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Minimum Distance Estimation of Possibly,Noninvertible Moving Average Models. Nikolay GOSPODINOV, Research Department Federal Reserve Bank of Atlanta Atlanta GA 30309 nikolay gospodinov atl frb org. Department of Economics Columbia University New York NY 10027 sn2294 columbia edu. This article considers estimation of moving average MA models with non Gaussian errors Information in. higher order cumulants allows identification of the parameters without imposing invertibility By allowing. for an unbounded parameter space the generalized method of moments estimator of the MA 1 model. is classical root T consistent and asymptotically normal when the MA root is inside outside and on the. unit circle For more general models where the dependence of the cumulants on the model parameters is. analytically intractable we consider simulation based estimators with two features First in addition to an. autoregressive model new auxiliary regressions that exploit information from the second and higher order. moments of the data are considered Second the errors used to simulate the model are drawn from a flexible. functional form to accommodate a large class of distributions with non Gaussian features The proposed. Downloaded by Columbia University at 05 35 20 July 2015. simulation estimators are also asymptotically normally distributed without imposing the assumption of. invertibility In the application considered there is overwhelming evidence of noninvertibility in the. Fama French portfolio returns, KEY WORDS Generalized lambda distribution GMM Identification Non Gaussian errors Noninvert. ibility Simulation based estimation, 1 INTRODUCTION coefficients which are often the objects of interest as shown by. Ferna ndez Villaverde et al 2007 using the permanent income. Moving average MA models can parsimoniously charac model Hansen and Sargent 1991 Lippi and Reichlin 1993. terize the dynamic behavior of many time series processes and Ferna ndez Villaverde et al 2007 among others empha. The challenges in estimating MA models are twofold First sized the need to verify invertibility because it affects how we. invertible and noninvertible MA processes are observationally interpret what is recovered from the data. equivalent up to the second moments Second invertibility re While economic analysis tends to only consider parameter. stricts all roots of the MA polynomial to be less than or equal values consistent with invertibility it is necessary in many sci. to one This upper bound renders estimators with nonnormal ence and engineering applications to admit parameter values. asymptotic distributions when some roots are on or near the in the noninvertible range For example in analysis of seismic. unit circle Existing estimators treat invertible and noninvertible and communication data noninvertible filters are necessary to. processes separately requiring the researcher to take a stand on recover the earth s reflectivity sequence and to back out the. the parameter space of interest While the estimators are super underlying message from a distorted one respectively A key. consistent under the null hypothesis of an MA unit root their finding in these studies is that higher order cumulants are nec. distributions are not asymptotically pivotal To our knowledge essary for identification of noninvertible models implying that. no estimator of the MA model exists which achieves identifi the assumption of Gaussian errors must be abandoned Lii and. cation without imposing invertibility and yet enables classical Rosenblatt 1992 approximated the non Gaussian likelihood. inference over the whole parameter space of noninvertible MA models by truncating the representation of. Both invertible and noninvertible representations can be con the innovations in terms of the observables Huang and Paw. sistent with economic theory For example if the logarithm of itan 2000 proposed least absolute deviations LAD estima. asset price is the sum of a random walk component and a station tion using a Laplace likelihood This quasi maximum likelihood. ary component the first difference or asset returns is generally QML estimator does not require the errors to be Laplace dis. invertible but noninvertibility can arise if the variance of the sta tributed but they need to have heavy tails Andrews Davis. tionary component is large While noninvertible models are not and Breidt 2006 2007 considered LAD and rank based esti. ruled out by theory invertibility is often assumed in empirical mation of all pass models which are special noncausal and or. work because it provides the identification restrictions without noninvertible autoregressive and moving average ARMA. which maximum likelihood and covariance structure based es. timation of MA models would not be possible when the data. are normally distributed Invertibility can also be used to narrow 2015 American Statistical Association. the class of equivalent dynamic stochastic general equilibrium Journal of Business Economic Statistics. July 2015 Vol 33 No 3,DSGE models as in Komunjer and Ng 2011 Obviously.
DOI 10 1080 07350015 2014 955175, falsely assuming invertibility will yield an inferior fit of the data Color versions of one or more of the figures in the article can be. It can also lead to spurious estimates of the impulse response found online at www tandfonline com r jbes. 404 Journal of Business Economic Statistics July 2015. models in which the roots of the autoregressive polynomial 2 IDENTIFICATION PROBLEMS IN MODELS. are reciprocals of the roots of the MA polynomials Meitz and WITH AN MA COMPONENT. Saikkonen 2011 developed maximum likelihood estimation. of noninvertible ARMA models with ARCH errors However Consider the ARMA p q process. there exist no likelihood based estimators that have classical L yt L et 1. properties while admitting an MA unit root in the parameter. space where L is the lag operator such that Lp yt yt p and the. This article considers estimation of MA models without im lag polynomial L 1 1 L p Lp has no common. posing invertibility We only require that the errors are non roots with L 1 1 L q Lq Here yt can be the. Gaussian but we do not need to specify the distribution Iden error of a regression model. tification is achieved by the appropriate use of third and higher. order cumulants In the MA 1 case appropriate means that Yt xt yt. multiple third moments are necessary as a single third moment where Yt is the dependent variable and xt are exogenous re. still does not permit identification In general identification of gressors In the simplest case when xt 1 yt is the demeaned. possibly noninvertible MA models requires using more uncondi data The process yt is causal if z 0 for all z 1 on. tional higher order cumulants than the number of parameters in the complex plane In that case there exist constants hj with. the model We make use of this identification result to develop. j 0 hj such that yt j 0 hj et j for t 0 1, generalized method of moments GMM estimators that are Thus all MA models are causal The process is invertible if. root T consistent and asymptotically normal without restricting z 0 for all z 1 see Brockwell and Davies 1991 In. the MA roots to be strictly inside the unit circle The estimators. Downloaded by Columbia University at 05 35 20 July 2015. control theory and the engineering literature an invertible pro. minimize the distance between sample based statistics and their cess is said to have minimum phase. model based analog When the model implied statistics have Our interest is in estimating MA models without prior knowl. known functional forms we have a classical minimum distance edge about invertibility The distinction between invertible and. estimator noninvertible processes is best illustrated by considering the. A drawback of identifying the parameters from the higher MA 1 model defined by. order sample moments is that a long span of data is required. to precisely estimate the population quantities This issue is yt et et 1 2. important because for general ARMA p q models the num. ber of cumulants that needs to be estimated can be quite large where et t and t iid 0 1 with 3 E t3 and 4. Accordingly we explore the potential of two simulation estima E t4 The invertibility condition is satisfied if 1 In that. tors in providing bias correction The first simulated method case the inverse of L has a convergent series expansion in. of moments SMM estimator matches the sample to the simu positive powers of the lag operator L Then we can express. lated unconditional moments as in Duffie and Singleton 1993 yt as L yt et with L j 0 L j. This infinite au, The second is a simulated minimum distance SMD estima toregressive representation of yt implies that the span of et. tor in the spirit of Gourieroux Monfort and Renault 1993 and its history coincide with that of yt which is observed by. and Gallant and Tauchen 1996 Existing simulation estima the. econometrician When 1 the inverse polynomial is, tors of the MA 1 model impose invertibility and therefore only j 0 L implying that yt is a function of future values. need the auxiliary parameters from an autoregression to achieve of yt which is not useful for forecasting This argument is often. identification We show that the invertibility assumption can be used to justify the assumption of invertibility It is however mis. relaxed but additional auxiliary parameters involving the higher leading to classify invertible processes according to the value of. order moments of the data are necessary In the SMD case alone Consider another MA 1 process yt represented by. this amounts to estimating an additional auxiliary regression yt et et 1 3. with the second moment of the data as a dependent variable. An important feature of the SMM and SMD estimators is that Even if in 3 is less than one the inverse of L L. errors with non Gaussian features are simulated from the gener is not convergent Furthermore the errors from a projection of. alized lambda distribution GLD These two simulation based yt on lags of yt have different time series properties depending. estimators also have classical asymptotic properties regardless on whether the data are generated by 2 or 3. of whether the MA roots are inside outside or on the unit Identification and estimation of models with an MA compo. circle nent are difficult because of two problems that are best under. The article proceeds as follows Section 2 highlights two stood by focusing on the MA 1 case The first identification. identification problems that arise in MA models Section 3 problem concerns at or near unity When the MA parame. presents identification results based on higher order moments ter is near the unit circle the Gaussian maximum likelihood. of the data Section 4 discusses GMM estimation of the MA 1 ML estimator takes values exactly on the boundary of the in. model while Section 5 develops simulation based estimators vertibility region with positive probability in finite samples This. for more general MA models Simulation results and an anal point probability mass at unity the so called pile up problem. ysis of the 25 Fama French portfolio returns are provided arises from the symmetry of the likelihood function around one. in Section 5 Section 6 concludes Proofs are given in the and the small sample deficiency to identify all the critical points. Appendix of the likelihood function in the vicinity of the noninvertibility. Gospodinov and Ng Minimum Distance Estimation of Possibly Noninvertible MA Models 405. boundary see Sargan and Bhargava 1983 Anderson and Take that this happens only over sufficiently small regions in the. mura 1986 Davis and Dunsmuir 1996 Gospodinov 2002 parameter space. and Davis and Song 2011 More precisely let be a K 1 parameter vector of in. The second identification problem arises because covariance terest where the parameter space is a compact subset of the K. stationary processes are completely characterized by the first and dimensional Euclidean space RK In the case of an ARMA p q. second moments of the observables The Gaussian likelihood model defined by 1 1 p 1 q 2 Let 0. for an MA 1 model with L 2 is the same as one with be the true value of and g G RL denote L L K. L 1 2 2 The observational equivalence of the covariance moments which can be used to infer the value of 0 Identifica. structure of invertible and noninvertible processes also implies tion hinges on a well behaved mapping from the space of to. that the projection coefficients in L are the same regardless the space of moment conditions g. of whether is less than or greater than one Thus cannot. be recovered from the coefficients of L without additional Definition 1 Let g RK RL be a mapping from to. assumptions g and let G g with G0 G 0 Then 0 is, This observational equivalence problem can be further globally identified from g if g is injective and is locally.
