Random autoregressive models: A structured overview
aa r X i v : . [ s t a t . M E ] S e p Random autoregressive models:A structured overview
Marta Regis ([email protected]) ∗ Paulo Serra ([email protected]) ∗∗ Edwin R. van den Heuvel ([email protected]) ∗ ∗ Eindhoven University of Technology, Department of Mathematics and Computer ScienceP.O. Box 513, 5600 MB Eindhoven, The Netherlands. ∗∗ Vrije Universiteit Amsterdam, Department of MathematicsDe Boelelaan 1111, 1081 HV Amsterdam, The Netherlands. ORCiD:0000-0003-4306-8673; email: [email protected] bstract Models characterized by autoregressive structure and random coefficientsare powerful tools for the analysis of high-frequency, high-dimensional andvolatile time series. The available literature on such models is broad, but alsosectorial, overlapping, and confusing. Most models focus on one propertyof the data, while much can be gained by combining the strength of variousmodels and their sources of heterogeneity.We present a structured overview of the literature on autoregressive mod-els with random coefficients. We describe hierarchy and analogies amongmodels, and for each we systematically list properties, estimation methods,tests, software packages and typical applications.
Keywords (Generalized) Random coefficient autoregressive models; (Gener-alized) Autoregressive conditional heteroskedasticity models; Autoregressive paneldata models; Time-series-cross-section models; Random coefficient panel models2
Introduction (Macro)economic studies are often characterized by a large number of units ob-served a relatively small number of times (e.g. population and income shares for alarge number of countries). On the other hand, data-sets in financial studies con-sist of high-frequency, volatile single-unit time series (e.g. daily trading volume ofIBM stock and annual change rates of world GDP). With the increased capabilityof storing information, data-sets are growing in all dimensions with economic vari-ables being monitored at higher frequency or for longer time, and financial timeseries being collected for multiple units. This makes the distinction between thetwo types of data more nuanced, and requires sophisticated statistical methods tohandle multiple sources of heterogeneity.Models have been proposed in the literature of the economic and financialfields to address pooling of short time series and handling volatility respectively.Although the two purposes seem very different, these models share, in most cases,an autoregressive structure and the inclusion of random coefficients. As a resultthese references constitute a very large literature, that is hard to navigate. Thetwo fields are hardly connected and the terminology is not unified, resulting inthe same model being defined differently or different mathematical models beingreferred to in the same way. For instance, the random coefficient autoregressive(RCA) model of Nicholls and Quinn (1982) used in financial applications and therandom coefficient autoregressive model of Liu and Tiao (1980) considered in bi-ological studies share the same name but are different. The first model aims atcapturing volatility in time for a single time series, while the goal of the second isto represent heterogeneity among units under the same modeling assumptions. Onthe other hand, the RCA of Liu and Tiao (1980) and the autoregressive panel datamodel of Nandram and Petruccelli (1997) are in fact the same model, althoughthey do not share the same name. Something similar happens for estimationmethods, that are often defined multiple times with different names, while havingidentical properties. In the introduction to their work, Chandra and Taniguchi(2001) emphasize that various approaches that developed independently in thefields of time series and panel data are often very similar. For instance, theyobserve that “estimating functions and generalized method of moments are essen-tially the same”. On the other hand, they refer to the model in hand as RCA,while it is in fact its generalized version. Some authors have pointed out the riskof confusion. For example Andˇel (1976) highlights the different sampling of therandom coefficients in the RCA model of Nicholls and Quinn (1982) from thatof Liu and Tiao (1980), while Hsiao (2014) compares models for panel data andRCA models, commenting on certain peculiarities. However, these resources areinsufficient to guide researchers and practitioners across such an extensive litera-ture. Furthermore, methods have also been investigated in theoretical statistical3nd econometric studies, and new developments and applications are now foreseenin medicine, engineering, psychology, sociology and politics. Also in this case,cross-referencing is not always guaranteed, contributing to additional duplication,confusion and fragmentation.With the present work we aim at giving a comprehensive, structured overviewof existing autoregressive models with random coefficients. We fill the gap be-tween two traditionally independent families of models that capture heterogeneityacross units and heterogeneity across time respectively (and can be linked to theeconomic and financial fields accordingly). This creates a solid basis for furtherdevelopments in handling multiple sources of heterogeneity, in line with recentpublications (Horv´ath and Trapani, 2016). For each model we illustrate advan-tages and shortcomings, properties, estimation methods, tests, software packagesand typical applications. We also investigate mutual analogies and differences.To accomplish this task, we define the random autoregressive moving average(RARMA), a unifying structure that captures all other models as particular cases.The RARMA is introduced here for classification purposes, to make the expositionclear. In fact, it provides a unified language to better connect and contrast existingmodels, and by using mathematical rigorousness it shows how well-known randomcoefficient models can be obtained from the general equation, by imposing certainrestrictions. We support this mathematical overview with a graph to visuallyrepresent the hierarchy and a table to have all models and their main characteristicsat a glance. This way, the explanation is visually and mathematically clear, andthus more easily accessible.Our overview guides the reader through the terminology used, the models ex-isting in the literature, and the choice of the most appropriate structure (withrelevant inference, tests and software packages) for specific types of data. Thisway, the RARMA structure creates a solid basis for further exploration and devel-opment of heterogeneous time series models for complex data.The paper is organized as follows: Section 2 presents a consolidated outline,presenting the RARMA and the model families existing in the literature. Thedefinition of the RARMA structure is provided in Section 3 together with itsassumptions. Then, following the structure given in Section 2, we move downthe hierarchy and focus on the two traditionally independent families of models:models for the heterogeneity in time (Section 4), and models for the heterogeneityacross units (Section 5). We state how to obtain each model equation from thegeneral one, gather related contributions from the literature, and discuss the mainproperties and estimation methods for these models. Section 6 is dedicated torecent model structures, capable of addressing both forms of heterogeneity. Finally,we end the work with a discussion in Section 7.4
Unified outline
The literature that we consider here is broad and heterogeneous, and the termi-nology often changes according to the field of application. This commonly causessome disorientation when first coming into contact with this literature, sometimesneedlessly isolating entire fields. Authors often propose already existing definitionsunaware of previous uses, or refer to the same object in different ways. In thissection we provide an overview of the terms which have been used in the literatureto describe both the data and the models. A complete list of all the definitionsintroduced in the past is beyond the scope of this work. However, the informa-tion provided here will considerably ease the reader’s navigation of the existingliterature. We also introduce the terminology that we use in the remainder of thisreview. Since a broad set of definitions already exists, we avoid introducing newterms unless strictly necessary. We stick to the existing nomenclature, even if itis somewhat unclear, and only modify it if the same term is used for two differ-ent objects, or make additions whenever we define a new object. The families ofmodels are then presented with the help of a graph, making the hierarchy clearat a first glance. In the remainder of the overview, the models in the graph areintroduced one by one and discussed.
