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. 404 Journal of Business amp Economic Statistics July 2015 models in which the roots of the autoregressive polynomial are reciprocals of the roots of the MA polynomials

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