elicited from a frequency domain identified if the matrix of partial derivatives G0 has full column. perspective If we take as rank, a starting point yt h L et j hj et j the frequency. response function of the filter is From Definition 1 1 and 2 are observationally equivalent. if g 1 g 2 that is g is not injective Section 3 1 shows. H hj exp i j H exp i in the context of an MA 1 model that second moments can. Downloaded by Columbia University at 05 35 20 July 2015. not be used to define a vector g that identifies without. where H is the amplitude and is the phase, response imposing invertibility However possibly noninvertible models. of the filter For ARMA models h z z z, j hj z can be identified if g is allowed to include higher order mo. The amplitude is usually constant for given and tends toward ments cumulants Sections 3 2 and 3 3 generalize the results to. zero outside the interval 0 For given a 0 the phase 0 MA q and ARMA p q models. is indistinguishable from 0 a for any 0,Recovering et from the second order spectrum. 3 1 MA 1 Model, This subsection provides a traditional identification analysis.
is problematic because S2 z is proportional to the amplitude of the zero mean MA 1 model Let 2 The data. H z 2 with no information about the phase The second yt are a function of the true value 0 For the MA 1 model. order spectrum is thus said to be phase blind As argued by Lii E yt yt 1 0 for j 2 Consider the population identification. and Rosenblatt 1982 one can flip the roots of z and z problem using only second moments of yt. without affecting the modulus of the transfer function With. g21 E yt yt 1 2, real distinct roots there are 2p q ways of specifying the roots g2. g22 E yt2 1 2 2,without changing the probability structure of yt. The moment vector g2 is the difference between the popula. tion second moments and the moments implied by the MA 1. 3 CUMULANT BASED IDENTIFICATION OF, model If the assumption that the data are generated by the. NONINVERTIBLE MODELS, MA 1 model is correct g2 evaluated at the true value. Econometric analysis on identification largely follows the pi of is zero g2 0 0 Under Gaussianity of the errors. oneering work of Fisher 1961 1965 and Rothenberg 1971 these moments fully characterize the covariance structure of. in fully parametric likelihood settings These authors recast the yt However g2 assumes the same value for 1 2. identification problem as one of finding a unique solution to and 2 1 2 2 For example if 1 0 5 2 1. a system of nonlinear equations For nonlinear models a suf and 2 2 2 0 25 g2 1 g2 2 Parameters that. ficient condition is that the Jacobian matrix of the first partial are not identifiable from the population moments are not con. derivatives is of full column rank See Dufour and Hsiao 2008 sistently estimable. for a survey However local identification is still possible if the The problem that the mapping g2 is not injective is typi. rank condition fails by exploiting restrictions on the higher order cally handled by imposing invertibility thereby restricting the. derivatives as shown in Sargan 1983 and Dovonon and Re parameter space to R 1 1 L2 H2 But there is still. nault 2013 To obtain results for global identification Rothen a problem because the derivative matrix of g with respect to. berg 1971 Theorem 7 imposed additional conditions to ensure is not full rank everywhere in R The determinant of. that the optimization problem is well behaved In a semipara 2. metric setting when the distribution of the errors is not specified G 4. identification results are limited but the rank of the derivative. matrix remains to be a sufficient condition for local identifica is zero when 1 This is responsible for the pile up problem. tion Newey and McFadden 1994 Komunjer 2012 showed discussed earlier Furthermore 1 lies on the boundary of. that global identification from moment restrictions is possible the parameter space As a consequence the Gaussian maximum. even when the derivative matrix has a deficient rank provided likelihood estimator and estimators based on second moments. 406 Journal of Business Economic Statistics July 2015. are not uniformly asymptotically normal see Davis and Dun If yt h L et and et t is a mean zero iid process we have. smuir 1996 Note however that the two problems with the. MA 1 model namely inconsistency due to nonidentification c 1 1 hi hi 1 hi 1 5. and nonnormality due to a unit root do not arise if there is prior i 0. knowledge about 2 We will revisit this observation in Section. where c e 0 0 0 denotes the th order cumulant of,et with 2 2 3 3 3 and 4 4 4 3 Thus the cu.
While the second moments of the data do not identify. mulants 3 measure the distance of et and hence of yt. 2 would the three nonzero third moments given by,from Gaussianity If 1 2 then c. E y 3 1 3 3 c is known as the diagonal slice of the th order. g3 g32 E yt2 yt 1 2 3 3 cumulant of yt, g33 2 3 Higher order cumulants are useful for identification of pos. E yt yt 1 3, sibly noninvertible models because the Fourier transform of. achieve identification The following lemma provides an answer c 1 2 1 is the th order polyspectrum. to this question 1, Lemma 1 Consider the MA 1 model yt et et 1 with S 1 1 H 1 H 1 H i. et t Suppose that t iid 0 1 with 3 E t3 Assume 6, that 0 3 0 and E t 3 Then Recovery of phase information necessarily requires that et has.