Data is characterized by three dimensions: the number of units/subjects n , thenumber of outcomes m , and the number of samples T . The relative sizes of thesedimensions can vary significantly, depending on the research study and application.The number of observed units n is typically large in economic and clinicalstudies. Here the focus is on pooling information from the n units, since thenumber of repeats T per unit is often limited. For this type of data, a varietyof terms has been introduced such as panel data (Hsiao, 2014), panel of timeseries (Franses, 2006), time series panel data (Nandram and Petruccelli, 1997) and time-series-cross-section data (Beck et al., 1998). All of these definitions aim tounderline the extension from traditional single unit time series to multiple units.Note however that in social sciences and medical research the term cross-sectional traditionally refers to the analysis of the population at a fixed time point – andthat this clashes with the meaning that this term takes in the context of timeseries. When considering the dimension m of the response variable, the terminologynormally used is uni-dimensional or univariate for m = 1 and multidimensional ormultivariate for m >
1. In some cases, m = 2 is distinguished from the multivariatecase with the term two-dimensional or bivariate . In the literature considered inthe present work, m -dimensional responses are rarely addressed.Finally, the third dimension is the number of observations T . Data with a high5umber of repeats on a single unit is normally the object of interest in finance, andis referred to as time series data . In these applications in fact, the number of unitsis often not mentioned and taken equal to one. The terms intensive/high-frequencylongitudinal also refer to repeated observations, and normally the distinction be-tween longitudinal and time series data is either in the number of repeats, whichis much larger for the latter, or in the number of observed units, which is largerfor longitudinal data.With the increasing amounts of data being collected, the three dimensions areoften all large at the same time, making the choice of the terminology not easy.Thus we propose the use of a very simple nomenclature, addressing the first two di-mensions by single/multiple-units and univariate/multivariate respectively. Then,since all of the data that we consider have more than two repeats, we will callit time series data . In this way, we can have for example a multiple-units uni-variate time series , a single-unit multivariate time series and so on. Furthermore,we propose a shortened version through which one can also specify precisely themagnitude of each dimension, namely ( n, m, T ) − data . Models for the analysis of temporal data are known in econometrics and statis-tics as time series models . In particular, we focus here on autoregressive models ,i.e. models regressing the outcomes on previous values of the same time series. Inthe economic literature, models with this structure are also defined as dynamicpanel data models to be distinguished from static panel data models which donot contain lagged dependent variables. Models used for repeated measurementsrecorded at lower frequency on multiple units are referred to as longitudinal models in statistics.All models treated in the overview are in Table 1, together with their acronyms.In the same table we explain how the definitions used here differ from the onesused in the literature, what the random effects are, and how they are sampled.Furthermore, we specify in which section the model is treated. We distinguish twofamilies of models (the blue dashed cluster on the left hand side, and red dottedbox on the right hand side of Figure 1 respectively) based on how the randomterms are sampled: models for heterogeneity in time have the coefficients that arestochastic processes in time, while models for heterogeneity across units sampletheir coefficients from a certain population described by a probability distribution.These two families do not usually overlap in the literature. The first is typical infinance and developed around the RCA model of Nicholls and Quinn (1982). Thesecond has grown in parallel to accommodate economic variables, has expandedtheoretically in econometrics and statistics, and towards biological and psycho-logical applications. Also the asymptotics vary substantially depending on the6esearch question and type of data in hand: asymptotic behavior of estimators isstudied with respect to the number of observations T −→ ∞ and fixed n when thefocus is on the heterogeneity in time or for increasing number of units n −→ ∞ and fixed amount of repeats when the focus is on pooling many units. Recentmodels addressing both heterogeneity in time and across units (random coefficientpanel data, and dynamic factor models) lie in the overlap of both families. Forthis and few other models, the asymptotic behavior is studied both with respectto time and unit.The acronyms in the nodes of Figure 1 correspond to those listed in Table 1.The arrows linking two nodes point from the more general model to a sub-modelthat has given rise at least to an independent publication, while the dashed lineshighlight looser connections and similarities. We introduce the random autoregres-sive moving average (RARMA) model to gather all the considered models undera single structure. Since the literature on the topic is extremely vast, we havelimited our overview to models fulfilling the following inclusion criteria:1. autoregressive structure,2. at least one parameter is randomly sampled from a distribution and3. discrete-time models.The gray nodes mark the models that are characterized by the three propertiesabove. For completeness we include also some structures which do not satisfy allthree properties, but are closely related to the main topic of the overview. Thesemodels are represented by white nodes in Figure 1, and shortly discussed in theremainder. 7 Random autoregressive models An m -variate time series { Y i,t } for unit i = 1 , . . . , n at time t = 1 , . . . , T i is said tofollow a random autoregressive model of order p – RAR( p ) if it satisfies Y i,t = p X k =1 ˜ g i,t,k Y i,t − k + X i,t β + Z i,t c i + e i,t , (1)where Y i,t and Y i,t − k are the m -vectors of response variables for unit i at time t and t − k respectively. The m × mr and m × ms block diagonal matrices X i,t and Z i,t are the design matrices at time t for the fixed and random effects respec-tively. The mr -vector β = [ β T , . . . , β Tm ] T and the ms -vector c i = [ c Ti, , . . . , c Ti,m ] T represent fixed and random effects, with β j and c i,j the vectors corresponding tothe j th outcome. In particular, c i is sampled from a distribution with zero meanand constant covariance matrix Σ c . The m × mp -matrices of auto- and cross-regression coefficients { ˜ g i,t } constitute a stochastic process indexed by i and t ,where ˜ g i,t = [˜ g i,t, , . . . , ˜ g i,t,p ]. Finally, the noise process { e i,t } is a sequence of m -variate random vectors with mean zero and covariance matrix Σ e i,t . No assumptionis made on serial and mutual correlation between { ˜ g i,t } and { e i,t } , while all therandom variables are independent across units. The random effects c i are alsoindependent of all other random coefficients,Cov ˜ g i,t c i e i,t = Σ g i ge c ge e i,t , with Σ ge a covariance matrix. Furthermore, the initial state Y i, is assumed to beof finite variance for any i .We make here a brief comment about the existence and identifiability of pro-cesses such as (1). Following Nicholls and Quinn (1982) and L¨utkepohl (2005),one can consider the simpler case of a first-order process on one unit since a vectorautoregressive (VAR) process of general order p can always be written as a VAR(1)model (L¨utkepohl, 2005): Y t = ˜ g t Y t − + X t β + Z t c i + e t . Without loss of generality we can set β = 0, and the random effects can be studiedseparately. It remains therefore to examine the process Y t = ˜ g t Y t − + e t . This is a VAR(1) model with a random autoregressive coefficient, so this repre-sentation ensures that if the coefficients ˜ g t almost surely satisfy certain conditions8 cronym Full model name Changes Section Random term