This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume 2, number 4

Volume Author/Editor: Sanford V. Berg, editor

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm73-4

Publication Date: October 1973

Chapter Title: A Survey of Stochastic Parameter Regression

Chapter Author: Barr Rosenberg

Chapter URL: http://www.nber.org/chapters/c9933

Chapter pages in book: (p. 381 - 397)

Annal ol Leonomic and Social fcasuri',ncni, 2/4. 97

A SURVEY OF STOCHASTIC PARAMETER REGRESSION

BY BARR ROSENBERG'

Several imporiwil models of stochastic parwneler uariai3On are described: randon,lr dispersed paronwiirs,.cequeniialfy r'urving parameters, stationart stochotic ptirunieters. cross-section time-series models, andthi' shilting regressions approach. Theories and methods of stochastic parameter regression arc' surie'd

I. STOCHASTIC PARAMETER REGRESSION

This article is a short survey of sonic of the more important issues in the theoryof regression with stochastically varying parameters. The stochastic parameterregression problem arises when regression coefficients vary unsystematically inthe familiar linear regression model

(1) .%'nr = + 11m = Xrbni + it = I, N. t = I.....T

where n connotes an individual within a cross section, t connotes a time period,and where one or the other of the subscripts will be suppressed when inappro-priate. The regression parameters or coefficients b1 .....b5, one of which may bethe intercept. are written with subscripts it and t to permit variation across indi-viduals and time periods. Many articles in the literature on stochastic parametershave provided arguments for the nonconstancy of coefficients across observa-tions, and it would be inappropriate to repeat these here. Suffice it to note that ifthe regression coefficients are to be regarded as the true partial derivatives of Vm

with respect to the regressors. x then it is improbable that these partial deriva-tives will be identical for two different observations.

Two types of parameter variation must he distinguished: svsu'inatie and.szochastic. In systematic variation, the individual parameter vectors may bewritten as b1 = f(., zr,) where the parameters . specify a functional form determin-ing as a function of observable variables z,,.. These observahies may includethe regressors xm themselves, if the true regression model is nonlinear, as well asother characteristics of the individual. When parameter variation is systematic,the regression problem is a (possibly nonlinear) regression on x, and ;, toestimate the parameters , and ordinary regression theory is applicable. For adiscussion of these matters, see Belsley (1973).

The stochastic parameter problem arises when parameter variation includesa component which is a realization of some stochastic process in addition towhatever component is related to observable variables. Thus, stochastic param-eter regression is a generalization of ordinary regression. Ideally, a model wouldbe so well specified that no stochastic parameter variation would be present, and

Assistant Professor of Business Administration, University of California, Berkeley. This researchwas supported by NSF grant GS 3306. This is. of necessity, a brief survey of the theory, and there mustinevitably be omissions and perhaps errors as well, for which the author offers his apologies.

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no eeneialiiatton would be flce(lell, hut the world N C\, han ideal. ] he distiric-lion between stochtstic aiid systematiC vaiiatiOi) is part Lcularlv important. beçiicin most models. stochastic parameter ai tanon isasstiiiied to he tiiicorrelatedwith ihe explanatory variables. Hence, these models are ilot intended to accountfor parameter variation resutting from misspecilication in a nonlInear modeldespite frequent assertions to this eflct iii the literature. Stochastic parameterregression should he employed as a supplement to analysis of systematic ania-lion, rather than as an alternative.

All of the stochastic parameter processes that have been considered ineconomics may be written in the general form:

(2) b,,, = 11(0)ni + B,,(D)q, ii = I N. I = 1

where ni is a U x I) vector of unknown parameters q is a large vector of stochasticterms. (p x I) say. having a proper joint distribution with mean ij(0) and variancematrix a2A(0). with a2 a scale parameter, and where the notation (0) indicatesthat these terms may depend ott a vector of stocluisflc spcclti(anon parameters 0.When these relations are substituted into the linear regression (1) and a!l ohser a-tions in the regression are combined, the result is a linear statistical model:

\\Tor, in matrix form,

= (0)m ± 'P(0)'q 4 U.