non Gaussian features In other words must exist and is. a g g2 g32 is not injective for any, Downloaded by Columbia University at 05 35 20 July 2015. nonzero for some 3 for recovery of the phase function see. Lii and Rosenblatt 1982 Lemma 1 Giannakis and Swami. b g g2 g3j for j 1 2 or 3 cannot locally identify, 1992 Giannakis and Mendel 1989 Mendel 1991 Tugnait. when 1 for any 2 and 3,1986 and Ramsey and Montenegro 1992. In Lemma 1 g3 and 2 3 are of the To establish that the MA q parameters are identifiable from. same dimension Part a states that there always exist cumulants of a particular order the typical starting point is to. 1 2 that are observationally equivalent in the sense generate identities that link the second and higher order cumu. that they generate the same moments For example 1 lants to the parameters of the model Different identities exist. 2 3 and 2 1 2 2 3 both imply the same for different choice of 1 2 1 Mendel 1991 provided. E yt yt 1 E yt2 E yt2 yt 1 Part b of Lemma 1 follows a survey of the methods used in the engineering literature One. from the fact that the determinant of the derivative matrix is zero of the simplest and earliest ideas is to consider the diagonal. at 1 As a result a single third moment cannot be guaran slice of the third order cumulants which implies the following. teed to identify both 3 and the parameters of the MA 1 model relation between the population cumulants and the q 1 vector. and 2 Global and local identification of at 1 requires of parameters 1 q 3. use of information in the remaining two third order moments. In particular the derivative matrix of g g2 g3 with re j c3 j 3 j2 c2 j c3 0. spect to 2 3 is of full column rank everywhere in j 1 j 0. including 1 However since g is of dimension five q 2q 7. this together with Lemma 1 implies that can only be over. identified if 3 0 The next subsection describes a general Define. procedure based on higher order cumulants for identifying the. 1 q 3 3 12 3 q2,parameters of MA q and ARMA p q models. 3 2 The MA q Model b c3 q c3 q 1 c3 0 c3 1, The insight from the MA 1 analysis that the parameters of c3 q 1 c3 q 0 0 0.
the model cannot be exactly identified but can be over identified. The system of Equation 7 can be expressed as, with an appropriate choice of higher order moments extends to. MA q models But for MA q models the moments of the A b 8. process are nonlinear functions of the model parameters and. The reason why 8 is useful for identification is that A. verifying global and local identification is more challenging. b is an over identified system of 3q 1 equations in 2q 1. Our analysis is built on results from the statistical engineering. unknowns The parameters are identifiable if can,literature. be solved from 8 Given that the derivative matrix of with. Let c 1 2 1 be the th 2 order cumulant of, respect to has rank q 1 the identification problem reduces. a zero mean stationary and ergodic process yt The second and. to the verification of the column rank of the matrix A given in. third order cumulants of yt are given by,the Appendix. c2 1 E yt yt 1,Lemma 2 Consider the MA q process yt et.
c3 1 2 E yt yt 1 yt 2 1 et 1 q et q where et t t iid 0 1 with. Gospodinov and Ng Minimum Distance Estimation of Possibly Noninvertible MA Models 407. 3 E t3 0 and E t 3 Let c denote the diag The p p Toeplitz matrix on the right hand side is a submatrix. onal slice of the th order cumulant of yt If c2 q and c3 q are of autocovariances and hence full rank Thus is identifiable. nonzero then the matrix A has full column rank 2q 1 By considering the spectrum at p frequencies identities can also. be derived in the frequency domain If et were non Gaussian. A proof is given in the Appendix Full rank of the ma the AR coefficients of an ARMA. trix A enables identification of and subsequently of pprocess. p can still be uniquely,determined from the equations i 0 j 0 i j c 1 i 2. This requires that the qth autocorrelation c2 q is nonzero j 3 1 0 for 3 and 1 2 q The idea of. and also that c3 q 0 In view of the definition of c3 q in using cumulants to identify the autoregressive parameters seems. 5 it is clear that skewness in et is necessary for identifica to date back to Akaike 1966 see Mendel 1991 p 281 and. tion of A similar idea can be used to analyze identifica Theorem 2 of Giannakis and Swami 1992. tion using fourth order cumulants defined as c4 1 2 3 The question then arises as to whether 1 p. E yt yt 1 yt 2 yt 3 c2 1 c2 2 3 c2 2 c2 3 1 1 q can be jointly identified from the third order cu. c2 3 c2 1 2 For example the diagonal slice of the fourth mulants alone The A b framework presented above re. order cumulants yields a system of equations given by quires that q is finite and hence does not work for ARMA p q. q models Assuming that the ARMA model has no common fac. i c4 i 2 4 3 i3 c2 i c4 9 tors hence it is irreducible the following lemma adapted from. i 1 i 0 Tugnait 1995 provides sufficient conditions for identifiability. As shown in the Appendix 1 q 2 4 3 1 of the parameters of ARMA p q models. q 2 4 3 2 4 3 13 2 4 3 q3 is iden, tifiable provided that 4 3 c2 q and c4 q are nonzero Lemma 3 Assume that the ARMA p q process 1 1 L. Downloaded by Columbia University at 05 35 20 July 2015. Nondiagonal slices of the fourth order cumulants were consid p L. yt 1 1 L q Lq et is irreducible and,i 0 i z 0 for z 1 where et t t. ered by Friedlander and Porat 1990 and Na et al 1995 satisfies. Giannakis and Mendel 1989 p 364 made use of the struc iid 0 1 with 3 E t3 0 and E t 3 Let c denote. ture of the A matrix to recursively compute the parameters in the diagonal slice of the th order cumulant of the MA p q. and hence This algorithm treats 3 12 3 q2 as process 1 1 L p Lp 1 1 L q Lq et and. free parameters when in fact they are not Although the method assume that c2 p q and c3 p q are nonzero Then the. is not efficient or practical for estimation the approach is one of parameter vector 1 p 1 q of the ARMA p q. the first to suggest the possibility of identification of MA models process is identifiable from the second and third cumulants of. using cumulants Tugnait 1995 subsequently obtained closed the MA p q process. form expressions for the MA parameters using c3 q and. autocovariances Friedlander and Porat 1990 proposed an op The thrust of the argument elaborated in the Appendix is. timal minimum distance estimation of the system 8 although that observational equivalence of the two ARMA p q process. this method still cannot separately identify the parameters 3 amounts to equivalence of two appropriately defined MA p q. or 4 and 2 As we will see below this approach is a re processes say z parameterized by p q vectors 1 and 2. stricted version of our proposed GMM method respectively But from Tugnait 1995 two MA p q pro. cesses are equivalent if c3z 1 p q 1 c3z 1 p q 2, for 0 1 p q We can now exploit results from the previ. 3 3 ARMA p q Model ous subsection Tugnait 1995 used information in the nondi. The previous two subsections have focused on MA q models agonal slices to isolate the smallest number of third and higher. because the p parameters in the autoregressive polynomial L order cumulants that are sufficient for identification of ARMA. can be easily identified Consider the ARMA p q model parameters. The representation A b provides a transparent way to. yt 1 yt 1 p yt p et 1 et 1 q et q see how higher order cumulants can be used to recover the pa. where et iid 0 2 If et were Gaussian one can exploit the rameters of the model without imposing invertibility However. fact that Eet q yt j 0 for j q or equivalently c2 1 this approach may use more cumulants than is necessary To see. p 1 why 5 implies that for an MA q process c3 q k 3 3 q k. k 0 c2 k k 1 0 for q q p This leads to the, system of equations and c3 q 0 3 3 q It immediately follows that k cc33 q k.
This so called C q k formula suggests that only q 1 third. c2 q 1 order cumulants c3 q for 0 q are necessary and suffi. c q 2 cient for identification of 1 q if 3 0 which is smaller. than the number of equations in the A b system, The key point in this section to highlight is that once non. c2 q p Gaussian features are allowed identification of noninvertible. models is possible from the higher order cumulants of the data. c2 q c2 q 1 c2 q p 1 1, c q 1 In practice we would want to use the covariance structure along. 2 c2 q 2 with identities based on the third and fourth order cumulants. Using information in the third or fourth cumulants alone would. be inefficient even though identification is possible This is. c2 q p 1 c2 q p, because the covariance structure would have been sufficient for. 10 identification if invertibility was imposed and the fourth order. 408 Journal of Business Economic Statistics July 2015. cumulants can be useful when the error distribution is near gest to consider. symmetric The identities considered shed light on which order. E yt yt 1 c2 1, cumulants are required for identification For example in the. MA 1 case the A b system E yt2 c2 0,E yt yt 1 c3 1.