HeterogeneityARCH Autoregressive conditional heteroscedasticity - 4.3 - timeARLM Autoregressive linear mixed - 5.1.2 effect ofcovariates unitARP Autoregressive panel data - 5.1.2 intercept unitBVAR Bayesian vector autoregressive - 5.2 AR coef unitCHARMA Conditional heteroscedasticity autoregressivemoving average - 4.4 AR coef timeDFM Dynamic factor - 6.2 AR coef timeGARCH Generalized autoregressive conditionalheteroscedasticity - 4.3 - timeGMB Generalized Markovian bilinear - 4.1.2 - timeGRCA Generalized random coefficient autoregressive - 4.1 AR coef timeHVAR Hierarchical vector autoregressive new 5.3 AR coef, effectof covariates unitRAR Random autoregressive new 3 AR coef, effectof covariates time and unitRARMA Random autoregressive moving average new 3 AR coef, effectof covariates time and unitRCA Random coefficient autoregressive - 4.1.1 AR coef timeRCAC Random coefficient autoregressive withcorrelated terms new 4.2 AR coef timeRCAP Random coefficient autoregressive panel data add paneldata todistinguishfrom RCA 5.1.1 AR coef unitRCARRS Random coefficient autoregressive regimeswitching - 4.5 AR coef timeRCEA Random coefficient exponentialautoregressive - 4.1.3 - timeRCP Random coefficient panel - 6.1 AR coef time and unitTSCS Time-series-cross-sectional - 5.1.3 AR coef unitTVAR Time-varying autoregressive - 4.6 AR coef timeUAR Unit-specific autoregressive new 5.1 AR coef, effectof covariates unit Table 1: For each model: acronyms and corresponding full names, together with the changes made to the originalname and the section in which the model is treated. Random term specifies which coefficients are random andheterogeneity specifies whether the coefficient is random in time and/or across units. ARMARARDFM RCPTVARGRCARCACRCARRSCHARMAGARCH ARCH RCA GMB RCEA HVAR UARARLMARP TSCSRCAPBVARRARMARARDFM RCP HVARTVAR
Heterogeneity in time Heterogeneity across units
Figure 1: Hierarchical structure of model families grouped by the type of heterogeneity they are suited for.The blue dashed line groups models for heterogeneity in time. The red dotted line clusters models for heterogeneityacross units. The acronyms in the graph’s node correspond to the full model specifications provided in Table 1. Thearrows point from the general structure to the particular case, while the dashed lines connect similar models in abroader sense. The white nodes are shortly treated in this overview, as they do not fulfil the inclusion criteria of thepresent work. namely those under which VAR processes exist), then also (1) will exist. Notehowever that (1) defines a much wider class of processes. Attesting to results inthe literature, the existence of particular instances of (1) has already been studied:Vanˇeˇcek (2007) considers the case in which the autoregressive coefficients are seri-ally uncorrelated, and uncorrelated with the residuals; L¨utkepohl (2005) assumesthe autoregressive coefficients to satisfy almost surely the conditions for existenceof VAR models, and to be uncorrelated both serially and of the residuals.Model (1) in its full specification is probably overparametrized. We obtainmeaningful submodels by introducing constraints on the model parameters. Forinstance, assume that there exists a filtration F t such that E (˜ g i,t |F t ) = g i , or (2) E (˜ g i,t |F t ) = g t , or (3) E (˜ g i,t |F t ) = γ. (4)In these cases, the resulting autoregressive coefficient, although being random,presents only subject-specific variability (2), or the way it changes in time is thesame across subjects (3), or it does not depend on either unit i or time t (4).Many other assumptions can be stated to derive existence and identifiability ofthe process (1), but this is outside the scope of the present work.In order to have a structure that includes all the models treated in the presentwork as particular cases, we extend the definition of RAR models to randomautoregressive moving average models – RARMA( p, q, r, s ) – with a state-spaceformulation very close to the one of Tsay (1987) (see Figure 1) Y i,t = p X k =1 γ k Y i,t − k + q X k =1 ϕ k w i,t − k + X i,t β + Z i,t c i,t + w i,t w i,t = r X j =1 f i,t,j w i,t − j + s X j =1 g i,t,j Y i,t − j + ℓ ,i,t ˆ Y i,t + e i,t . (5)Here γ = [ γ , . . . , γ p ] T and ϕ = [ ϕ , . . . , ϕ q ] T are constant autoregressive and mov-ing average coefficients respectively, f i,t = [ f i,t, , . . . , f i,t,r ] T , g i,t = [ g i,t, , . . . , g i,t,s ] T and ℓ ,i,t are stochastic processes indexed by i and t with mean zero, andCov c i,t g i,t f i,t ℓ ,i,t e i,t = Σ c g i ge f l
00 Σ ge e i,t . Furthermore, ˆ Y i,t − denotes the conditional expectation of Y i,t given F t − . Stabilityand stationarity of the process in (5) have been proven by Tsay (1987) when11ome filtration satisfying (3) is imposed, and X i,t = Z i,t = 0 ∀ i, t . Note that aRARMA( p, , , p ) model with ℓ ,i,t = 0 a.s. is in fact a RAR( p ) model, where˜ g i,t,k = g i,t,k + γ i,k . Similarly we will show that also the other models presented inthis overview are particular cases of the general RARMA structure (see Figure 1). This section deals with models suited for (1 , m, T ) − data, typically with T large,grouped by the blue dashed box on the left hand side of Figure 1, and characterizedby parameters that are random in time. Since in all models treated here n = 1,we drop the subscript i . It is worth mentioning that such models have also beentreated in the closely related field of statistical tracking, but we do not pursue thisconnection here (c.f. Kushner and Yin (2003) for an overview). With the notation and distributional assumptions introduced in Section 3, thegeneralized random coefficient autoregressive (GRCA) model of order p can bewritten as Y t = p X k =1 ( γ k + g t,k ) Y t − k + e t . (6)It is in fact a RARMA( p, , , p ) model with ℓ ,t = 0 a.s. and X t = Z t = 0 for every t . The autoregressive coefficients g t = [ g t, , . . . , g t,p ] T and errors e t are seriallyuncorrelated, but they are allowed to be mutually correlated. In the literatureGRCA models are often referred to as random coefficient autoregressive (RCA)models (Conlisk, 1974, 1976; Chandra and Taniguchi, 2001; Hill and Peng, 2014),although RCA models are in fact a particular case of GRCAs (c.f. Section 4.1.1)).The GRCA includes also the (generalized) Markovian bilinear ((G)MB) model andthe random coefficient exponential autoregressive (RCEA) models for particularchoices of the autoregressive coefficients (Hwang and Basawa, 1998). Such modelshave been developed for econometric and financial applications , but have beenlater inherited in engineering (e.g. hydrology, metrology and telecommunication)and biology (e.g. dynamic population models) for their flexibility in modellingoccasional sharp spikes.Necessary and sufficient conditions for stability of GRCA models were derivedin early references by Conlisk (1974, 1976). For inference on γ various methodshave been proposed: maximum likelihood, shown to lead to locally asymptotically12ormal and asymptotically optimal estimators (Hwang and Basawa, 1997), condi-tional and weighted conditional least squares (CLS and WLS) (Hwang and Basawa,1998), and the estimated optimal estimators (EOE) of Chandra and Taniguchicombining Godambe’s optimal estimating functions with CLS and the method ofmoments. In terms of efficiency, WLS has been shown to outperform CLS (Hwang and Basawa,1998), and to be identical to the EOE (Chandra and Taniguchi, 2001).More recent publications have proposed estimation methods that can estimateboth γ and the covariance parameters, previously treated as nuisance. Fink and Kreiss(2014) propose bootstrap methods based on quasi-maximum likelihood (QML),while Zhao and Wang (2012) introduce empirical likelihood (EL) estimators, de-riving also the asymptotic distribution of the estimators and a non-parametricversion of Wilk’s theorem. The advantages of EL on WLS include robustnessagainst heteroskedasticity, distribution-adapted confidence intervals and avoidingthe estimation of the asymptotic covariance matrix.For the estimation of non-stationary GRCAs (and analogously for gen-eralized autoregressive conditional heteroskedasticity (GARCH) models, c.f. Sec-tion 4.3), Truquet and Yao (2012) introduce QML estimation, and prove its asymp-totic