To see that the model differs little from the traditional regression model,notice that it may be rewritten as:

y = (0)m + (P(0):1)().

Thus, the model is a regression with an augmented vector oF stochastic termsincluding both stochastic parameters and disturbances. In contrast to ordinaryregression, where the disturbances are customarily eliminated as nuisance param-eters at the first opportunity, in stochastic parameter regression the estimatorsfor the stochastic terms are followed through carefully. However, as Theit hasshown (1971, Ch. 5). there are implied estimators for the stochastic terms in theordinary regression model, and Theil's BLUS procedure is an instance of thegeneral procedures for optimal estimation of the stochastic terms in a linearstatistical model such as (4).

The regression may he written in yet another way as:

y - 'P(0)i(0) = (0)m H- V.where

= (P(0):I)("__0)).

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.Vl x A11(9) 1x11 B11(0) 1u1

11 2

)

x'1 Al2(0) 2 B1 2(0),11+

U1,

.Ass:me that L(tn . ',arn 2vffl) Los (Ill)) U. Then. L() U.

var )v) -J2(q'(0)A(0)'P(9) (0)) r2V(0). Thus, the vector has the proper-ties ofa roscedai:e di\tuI hance ector, and th . h1tiL partmctei leg: eiuiimodel, insofar as it implies an estimator for ni, is seen to he 110 more ihan an ordiri-arv regression with a complicated covariance among the ''disturbances.

II. Siouisric IARAMI:TER r\4ODfi.S

Before discussing the several stochastic parameter models that hate beenproposed in the literature, the concept ol' the Markovian Canonical Form for amodel, which will he needed in the discussion, will be introduced.

I. VIarkoi',apj ('ano,iicul Forni

A stochastic parameter regression model is in Markoiian ('anonical FormMC'F). when the (possibly transformed) set of parameter vectols. b b.satisfy the following conditions:

(if The observations of tile dependent variables appear in linear regressionsof tile form v = 1P(0)b + u. such that for i coy (ut, u') 0.

(ii) The joint distribution of tile parameters may be represented by priordistributions for the parameter vectors, which are uncorrelated acrossdifferent parameter vectors: and by a set of linear transition relations, eachlinking sonic pair of parameter vectors in the form b = + d1.where each stochastic parameter s/nj t vector d1 has a specified mean andvariance which may depend on 0, and where each shift vector is i,in-correlated with all other shift vectors and with all disturbances.

A niodel in MCF may he represented visually by a graph. in which the erticcscorrespond to the parameter vectors, and the links between vertices to the transi-tion relations. For instance, in a time series regression in which disturbances areserially uncorretated and parameters follow a first-order Markov process. so thatsuccessie parameter shifts are uncorrelated. the model is in MCI represented bythe graph in Figure 1.

b, b,

Figurc I

A model that is not ininiediately in MCI may often he transformed into thisform by adjoining to the parameter vector slate rariahies that transmit the correla-tion in the model. To clarify this, consider tile familiar fixed-parameter modelwith fIrst-oi-der autoregressive disturbance process given by u, = pa, - - '.where E(e,) = 0. var (e,) = a. By regarding the disturbance U, as tile ''stochasticparameter." the model is represented in MCI, with graph as in Figure I .as follows:

Regrcsioii ,'e!alwnsb

1=1.....TU,

Prior distribution

) = 0. var (ii, ) = (r( I - P1

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I

Tra;isition relolion.s

U, -+ , t I I

= 0, var [e',]

This illustration shows two interesting characteristics of stochastic parameterregression. For one, the distinction between disturbances and stochastic param-eters is purely arbitrary. Secondly, there may, in fact, he no trite disturbances atall; when u becomes a parameter, no additive disturbance remains.

