E cient Likelihood Evaluation of State-Space Representations

Transcription

1 E cient Likelihood Evaluation of State-Space Representations David N. DeJong, Roman Liesenfeld, Guilherme V. Moura, Jean-Francois Richard, Hariharan Dharmarajan University of Pittsburgh Universität Kiel VU University Bates White LLC August 2010 DLMRD () EIS Filtering August / 41

2 Objective Likelihood evaluation and ltering for state-space representations featuring departures from: Linearity Normality DLMRD () EIS Filtering August / 41

3 Motivation In the linear/normal case, exact likelihood evaluations are available analytically via the Kalman lter. However, linear/normal characterizations of economic phenomenon are often inadequate or inappropriate, thus necessitating the implementation of numerical approximation techniques known as sequential Monte Carlo (SMC) methods. Example: In working with DSGE models, linear approximations are problematic for conducting likelihood analysis (Fernandez-Villaverde and Rubio-Ramirez, 2005 JAE; 2009 REStud) DLMRD () EIS Filtering August / 41

4 Sketch of Literature on SMC Methods SMC methods employ importance sampling densities to construct numerical approximations of integrals that arise in pursuit of likelihood evaluation and ltering. Typically, importance samplers are based on discrete approximations of ltering densities. The individual elements of these samplers are known as particles; the approximations they represent collectively are known as particle swarms. DLMRD () EIS Filtering August / 41

5 SMC Methods, cont. Baseline methods construct time-t approximations of ltering densities absent information on the time-t observables y t. Such methods are termed as being unadapted. Leading examples include Handschin and Mayne, 1969 Intl. J. of Control; Handschin, 1970 Automatica; Gordon, Salmond, and Smith, 1993 IEEE Proceedings. Baseline methods are relatively easy to implement, and yield unbiased estimates; however, they can be numerically ine cient. Re nements seek to achieve improvements in numerical e ciency by taking y t into account in constructing time-t samplers. The pursuit of such improvements is known as adaption. A prominent example of an adapted algorithm is the auxiliary particle lter of Pitt and Shephard, 1999 JASA. DLMRD () EIS Filtering August / 41

6 SMC Methods, cont. To date, adaption has been pursued subject to the constraint that the discrete support of the ltering density constructed in period t 1 is taken as given and xed in period t. We refer to the imposition of this constraint as the pursuit of conditional adaption. The approach to ltering we propose here is implemented absent this constraint: our objective is to pursue unconditional adaption. DLMRD () EIS Filtering August / 41

7 SMC Methods, cont. Speci cally, we use continuous approximations of ltering densities as an input to the construction of time-t importance samplers designed to generate optimal (in terms of numerical e ciency) global approximations to targeted integrands. The approximations fully account for the information conveyed by y t, and are constructed using the methodology of e cient importance sampling (EIS) developed by Richard and Zhang, 2007 J. of Econometrics. Resulting likelihood approximations are continuous functions of model parameters, greatly enhancing the pursuit of parameter estimation. DLMRD () EIS Filtering August / 41

8 State Space Representations State-transition equation: s t = γ(s t 1, Y t 1, υ t ) Associated density: Measurement equation: f (s t js t 1, Y t 1 ) y t = δ (s t, Y t 1, u t ) Associated density: Initialization: f (y t js t, Y t 1 ) f (s 0 ) DLMRD () EIS Filtering August / 41

9 State Space Representations, cont. Objective: evaluate the likelihood function f (Y T ) = T t=1 f (y t jy t 1 ), where f (y 1 jy 0 ) f (y 1 ). Time-t likelihoods are evaluated via marginalization of measurement densities: Z f (y t jy t 1 ) = f (y t js t, Y t 1 ) f (s t jy t 1 ) ds t. Marginalization requires the evaluation of f (s t jy t 1 ): Z f (s t jy t 1 ) = f (s t js t 1, Y t 1 ) f (s t 1 jy t 1 ) ds t 1, where f (s t jy t ) = f (y t, s t jy t 1 ) f (y t jy t 1 ) = f (y tjs t, Y t 1 ) f (s t jy t 1 ). f (y t jy t 1 ) DLMRD () EIS Filtering August / 41