normality and consistency. Hill and Peng (2014) create a unified frameworkin the estimation of GRCA for stationary and non-stationary, possibly trendedtime series, either with or without random coefficients. This is achieved by com-bining EL with the WLS score equation.The literature on tests for GRCA , moreover, includes tests for stationarityand ergodicity in GRCA(1) (Zhao and Wang, 2012; Zhao et al., 2015), and forparameters change (Zhao et al., 2013, 2014). In another publication an approachto variable selection is also proposed (Zhao et al., 2018). Assuming g t to be independent of the error e t in (6), this equation leads to therandom coefficient autoregressive (RCA) model of Nicholls and Quinn (1982), aspreviously pointed out by Hwang and Basawa (1998). This structure constitutesone of the first attempts to capture the heterogeneity typical of time series infinancial applications . As for GRCA, this model has then been used in manyother fields where random perturbations are present, such as in macroeconomy, en-gineering (e.g. car vibrations or ship rolling), biology (e.g. brain-waves recordings)and meteorology. Stability, stationarity and ergodicity in RCA models have been largelystudied in early publications by Andˇel (1976); Nicholls and Quinn (1981, 1982);Feigin and Tweedie (1985). Stability and ergodicity are then classical assump-tions when dealing with estimation of RCA . Least squares estimation has beenlargely used (Nicholls and Quinn, 1981, 1982; Tsay, 1987), and is often taken as13enchmark for modern advances. Furthermore, since LS estimates are stronglyconsistent and under suitable conditions they obey the central limit theorem, theyoften serve as starting point for iterative schemes such as maximum likelihood(ML) (Nicholls and Quinn, 1982; Tjøstheim, 1986; Allal and Benmoumen, 2013),or in combination with other procedures such as estimating functions (EF) (Thavaneswaran and Abraham,1988; Abdullah et al., 2011) and bootstrap methods (Pr´aˇskov´a, 2003; Fink and Kreiss,2013). Note that for ML estimation, autoregressive coefficients and residuals arealso typically assumed to be jointly normal. Furthermore, for RCA models, the EFis equivalent to WLS (Abdullah et al., 2011). Schick (1996) and Koul et al. (1996)derive an efficient adaptive locally asymptotically minimax estimator for γ . Qian(1996) propose minimum distance estimators, while Ghahramani and Thavaneswaran(2009) combine LS’s and least absolute deviations (LAD)’s estimating functionsto estimate model volatility (both for RCA and for GARCH models, c.f. 4.3).Aue et al. (2006) use QML to estimate the parameters of a scalar RCA(1), andimpose only minimal conditions on the sequences of random coefficients and resid-uals. Under suitable conditions, they derive strong consistency and asymptoticnormality for these estimates. Recently the generalized moment estimator and theWhittle estimator (involving an approximation to the likelihood function) havealso been proposed (Shitan et al., 2015; Bibi, 2016).Inference for models that fall under the RARMA structure normally requiresfavourable assumptions, such as stationarity and limited second order moments.However, especially for RCA models, the literature is broad and deals also withnon-standard assumptions. Authors have explored estimation of non-stationaryRCAs , and as for the GRCA model, proposed QML estimation. Although thevariance of the initial error cannot be estimated, weak consistency and asymptoticnormality of QML estimates can be proven (Berkes et al., 2009; Aue and Horv´ath,2011). In the latter reference, the method is validated through a Monte Carlo simu-lation study and applied to real financial data. The Bayesian approach also enablesthe analysis of non-stationary time series, and allows the inclusion of prior infor-mation (Diaz, 1990; Yang, 1995; Barnett et al., 1996; S´afadi and Morettin, 2003;Wang and Ghosh, 2008, 2009; De, 2014). The comparison between the Bayesianapproach of Wang and Ghosh (2009) and the QML approach applied to the samedaily stock transaction volume data leads to a non-decisive result on the real data-set, but to the conclusion of a better performance of the frequentist approach onthe simulated data (Aue and Horv´ath, 2011).For RCAs, authors have also considered the case of infinite error variance ,and proposed (smoothed) LAD for inference, consistent and asymptotically nor-mal (Thavaneswaran and Peiris, 2001, 2004), and providing more precise estimatesthan least squares in the case of infinite variance (Goryainov and Goryainova,2016). The CLS estimator of the autoregressive coefficient is also shown to be14symptotically normal when the second moment of the innovation is infinite undersome weak conditions (Fu and Fu, 2015).Recently a test for strict stationarity of RCAs has been proposed by Trapani(2020), while other tests exist for parameter changes (Pr´aˇskov´a, 2015; Li et al.,2015b,a), and for the randomness of the coefficients (Lee, 1998; Akharif et al.,2003; Horv´ath and Trapani, 2019) – the latter publication in particular tests for anull on the boundary of the parameter space.In econometrics, testing for unit root in autoregressive models is of partic-ular interest (Nelson and Plosser, 1982; Phillips, 1988). This naturally extendsto RCA models, where the unit root assumption reduces to γ = 1, togetherwith the alternatives of near-stationarity ( γ →
1) and mild explosivity ( γ >
1) (Leybourne et al., 1996; Sollis et al., 2000; Aue, 2008), and it closely relatesto the detection of bubbles (Psaradakis et al., 2001; Banerjee et al., 2020). Manyreferences report tests, inference methods and asymptotic consequences of thethree regimes, determined by how γ approaches the unit value (Wang and Ghosh,2008; Nagakura, 2009a,b; Aue, 2008). Note furthermore that also in this case,there exists a completely separated literature dealing with the so-called stochas-tic unit root (STUR) model, explosive random coefficient autoregressive (ERCA)model, or near-explosive random coefficient autoregressive model (NERC) withvery different acronyms while in fact all are RCAs with particular γ ’s. Take g t,k = θ k e r k t in (6), with θ = [ θ , . . . , θ p ] T a vector of constants and r =[ r , . . . , r p ] T a vector of non-negative integers. The resulting model is the so-calledgeneralized Markovian bilinear (GMB) model, Y t = p X k =1 ( γ k + θ k e r k t ) Y t − k + e t , (7)of which the Markovian bilinear model is a particular case when r k = 1 ∀ k (Tong,1981; Feigin and Tweedie, 1985; Cline and Huay-min, 2002). Note that althoughGMB models have a random autoregressive coefficient, this coefficient is not sam-pled from a different stochastic process, as it is for RCA models (c.f. Section 4.1.1).Rather, the autoregressive coefficient is function of the noise term.15 .1.3 Random Coefficient Exponential Autoregressive models (RCEA) Take g t,k = ( θ k + θ k exp( − θ k e t )) e t in (6), where θ j = [ θ j , . . . , θ p j ] T , j = 1 , , Y t = p X k =1 (cid:0) γ k + ( θ k + θ k exp( − θ k e t )) e t (cid:1) Y t − k + e t , (8)known in literature as random coefficient exponential autoregressive (RCEA) model (Hwang and Basawa,1998; Priestley, 1980). As in GMB models, the heteroskedasticity in the autore-gressive coefficient is inherited from the error term. In practical applications for example encountered in economy and finance (e.g. vol-ume transaction data), the assumption of independence in time of the coefficientsoften does not hold. Models that account for some time-dependence among therandom parameters have thus been introduced. These models also extend theRCA model (Section 4.1.1), but are not particular cases of the GRCA. In factthey are again derived from (1) without the subscript for the unit, by setting X t = Z t = 0 ∀ t , but contrarily to GRCA they assume independence between noiseand autoregressive coefficients, and allow serial correlation for one of the two ran-dom variables. The acronym introduced here for this family is RCAC – randomcoefficient autoregressive models with correlated terms. An example of RCAC model relaxes the assumption of serial independence ofthe autoregressive coefficients, by assuming the random coefficient of the scalarRCA(1) to be g t, = α z t + α z t − , with α , α = 0 constant coefficients and z , . . . , z t independent random variables with zero means and the same variance σ z ,independent of both Y and { e t } (Koubkov´a, 1982). In this publication, conditionsfor stationarity , covariance function and best linear predictions are obtained. In the dissertation, Vanˇeˇcek (2007) extends the results for RCA models undertypical assumptions (c.f. Section 4.1.1) to RCAs with the noise being an ergodic,16trictly stationary martingale difference sequence with respect to the previous ob-servations. Although the author refers to it as a generalized random coefficientautoregressive model, this should not be confused with model (6) as pointed outin the second chapter of the dissertation, since the autoregressive coefficient is in-dependent of the residuals. The author proposes a new functional estimator forRCA( p ) and multivariate RCA(1) under this assumption, extending the methodoriginally proposed by Schick (1996) for RCA(1) models. The newly introducedestimator is strongly consistent and asymptotically normal. With an extensivesimulation study the author compares LS, WLS, ML and the new functional es-timator in terms of efficiency and asymptotic variance, and concludes that theWLS seems the optimal choice. The code for parts of the implementation inR (R Core Team, 2019) is provided in the publication. A study comparing theperformance of the LS method on RCA models with correlated and uncorrelatederror sequence is presented by Araveeporn (2013). From the simulation study, theauthor concludes that accounting for correlation in residuals improves the resultsonly when the data oscillates, and for real data the model with autocorrelationsoutperforms the simpler model with serially independent error sequence. A significant contribution to the literature for modeling stochastic volatility infinancial applications is given by the family of generalized autoregressive con-ditional heteroskedasticity models of Bollerslev (1986). With the notation of theprevious sections a GARCH( p, q ) model is defined as Y t |F t − ∼ N ( Xβ, h t ) h t = α + p X k =1 α k w t − k + q X k =1 β k h t − k w t = Y t − Xβ (9)where F t − is a filtration, and α , α k , and β k are coefficients to be estimated. Thismodel is suited for (1 , m, T ) − data, with m possibly larger than one (Bauwens et al.,2006). This model can also be seen as a RARMA(0 , , ,
0) model with Z t = 0 ∀ t and ℓ ,t = 0 almost surely. Thus w t = e t , and e t is normally distributed withvariance h t . As for GMB and RCEA models (Sections 4.1.2 and 4.1.3), theheteroskedasticity in time comes from the noise and not from sampling from adifferent distribution. We mention this family shortly for completeness, and re-fer to other references that deal in depth with this model (Bauwens et al., 2006;Francq and Zakoian, 2011), and mention McCullough and Renfro (1998); Brooks171997); Brooks et al. (2003); Ghalanos (2014, 2019); GAR; Kim (1993); Boffelli and Urga(2016) for what concerns software implementation . Autoregressive conditional heteroskedasticity models (ARCH)
The par-ticular case of (9) with q = 0 constitutes the autoregressive conditional het-eroskedasticity (ARCH) model of order p of Engle (1982). Here the covariancestructure is only function of the noise at previous states, h t = α + P pk =1 α k w t − k , imposing shorter memory to the process.Similarly to GARCH the heterogeneity in ARCH is obtained by incorporatingthe error term in the definition of some parameters, and we refer to publicationsspecifically on the topic for further details (Bollerslev et al., 1992; Degiannakis and Xekalaki,2004; Gouri´eroux, 2012), while for software packages one can refer to the onesmentioned for fitting GARCH models (c.f. Section 4.3). It is however interestingto note that RCA and ARCH models share second order properties and can bestudied in parallel (Tsay, 1987; Wolff, 1988; Chandra and Taniguchi, 2001). After the introduction of the ARCH model (Section 4.3), many authors investi-gated its relationship with the RCA model (Section 4.1.1), and their contributionin econometric applications , such as in spot rate predictions. In fact, the first canbe rewritten with the structure of the second, and they possess the same secondorder properties (Tsay, 1987; Wolff, 1988). The introduction of conditional het-eroskedasticity autoregressive moving average (CHARMA) models by Tsay (1987)is motivated by the consideration that neither RCA nor ARCH have parsimoniousdefinitions. CHARMA models are defined by the equations γ ( B ) Y t = ϕ ( B ) w t f t ( B ) w t = ℓ ,t ˆ Y t − + g t ( B ) Y t + e t (10)where B is the backwards operator, and are in fact RARMA( p, q, r, s ) with X t = Z t = 0 , ∀ t , and with both serially and mutually independent random terms hav-ing constant variances. Estimation can be carried out via LS, and appropriatemoment conditions ensure asymptotic normality of the estimators. Multivariateextensions of the CHARMA model are discussed by Ahn and Reinsel (1990). Notethat RCA and ARCH are also particular cases of CHARMA (Tsay, 1987).18 .5 Random coefficient autoregressive regime switching mod-els (RCARRS)
For the particular application of estimating time-varying hedges ratios, manymodels have been explored – in particular, under the RARMA family, the bivariateGARCH and the RCA models (Bera et al., 1997). The so-called random coefficientautoregressive regime switching (RCARRS) model of Lee et al. (2006) was intro-duced by combining the flexibility of RCA models with the state-dependence prop-erties of Markov regime switching (MRS) models (Alizadeh and Nomikos, 2004).It was originally formulated as S t = α s t + ζ t F t + ε t,s t (11)( ζ t − ¯ ζ ) = φ ( ζ t − − ¯ ζ ) + ν t , (12)where S t and F t are the spot and future values at time t , α s t is a state-dependentconstant parameter, ζ t is the autoregressive coefficient with steady-state ¯ ζ . Thetwo iid sequences ε t,s t and ν t with variances σ ε st and σ ν respectively are also mutu-ally independent. The state equation on ζ t has autoregressive coefficient | φ | < ζ t sequence. The parameters depend on a latent vari-able s t following a two-state first-order Markov-switching process with unknowntransition probabilities p and p . For estimation , Lee et al. propose ML whilerecurring to Kims filter (Kim (1994); a combination of the extended Kalman andHamilton filters) after an appropriate transformation. From a comparison of themodels proposed for this application, Lee et al. (2006) conclude that RCA has thebest in-sample (on the data used to estimate model parameters) performance, whileRCARRS the best out-of-sample (on new data) performance for the case study athand. With a small extension to state-dependent variables, the RCARRS modelof order p can be seen as a RARMA(1 , , ,
0) model with γ k = φ k , ϕ k = − φ k , X t = [1 , F t ], Z t = F t , β = [(1 − φ ) α s t , ¯ β ] T and c t = ν t . Furthermore, s t ∈ { , } determines the distribution being sampled, α s t = α { s t = 0 } + α { s t = 1 } ,ε t,s t = ε t, { s t = 0 } + ε t, { s t = 1 } , where α and α are two constant values, ε t, ∼ N (0 , σ ) and ε t, ∼ N (0 , σ ). The stream of publications on Bayesian time-varying autoregressive (TVAR) mod-els (Prado and West, 1997; Prado et al., 2000) developed completely independentlyfrom the remaining RARMA literature. The Bayesian framework, together withthe non-stationary time series in hand, may be the reason for the non-existence19f references to other publications also dealing with heteroskedastic time seriestreated in other sections. In fact, this dynamic linear regression model can bederived from the general equation (1) similarly to GRCA models. On the otherhand, TVAR models allow order uncertainty and multiple assumptions on therandom parameters. Such models find application in various modern contexts,such as biomedical signal processing (including the analysis of multiple electroen-cephalographic traces) and communications. Prado et al. (2000) consider a modelfor (1 , , T ) − data with the autoregressive coefficient following a Gaussian randomwalk. The variance of the noise term σ t = σ t − ( δ/η t ) depends on η t ∼ Beta ( a t , b t ),allowing for heteroskedasticity. Furthermore, η t is assumed to be serially indepen-dent and independent of e t and ζ t . Also this model is a special case of our generalstructure. With the mentioned choices, it is a RARMA(0 , , ,
