiles, eξ t, eξ t 1 and eξ t+1, and similarly for t = 1 and T. If a solution is located on an observation, that is eξ t = y t, then IQ (y t ξ t ) is not de ned as the check function is not di erentiable at zero. Harvey (University of Cambridge) QES December / 40

12 In a Gaussian model, (9), a little algebra leads to the classic Wiener-Kolmogorov (WK) formula for a doubly in nite sample. For the AR(1) model eµ t = µ + g g y (y t µ) where µ = E (µ t ), g = σ 2 η/((1 φl)(1 φl 1 )) is the autocovariance generating function (ACGF) of µ t, L is the lag operator, and g y = g + σ 2 ε. The WK formula has the attraction that, for simple models, g y can be written in terms of the reduced form and g/g y can be expanded to give an explicit expression for the weights. Here the reduced form of y t is an ARMA(1, 1) model and q µ θ eµ t = µ + φ(1 θ 2 θ jjj (y t+j µ) (11) ) j= where q µ = σ 2 η/σ 2 ε and h θ = (q µ φ 2 )/2φ q µ φ 2 i 2 1/2 4φ 2 /2φ. This expression is still valid for the random walk except that µ disappears because the weights sum to one. Harvey (University of Cambridge) QES December / 40

13 In order to proceed in a similar way with quantiles, we need to take account of corner solutions by de ning the corresponding IQ 0 ts as the values that give equality of the associated derivative of J. Then we can set J/ ξ t in (10) equal to zero to give eξ t ξ g = 1 ω IQ(y t eξ t ) (12) for a doubly in nite sample with ξ known. Using the lag operator yields eξ t = ξ + σ2 η ω j= φ jjj 1 φ 2 IQ(y t+j eξ t+j ) (13) Note that if the observations are multiplied by a constant, then the quantile is multiplied by the same constant, as are ω and σ 2 η. By de ning the quasi signal-noise ratio as q = σ 2 η/ω 2, it remains scale invariant. Harvey (University of Cambridge) QES December / 40

14 An expression for extracting quantiles that has a similar form to (11) can be obtained by adding (eξ t ξ )/ω 2 to both sides of (12) and re-arranging to give eξ t = ξ + g h e gy ω ξ t ξ + ωiq(y t eξ t )i (14) where gy ω = g + ω 2.This is not an ACGF but it can be treated as though it were. Thus we obtain eξ t = ξ + q τ θ φ(1 θ 2 θ jjj [eξ t+j ξ + ωiq(y t+j eξ t+j )] (15) ) j= where θ is as de ned for (11) but with q µ replaced by q = σ 2 η/ω 2. For the random walk, the weights sum to one, so ξ drops out giving eξ t = 1 θ 1 + θ θ jjj [eξ t+j + ωiq(y t+j eξ t+j )] (16) j= Harvey (University of Cambridge) QES December / 40

15 Non-parametric smoothing Changing quantiles may be estimated non-parametrically, as in Yu and Jones (1998), by minimising a local check function to give an estimator, bξ t, that satis es h K (j/h)iq(y t+j bξ t ) = 0 (17) j= h where K (.) is a weighting kernel, h is a bandwidth.with IQ(y t+j bξ t ) de ned appropriately if y t+j = bξ t. Proposition If the same kernel and bandwidth are used for di erent quantiles, they cannot cross (though they may touch). Harvey (University of Cambridge) QES December / 40

16 It is interesting to compare the above weighting scheme with the one implied by the random walk model. If eξ t+j in (16) is constant, it satis es (17) with K (j/h) replaced by θ jjj so giving an (in nite) exponential decay. The time series model determines the shape of the kernel while the q parameter plays the same role as the bandwidth. An important advantage of the more model-based approach for forecasting is that it automatically determines a weighting pattern at the end of the sample that is consistent with the one in the middle. Harvey (University of Cambridge) QES December / 40