10 Particle Filters: General Principle Period-t computation inherently requires the evaluation of Z Z f (y t jy t 1 ) = f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) bf (s t 1 jy t 1 )ds t 1 d Particle lters rely upon approximations in the form of a mixture-of-dirac measures associated with the period-(t which is xed in period-t : fs i t 1 gn i=1 bf (s t 1 jy t 1 ) = N ω i t 1 δ s i t 1 (s t 1 ), i=1 1) swarm where δ s i t 1 (s) denotes the Dirac measure at point st i 1, and ωi t 1 the weight associated with particle st i 1. This approximation e ectively solves the (inner) integration in s t 1, yielding f (y t jy t N Z 1 ) = ω i t 1 f (y t js t, Y t 1 ) f s t jst i 1, Y t 1 dst. i=1 DLMRD () EIS Filtering August / 41

11 Unadapted Filters Period-t algorithm: Inherit bf (s t 1 jy t 1 ), represented using fω i t 1, si t 1 gn i=1, from the period-(t 1) step. Approximate f (s t jy t 1 ) : for each st i 1, draw si t from f s t jst i 1, Y t 1, yielding bf (y t jy t 1 ) = Approximate bf (s t jy t ) as bf (s t jy t ) = N ω i t 1 f y t jst, i Y t 1. i=1 N ω i t δ s i t (s t ), i=1 where the (posterior) weights ω i t obtain from the (prior) weights ω i t 1 by application of Bayes theorem: ω i t = ω i t 1 f y tjst, i Y t 1 bf (y t jy t 1 ). DLMRD () EIS Filtering August / 41

12 Conditional Adaptation The measurement density incorporates the assumption that y t is independent of s t 1 given (s t, Y t 1 ); this implies f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) = f (s t js t 1, Y t ) f (y t js t 1, Y t 1 ) When this factorization is analytically tractable, it is possible to achieve conditionally optimal adaption: Z Z f (y t jy t 1 ) = f (s t js t 1, Y t ) f (y t js t 1, Y t 1 ) bf (s t 1 jy t 1 ) ds Z = f (y t js t 1, Y t 1 ) bf (s t 1 jy t 1 ) ds t 1 = N ω i t 1 f y t jst i 1, Y t 1. i=1 DLMRD () EIS Filtering August / 41

13 Conditional Adaptation: Implementation To implement, for each particle st i 1, draw a particle si t from f s t jst i 1, Y t. The corresponding weights are given by ω i t = ω i t 1 f y tjst i 1, Y t 1. bf (y t jy t 1 ) Key di erence relative to unadapted lters: the draws of s t are conditional on y t. Since ω i t does not depend on st, i but only on st i 1, its conditional variance is zero given fst i 1 gn i=1. This is referenced as the optimal sampler following Zaritskii et al., 1975 Automation and Remote Control; Akaski and Kumamoto, 1977 Automatica. Since the factorization f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) = f (s t js t 1, Y t ) f (y t js t 1, Y t 1 ) is tractable only in special cases, this sampler represents a theoretical rather than an operational benchmark. DLMRD () EIS Filtering August / 41

14 Approximate Conditional Optimality Attempts at approximating conditional optimality follow from the interpretation of f (y t jy t 1 ) = N Z ω i t 1 f (y t js t, Y t 1 ) f s t jst i 1, Y t 1 dst i=1 as a mixed integral in (s t, k t ), where k t denotes the index of particles, and follows the multinomial distribution MN N, fω i t 1 i=1 gn. The likelihood integral may then be evaluated via importance sampling, relying upon a mixed density kernel of the form γ t (s, k) = ω k t 1 p t (s, k) f s t jst k 1, Y t 1. Pitt and Shephard (1993 JASA) pursue conditional optimality by specifying p t (s, k) as p t (s, k) = f y t jµ k t, Y t 1, µ k t = E (s tjst k 1, Y t 1 ). DLMRD () EIS Filtering August / 41

15 Unconditional Optimality Returning to the period-t likelihood integral Z Z f (y t jy t 1 ) = f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) bf (s t 1 jy t 1 )ds t 1 d consider the theoretical factorization f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) bf (s t 1 jy t 1 ) = f (s t, s t 1 jy t ) f (y t jy If analytically tractable, f (s t, s t 1 jy t ) would be the unconditionally optimal (fully adapted) sampler for the likelihood integral, as a single draw from it would produce an estimate of f (y t jy t 1 ) with zero MC variance. The period-t ltering density would then obtain by marginalization with respect to s t 1 : Z f (s t jy t ) = f (s t, s t 1 jy t ) ds t 1. DLMRD () EIS Filtering August / 41