1) model with X t = Z t = 0 ∀ t and ℓ ,t = 0 almost surely, g t following a random walk and { e t } having variance Σ e i,t = σ t as defined in the paper and rewritten above. In this section we describe the family of models delimited by the red dotted line onthe right hand side of Figure 1. The common feature among these models is thatthey pool information from n time series assumed to follow the same underlyingprocess. This is important when the number of observations T in ( n, m, T )-datais limited. Vector autoregressive (VAR) models are often employed to considermultiple responses simultaneously, but they are characterized by a large numberof parameters which makes the estimation challenging. Bayesian vector autore-gressive (BVAR) models aim to overcome this issue by pooling information fromdifferent units, but, as noted by Nandram and Petruccelli (1997), they requiremore restricting assumptions on the priors when there is a large number of theseshort time series. In fact, many of the models treated in this section were in-troduced to extend the flexibility of BVAR models and enable a more efficientanalysis of a large number of short time series by introducing random coefficients.The randomness can be found either in the dynamic part (i.e. in the autoregressivecoefficient) or in the static part (i.e. random effects), depending on the purpose.Publications about models in this family find application in the economic,sociological, biological, agricultural, international relations and industrial fields,where multiple time series with similar behavior are available. There exist var-ious overviews on the topic (Franses, 2006; Horv´ath and Wieringa, 2008; Hsiao,2014; Krishnakumar, 2012), but their scope is limited to a single field of appli-cation and related publications. Horv´ath and Wieringa (2008) summarize variousapproaches specific to marketing applications, giving the appropriate estimationmethod, drawbacks and references for each. All of the models that they investigate20re particular cases of the RARMA structure, accommodating different levels ofheterogeneity, and they are compared both via a simulation study and the appli-cation to a real case study. Franses (2006) focuses on marketing applications, andbeside reviewing existing models, proposes a series of possible extensions combiningwell-known features – some of which already exist in different fields of application.Also the overview of Hsiao (2014) focuses on models for economic applications,ranging from static random effects models to those including autoregressive struc-ture.Since the existing literature is wide and diverse, but at the same time it showsmany similarities, we introduce the unit-specific autoregressive (UAR) model toorganize the right hand side of Figure 1. We state how to set the parameters ofthe UAR structure to obtain the remaining models as particular cases, showingtheir hierarchy and listing the main related publications. With the notation of Section 3, the general unit-specific autoregressive (UAR)model of order p can be written as Y i,t = p X k =1 γ k Y i,t − k + X i,t β + Z i,t c i + w i,t ,w i,t = r X j =1 f i,j w i,t − j + s X j =1 g i,j Y i,t − j + e i,t . (13)It is in fact a RARMA( p, , r, s ) model where the random coefficients are indexedonly by i and are independent, and ℓ ,i,t = 0 almost surely. In (13) take X i,t = Z i,t = 0 ∀ i, t , and f i,j = 0 almost surely Y i,t = p X k =1 γ k Y i,t − k + w i,t ,w i,t = s X j =1 g i,j Y i,t − j + e i,t . (14)The resulting equation is the random coefficient autoregressive model, which hasbeen widely studied and applied mainly in economic and biological applications ,for its ability to pool multiple time series (Liu and Tiao, 1980; Tiao, 1993). Since21here is no agreement in the definitions, we refer to this structure as random co-efficient autoregressive panel data (RCAP) model, to keep the analogy with RCAmodels (Section 4.1.1). The residuals are assumed to be a series of identicallydistributed normal random variables with variance Σ e i,t . In most cases the co-variance matrix is diagonal and constant, but there exist exceptions (Pai et al.,1994; Nandram and Petruccelli, 1997), and only one reference makes the explicitassumption for the independence between autoregressive coefficients and errorterm (S´afadi and Morettin, 2003). Estimation for RCAP models is often addressed with Bayesian approaches.Robinson (1978) proposes a way to estimate moments (with consistent and asymp-totically normal estimators) that can be used as prior information. Liu and Tiao(1980) show that the prior on the parameters of the distribution of the randomcoefficients has progressively less influence as the number of units increases. Fora RCAP(1), they derive the theoretical posterior distribution of the coefficientsin case of beta distributed AR coefficients, discuss how the model can be ap-plied to seasonal data and how it can be extended to second order autoregres-sion. Li and Hui (1983) propose empirical Bayes estimates for RCAP( p ), with asimpler implementation and no need for the prior distribution of the coefficientscompared to the method of Liu and Tiao (1980), and show that the newly pro-posed approach outperforms LS estimation in most of the cases. In fact, thelimit distributions of empirical Bayes, Bayes and frequentist estimates are asymp-totically equivalent (Kim and Basawa, 1992). The independent multivariate stu-dent’s t–inverse gamma prior is shown to lead to the best results among threepriors in a genetic study, while on simulated data, the method performs wellwith all the tested priors (S´afadi et al., 2011). Authors have also developed aMetropolis-within-Gibbs sampling algorithm and two algorithms based on MCMC(only one requiring stationarity) (Pai et al., 1994; Nandram and Petruccelli, 1997).Nandram and Petruccelli (1997) state that their method performs better thanVAR or BVAR, both on stationary and non-stationary time series, and analyzethe same case study investigated by Liu and Tiao (1980). Other recent publica-tions use model-based approaches similar to the ones described in this section forclustering (Nascimento et al., 2012; Wang et al., 2012; Nascimento et al., 2016). When the random autoregressive and moving average coefficients are null almostsurely, (13) reduces to the autoregressive linear mixed ARLM( p ) model of order p of Funatogawa et al. (2007), Y i,t = p X k =1 γ k Y i,t − k + X i,t β + Z i,t c i + e i,t . (15)22ere the autoregressive coefficient is constant, and the heterogeneity is due tothe random effects c i . The assumptions on the random coefficients are thesetraditional in linear mixed models (LMM), c i and e i,t are assumed to be inde-pendent across units i , both iid normal with constant variances Σ c and Σ e re-spectively. The authors introduce this model for the specific application of fit-ting dose-response profiles in clinical trials showing initial sharp changes, decreas-ing rates of change, and finally approaching random patient-specific asymptotes.Funatogawa et al. (2007) discuss the properties of the ARLM model, and via asimulation and case study they conclude that their model outperforms previousapproaches for this specific application. Funatogawa et al. (2008b) focus on theeffect of drop-outs on the asymptotes’ estimates and consider estimation in caseof unequally spaced measurements in time (Funatogawa and Funatogawa, 2012a).The same authors also introduce the bivariate model and analyze the case in whichthe dose is based on previously observed responses (Funatogawa et al., 2008a;Funatogawa and Funatogawa, 2012b). Estimation is performed via ML, shown tobe consistent, in some cases combined with a state-space representation to enableKalman filter estimation (Funatogawa and Funatogawa, 2008). Funatogawa et al.(2007) propose a reparametrization of (15) that makes the connection with tra-ditional LMM more evident, allows the extension to higher order autoregression,and the implementation in standard software for LMM. This method doesnot work when intermittent (i.e., followed by observed values) missing values arepresent.