17 State Space Form The state space form (SSF) for a univariate time series is: y t = z 0 t α t + ε t, Var(ε t ) = σ 2 t, t = 1,..., T (18) α t = T t α t 1 +η t, Var(η t ) = Q t where α t is an m 1 state vector, z t is a non-stochastic m 1 vector, σ 2 t is a non-negative scalar, T t is an m m non-stochastic transition matrix and Q t is an m m covariance matrix. The speci cation is completed by assuming that α 1 has mean a 1j0 and covariance matrix P 1j0 and that the serially independent disturbances ε t and η t are independent of each other and of the initial state. Harvey (University of Cambridge) QES December / 40

18 Consider the criterion function J = T ht 1 ρ(y t zt 0 α t ) t=1 T 1 2 η 0 t Q 1 1 t η t t=2 2 (α 1 a 1j0 ) 0 P 1 1j0 (α 1 a 1j0 ), (19) where ρ(y t zt 0 α t ) is as in (2) or (3), with zt 0 α t equal to ξ t (τ) or µ t (ω), Q t and P 1j0 are are assumed positive de nite matrices as in (18) and h t is a non-stochastic sequence of postive scalars. Suppose that the initial state and the ηt 0 s are normally distributed. For a Gaussian model of the form (18), logarithm of the joint density of the observations and the states is, ignoring irrelevant terms, given by J with ρ(y t zt 0 α t ) = (y t µ t (0.5)) 2 and h t = 2σ 2 t. Di erentiating J with respect to to each element of α t gives a set of equations, which, when set to zero and solved gives the minimum mean square error estimates of α t. These they may be computed e ciently by the Kalman lter and associated smoother (KFS) as described in Durbin and Koopman (2001, pp ). If all the elements in the state are nonstationary and given a di use prior, the last term in J disappears. An algorithm is available as a subroutine in the SsfPack set of programs within Ox; see Koopman et al (1999). Harvey (University of Cambridge) QES December / 40

19 We can think of (19) as a criterion function that provides the basis for computing a quantile or expectile subject to a set of constraints imposed by the time series model for the quantile or expectile. For expectiles di erentiating J gives J = z 1 (2/h 1 )IE (y 1 z α 1α 0 1 ) P 1 1j0 (α 1 a 1j0 ) + T2Q (α 2 T 2 α 1 ) 1 J = z t (2/h t )IE (y t zt 0 α t ) Qt 1 (α t T t α t 1 ) + T α t+1q 0 t+1 1 (α t+1 T t t=2,..., T 1, J = z T (2/h T )IE (y T zt 0 α α T ) QT 1 (α T T T α T 1 ). T where IE (y t µ t (ω)) = jω I (y t µ t (ω) < 0)j (y t µ t (ω)), t = 1,..., T. (21) The smoothed estimates, eα t, satisfy the equations obtained by setting these derivatives equal to zero. Harvey (University of Cambridge) QES December / 40

20 Let h t = g t /κ, where κ is a constant, the interpretation of which will become apparent. For any expectile, adding and subtracting z t gt 1 zt 0 α t to the equations in (20) allows the rst term to be written as z t g 1 t [z 0 t α t + 2κIE y t z 0 t α t ] zt g 1 t z 0 t α t, t = 1,..., T. (22) This suggests that we set up an iterative procedure in which the estimate of the state at the i-th iteration, eα (i) t, is computed from the KFS applied to a set of synthetic observations constructed as (i 1) by t = ztbα 0 (i 1) t + 2κIE y t ztbα 0 (i 1) t. (23) The iterations are carried out until the bα (i)0 t s converge whereupon eµ t (ω) = zteα 0 t. Harvey (University of Cambridge) QES December / 40

21 For quantiles, the rst term in each of the three equations of (20) is given by z t ht 1 IQ(y t zt 0 α t ), where IQ(y t ξ t (τ)) = τ 1, if yt < ξ t (τ) τ, if y t > ξ t (τ) t = 1,..., T, (24) and the synthetic observations in the KFS are (j 1) by t = ztbα 0 (j 1) + κiq y t ztbα 0 (j 1), t = 1,..., T (25) t However, the possibility of a solution where the estimated quantile passes through an observation means that the algorithm has to be modi ed somewhat; see De Rossi and Harvey (2006). t Harvey (University of Cambridge) QES December / 40