16 Unconditional Optimality, cont. Our goal: approximate unconditional optimality by constructing importance samplers in (s t 1, s t ) for the likelihood integral Z Z f (y t jy t 1 ) = f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) bf (s t 1 jy t 1 )ds t 1 d The goal is pursued via the principle of e cient importance sampling (EIS). DLMRD () EIS Filtering August / 41

17 EIS (Richard and Zhang, 2007 JoE) Let ϕ t (λ t ) denote the integrand with λ t = (s t 1, s t ). f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) bf (s t 1 jy t 1 ), Implementation of EIS begins with the pre-selection of a parametric class K = fk (λ t ; a t ) ; a t 2 Ag of analytically integrable auxiliary density kernels. The corresponding density functions (IS samplers) and IS ratios are given respectively by g(λ t ja t ) = k(λ Z t; a t ), χ(a t ) = k(λ t ; a t )dλ t, χ(a t ) ϕ ω t (λ t ; a t ) = t (λ t ) g t (λ t ja t ). DLMRD () EIS Filtering August / 41

18 EIS, cont. Objective: select ba t 2 A to minimize the MC variance of the IS ratio over the full range of integration. A near-optimal value ba t obtains as the solution to Z (ba t, bc t ) = arg min (a t,c t ) [ln ϕ t (λ t ) c t ln k(λ t ; a t )] 2 g(λ t ja t )dλ t, where c t denotes an intercept meant to calibrate the ratio ln (ϕ t /k). This represents a standard least squares problem, except that the auxiliary sampling density depends upon a t. This is resolved via the speci cation of an initial value ba 0 t, and the search for a xed point solution via iterations on (ba l+1 t, bc t l+1 ) = arg min (a t,c t ) R i=1 h ln ϕ t (λ i t,l ) c t ln k(λ i t,l; a t )i 2. DLMRD () EIS Filtering August / 41

19 The EIS Filter Having obtained the xed-point solution ba t, the likelihood EIS estimate is given by bf (y t jy t 1 ) = 1 S S ω t st i 1, s i t; ba t, i=1 ϕ ω t (λ t ; a t ) = t (λ t ) g t (λ t ja t ), where fst i 1, si tg S i=1 denotes i.i.d. draws from the EIS sampler g (s t 1, s t jba t ). A period-t ltering density approximation is then given by the marginal of g in s t. : Z bf (s t jy t ) = g (s t 1, s t ; ba t ) ds t 1. DLMRD () EIS Filtering August / 41

20 Initialization The selection of a good initial sampler g t s t, s t 1 jba 0 t is critical for achieving reliable convergence to an e ective nal sampler g t (s t, s t 1 jbat ). We rely upon local Taylor Series expansions to construct initial Gaussian samplers. This is similar to the procedure proposed by Durbin and Koopman (1997) whereby (local) Gaussian approximations are used as importance samplers to evaluate the likelihood function of non-gaussian state space models. Critical di erence: we use these local approximations to construct starting values for fully iterated global EIS approximation. DLMRD () EIS Filtering August / 41

21 Algorithmic Summary of the EIS Filter Propagation: Inheriting bf (s t 1 jy t 1 ) from period (t 1), obtain the integrand ϕ t (s t 1, s t ) = f (y t js t, Y t 1 ) f (s t js t 1, Y t 1 ) bf (s t 1 jy t 1 ). EIS Optimization: Construct an initialized sampler g t s t 1, s t jba 0 t, and obtain the optimized parameterization ba t as the solution to (ba l+1 t, bc l+1 t ) = arg min (a t,c t ) R i=1 h ln ϕ t (λ i t,l ) c t ln k(λ i t,l; a t )i 2. DLMRD () EIS Filtering August / 41

22 Algorithmic Summary, cont. Likelihood integral: Obtain draws st i 1, si t g t (s t 1, s t jbat ), and approximate bf (y t jy t 1 ) as N i=1 from bf (y t jy t 1 ) = 1 S S ω t st i 1, s i t; ba t. i=1 Filtering: Approximate bf (s t jy t ) as Z bf (s t jy t ) = g (s t 1, s t ; ba t ) ds t 1. Continuation: Pass bf (s t jy t ) to the period-(t + 1) propagation step and proceed through period T. DLMRD () EIS Filtering August / 41