The essential characteristic of the MCF is the absence of correlation acrossregression observations and transition relations. The regression observationsand/or prior distribution for any parameter vector then provide anwont ofinforma-tion about that parameter vector, with random components uncorrelated withother stochastic terms in the model, which may be processed by the ruies ofordinary fixed-parameter regression. Since the linear transition relations betweenparameters are uncorrelated with the atoms of information, the atoms may becombined by applying, at each transition, the rule for a given linear transforma-tion with an uncorrelated additive stochastic shift. All information in the regres-sion may be combined by recursive application of this procedure of stepwisecomposition, incorporating the information for each parameter vector by ordinaryregression, and combining information by linear stochastic transformation orextrapolation. The process may be visualized as a tracing out of the links in thegraph representing the MCF, with the step along each link being an extrapolationof information collected at the previous vertex, and with the action at each vertexbeing the combination of the atom of regression information for that parametervector with the extrapolated prior information.

This approach to deriving estimators in the stochastic regression problemmay be termed atornistic (Rosenberg, i968b). It contrasts with the alternativeholistic approach of direct derivation from the general linear form (3). Whateverthe method b which the estimators are derived, the great significance of the MCFlies in the fact that the matrix algebra required for the atornistic regressions andfor the extrapolations is never of greater dimension than tile dimension of theparameter vectors b.2 Hence, it follows that an MCF with parameter vectors oflow dimension is a sufficient condition for estimation in the model to he computa-tionafly feasible. In addition, when the model cannot be exactly l)laced in MCF,an approximation in MCF often suggests an approximate computationalprocedure.

B. Randomly Dispersed Parwneter Models

The term "random," as opposed to "stochastic." is reserved for the specificmodel in which regression parameter vectors are random drawings from acommon multivarjate distribution, with mean vector m and variance matrix

2 For instance, the models proposed by Cooley and Prescott (l973c; and Sarris 119731 are easilytransformed to have the MCF in Figure I; with this insight, application of the formulae in Rosenberg((1968a) (l973c. Sect. II)) allow the N-dimensional matrix operations proposed by these authors to beavoided entirely. There is a substantial savings in required computations when N is large )inore than 4orders of magnitude in the "Capital Market Application" mentioned by Cooley and Prescott)

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L

Figure 2

When only the constant term varies, the model becomes the analysis olcovariancewith random effects. If the parameters are represented as b = m + i, n = I.....N,where m is the population mean vector, the process is in the general form (2).When disturbances are uncorrelated across the individual parameter vectors,the model falls immediately into the MCF in Figure 2.

The model is clearly applicable to a random sample of individuals in whichonly one observation is taken on each individual. When there are repeated observa-tions on an individual, the model applies if successive parameter vectors for theindividual are independently drawn, as assumed by Fisk (1967), or if the successiverealizations for each individual are identical, as assumed by Rao (1965b).

Several early studies examined the case where only a single observationcorresponded to each parameter vector. Rubin [1950] wrote down the likelihoodfunction, Theil and Mennes (1959) and Hildreth and 1-buck (1968) studied estima-tion of m and , assuming Li to be diagonal, and Fisk (1967) studied this casewith general Li. Rao (1965a, pp. 192-193: 1965b) and recently Efron and Morris(1972) studied the second case where multiple observations are generated by eachindividual parameter vector, but with the assumption that the regressor matrix isof rank k and identical for all individuals. Swamy (1970, 1971, and 1973) permitteddiffering Xr but retained the requirement of rank equal to k. Rosenberg (1973a)relaxed the rank k assumption. Applications of the model appear in Swamy(1971), Rosenberg and McKibben (1973), and Sheiner, Rosenberg, and Metmon(1972). In the last, the regressions are nonlinear, and an approximation suggestedby the MCF is employed.

C. Sequential or Markor Parameter Models

In these models, parameters follow a first-order Markov process, or moregenerally, an autoregressive or moving-average process of low order. The model isnaturally applied to a time series, with the stochastic parameter process intro-ducing random drift in the parameters. The process is represented as b,1 ='Jib, + j,, r = I,. . -, T. In order to place it in the general form (2), the parametervector in the initial period may be chosen as the unknown vector m, and theparameters may be represented as b, = 'Jm + t = 1,..., T.The MCF was given in Figure 1. This general model was first analyzed by Kalman(1960) and Kalnian and Bucy (1961), who originated an extensive literature incontrol theory and the applied physical sciences, in which the optimal estimationmethods are often referred to as the Wiener-Kalman or Kalmnan-Bucy filter.