0 c2 1 0 c3 1 E yt3 c3 0,c 1 c 0 c 1,3 2 2 c3 0 2,E yt yt 1 c3 1. c3 0 c2 1 c2 0 c3 1,c3 1 0 c2 1 0, tells us that the third order cumulants c3 1 c3 0 c3 1 will m 2 3. be needed to identify the MA 1 parameters This is used to 1 3. guide estimation which is the subject of the next section 3 3. Note that the equations in 11 are particular linear combinations. 4 GMM ESTIMATION of the moment conditions in 14 The conditions in Lemma 2. that c2 1 0 and c3 1 0 correspond to the conditions 0. The results in Section 3 suggest to estimate the parameters of and 3 0 in Lemma 1. ARMA p q models by matching second and higher order cu. Downloaded by Columbia University at 05 35 20 July 2015. mulants Friedlander and Porat 1990 p 30 proposed a two step Proposition 1 Consider the MA 1 model Suppose that. procedure for estimating ARMA p q models where the AR pa in addition to the assumptions in Lemma 1 we have that. rameters are obtained first from the autocovariances spectrum E et 6 and 0 is in the interior. of the compact param, of the process and the MA parameters are then estimated from eter space Also assume that T g 0 N 0 and. g converges uniformly to G over, the filtered process using information in the higher order cu. mulants a similar estimation strategy has been proposed by an Then is T consistent with asymptotic distribution given by. anonymous referee Our proposed estimation strategy is similar 13. in spirit but it estimates all of the unknown parameters in one The derivative matrix G g. is of full column rank ev, step erywhere in even at 1 As a result this GMM estimator.
Let gt be conditions characterizing the model parame is root T consistent and asymptotically normal. terized by and such that at the true value 0 E gt 0. 0 Given data y y1 yT one can construct g,t 1 gt the sample analog of g E gt Let. 4 1 Finite Sample Properties of the GMM Estimator, denote a consistent estimate of the positive definite matrix. To illustrate the finite sample properties of the GMM esti. limT var T g 0 The optimal GMM estimator of, mator data with T 500 observations are generated from an. is defined as,MA 1 model yt et et 1 and et t where t is iid 0 1. and follows a GLD which will be further discussed in Section. arg min g 12 5 1 For now it suffices to note that GLD distributions can be. characterized by a skewness parameter 3 and a kurtosis parame. Full rank of the derivative matrix G g, evaluated at 0 is ter 4 The true values of the parameters are 0 5 0 7 1 1 5.
sufficient for 0 to be a unique solution to the system of nonlinear and 2 1 3 0 0 35 0 6 and 0 85 and 4 3 The re. equations characterized by G 1 g 0 The full rank sults are invariant to the choice of Lack of identification. condition in the neighborhood of 0 is also necessary for the of arises when 3 0 and weak to intermediate identifi. estimator to be asymptotically normal Under the assumptions cation occurs when 3 0 35 and 0 6 Unreported numerical. given by Newey and McFadden 1994 results revealed that the estimator based on the moment condi. tions 14 possesses substantially better finite sample properties. d than the estimator based on 11 We only consider the finite. 0 N 0 G 0 1 G 0 1, sample properties of the estimator for the MA 1 model when. the orthogonality conditions are both necessary and sufficient. Consistent estimation of possibly noninvertible ARMA p q for identification. models depends on the choice gt We consider three possi Table 1 presents the mean the median and the standard. bilities beginning with a classical GMM estimator deviation of three estimators of over 1000 Monte Carlo. For the MA 1 model let 2 3 be the parameters replications The first is the GMM estimator of 2 3. to be estimated and define which uses 14 as moment conditions The second is the. infeasible GMM estimator based on 14 but assumes 2. g m 0 m is known and estimates only 3 As discussed earlier. fixing 2 solves the identification problem in the MA 1. where m 0 T1 Tt 1 mt 0 is a consistent estimate of model and by not imposing invertibility 1 is not on the. E mt 0 The identification results in Lemma 1 and 14 sug boundary of the parameter space for We will demonstrate. Gospodinov and Ng Minimum Distance Estimation of Possibly Noninvertible MA Models 409. Table 1 GMM and Gaussian QML estimates of from MA 1 model with possibly asymmetric errors. GMM estimator Gaussian QML estimator Infeasible GMM estimator. 0 Mean Med P,1 Std Mean Med P,1 Std Mean Med P, 0 5 1 392 1 692 0 578 0 790 0 500 0 502 0 000 0 040 0 489 0 486 0 000 0 071. 0 7 1 152 1 117 0 564 0 428 0 700 0 701 0 000 0 033 0 674 0 675 0 000 0 084. 1 0 1 057 1 004 0 509 0 279 0 965 0 971 0 063 0 028 0 970 0 974 0 386 0 082. 1 5 1 144 1 105 0 547 0 467 0 666 0 667 0 000 0 034 1 473 1 471 1 000 0 073. 2 0 1 353 1 600 0 563 0 783 0 500 0 501 0 000 0 040 1 969 1 967 1 000 0 081. 0 5 0 823 0 518 0 223 0 615 0 500 0 500 0 000 0 040 0 488 0 484 0 000 0 071. 0 7 0 903 0 773 0 262 0 368 0 699 0 700 0 000 0 033 0 675 0 673 0 000 0 085. 1 0 1 057 1 020 0 543 0 264 0 964 0 969 0 053 0 028 0 972 0 976 0 377 0 081. 1 5 1 367 1 427 0 808 0 414 0 666 0 667 0 000 0 034 1 475 1 474 1 000 0 073. 2 0 1 757 1 950 0 827 0 642 0 500 0 501 0 000 0 040 1 971 1 969 1 000 0 080. 0 5 0 552 0 493 0 034 0 260 0 500 0 501 0 000 0 040 0 488 0 485 0 000 0 071. 0 7 0 738 0 690 0 062 0 203 0 699 0 700 0 000 0 033 0 677 0 673 0 000 0 085. 1 0 1 042 1 009 0 528 0 237 0 964 0 968 0 048 0 028 0 975 0 982 0 389 0 077. Downloaded by Columbia University at 05 35 20 July 2015. 1 5 1 514 1 527 0 964 0 307 0 666 0 667 0 000 0 034 1 478 1 478 1 000 0 069. 2 0 1 986 2 039 0 969 0 423 0 500 0 501 0 000 0 040 1 975 1 973 1 000 0 076. 0 5 0 511 0 487 0 003 0 121 0 500 0 500 0 000 0 040 0 489 0 485 0 000 0 069. 0 7 0 688 0 674 0 003 0 118 0 699 0 699 0 000 0 033 0 678 0 677 0 000 0 084. 1 0 1 012 0 999 0 496 0 187 0 964 0 966 0 046 0 027 0 978 0 987 0 416 0 072. 1 5 1 556 1 544 0 997 0 268 0 666 0 667 0 000 0 034 1 482 1 483 1 000 0 063. 2 0 2 025 2 043 0 993 0 366 0 500 0 501 0 000 0 040 1 980 1 979 1 000 0 070. NOTES The table reports the mean median med probability that 1 and standard deviation std of the GMM Gaussian quasi maximum likelihood QML and infeasible. GMM estimates of from the MA 1 model yt et et 1 where et t and t iid 0 1 are generated from a generalized lambda distribution GLD with a skewness. parameter 3 and no excess kurtosis The sample size is T 500 the number of Monte Carlo replications is 1000 and 1 The GMM estimator is based on the moment conditions. E yt yt 1 2 E yt2 1 2 2 E yt2 yt 1 2 3 3 E yt3 1 3 3 3 E yt yt 1. 3 3 The infeasible GMM estimator is based on the same set of moment conditions. but with 1 assumed known Both GMM estimators use the optimal weighting matrix based on the Newey West HAC estimator with automatic lag selection. that our proposed GMM estimator has properties similar to functions of the standardized GMM estimator t statistic of. this infeasible estimator The third is the Gaussian quasi ML for the MA 1 model with GLD errors and a skewness parameter. estimator of 2 with invertibility imposed which is used of 0 85 strong identification The sample size is T 3000 and. to evaluate the efficiency losses of the GMM estimator for 0 5 1 1 5 and 2 Overall the densities of the standardized. values of in the invertible region 0 5 and 0 7 GMM estimator appear to be very close to the standard normal. The results in Table 1 suggest that regardless of the degree density for all values of The coverage probabilities of the 90. of non Gaussianity the infeasible estimator produces estimates confidence intervals for 0 5 0 7 1 1 5 and 2 are 91 8. of that are very precise and essentially unbiased Hence fix 90 5 92 6 89 5 and 92 9 respectively. ing solves both identification problems without the need of. non Gaussianity although a prior knowledge of is rarely avail. 