23 Performance We demonstrate the performance of the EIS lter relative to the (unadapted) bootstrap particle lter of Gordon, Salmond, and Smith (1993 IEEE Proceedings). Application is to four data sets: two arti cial/actual pairs associated with two DSGE models. Model 1: the two-state RBC model used by Fernandez-Villaverde and Rubio-Ramirez (2005 J. Applied Econometrics) to demonstrate the bootstrap particle lter. Model 2: a six-state version of the small open economy model fashioned from Mendoza (1991 AER), Schmitt-Grohe and Uribe (2003 J. Int l Economics). DLMRD () EIS Filtering August / 41

24 Performance, cont. Each data set o ers a unique challenge: RBC Model, Arti cial Data: highly informative measurement densities RBC Model, Actual Data: outliers (1974:IV, 1980:II) SOE Model, both data sets: relatively high-dimensional state space, relatively signi cant departures from linearity in the state-transition equations. DLMRD () EIS Filtering August / 41

25 RBC Model Representative household s problem: subject to max U = E 0 β t t=0 y t = z t kt α nt 1 α, 1 = n t + l t, y t = c t + i t, c ϕ t l 1 k t+1 = i t + (1 δ)k t, t 1 φ ϕ 1 φ z t = z 0 e gt e ω t, ω t = ρω t 1 + ε t., DLMRD () EIS Filtering August / 41

26 Example, cont. State Transition Equations: 1 + g k 0 (k t, z t ) 1 α = i(k t, z t ) + (1 δ)k t log z t = (1 ρ) log(z 0 ) + ρ log z t 1 + ε t. Observation Equations: x t = x(k t, z t ) + u x,t, x = y, i, n, u x,t N(0, σ 2 x ). DLMRD () EIS Filtering August / 41

27 SOE Model Representative household s problem: ct ϕ max U = E 0 θ t ω 1 n ω 1 γ t 1 t, ω > 0, γ > 0, t=0 1 γ θ t+1 = β ( ec t, en t ) θ t, θ 0 = 1, β ( ec t, en t ) = 1 + ec t ω 1 en t ω ψ, ψ > 0, where ( ec t, en t ) denote average per capita consumption and hours worked, subject to DLMRD () EIS Filtering August / 41

28 SOE Model, cont. x t = A t k α t n 1 a t d t+1 = (1 + r t ) d t x t + c t + i t + φ 2 (k t+1 k t ) 2 k t+1 = v 1 t i t + (1 δ) k t ln A t+1 = ρ A ln A t + ε At+1, ε At iidn(0, σ 2 ε A ) ln r t+1 = (1 ρ r ) ln r + ρ r ln r t + ε rt+1, ε rt iidn(0, σ 2 ε r ) ln v t+1 = ρ v ln v t + ε vt+1, ε vt iidn(0, σ 2 ε v ) ln ϕ t+1 = ρ ϕ ln ϕ t + ε ϕt+1, ε ϕt iidn(0, σ 2 ε ϕ ). Observables: y 0 t = (x t, c t, i t, n t ). DLMRD () EIS Filtering August / 41

29 Experiment 1: Bias in the EIS Filter? For each data set, generate 100 date-by-date log-likelihood approximations (using 100 di erent sets of random numbers) using the BP lter, N = 1, 000, 000. Given the unbiasedness of the BP lter, these approximations serve as a benchmark for judging the EIS lter. Next, generate 100 log-likelihood approximations using the EIS lter (N = R = 100 for the RBC model, N = R = 200 for the SOE model). Calculate the di erence in approximations for each of the 10,000 possible combinations of likelihood values, and searched for instances in which di erences were signi cantly di erent from zero. Result: in all instances, zero lies between the 5 th and 95 th percentiles of resulting boxplots. I.E., we cannot reject the null that di erences between estimators merely re ect numerical error. DLMRD () EIS Filtering August / 41

30 Example 1, cont. DLMRD () EIS Filtering August / 41

31 Experiment 2: Comparison of Numerical E ciency Once again, for each data set, generate 100 date-by-date log-likelihood approximations (using 100 di erent sets of random numbers) using the BP and EIS lters. Objective: compare the numerical standard errors associated with the lters. BP Filter: N = 60, 000 for the RBC model (following F-V/R-R), N = 150, 000 for the SOE model. Computational times range from seconds (RBC, arti cial) to seconds (SOE, actual). EIS Filter: N = R = 100 for the RBC model, N = R = 200 for the SOE model. Computational times range from 0.55 seconds (RBC, arti cial) to 2.18 seconds (SOE, actual). Result: Tremendous e ciency gains associated with the EIS lter. DLMRD () EIS Filtering August / 41