A range of studies have introduced applications of this model into economicsand statistics: Rosenberg (1967, 1968a), Teräsvirta (1970), Duncan and Horn(1972), Bowman and LaPorte (1972). Where only the constant or intercept termvaries, the special case of the odaptire regression model arises: Cooley (1971)and

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('ooley and Prescot 1(1 973a. I 973h). The further pec tal case ol time eiie' a iitiv',kwhere a time ai ing constant is the only term present. ha. bc hdic:dsisely : Swerhng (1959), M uth (196W. Pa rten I 961 L lx a tid ikis (I1970). Another family of special cases was studied in depth b RiO ii(l 963 ).Secalso, the articles in this issue h Coolev and Prescott I 97k), Sarris p73), andRosenberg )1973c).

D.Sitionwv StocIia.sti( Pw'onien'r I oiIeL

Burnett and (iuthrie (1970) introduced the eenci'aI iitodcl '. here parametervariation over time follows a stationar stochastic process, Althouh this modelis of formal interest, it sutkrs froni the defect that it Cititilot, m general, he trans-formed into Markovian Canonical Form wit Ii a parameter ' ecior of low (limen-sion, so that the computational problems associated with it arc horrendous

E. Cro.s.s-Seciio,i 'I 'ime-Seric. A'Iodt'L

Statisticians have percei ed the need ioi responsiveness to Itriime1ei' vaija-tion most clean in cross-section time-series analysis, not necessarily because it isintrinsically any more necessary here than in the analysis of a sinele cross sectionor a single time series. hut rather because the greater number of obser ations allowsmore degrees of freedom to deal with the problem. The traditional method allowsthe intercept in the regression to vary randomly over time (time effects) and acrossthe population (individual etkcts). An extensive literature on this subject, which isan interesting case of stochastic parameter regression, exists. Some importantrecent contributions are Tiao and 'Tan (1965. 1966). Chetty (196$). Wallace andHussain (1969). Amemiya (1971), Maddala (1971), Swami' (1971. Ch. 3). Swamand Arora (1972). Swam)' has applied the random parameter model to a crosssection of time series, assuming that individual parameters arc fixed oer time. Hehas also anaiyted the case where a random time effect occurs each period : theseparameters must he adjoined to the individual paranieter vectors to bring themodel into the MCF !fl Figure 2. Hsaio (1972. 1973) has proposed an extension.in hich the parameter vectors are the sum of random individual parametervectors and random time vectors, so that the traditional procedure applied to theintercept is generalized to the entire parameter vector (I 972a) The MCF for thismodel cannot be fully simplified, and Hsaio's decomposition of the computationalformulae requires a matrix inversion of rank Nk (1973, Eq. A 12). an inkasihkcomputation in a large cross section. Therefore, an approximation. presumablybased on approxiniate decomposition of the model into a randomli dispersedprocess oer lime. Superimposed on another across the population. is required.Another article in this issue proposes a paranieu'r flIOdCI. in whichindividtjiI pararnetem' ectors follow first-order Marko processes stiiordinatedto a tendency to converge to the population norm. The MCI" for this model, givenin Rosenberg (197k, Figure 1) in terms of parameter \ectors of order A'k. can heaccurately approximited by a simplified MC 'F' of order k. thereby rendering thecomputations feasible.

3X6

I". Switching Propneu'r 1ode1

Quandt and Goldfeld 1973) have continued Quandt's earlier study (1958) ofthe model where a parameter vector switches stochastically between two values.The model is applicable, for instance, if supply and demand functions alternatelydominate in a market. Any of the parameter processes introduced thus far couldhe formulated as a switching parameter model. However, the estimation methodsgenerally considered for these processes are intended for parameter variation witha continuous distribution, and are less efficient in the switching parameter problemthan the methods developed by Quandt and Goldfeld to exploit the binary distribti-lion.