Autoregressive panel data models (ARP)
The autoregressive (or dynamic)panel data (ARP) model, Y i,t = p X k =1 γ k Y i,t − k + X i,t β + c ,i + w i,t (16)is a special case of (15) with random intercept only, although no cross referencesexist between the two model families, applied in contexts with ( n, , T ) − data withsmall T . This structure is particularly popular in econometric, economic andpsychology applications , where typically one has ( n, , T ) − data with small T . Estimation and identifiability are discussed in two consecutive publicationsby Anderson and Hsiao (1981, 1982). The authors also investigate the consis-tency of ML estimates and asymptotic properties as both T and n diverge toinfinity, evaluating the influence of initial conditions. Anderson and Hsiao (1982)extend model (16) allowing for the inclusion of both time-invariant ( serial corre-lation model ) and time-varying ( state dependence model ) exogenous variables andshow which parameters can be estimated. Their work fits the general frameworkof MaCurdy (1982), structuring error models. For estimation, the generalized23ethod of moments (GMM) estimator has large finite sample bias and poor preci-sion because the series are highly autoregressive and the number of observations istypically moderately small. Various alternative approaches exist, often exploitinginstrumental variables. A comparative study illustrates how a modified versionof the least squares dummy variable estimator (i.e. including dummy variables toeliminate individual effects) can outperform GMM in terms of asymptotic vari-ance, while achieving small bias (Kiviet, 1995). Furthermore, various modifiedversions of the GMM have been investigated, such as the Arellano-Bond esti-mator and the system GMM (Arellano and Bond, 1991; Ahn and Schmidt, 1995;Blundell and Bond, 1998, 2000; Blundell et al., 2001; Bond, 2002). Few referencesstudy Bayesian inference in ARP models under either standard and non-Gaussianassumptions (Hirano, 2002; Ju´arez and Steel, 2010).The fundamental problem of testing the presence of individual effects (Holtz-Eakin,1988; Arellano and Bond, 1991), serial correlation (Wooldridge, 2002; Drukker et al.,2003) and unit roots (Levin et al., 1992) in dynamic panel data models is addressedby many authors to enable the choice of the most parsimonious structure and totest stationarity.The number of publications studying (mostly economic) applications of model (16)is extremely large. More information on model selection and estimation methodscan be found in comparative papers such as the ones from Judson and Owen (1999)and Bond (2002). The large interest in these models is reflected in the number of software packages available for simulation and inference. We mention for refer-ence the plm package (Croissant et al., 2008), the cquad package (Bartolucci and Pigini,2015) allowing dynamic binary panel data, the OrthoPanels package (Cubranic et al.,2019) using the orthogonal reparametrization approach, and the panelvar pack-age (Sigmund and Ferstl, 2017). Time-series-cross-sectional (TSCS) models of Beck and Katz (1995) constitute aniche independent of the rest of the literature. The main reason for this is probablythe type of data being analyzed, large in n and T , and the asymptotics, conse-quently studied with respect to both dimensions. The model is widely applied inpolitical economy and can be stated in the following state-space form Y i,t = X i,t β + w i,t w i,t = r X j =1 f i,j w i,t − j + e i,t . (17)where e i,t are independent identically distributed random variables with zero mean.Note that (17) is in fact a UAR model with g i = 0 a.s., γ k = 0 ∀ k and Z i,t = 0 ∀ i, t .24or estimation Beck and Katz (1995) propose the so-called Parks method, warn-ing that the generalized least squares (GLS) method underestimates variability inTSCS data unless T ≫ n . Feasible GLS (FGLS) is discussed as an alternative.The authors conduct a Monte Carlo simulation study to show the importanceof including individual autoregressive coefficients in small data-sets, and suggestthe use of panel-corrected standard errors which take heterogeneous serial corre-lations into account. Beck et al. (1998) extend the model to time series with abinary response variable and a large number of observed units. The authors alsoalert researchers that taking correlation in time into account is crucial for properinference, and show the consequences of neglecting correlations by looking at pre-viously published studies where correlations were ignored (Beck et al., 1998; Beck,2001a,b). Podest`a (2006) compares the goodness of fit of various models on thespecific case study of Welfare State development. Bayesian vector autoregressive (BVAR) models are the traditional vector autore-gressive models, when the autoregressive coefficients are estimated via Bayesianapproaches by setting a prior (i.e. shrinkage methods). This, jointly with the like-lihood of the data, returns a posterior distribution for the coefficients. Bayesianmethods are shown to be appropriate in the estimation of large dynamic mod-els (Ba´nbura et al., 2010). A BVAR model of order p is in fact a RARMA( p, , , X i,t = Z i,t = 0 ∀ i, t , ℓ ,i,t = 0 a.s. and serially independent residuals.However, BVAR models do not fulfill our inclusion criteria since the randomnesson the coefficients is imposed for estimation, while we consider models where thecoefficients are random by assumption (inclusion criteria 2.). We refer thus to otherreferences for more details (Litterman, 1986; De Mol et al., 2008; Ba´nbura et al.,2010; Wo´zniak, 2016). In recent years, studies in psychology and behavioral science have started combin-ing dynamic models for temporal behavior with hierarchical structures to accountfor both inter- and intra-individual variability. These models address data whereall dimensions ( n, m, and T ) are large, and developed almost independently ofthe remaining literature. Since a definition does not exist and the models consid-ered here vary also in terms of structure, we collect them under the general termhierarchical vector autoregressive (HVAR) models - highlighting the multivariateoutcome and the hierarchical definition, without forcing a specific model structure.This includes, for instance, the multilevel VAR model of Bringmann et al.(2013), that is in fact a RCAP (14) suited for ( n, m, T )-data. Given the field25f application and the fact that this model constitutes the basis for a networkapproach, we treat it separately. It is also interesting to notice the analogy withBVAR models – in the sense that the model equations are the same, but here theauto- and cross-regression coefficients are random for the model assumptions, and estimation is based on pseudo-likelihood. The model of Bringmann et al. (2013)defines subject-specific equations that enable comparisons across groups.Another example in this model family is the hierarchical state space modelof Lodewyckx et al. (2011) (c.f. Oravecz et al. (2011); Ranganathan et al. (2014)for the continuous-time version), a dynamic linear model with subject-specificparameters, that can be rewritten as Y i,t = Z i,t c i,t + w i,t w i,t = p X j =1 f i,j w i,t − j + e i,t , with the notation introduced in Section 3. It is in fact a RARMA(0 , , p,