22 Estimates of time-varying quantiles and expectiles obtained from the smoothing equations of the previous sub-section can be shown to satisfy properties that generalize the de ning characteristics of xed quantiles and expectiles. It is assumed that the state has been arranged so that the rst element represents the level and that (without loss of generality) the rst element in z t has been set to unity. Let the rst derivative, with respect to α t, of the second term of J be written j2 0 = T t=1 A t α t, where the Ats 0 are m m matrices. Lemma For a model in SSF with a di use prior on the initial state, a su cient condition for the rst element in the vector j2 0 to be zero is that the rst column of T t I consists of zeroes for all t = 2,..., T. Harvey (University of Cambridge) QES December / 40

23 If h t is time-invariant and the conditions of the Lemma hold, the estimated quantiles satisfy the fundamental property of sample time-varying quantiles, namely that the number of observations that are less than the corresponding quantile, that is y t < eξ t (τ), is no more than [T τ] while the number greater is no more than [T (1 τ)]. Harvey (University of Cambridge) QES December / 40

24 Proposition If the distribution of y is time invariant when adjusted for changes in location and scale, and is continuous with nite mean, the population τ quantiles and ω expectiles coincide for ω satisfying ω = R ξ(τ) (y ξ(τ))df (y) R ξ(τ) (y ξ(τ))df (y) R ξ(τ) (y ξ(τ))df (y) where F (y) is the cdf of y. Assuming this to be the case, eµ t (ω) is an estimator of the eτ quantile, ξ t (eτ), where eτ is de ned as the proportion of observations for which y t < eµ t (ω), t = 1,..., T. When used in this way we will denote the estimator eµ t (ω) as eµ t (eτ). However, it will not, in general, coincide with the time-varying eτ quantile estimated directly since it weights the observations di erently. In particular, it is unlikely to pass through any observations. Harvey (University of Cambridge) QES December / 40

25 Parameter estimation The smoothing algorithms depend on parameters that can be estimated by cross validation. For time-varying quantiles, the function to be minimized is T CV (τ) = ρ τ (y t eξ ( t) t ) (26) t=1 where eξ ( t) t is the smoothed value at time t when y t is dropped; see De Rossi and Harvey (2006). A similar criterion, CV (ω), may be used for expectiles. In a time invariant model with quantiles or expectiles following a random walk, Q t is a scalar equal to σ 2 η(τ) or σ 2 η(ω). We would like a suitable parameterization in terms of a quasi signal-noise ratio that is scale invariant. For the mean it will be recalled that h t = 2σ 2 t and so in a time invariant model the usual signal-noise ratio, σ 2 η/σ 2, implies that g t = σ 2 and κ = 0.5 in (22). A similar normalization can be applied for other expectiles so the quasi signal-noise ratios are q ω = σ 2 η(ω)/σ 2. Hence the iterations are based on (22) with g t set to one and κ = 0.5. For Harvey quantiles, (University of Cambridge) σ 2 /h is not scale invariant. QES We therefore consider Decemberthe 2007 quasi 25 / 40

26 Nonparametric regression with cubic splines A slowly changing quantile can be estimated by minimizing the criterion function ρ τ fy t ξ t g subject to smoothness constraints. The cubic spline solution seeks to do this by nding a solution to Z min ρ τ fy t ξ(x t )g + λ 2 fξ 00 (x)g 2 dx (27) T t=1 where ξ(x) is a continuous function with square integrable second derivative, 0 x T and x t = t. The parameter λ 2 controls the smoothness of the spline. We show the same cubic spline is obtained by quantile signal extraction. The SSF allows irregularly spaced observations to be handled since it can deal with systems that are not time invariant. The form of such systems is the implied discrete time formulation of a continuous time model. This generalisation allows the handling of nonparametric quantile and expectile regression by cubic splines when there is only one explanatory variable. The observations, which may be from a cross-section, are arranged so that the values of the explanatory variable are in ascending order. Harvey (University of Cambridge) QES December / 40