32 Example 2, cont. RBC Model BP Filter EIS Filter Initial Sampler Mean NSE Mean NSE Mean NSE Art e Act SOE Model BP Filter EIS Filter Initial Sampler Mean NSE Mean NSE Mean NSE Art Act Table: DLMRD () EIS Filtering August / 41

33 Example 2, cont. DLMRD () EIS Filtering August / 41

34 Experiment 3: Are Results Data-Set Speci c? Repeat Experiment 2 for 100 arti cial data sets generated from the four model parameterizations represented in the previous experiments (EIS lter, only). This yields a distribution of NSEs, indicating whether the NSEs in the previous table are somehow unusual. In addition, we construct sampling (statistical) errors by calculating the standard deviation of likelihood estimates obtained across data sets using the EIS lter implemented with a single set of common random numbers. Result: the NSEs reported in Table 2 only appear unusual for the RBC model, actual data set, which features the two signi cant outliers. Also: statistical standard errors dominate NSEs (by two to ve orders of magnitude). DLMRD () EIS Filtering August / 41

35 Example 3, cont. RBC Model SSE NSE Mean Std. Dev. Arti cial Data e-4 1.2e-4 Actual Data e-4 SOE Model SSE NSE Mean Std. Dev. Arti cial Data Actual Data Notes: SSE stands for statistical standard errors, which were computed as standard deviations of log-likelihood values across 100 alternative data sets. NSE denotes numerical standard errors. Table: DLMRD () EIS Filtering August / 41

36 Experiment 4: Continuity of log-likelihood surfaces Generate log-likelihood surfaces by allowing each model parameter to vary individually above and below its ML estimate, holding all additional parameters xed at their ML estimates. For each parameter combination, obtain log-likelihood approximations using the same set of CRNs, to eliminate numerical error. Result: surfaces associated with the BP lter are discontinuous; those associated with the EIS lter are continuous. Figure 3: SOE model, actual data set. DLMRD () EIS Filtering August / 41

37 Experiment 4, cont. DLMRD () EIS Filtering August / 41

38 Experiment 5: Outliers, and Bias Redux For RBC model, arti cial data set, generate 12 variations by inserting an outlier in the second observation of one of the observables, keeping the remaining variables xed at their original values. Four outliers were generated for each variable: two deviated by 4 sample standard deviations from the sample mean, and two deviated 8 sample standard deviations from the sample mean. For each new data set (as well as for the original), log-likelihood values were calculated for periods 1 and 2 using the BP and EIS lters, and also the Gauss-Chebyshev quadrature method implemented with 250 nodes along all three dimensions of integration, for a total of = 15, 625, 000 nodes. By evaluating the rst two periods only, implementation of the quadrature method is feasible, and provides a near-exact value of targeted log-likelihoods. Result: the EIS lter remains free of bias, and its associated NSEs are fairly uniform across data sets. The performance of the BP lter deteriorates in the presence of bias. DLMRD () EIS Filtering August / 41

39 Example, 5 cont. BP Filter EIS Filter t = 1 t = 2 t = 1 t = 2 Mean NSE Mean NSE Mean NSE Mean x Mean NSE Mean NSE Mean NSE Mean i Mean NSE Mean NSE Mean NSE Mean n 0 DLMRD () EIS Filtering August / 41 0

40 Summary 1 Particle-based lters are easy to implement and produce unbiased likelihood estimates. 2 However, they are prone to numerical ine ciency, induce spurious discontinuities in likelihood surfaces, and at best admit e orts towards approximating conditional adaptation. 3 In turn, the EIS lter implements continuous approximations of ltering densities, enabling the pursuit of unconditional adaption. 4 The EIS algorithm produces global approximations of targeted integrands, yields signi cant gains in numerical e ciency, and produces likelihood surfaces that are continuous in model parameters. DLMRD () EIS Filtering August / 41

41 Extensions The foregoing results were obtained using Gaussian approximations of ltering densities. While such approximations proved su cient in the applications to DSGE models we have considered, they are clearly not appropriate in general. We are currently working to develop operational EIS samplers that are more exible than those drawn from the exponential family of distributions. One such extension entails the development of an EIS procedure to construct global mixtures of Gaussian samplers; under this approach, EIS optimization is pursued via non-linear least squares implemented using analytical derivatives. The goal is to facilitate EIS implementations using highly exible samplers that will prove e cient in applications involving even the most challenging of targeted integrands. DLMRD () EIS Filtering August / 41