0) modelwith ℓ ,i,t = 0 a.s. and an additive structure for e i,t = z i X t,i + v t . The authorstake a Bayesian approach for the estimation . Furthermore, hierarchical Bayesianmodels have received particular attention and they are shown to be well suitedfor studies in psychology (Shiffrin et al., 2008; Lee, 2011; Ranganathan et al.,2014). In fact, also software packages exist (e.g. BVAR (Kuschnig and Vashold,2020)). Adolf et al. (2014) have also introduced a test to check whether inter- andintra-individual model structures are equivalent. In this section we describe models that can accommodate heterogeneity both intime and across units. The availability and dimension of data-sets have increasedsignificantly in the recent years, and require methods capable of addressing multi-ple properties of the data simultaneously. In this direction, the random coefficientpanel model has been recently developed by (Horv´ath and Trapani, 2016). Typicalexample applications include macroeconomic and financial studies, but also themodern challenge of analysing multiple channels of electroencephalography (EEG)data and speech signals. Also the dynamic factor model of Prado et al. (2001) aimsat capturing multiple dynamics present in the data and is particularly suited forthe analysis of EEG data. 26 .1 Random coefficient panel models (RCP)
With the notation of Section 3, a random coefficient autoregressive panel (RCP)model (Horv´ath and Trapani, 2016) is defined as Y i,t = p X k =1 γ k Y i,t − k + X i,t β + Z i,t c i + w i,t ,w i,t = s X j =1 g i,t,j Y i,t − j + e i,t , (18)which is in fact a RARMA( p, , , p ) (5) model with ℓ ,i,t = 0 almost surely, theautoregressive coefficients independent of the residuals and all the stochastic termsserially independent, and independent across units. Note that the nomencla-ture could be confused with that of autoregressive panel data models, treatedin Section 5.1.2, but we make the choice here not to introduce additional terms.Model (18) is suited for ( n, , T ) − data, and is capable of addressing heterogene-ity both in time and across units, with subject-specific stochastic autoregressivecoefficients and the following factor structure for the error term e i,t = z i,t + χ i v t . (19)Here, the term v t has zero mean and unit variance, while χ i is independent acrossunits and independent of the other random terms. Random coefficient panel mod-els include both RCA and RCAP models as particular cases for some filtration, X i,t = Z i,t = 0 , ∀ i, t and no structure is imposed on e i,t .In their publication, Horv´ath and Trapani (2016) show that the unit root prob-lem exists only in case T −→ ∞ , and prove that the WLS estimator is asymptot-ically normal. Furthermore they illustrate that this estimator performs well alsofor relatively small panel data (both in terms of T and n ) via a simulation study.A macroeconomic and a financial applications, motivating the study, are includedin the publication. Bayesian TVAR models (c.f. Section 4.6) are suited for single non-stationary timeseries. These models have been extended to pool information from multiple non-stationary time series, and to understand the cross and the spatio-temporal rela-tionships. These dynamics are of interest, for example, in the analysis of multiplechannels of EEG data and speech signals. In dynamic factor models (DFM) thestate follows a dynamic linear or TVAR model (West and Harrison, 2006), whilethe observation equation can assume various forms. Again, as for the TVAR27odel, multiple assumptions are possible. One example is the regression withfixed factor weights to adjust the effect of the latent variable for different chan-nels (Prado and West, 1997; Prado et al., 2001), Y i,t = χ i x t + ν i,t ,x t = p X j =1 φ t,j x t − j + η t ,φ t = φ t − + w t . (20)It is in fact a RARMA(0 , , p, p ) model with g i,t = − f i,t = φ t , assumed to follow arandom walk, and a multiplicative structure for the residuals ( e i,t = χ i η t ) similarto (19) of Horv´ath and Trapani (2016). In this case, however, χ i ’s are fixed factors,and thus the inclusion of this class into the models for the heterogeneity acrossunit and time is questionable. It is also interesting to note the analogy with HVARmodels, but here the autoregressive coefficients are random in time while in HVARthey are random samples from the population. In this paper we presented a structured overview of models with autoregressivestructure and random coefficients. The existing literature, broad and disconnected,is structured and made accessible also to the reader who is new to the field.First we provided a concise summary of the terminology used for data-sets andmodels. Then we introduced the RARMA structure to provide a unified language,and to show the hierarchy existing among models in a formal way. Through amathematical approach, we stated the simplifying assumptions necessary to getthe nested models from the more complex ones. This way, also similarities anddifferences are shown explicitly. The exposition is supported by a graph and atable, that help visualizing the hierarchy and getting the variety of existing modelsat a glance.Since the literature is fragmented, bringing together traditionally independentfields will boost the research in each field, exploiting the achievements of others.The present work supports this evolution, by also providing an overview of prop-erties, existing estimation methods and tests that can be exploited and expandedin different directions to adapt to various types of data. This will also avoid repli-cation of results, which is quite common in this context. In each section, whereavailable, software packages or code made available with publications are listed.However, it is apparent that while the literature about these models is extremelybroad, it is not supported by a corresponding availability of implementations, fur-ther limiting the reproducibility and the possibility of advances in the field. Some28ubfields like (G)ARCH and panel data models are well developed also in terms ofsoftware, but future contributions are needed in other fields like (G)RCA.We limited this review to autoregressive models with random parameters fordiscrete-time data, but extensions in multiple directions are possible. For exam-ple, it would be interesting to extend this work to other autoregressive models,e.g. models with deterministically time-varying parameters. It would also be valu-able to extend the overview to continuous-time models. Further research shouldfocus furthermore on the analysis of the newly introduced RARMA model, check-ing identifiability and proposing suitable estimation methods. Hopefully the re-sults will move towards open source implementations, differently from the currentliterature – which is vast and not easily reproducible.
Acknowledgements
This research was performed within the framework of the strategic joint researchprogram on Data Science between TU/e and Philips Electronics Nederland B.V.
Conflict of interests
No potential conflict of interest was reported by the authors.
Data availability statement
Data sharing is not applicable to this article as no new data were created oranalyzed in this study.
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