27 Bosch, Ye and Woodworth (1995) propose a solution to cubic spline quantile regression that uses quadratic programming. Unfortunately this necessitates the repeated inversion of large matrices of dimension up to 4T 4T. This is very time consuming. Our signal extraction appears to be much faster (and more general) and makes estimation of the smoothing parameter (quasi signal-noise ratio) a feasible proposition. The fundamental property of quantiles continues to hold with irregularly spaced observations. All that happens is that the SSF becomes time-varying. If there are multiple observations at some points then n, the total number of observations, replaces T, number of distinct points, in the summation. Harvey (University of Cambridge) QES December / 40

28 An example of cubic spline regression is provided by the motorcycle data, which records measurements of the acceleration, in milliseconds, of the head of a dummy in motorcycle crash tests. The observations are irregularly spaced and at some time points there are multiple observations. Harvey and Koopman (2000) highlight the stochastic trend connection. Figure 2 shows the cubic spline expectiles obtained using the value of σ 2 ζ /σ2 = 0.07 computed by CV for the mean. The graph gives a clear visual impression of the movements in level and dispersion. If we count the number of observations below each expectile, they can be interpreted as quantiles if we are prepared to assume that the shape of the distribution is time invariant. Harvey (University of Cambridge) QES December / 40

29 Obs time ω =0.25 q=0.07 ω =0.75 q=0.07 time time ω =0.05 q=0.07 ω =0.5 q=0.07 ω =0.95 q= Figure: Cubic spline expectiles tted to the motorcycle data. The parameter q µ is estimated by cross validation. Harvey (University of Cambridge) QES December / 40

30 Conclusions ( so far) Time-varying quantiles and expectiles are GOOD THINGS and provide information on various aspects of a time series, such as dispersion and asymmetry. Time-varying quantiles can be tted iteratively applying a suitably modi ed state space signal extraction algorithm. The algorithm for time-varying expectiles is much faster as there is no need to take account of corner solutions. Satisfy the de ning property of xed quantiles in having the appropriate number of observations above and below it, while expectiles satisfy properties that generalize the moment conditions associated with xed expectiles. Our model-based approach means that time-varying quantiles and expectiles can be used for forecasting. As such they o er an alternative to methods such as those in Engle and Manganelli (2004) and Granger and Sin (2000), that are based on conditional autoregressive models. Equivalent to tting a cubic spline. Because the state space form can handle irregularly spaced observations, the proposed algorithms are easily adapted to provide a viable means of computing spline-based Harvey (University of Cambridge) QES December / 40

31 But estimating the signal noise ratio for quantiles takes a long time Harvey (University of Cambridge) QES December / 40

32 The dual: tracking the distribution Harvey (University of Cambridge) QES December / 40

33 A di erent, but complementary approach, is to pre-assign a value ξ and then construct a binary series, I (y t ξ). At any point in time E (I (y t ξ) = τ t, t = 1,..., T (28) If the quantiles are xed and known, setting ξ = ξ(τ) means that E (I t ) = τ. If the distribution changes, a discount parameter, ω, leads to where eτ t+1jt = a t+1jt / a t+1jt + b t+1jt, t = 1,..., T, a t+1jt = ωa tjt 1 + ωi t, t = 1,..., T (29) b t+1jt = ωb tjt 1 + ω(1 I t ) (30) with a 1j0 = b 1j0 = 1. Note that eτ t+1jt = Pr (I t+1 (τ) = 1jI j (τ), j = 1,..., t) = eτ tjt. Harvey (University of Cambridge) QES December / 40