5 SIMULATION BASED ESTIMATION, able in practice By construction the Gaussian QML estimator. imposes invertibility and works well when the true MA param A caveat of the GMM estimator is that it relies on pre. eter is in the invertible region but cannot identify the parameter cise estimation of the higher order unconditional moments but. values in the noninvertible region While for 3 0 35 the iden finite sample biases can be nontrivial even for samples of mod. tification is weak and the estimates of are somewhat biased erate size This can be problematic for GMM estimation of. for higher values of the skewness parameter the GMM estimates ARMA p q models since a large number of higher order terms. of are practically unbiased needs to be estimated To remedy these problems we consider. Table 1 also presents the empirical probability that the partic the possibility of using simulation to correct for finite sample. ular estimator of is greater than or equal to one which provides biases see Gourieroux Renault and Touzi 1999 Phillips 2012. information on how often the identification of the true parameter Two estimators are considered The first is a simulation analog. fails The Gaussian QML estimator is characterized by a pile of the GMM estimator and the second is a simulated minimum. up probability at unity which can be inferred from P 1 distance estimator that uses auxiliary regressions to efficiently. when 0 1 as argued before Even when 3 0 35 the GMM incorporate information in the higher order cumulants into a. estimator correctly identifies if the true value of is in the in parameter vector of lower dimension Both estimators can ac. vertible or the noninvertible region with high probability This commodate additional dynamics kurtosis and other features of. probability increases when 3 0 85 the errors, Finally to assess the accuracy of the asymptotic normal Simulation estimation of the MA 1 model was considered in. ity approximation in Proposition 1 Figure 1 plots the density Gourieroux Monfort and Renault 1993 Michaelides and Ng. 410 Journal of Business Economic Statistics July 2015. Downloaded by Columbia University at 05 35 20 July 2015. Figure 1 Density functions of the standardized GMM estimator t statistic of based on data T 3000 generated from an MA 1 model. yt et et 1 with 0 5 1 1 5 2 and et iid 0 1 The errors are drawn from a generalized lambda distribution with zero excess kurtosis. and a skewness parameter equal to 0 85 For the sake of comparison the figure also plots the standard normal N 0 1 density. 2000 Ghysels Khalaf and Vodounou 2003 and Czellar and beta gamma and Weibull distributions The second advantage. Zivot 2008 among others but only for the invertible case All is that it is easy to simulate from The percentile function is. of these studies use an autoregression as the auxiliary model For given by. 0 5 and assuming that 2 is known Gourieroux Monfort. and Renault 1993 found that the simulation based estimator u 1 1 U 3 1 U 4 2 15. compares favorably to the exact ML estimator in terms of bias. and root mean squared error Michaelides and Ng 2000 and where U is a uniform random variable on 0 1 1 is a lo. Ghysels Khalaf and Vodounou 2003 also evaluated the prop cation parameter 2 is a scale parameter and 3 and 4 are. erties of simulation based estimators with 2 assumed known shape parameters To simulate t a U is drawn from the uni. Czellar and Zivot 2008 reported that the simulation based es form distribution and 15 is evaluated for given values of. timator is relatively less biased but exhibits some instability and 1 2 3 4 Furthermore the shape parameters 3 4 and. the tests based on it suffer from size distortions when 0 is close the location scale parameters 1 2 can be sequentially eval. to unity see also Tauchen 1998 for the behavior of simulation uated Since t has mean zero and variance one the parameters. estimators near the boundary of the parameter space 1 2 are determined by 3 4 so that t is effectively char. acterized by 3 and 4 As shown in Ramberg and Schmeiser. 1975 the shape parameters 3 4 are explicitly related to. 5 1 The GLD Error Simulator the coefficients of skewness and kurtosis 3 and 4 of t see. the Appendix A consequence of having to use the GLD to. The key to identification is errors with non Gaussian features simulate errors is that the parameters 3 and 4 of the GLD. Thus in order for any simulation estimator to identify the pa distribution must now be estimated along with the parameters. rameters without imposing invertibility we need to be able to of yt even though these are not parameters of interest per se In. simulate non Gaussian errors t in a flexible fashion so that yt practice these GLD parameters are identified from the higher. has the desired distributional properties order moments of the residuals from an auxiliary regression. There is evidently a large class of distributions with third. and fourth moments consistent with a non Gaussian process. that one can specify Assuming a particular parametric error 5 2 The SMM Estimator. distribution could compromise the robustness of the estimates. We simulate errors from the GLD 1 2 3 4 considered Define the augmented parameter vector of the MA 1 model. in Ramberg and Schmeiser 1975 This distribution has two as 2 3 4 Our SMM estimator is based on. appealing features First it can accommodate a wide range of. values for the skewness and excess kurtosis parameters and it g mt 0 m 16. includes as special cases normal log normal exponential t T t 1 T S t 1 t. Gospodinov and Ng Minimum Distance Estimation of Possibly Noninvertible MA Models 411. where mt 0 is evaluated on the observed data y based on inversion of the distance metric test without an ex. y1 yT and mSt is evaluated on the data yS plicit computation of the variance matrix Avar It should be. y1S yTS yTS S of length T S S 1 simulated for a stressed that despite the choice of a flexible functional distribu. candidate value of Essentially the quantity m which tional form for the error simulator our structural model is still. is chosen to summarize the dependence of the model on the correctly specified This is in contrast with the semiparametric. parameters is approximated by Monte Carlo methods indirect inference estimator of Dridi Guay and Renault 2007. It remains to define mt 0 In contrast to GMM estimation They considered partially misspecified structural models and. we now need moments of the innovation errors to identify 3 thus required an adjustment in the asymptotic variance of the. and 4 The latent errors are approximated by the standardized estimator. residuals from estimation of an AR p model,yt 0 1 yt 1 p yt p t 5 3 The SMD Estimator.
For the MA 1 model the moment conditions given by Higher order MA q models and general ARMA p q mod. els can in principle be estimated by GMM or SMM But. mt 0 yt yt 1 yt2 yt2 yt 1 yt3 yt yt 1,yt3 yt 1 yt yt 1. as mentioned earlier the number of orthogonality conditions. yt4 t3 t4 17 increases with p and q Instead of selecting additional mo. ment conditions we combine the information in the cumu. reflect information in the second third and fourth order cu lants into the auxiliary parameters that are informative about. mulants of the process yt as well as skewness and kurtosis of 0 arg min QT y and. the parameters of interest Let, Downloaded by Columbia University at 05 35 20 July 2015. the errors arg min QT y be the auxiliary parameters. To establish the consistency and asymptotic normality of the estimated from actual and simulated data QT denotes the ob. SMM estimator we need some additional notation and reg. jective function of the auxiliary model and is a consistent. ularity conditions Let Fe denote the true distribution of the estimate of the asymptotic variance of Our simulated mini. structural model errors and be the class of GLDs mum distance SMD based on. Proposition 2 Consider the MA 1 model and let,limT var T g In addition to the assumptions in. Lemma 1 assume that Fe E et 8 sup G shares the same asymptotic properties as the SMM estimator in. G 0 0 is in the interior of the compact parameter Proposition 2 The SMD estimator is in the spirit of the indi. d rect inference estimation of Gourieroux Monfort and Renault. space and T g 0 N 0 Then, 1993 and Gallant and Tauchen 1996 Their estimators require. that the auxiliary model is easy to estimate and that the mapping. T 0 N 0 1 1, S from the auxiliary parameters to the parameters of interest is.