III. TuE IMPOR1AN('l OF REspoNsivENEss ]O STOcIIAsTII' PARAMETER

VARIA1 ION

In the general stochastic parameter model (2), the estimation problem can heviewed as consisting of three parts: (i)estiniation of the unknown parameter vectorm, (ii) estimation of the stochastic parameter vector t, and (iii) estimation of thestochastic specification, c2 and 0. For instance, in the random parameter model. a

(i) corresponds to estimation of the population mean parameter vector. (ii) toestimation of the individual parameter vectors, and (iii) to estimation of the dis-persion of the individual parameter vectors. In the sequential parameter model, (i)corresponds to estimating the initial value of the parameter vector (or. by a trans-formation, to estimation of the current value), (ii) corresponds to estimating thehistory of parameter realizations, and (iii) to estimating the Marko parameterprocess.

It is clear that (iii) can only be accomplished in the context of a randomparameter model. Thus, when the process of parameter variation is of interest (as.for instance, if stochastic variation in corporate return on equity reflects competi-tive forces and the magnitude of these forces is to be determined), a randomparameter model is essential.

With regard to estimating m. it is clear from (5) that, so long as any nonzeromean in the stochastic parameters is adjusted for, an ordinary least squaresOLS) regression for rn will be defectie only insofar as it ignores heteroscedasti-

city: thus, OLS will be unbiased hut inefficient, and the advantage of a randomparameter context consists in allowing the heteroscedasticity to be identified sothat Aitken's Generalized Least Squares may be applied. This improvement inefficiency appears to be substantial in simulations (e.g., Cooley and Prescott(1973a), Rosenberg (1973c)). Equally important, in the presence of parametervariation, OLS sampling theory severely understates parameter estimation errorvariance. Thus, recognition of the correct specification removes a downwardbias in estimated error variance.just as in the use of the familiar Cochrane-Orcutttransformation when disturbances are serially correlated.

Inappropriate use of a fixed-parameter model causes even more severe prob-lems in estimating the individual parameter vectors, because stochastic param-eter variation introduces random components in these vectors. Because thefixed-parameter model ignores these random components, the inefficiency and

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invalidity of OLS are severe in the simulations reported in Rosenberg (IOLS error variance rises to fIve times the cilicient variance, and OLS simplintheory underestiniates OLS error variance by a htctor of twenty or more. In thestochastic parameter model, minimum mean square estjmatton is achieved byattributing a proportion of the residuals y - (0)th to the stochastic parametersfamiliar econometric methods such as Theil's BLUS procedure and ihe estima-tion of the latest residual in the autoregressive disturbance model preparatory toforecasting, are special cases of this method.

If parameter variation is ignored in estimation, the estimate for iti will tendto be an average of the realized individual parameter vectors. From this perspec-tive, responsiveness to parameter variation s seen to he (i) valuable in achieving amore efficient estimator for this average, with more valid sampling theory: (ii)critically important in estimating the realized values of the individual parametersas distinct from the average, and (iii) essential in analyzing the stochastic parameterprocess.

As an illustration, consider the medical problem of estimating the phvsio.logical parameters describing an individual's response to a drug, with the purposeof recommending a correct dosage. These parameters will vary svsLematicallacross the patient population in relation to measurable patient characteristicssuch as weight, and will, in addition, vary stochastically about these Systematicpredictions in a manner that can be modeled, at a first approximation, by therandom parameter model, assuming individual stochastic parameters to be fixedover time. Then one or more observations on the individual's response to the drugallow this random component to be estimated, thereby allowing dosages nioreappropriate to the individual to be prescribed. This approach has been appliedwith substantial success in a program under way at the University of CaliforniaMedical Center, San Francisco (Sheiner, et al. (1972)). The method providessuperior estimates of the population mean parameters and of the systematicparameter variation,3 improved predictions of individual parameters, andmedically useful descriptions of the extent of stochastic parameter \ arcation.

