[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83770-en":3,"doc-seo-83770-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83770,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Likelihood Geometry of Moving Average and Autoregressive Processes","This paper studies maximum likelihood estimation for moving average (MA) time series models through the lens of algebraic statistics and algebraic geometry, emphasizing the structure and count of solutions to the likelihood equations. It classifies critical points leading to non-invertible MA models and analyzes composite likelihood alternatives. The framework is extended to autoregressive (AR) processes, with algebraic closed-form parameter formulas when possible, and simulation comparisons between numerical algebraic geometry and traditional optimization methods.","arXiv :2607 .04046v1 [math . ST] 4 Jul 2026  \nLikelihood Geometry of Moving Average and Autoregressive Processes  \nCarlos Am´endola Gabriel Riffo  \nAbstract  \nWe study the problem of maximum likelihood estimation for moving average (MA) time series models from the perspective of algebraic statistics, with a focus on the structure and number of solutions of their likelihood equations. Of particular interest is to classify the critical points that lead to non-invertible models. We consider the composite likelihood as an alternative estimation method and analyze its critical points. We extend our algebraic analysis to autoregressive processes (AR) . We provide algebraic closed form formulas for the parameters when possible. We also explore in simulations how methods from numerical algebraic geometry perform against traditional optimization for these models.  \n1 Introduction  \nMoving average (MA) and autoregressive (AR) models play a central role in time series analysis due to their wide range of applications, including modeling financial returns [BJRL15], analyzing climate and environmental data [Wil11], and processing engineering signals [OSB99] . From an algebraic perspective, [AP22] introduced the concept of autocovariance varieties for MA processes, which represent all possible autocovariance vectors generated by a process of fixed order. This algebraic-geometric viewpoint provides a natural framework for studying parameter identifiability, analyzing the structure of critical points, and enumerating solutions to the maximum likelihood equations. Similarly, AR processes can be analyzed using their autocovariance structure and characteristic polynomials, allowing for a unified algebraic treatment of parameter estimation problems.  \nDefinition 1.1 . Let q ∈ N. A moving average process of order q, denoted by MA(q), is defined as a sequence of random variables (Yt)t∈Z given by  \nq  \nYt =X ak Zt−k ,  \nk=0  \nwhere ak ∈ R and (Zt)t∈Z are i.i. d standard normal random variables.  \nIn other words, each observation Yt is a linear combination of q + 1 finitely many innovations. The autocovariance function of the process, γ : Z → R is  \n􀀸  \nγ (h) := Cov(Yt+h, Yt ) = Cov(Yh , Y0 ) =􀀼  \n􀀺  \nPqk0|h| akak+h 0  \nif 0 ≤ |h| ≤ q  \nif |h| > q.  \nFormally, the process is said to be invertible if it admits a representation as an autoregressive process of infinite order. To characterize invertibility, one can associate to the MA(q) process the characteristic polynomial  \nq  \nΘ(x) =X ak xk , (1)  \nk=0  \nwhose roots z 1 , . . . , zq determine the invertibility of the process via the spectral factorization identity  \nσ 2 Θ(x)Θ(x−1) =X γ(t) xt.  \nt∈Z  \nSince Θ(x)Θ(x−1) is invariant under the substitution zi →7 z1 , any root configuration yieldsan equivalent autocovariance structure. The process is invertible under the ai parametrization if all roots lie outside the unit disk, i.e. , |zi | > 1 for all i. The process still admits an invertible representation as long as all roots satisfy |zi |  1. Consequently, the process is non-invertible if and only if at least one root satisfies |zi | = 1 .  \nAutoregressive processes are defined as follows.  \nDefinition 1.2 . Let p ∈ N. An autoregressive process of order p, denoted by AR(p), is defined as a sequence of random variables (Xt)t∈Z given by  \np  \nXt =X ϕiXt−i + σZt , (2)  \ni=1  \nwhere ϕi ∈ R, σ > 0 and (Zt)t∈Z are i.i. d standard normal random variables.  \nEach Xt depends linearly on its p previous values, and its autocovariance function satisfies the classical Yule–Walker equations [BD09], which provide a direct link between the parameters ϕ = (ϕ1 , . . . , ϕp ) and the covariance structure. The process is always causal (invertible), but some parameter choices make the process non-stationary [BD09] .  \nIn both settings, estimation of the parameters from observed data is a central problem. Let y := (y1 , . . . , yn)⊤ and x := (x1 , . . . , xn)⊤ denote finite samples for t = 1 , 2 ,   , n from MA(q) and AR(p","cbCaiiAY7yMPtbxW","https://ap.wps.com/l/cbCaiiAY7yMPtbxW","pdf",1091548,5,1,37,"English","en",105,"# Introduction\n## MA(q) and invertibility via characteristic polynomials\n## AR(p) and stationarity considerations\n## Maximum likelihood formulation and complexity (ML-degree)\n## Algebraic approach and computational tools","[{\"question\":\"What is the main focus of the paper on MA models?\",\"answer\":\"The paper focuses on maximum likelihood estimation for moving average (MA) time series using algebraic statistics, specifically studying the structure and number of solutions to the likelihood equations and classifying critical points that produce non-invertible models.\"},{\"question\":\"How does the paper characterize invertibility for an MA(q) process?\",\"answer\":\"Invertibility is characterized using the roots of the MA characteristic polynomial Θ(x): the process is invertible when all roots lie outside the unit disk (and non-invertible when at least one root satisfies |zi| = 1).\"},{\"question\":\"What extension is made beyond moving average processes?\",\"answer\":\"The paper extends the algebraic analysis to autoregressive processes (AR), applies similar likelihood-equation critical point reasoning, provides closed-form parameter formulas when possible, and compares numerical algebraic geometry methods to traditional optimization in 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is the main focus of the paper on MA models?","Question",{"text":76,"@type":77},"The paper focuses on maximum likelihood estimation for moving average (MA) time series using algebraic statistics, specifically studying the structure and number of solutions to the likelihood equations and classifying critical points that produce non-invertible models.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the paper characterize invertibility for an MA(q) process?",{"text":81,"@type":77},"Invertibility is characterized using the roots of the MA characteristic polynomial Θ(x): the process is invertible when all roots lie outside the unit disk (and non-invertible when at least one root satisfies |zi| = 1).",{"name":83,"@type":74,"acceptedAnswer":84},"What extension is made beyond moving average processes?",{"text":85,"@type":77},"The paper extends the algebraic analysis to autoregressive processes (AR), applies similar likelihood-equation critical point reasoning, provides 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