well defined We use such a mapping to collect information in. the unconditional cumulants into a lower dimensional vector of. auxiliary parameters to circumvent direct use of a large number. Consistency follows from identifiability of and the higher of unconditional cumulants. order cumulants play a crucial role In our procedure 3 and 4 We consider least square LS estimation of the auxiliary. are defined in terms of 3 and 4 Thus 3 and 4 are crucial for regressions. identification of and 2 even though they are not parameters. of direct interest yt 0 1 yt 1 p yt p t 19a, A key feature of Proposition 2 is that it holds when is less yt2 c0 c1 1 yt 1 c1 r yt r 2. than equal to or greater than one In a Gaussian likelihood. setting when invertibility is assumed for the purpose of iden c2 r yt r. tification there is a boundary for the support of at the unit. circle Thus the likelihood based estimation has nonstandard with an appropriate choice of p and r Equation 19a has been. properties when the true value of is on or near the boundary used in the literature for simulation estimation of MA 1 mod. of one In our setup this boundary constraint is lifted because els when invertibility is imposed and often with 2 assumed. identification is achieved through higher moments instead of known We complement 19a with the regression defined in. imposing invertibility As a consequence the SMM estimator 19b The parameters of this regression parsimoniously sum. has classical properties provided that and enable iden. 3 4 marize information in the higher moments of the data Compared. tification to the SMM in which the auxiliary parameters are unconditional. Consistent estimation of the asymptotic variance of can moments the auxiliary parameters are based on conditional. proceed by substituting a consistent estimator of and evalu moments Equation 19b also provides a simple check for the. ating the Jacobian G numerically The computed standard prerequisite for identification If the c coefficients are jointly. errors can then be used for testing hypotheses and constructing zero identification would be in jeopardy. confidence intervals Inference on the MA parameter of interest Let 3 and. 4 denote the sample third and fourth moments, can also be conducted by constructing confidence intervals of the ordinary LS OLS residuals in 19a The auxiliary. 412 Journal of Business Economic Statistics July 2015. Table 2 SMM and SMD estimates of from MA 1 model with asymmetric errors. Mean Med P,1 Std Mean Med P, 0 0 5 0 488 0 484 0 001 0 054 0 503 0 503 0 000 0 043. 0 0 7 0 693 0 688 0 000 0 083 0 705 0 703 0 002 0 053. 0 1 0 0 949 0 988 0 421 0 137 0 973 0 982 0 406 0 089. 0 1 5 1 563 1 520 0 962 0 280 1 482 1 493 0 980 0 104. 0 2 0 1 903 1 959 0 940 0 337 1 996 1 995 0 988 0 180. 0 0 5 0 437 0 426 0 013 0 064 0 549 0 479 0 062 0 055. 0 0 7 0 648 0 636 0 006 0 099 0 748 0 687 0 143 0 059. 0 1 0 0 929 0 959 0 318 0 160 1 068 1 087 0 790 0 117. 0 1 5 1 573 1 561 0 940 0 274 1 483 1 486 0 983 0 113. 0 2 0 1 861 1 956 0 883 0 374 1 926 1 924 0 978 0 240. NOTES The table reports the mean median med probability that 1 and standard deviation std of the SMM estimates of from the MA 1 model yt et et 1 where. et t t t iid 0 1 are generated from a generalized lambda distribution GLD with a skewness parameter 3 0 85 and no excess kurtosis and t 1 or t2 0 7 0 3et 1. The sample size is T 500 and the number of Monte Carlo replications is 1000 The SMM estimator is based on the moment conditions mSMM t defined in 17 and the SMD estimator. is based on the auxiliary parameter vector SMD defined in 19c The SMM and SMD estimators use the optimal weighting matrix based on the Newey West HAC estimator. Downloaded by Columbia University at 05 35 20 July 2015. parameter vector based on the data is 5 4 Finite Sample Properties of the Simulation Based. Estimators, To implement the SMM and SMD estimators we simulate T S. errors from the generalized lambda error distribution Larger. The parameter vector S is analogously defined except values of S the number of simulated sample paths of length. that the auxiliary regressions are estimated with data simulated T tend to smooth the objective functions which improves the. for a candidate value of identification of the MA parameter As a result we set S 20. Figure 2 Logarithm of the objective function of simulation based estimator of and based on data T 1000 generated from an MA 1. model yt et et 1 with 0 7 and et iid 0 1 The errors are drawn from a generalized lambda distribution with zero excess kurtosis. and a skewness parameter equal to 0 0 35 0 6 and 0 85. Gospodinov and Ng Minimum Distance Estimation of Possibly Noninvertible MA Models 413. Table 3 SMD SMM and Gaussian QML estimates of and from an ARMA 1 1 model with exponential mixture of normals errors. Errors Estimator Mean Med Std P,1 Mean Med Std,Exponential errors.
0 1 5 0 0 5,SMD 1 552 1 489 0 544 0 954 0 493 0 501 0 162. SMM 1 497 1 480 0 378 0 994 0 496 0 504 0 109, Gaussian QML 0 652 0 686 0 206 0 000 0 482 0 511 0 217. SMD 2 039 2 001 0 626 0 976 0 483 0 490 0 134,SMM 1 919 1 958 0 648 0 967 0 473 0 501 0 194. Gaussian QML 0 011 0 010 0 571 0 000 0 011 0 003 0 567. Mixture errors,0 1 5 0 0 5,SMD 1 501 1 480 0 415 0 967 0 505 0 516 0 137. SMM 1 277 1 379 0 671 0 805 0 444 0 512 0 319, Gaussian QML 0 660 0 688 0 168 0 000 0 487 0 510 0 186.