IV. ESTIMATION OF THE STOCHASTIC Sucu.icAriç)\The estimation problem in a linear stochastic parameter regression breaksnaturally into two stages: (i) estimation of the stochastic specificatioti parameters.0: and (ii) estimation of the scale parameter a2, of the unknown parameters iii.and of the stochastic parameters q. The break occurs because the estimators for thelatter parameters, conditional on 0, can be expressed analytically, whereas mostestimators for 0 must be computed by iterative procedures, Note that a2 is treatedas a scale parameter for all second moments (thosc of the parameter distributionas well as those of the disturbances), so that 0 specifies only the relative niagni-tudes of these moments This allows the problem of estimating 0 to he reduced byone dimension and is therefore convenient Estimation of a2, conditional on 0. isalways a straightforv matter, exactly analogous to the methods for an ordin-ary regression: moreover, this computation is always iniplcit in calculation of

inappropriate use of the tlxed.parameer assurnptioi in ibis case ield hiaed a well ,ts mcii-c!ent estimators hecaus the regression models arc nonlinear

3g8

the sample likelihood or posterior distribution for 0, so that it need not be treatedseparately. (It ic a natural alternative to include a2 in an expanded 0 vector, andthe analysis could be carried out in this way if desired.)

In this section, the problem of estimating 0 will be discussed. Let R5 be theregion of admissible values. Notice that there are two classes of criteria for optimalestimation of 0: those which are specific to 0, and those which refer to the proper-ties of the estimators for m, 'i implied by the estimator of 0.

When stochastic terms are normally distributed, two natural estimators toconsider are the Maximum Likelihood estimator (MLE) and the Bayesian. InRosenberg (I 973b), the likelihood function and Bayesian posterior for Dare derivedfor the general model (3). The forrnu!ae to compute these for each 0 also yield theMLE and Bayes posterior distributions for a2, m, and ii, conditional on 0. Maxi-nium Likelihood estimation may be accomplished by a search over R0, evaluatingthe likelihood function at each point. Bayes estimation is accomplished J,ynumerical integration, with respect to the posterior distribution for 0, again by aniteration over R0. Computations at each step of the iteration are of the same orderas required for estimation of m, conditional on 0. Therefore, if the latter computa-tion is feasible, as will be discussed in Section V, then the only computationalproblem arising is the need for repeated computation of the formulae for many 0values in R. For an application of these methods to the sequential parameterproblem, in which the MCF is exploited to simplify the computations for any 0.see Rosenberg (1973c, Sections IlA, IIB). Coolev and Prescott have taken the same

approach to the special case of the Adaptive Regression Problem (l973a. 1973h).4For the randomly dispersed parameter problem, see Fisk (1967, Sections, 5, 6),Swamy (1971, p. 111), Rosenberg (l973a).

The author's experience with these methods is quite favorable. In both thesequential and the random parameter models, the likelihood function has beenwell behaved, without extrema other than the global maximuni, and convergencehas been achieved without difficulty. The computations are quite feasible on third-generation computers convergence in a nonlinear random parameter model withseven elements in 0 and roughly 500 observations is routinely accomplished in lessthan a minute on the CDC 7600, using a modified FletcherPowell algorithm.Moreover, the estimates for the stochastic specification parameters and for theregression parameters in several enipirical applications have been consistent witha priori expectations. aithough the latter were not included in the estimationprocedure in any way. This is perhaps the most robust test of any method.

The optimal properties of Bayesian methods, given that the prior distributionis appropriate, are well known. The asymptotic optimality of some MaximumLikelihood estimators is also well known, and Anderson (1970) extends theseoptimal properties to the case where the variance matrix V(0) is linear in 0. He alsonotes that if a consistent initial estimator of 0 is available in this case. a singleiteration of the NewtonRaphson procedure applied with these initial estimates

Hoever, in Cootey and Prescott (1973c there appears to be a misunderstanding as to theexistence of a general solution. CP appear to assert that their mode! is not a member of the family ofmodels with sequential MCF as in Figure 1. As a matter of fact, the model is easily transformed to havethe MCF in Figure 1, and the likelihood function for the model in this form is given in Rosenberg(1973c, Sect. II).