34 The log-likelihood function is log L(ω) = T t=1 fi t ln eτ tjt 1 + (1 I t (τ)) ln(1 eτ tjt 1 )g, (31) where eτ 1j0 = 1/2. The ltered estimates are an EWMA in the indicators, that is eτ t+1jt = eτ tjt = t 1 Σ j=0 ωj I t j + ω t t 1 Σ j=0 ωj + 2ω t ' (1 ω) t 1 Σ j=0 ωj I t j with the terms in ω t not present if a 1j0 = b 1j0 = 0. It is reasonable to construct a two-sided smoothed estimator of the same form. This may be done using the smoother for a Gaussian random walk plus noise with q = (1 ω) 2 /ω. In the middle of a large sample eτ tjt = 1 ω 1 + ω Σ j ωj I t j (32) Harvey (University of Cambridge) QES December / 40

35 This is also the same problem as forecasting the sign of a variable, or more generally y t ξ, where ξ is some pre-assigned value. Indeed it can be used for any binary series eg I t = 1 if Cambridge win the boat race. Here ω = (and q = 0.017) 1.0 boat Level Harvey (University of Cambridge) QES December / 40

36 By forming a whole grid a changing distribution function can be estimated and tracked. There are N categories de ned by the boundaries, ξ 1,..., ξ N 1,. Let I t,j = 1 if y t ξ j, j = 1,..., N 1 and zero otherwise, ie I (y t ξ j ), and de ne I N,t = 1 and I 0,t = 0. Then a j,t+1jt = ω j a j,tjt 1 + ω j I j,t, j = 1,..., N, t = 1,..., T (33) with a j,1j0 = 1, j = 1,..., N, and eτ j,t+1jt = eτ j,tjt = Σ j i=1 a i,t+1jt /Σ N j=1 a j,t+1jt. The eτ 0 j,tjts lie in the range [0,1] by construction and, of course, eτ N,tjt = 1. When the ω 0 s are the same, the estimated series of probabilities, eτ j,tjt, cannot cross. Harvey (University of Cambridge) QES December / 40

37 The Dirichlet-multinomial or Polya predictive distribution leads to the log-likelihood function where ln L(ω 1,.., ω N ) = T ln `t t=1 ln `t = ln Γ(Σ N j=1a j,tjt 1 ) Σ N j=1 lnfγ(i j I j 1 + a j,tjt 1 )/Γ(a j,tjt 1 )g, (34) The binary likelihood, (31), is obtained when N = 2. Figure 4 shows a graph of estimates of the discount factors for GM with ξ 1,..., ξ N 1 set to the 5%, 10%,..., 95% quantiles from the raw (empirical) distribution. As expected there is very little discounting at and around the centre of the distribution. Quadratic tted by OLS. This suggests that a general approach for estimating ω 0 s as a slowly changing function of τ is to maximize the likelihood, (34), wrt a, b and c in the quadratic ω(τ) = a + b + cτ 2. May set ω = 1 and/or impose symmetry. Harvey (University of Cambridge) QES December / 40

38 Figure: GM - quadratic tted to estimated discount factors Harvey (University of Cambridge) QES December / 40

39 Time-varying quantiles may be extracted from the eτ 0 j,tjts as follows. Suppose an estimate of ξ t (τ) is required, and that eτ k 1,tjt τ eτ k,tjt for some k = 1,.., N. Linear interpolation then yields bξ t (τ) = τ eτ k 1,tjt eτ k,tjt eτ k 1,tjt fξ k ξ k 1 g + ξ k 1, t = 1,..., T What is the relationship between these estimates and those obtained directly? If eξ t+j (τ) = ξ(τ) is constant, then IQ(y t+j ξ(τ)) = τ I t+j (y t ξ(τ)), and so the condition j= θ jjj IQ(y t+j ξ(τ)) = 0 implied by tting the time-varying quantile implies in turn that j= θ jjj I t+j (y t ξ(τ)) = τ. A comparison with (32) shows that ω = θ. Harvey (University of Cambridge) QES December / 40

40 So tracking the distribution may be better - but really only viable when there is no trend in mean or variance. *********** THE END ********************* Harvey (University of Cambridge) QES December / 40