SMD 1 728 1 723 0 498 0 978 0 570 0 580 0 191, Downloaded by Columbia University at 05 35 20 July 2015. SMM 1 537 1 678 1 015 0 785 0 457 0 516 0 380, Gaussian QML 0 012 0 003 0 563 0 000 0 009 0 001 0 558. NOTES The table reports the mean median med standard deviation std and the probability that P. 1 of the SMD SMM and Gaussian QML estimates of and from the. ARMA 1 1 model 1 L yt 1 L et where et t and t is an exponential random variable with a scale parameter equal to one exponential errors or a mixture of normals. random variable with mixture probabilities 0 1 and 0 9 means 0 9 and 0 1 and standard deviations 2 and 0 752773 respectively mixture errors The exponential errors are recentered. and rescaled to have mean zero and variance one The sample size is T 500 and the number of Monte Carlo replications is 1000. although S 20 seems to offer even further improvement es distribution the SMM estimator of exhibits only a small bias. pecially for small T but at the cost of increased computational for some values of e g 0 2 While there is a positive. time The SMM and SMD estimators both use p 4 SMD probability that the SMM estimator will converge to 1 in. additionally assumes r 1 in the auxiliary model 19b stead of especially when is in the noninvertible region. As is true of all nonlinear estimation problems the numerical this probability is fairly small and it disappears completely for. optimization problem must take into account the possibility of larger T not reported to conserve space When the error distri. local minima which arises when the invertibility condition is bution is misspecified GLD errors with ARCH structure the. not imposed Thus the estimation always considers two sets properties of the estimator deteriorate the estimator exhibits a. of initial values Specifically we draw two starting values for larger bias but the invertible noninvertible values of are still. one from a uniform distribution on 0 1 and one from a identified with high probability However the SMD estimator. uniform distribution on 1 2 with the starting value for set provides a substantial bias correction efficiency gain and iden. equal to y2 1 2 for each of the starting values for The tification improvement Interestingly in terms of precision the. SMD estimator appears to be more efficient than the infeasible. starting values for the shape parameters of the GLD 3 and 4. estimator in Table 1 for values of in the invertible region The. are set equal to those of the standard normal distribution with. SMD estimator continues to perform well even when the error. 3 0 and 4 3 In this respect the starting values of. simulator is misspecified, 3 and 4 contain little prior knowledge of the true parameters. Figure 2 illustrates how identification depends on skewness. MA 1 In the first experiment data are generated from. by plotting the log of the objective function for the SMD estima. yt et et 1 et t t tor averaged over 1000 Monte Carlo replications of the MA 1. model with 0 7 and 1 The errors are generated from. where t iid 0 1 is drawn from a GLD with zero excess GLD with zero excess kurtosis and three values of the skewness. kurtosis and a skewness parameter 0 85 with i t 1 parameter 0 0 35 0 6 and 0 85 In evaluating the objective. or ii t 0 7 0 3et 1, ARCH errors The sample size is function the values of the lambda parameters in the GLD are. T 500 the number of Monte Carlo replications is 1000 and set equal to their true values The first case no skewness cor. takes the values of 0 5 0 7 1 1 5 and 2 Note that the responds to lack of identification and there are two pronounced. structural model used for SMM and SMD does not impose the local minima at and 1 As the skewness of the error distri. ARCH structure of the errors that is the error distribution is bution increases the second local optima at 1 flattens out and. misspecified This case is useful for evaluating the robustness it almost completely disappears when the error distribution is. properties of the proposed SMM and SMD estimators highly asymmetric. Table 2 reports the mean and median estimates of the ARMA 1 1 In the second simulation experiment data are. standard deviation of the estimates for which identification is generated according to. achieved and the probability that the estimator is equal to or. greater than one When the errors are iid drawn from the GLD yt yt 1 et et 1 20. 414 Journal of Business Economic Statistics July 2015. where et is i a standard exponential random variable with a portfolios formed on the ratio of book equity to market equity. scale parameter equal to one which is recentered and rescaled The size book to market breakpoints are the NYSE quintiles. to have mean zero and variance 1 or ii a mixture of normals and are denoted by small 2 3 4 big low 2 3 4 high in. random variable with mixture probabilities 0 1 and 0 9 means Table 4. 0 9 and 0 1 and standard deviations 2 and 0 752773 respec Table 4 presents the sample skewness and kurtosis as well as. tively The second error distribution is included to assess the the estimates and the corresponding standard errors in paren. robustness properties of the simulation based estimator to error theses below the estimate for each estimator and portfolio re. distributions that are not members of the GLD family turn All of the returns exhibit some form of non Gaussianity. We consider two parameterizations that give rise to a causal which is necessary for identifying possible noninvertible MA. process with a noninvertible MA component The first param components The Gaussian QML produces estimates of the MA. eterization is 0 5 and 1 5 The second parameteri coefficient that are small but statistically significant with a few. zation 0 5 and 2 produces an all pass ARMA 1 exceptions in the big size category The SMM relaxes the. 1 process which is characterized by 1 This all pass invertibility constraint and delivers somewhat higher estimates. process possesses some interesting properties see Davis 2010 of the MA parameter but most of these estimates still fall in the. First yt is uncorrelated but is conditionally heteroscedastic Sec invertible region By contrast the SMD estimator suggests that. ond if one imposes invertibility by letting and scale up all of the 25 Fama French portfolio returns appear to be driven. the error variance by 1 2 the process is iid and the AR and. MA parameters are not separately identifiable Imposing invert. ibility in such a case is not innocuous and estimation of the Table 4 SMD SMM and Gaussian QML estimates of MA 1 model. for stock portfolio returns, parameters of this model is quite a challenging task.
Downloaded by Columbia University at 05 35 20 July 2015. Table 3 presents the finite sample properties of the SMD and Skewness Kurtosis QML SMM SMD. SMM estimators for the ARMA 1 1 model in 20 using the. same auxiliary parameters and moment conditions for the esti Low 0 039 5 244 0 155 4 711 4 325. 0 028 0 650 0 470, mation of MA 1 For comparison we also include the Gaussian 2 0 030 6 136 0 160 0 273 4 043. 0 027 0 028 0 417, quasi ML estimator The SMD estimates of appear unbiased Small 3 0 132 5 889 0 179 0 287 3 802. for the exponential distribution and are somewhat downward 0 032 0 027 0 348. biased for the mixture of normals errors But overall the SMD 4 0 164 6 131 0 180 4 754 4 092. 0 034 0 538 0 455, estimator identifies correctly the AR and MA components with High 0 208 6 464 0 241 3 368 2 944. 0 032 0 349 0 254, high probability The performance of the SMM estimator is Low 0 318 4 677 0 144 0 212 3 694. also satisfactory but it is dominated by the SMD estimator The 0 032 0 027 0 289. Gaussian QML estimator imposes invertibility and completely 2 0 419 5 551 0 143 0 219 3 880. 0 035 0 697 0 384, fails to identify the AR and MA parameters when 0 5 2 3 0 458 6 105 0 153 0 251 3 763.
0 035 0 026 0 312, and 2 Even with a misspecified error distribution and 4 0 439 6 148 0 156 0 241 4 120. 0 035 0 025 0 394, a fairly parsimonious auxiliary model the finite sample proper. High 0 414 6 186 0 166 0 232 3 745, ties of our proposed simulation based estimators remain quite 0 030 0 027 0 306. attractive Low 0 371 4 701 0 117 0 178 3 001,0 030 0 022 0 162. 2 0 506 5 936 0 151 0 278 3 702,0 035 0 022 0 323, 5 5 Empirical Application 25 Fama French Portfolio 3 3 0 510 5 324 0 146 4 884 3 553.
0 034 0 386 0 272,Returns 4 0 276 5 314 0 142 0 246 3 537. 0 034 0 026 0 283,Noninvertibility can be consistent. with economic theory For High 0 305 6 081 0 154 4 981 3 875. example suppose yt Et s 0 033 0 488 0 340, s 0 xt s is the present value of Low 0 234 4 933 0 104 0 168 3 338. xt et et 1 As shown by Hansen and Sargent 1991 0 033 0 022 0 180. the solution yt 1 et et 1 h L et implies that the 2 0 585 6 135 0 143 0 203 3 649. 0 034 0 022 0 416,root of h z is 1, which can be on or inside the unit cir 4 3 0 503 6 348 0 140. cle even if 1 If there is no discounting and 1 yt 4 0 231 4 930 0 092 0 212 4 045. has an MA unit root when 0 5 and h L is noninvertible 0 035 0 020 0 300. High 0 193 5 385 0 118 0 244 4 680, in the past whenever 0 5 Note that even if an autore 0 032 0 021 0 405.
gressive processes is causal it is still possible for the roots of Low 0 253 4 565 0 065 0 106 5 078. 0 030 0 028 0 799, to be inside the unit disk 2 0 362 4 677 0 052 0 157 5 602. 0 034 0 024 0 686, Present value models are used to analyze variables with a for. Big 3 0 264 5 209 0 035 0 100 6 457, ward looking component including stock and commodity prices 0 031 0 029 1 119. We estimate an MA 1 model for each of the 25 Fama French 4 0 168 4 608 0 025 0 125 6 206. 0 032 0 021 1 140, portfolio returns using the Gaussian QML and the proposed High 0 200 4 002 0 072 0 140 4 803. 0 032 0 020 0 611, SMM and SMD estimators The data are monthly returns on.
NOTES The table reports the SMD SMM and Gaussian quasi ML estimates and standard. the value weighted 25 Fama French size and book to market errors in parentheses below the estimates for the MA 1 model yt et et 1 where. ranked portfolios from January 1952 until August 2013 from et iid 0 2 and yt is one of the 25 Fama French portfolio returns The first two columns. report the sample skewness and kurtosis of yt The standard errors for SMM and SMD are. Kenneth French s website The portfolios are the intersec constructed using the asymptotic approximation in Proposition 2. tions of five portfolios formed on size market equity and five. Gospodinov and Ng Minimum Distance Estimation of Possibly Noninvertible MA Models 415. by a noninvertible MA component The results are consistent Identification of MA q Models Using Third Order Cumulants The. with the finding through simulations that the SMD is more capa system of equations can be expressed as A b where. ble of estimating in the correct invertibility space The SMD. estimates are fairly stable across the different portfolio returns A C. with a slight increase in their magnitude and standard errors for. the big size portfolios Also a higher precision of the MA. estimates is typically associated with returns that are character 0 0 0. ized by larger departures from Gaussianity Overall our SMD 3 0. method provides evidence in support of noninvertibility in stock c3 q 1 c3 q 0. c3 q 1 c3 q 2 c3 0,6 CONCLUSIONS,0 c2 q c2 q 1 c2 1. This article proposes classical and simulation based GMM 0 0 c2 q c2 2. estimation of possibly noninvertible MA models with non E. Gaussian errors The identification of the structural parameters. is achieved by exploiting the non Gaussianity of the process 0 0 0 c2 q. through third order cumulants This type of identification also. c3 q c3 q 1 c3 1, removes the boundary problem at the unit circle which gives 0. Downloaded by Columbia University at 05 35 20 July 2015. rise to the pile up probability and nonstandard asymptotics. of the Gaussian maximum likelihood estimator As a conse. quence the proposed GMM estimators are root T consistent 0 0 c3 q. and asymptotically normal over the whole parameter range pro. vided that the non Gaussianity in the data is sufficiently large to c2 q 0 0 0. c q 1 c2 q 0,ensure identification 2 0, Other research questions arise once the assumption of invert c2 q 2 c2 q 1 c2 q 0. ibility is relaxed A potential problem with the GMM estimator. is that the number of orthogonality conditions can be quite large. This is especially problematic for ARMA p q models Ideally c2 0 c2 1 c2 2 c2 q. the orthogonality conditions should be selected or weighted in c2 1 c2 0 c2 1 c2 q 1. an optimal fashion More generally how to determine the lag. length of the heteroskedasticity and autocorrelation consistent. HAC estimator without imposing invertibility remains a topic c2 q c2 q 1 c2 q 2 c2 0. for future research The rank of the 3q 1 2q 1 matrix A is the sum of the column. rank of the submatrix consisting of B and D and the rank of the. submatrix consisting of C and E The rank of the first subblock is. APPENDIX PROOFS determined by the rank of the q q square matrix D which is q if. c3 q 0 The rank of C is determined by the rank of the square. Proof of Lemma 1 The result in part a follows im matrix C1 which is q 1 if c2 q 0 The full rank result follows. mediately by noticing that g 1 and g 2 where g from the assumption that c2 q and c3 q are nonzero. E yt yt 1 E yt2 E yt2 yt 1 are observationally equivalent for 1 Since c2 q 0 and c3 q 0 from the assumptions of Lemma 2. 2 3 and 2 1 2 2 3 For part b let us define the the triangular matrices C1 and D have column ranks of q 1 and q. derivative matrix of g g2 g3 as respectively Therefore A has a full column rank of 2q 1 and the. parameter vector can be obtained as a unique solution to the. system of Equation 8 Since the derivative matrix of given by. 1 2 0 1 0 0 0,G 2 0 0 1 0,3 2 3 3 1 3 3 1 3 3, with G 1 2 i for i 3 4 or 5 denoting its corresponding 3 3 block 3 1 0 0 1. Direct calculations of the determinants give G 1 2 3 1 2 2 5 0 2 3 2 0 22. G 1 2 4 1 2 1 3 5 and G 1 2 5 1 2 5 which,are all zero at 1.
0 0 3 q q2, Proof of Lemma 2 Giannakis and Mendel 1989 solved from the. system of overdetermined equations but did not establish uniqueness is of full column rank the parameter vector of interest. of the solution The argument for identification from third and fourth 1 q 3 is identifiable. order cumulants i e Equation 7 and fourth order cumulants i e Identification of MA q Models Using Fourth Order Cumulants The. Equation 9 are similar We begin with 7 MA q model implies the following relation between the diagonal slices. 416 Journal of Business Economic Statistics July 2015. of the fourth order cumulants and the q 1 vector of parameters The GLD Distribution The two parameters 3 4 are related to 3 and. 1 q 2 4 3 4 as follows see Ramberg and Schmeiser 1975. c 3ab 2a 3,i c4 i 4 3,i3 c2 i c4 0 32,d 4ac 6a 2 b 3a 4. q 2q A 1 4,Define where a 1,3 1 4 2 B A 2 c 1 3 3 3Beta 1 2 3 1. 1 q 2 4 3 2 4 3 13 4 3Beta 1 3 1 2 4 1 3 1,1 3 3 1 4 6Beta 1 2 3 1 2 4 4Beta 1 3. and Beta denotes the beta function,b c4 q c4 q 1 c4 0 c4 1.
c4 q 1 c4 q 0 0 0,0 0 0 c2 q 0 0 0,c q c2 q 1 c2 q 0. c4 q 1 c4 q c2 q 2 c2 q 1 c2 q 0, Downloaded by Columbia University at 05 35 20 July 2015. c 4 q 1 c4 q 2 c4 0 c2 q c2 q 1 c2 q 2 c2 0,c q c4 q 1 c4 1 c2 q c2 q 1 c2 1. 0 c4 q c4 2 0 0 c2 q c2 2,0 0 c4 q 0 0 0 c2 q, Then the system of Equation A 1 can be expressed as. A b ACKNOWLEDGMENTS, The authors thank the Editor an Associate Editor two anony.
and the identification of and follows similar arguments as those. for the third order cumulants mous referees Prosper Dovonon Anders Bredahl Kock Ivana. Komunjer and the participants at the CESG meeting at Queen s. University for useful comments and suggestions The second. Proof of Lemma 3 The proof follows some of the arguments in the author acknowledges financial support from the National Sci. proof of Theorem 1 in Tugnait 1995 Consider two ARMA p q mod ence Foundation SES 0962431 The views expressed here are. els 1 L yt 1 L et and 2 L yt 2 L et which can be rewritten. as zt 2 L 1 L et 1 L et and zt 1 L 2 L et 2 L et the authors and not necessarily those of the Federal Reserve. where zt a1 L a2 L yt Let 1 1 1 1 p 1 1 1 q and Bank of Atlanta or the Federal Reserve System. 2 2 1 2 p 2 1 2 q Note that zt is an MA p q pro, cess since 1 2 L 1 L and 2 1 L 2 L are polynomi Received March 2013 Revised May 2014. als of order p q As in Lemma 2 we can write,A 1 1 3 b References. Akaike H 1966 Note on Higher Order Spectra Annals of the Institute of. Statistical Mathematics 18 123 126 407, where A and b are functions of second and third cumulants of zt But Anderson T W and Takemura A 1986 Why Do Noninvertible Esti. from Lemma 2 there exists a unique solution to the system of equations mated Moving Average Models Occur Journal of Time Series Analysis 7. A 3 b Hence there is a one to one mapping between A b 235 254 405. and 3 and the two ARMA models are identical in the sense that Andrews B Davis R and Breidt F J 2006 Maximum Likelihood Esti. 1 2 Therefore 1 p 1 q is identifiable from mation of All Pass Time Series Models Journal of Multivariate Analysis. 97 1638 1659 403, the second and third cumulants used in constructing A and b provided. 2007 Rank Based Estimation of All Pass Time Series Models The. that c2 p q 0 and c3 p q 0 Annals of Statistics 35 844 869 403. Brockwell P J and Davies R A 1991 Time Series Theory and Methods. 2nd ed New York Springer Verlag 404, Proof of Proposition 1 The results in Section 3 ensure global and local Czellar V and Zivot E 2008 Improved Small Sample Inference for Efficient.
identifiability of 0 The consistency of, follows from the identifia Method of Moments and Indirect Inference Estimators Mimeo University. bility of 0 and the compactness of Taking a mean value expansion of Washington 410. of the first order conditions of the GMM problem and invoking the Davis R 2010 All Pass Processes With Applications to Finance 7th Inter. central limit theorem deliver the desired asymptotic normality result national Iranian Workshop on Stochastic Processes Tehran Iran Plenary.


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