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a

as a starting point will then provide asymptotically LlhCieflt estimators ('001ev(1971) and Cootey and Prescott (1 972h) dci i thL ds) iliptotic properties 01 theparameter shift variance estimator in the adaptive regression, which is a specjtlcase of Anderson's problem. Notice, however, that the tfluilSitj0fl matrices ui theMCF enter nonlinearly into the matrix V(0) asymptotic CI1jciec when V isnonlinear in 0 has not been demonstrated, to the author's knowledge, although thedemonstration should be relatively straightforward. See Kusimer (1967) andKashyap (1970) in this context. The asymptotic sampling properties of the Bayesianestimators are essentially equivalent to those of the Maximum Likelihood cstinitor (see, e.g.. Johnson (1967) and Zellner (1971, p. 31)).

On the other hand, the small sample properties of Maximum Likelihoodestimators are not necessarily optimal, nor are the Bayesian estimators' propertiesnecessarily optimal, in a sampling theory sense, ii the prior distribution i notbelieved. Efron and Morris (1972. 1973) approach this problem. For the randomlydispersed parameter model, with identical regressor matrices for all indjvjdilsthey are able to deduce the sampling properties of the regression parameterestimators implied by a class of estimators for the dispersion matrix and todeduce from this the implications of alternative prior distributions for elements of0. Aithough substantial generalizations will be needed before these results can bebrought to bear on the complex regression prob!ems usually encountered ineconometrics, this approach is promising.

Within the class of estimators considered by EM are Sonic estimators of 0that are simple quadratic functions of the dependent variables, possibly truncatedby setting to zero any negative estimated variances. These estimators fall withina general category of possibly truncated quadratic or iterative quadratic estinia-tors of the variances in 0, with the number of iterations, if there are any, beingsmall. Within this general category, the simplest estimators are regressions of thcform ê = 0, where é represents the squared values of residuals (or a transforma-tion of these residuals) from (5), and X represents the squares and cross productsof the regressors in (5). Fisk (1967) pioneered this method and Hildrcth and Houck(1968) developed it independently. Where only a single observation on eachparameter vector arises in the random model with normall) distributed stochasticterms, a single iteration of this method is asyinptoticillv efficient, as Hildrethand Houck have shown, and as follows from Amemiya's general results onregression where the variance of the dependent variable is proportional to thesquare of its expectation, with the dependent variable following a gamma distribti-tion (1973). In more coniplicated stochastic parametcr regression niodels, how-ever, a complicated heteroscedasticity in this "second moment regression"appears, and the method can no longer be made asvmptoticall' efficient withoutarduous computations Nevertheless because of its computational simplicity, itmay be proposed as a quick initia! estimator for a subsequent Maximum Likeli-hood procedure, as in Rosenberg (l973c, Section tic).

Analogous to these "second moment regression" estimators are a set ofquadratic estimators for variances that have been conceived for \arious models.which have the virtue of unbjasedness but not necessarily any virtue of small-sample or even asymptotic optiniality. Among these are Swaniy's suggestedestimator of for the randomly dispersed model with differing indivithial regres-

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sor matrices of full rank (l)7O), and Anderson's suggested initial estimator forthe ieneral covariance matrix \'(0) that is linear in 0 (1971), Eq. (2.16)). See alsoArora (1973).

Rao (1970, 1971, 1972) has investigated quadratic estimators that are tin-biased and optimal, either in the sense of minimum variance MlVQUE) or in theweaker but coniputationally more accessible sense of minimum norm (MINQUE).For both classes, optimality is defined with respect to a norm depending on 0,so that construction of the optimal estimator for 0 requires, in principle, that 0be known. If a guess at 0 is used to define the estimator, the estimator will beunbiased but not efficient. If 0 yielded by the first iteration is used to define theoptimal estimator for a second iteration, the estimator may become more efficient.but the property of unhiasedness is lost. Rao's approach is presumably bestcarried out as an iterative estimator. Rao (1972) has provided some generallemmas to aid in deriving the MINQUE estimators for a general linear model.MINQUE is very closely related to Anderson's iterative estimator. Anderson'sproblem where V(0) is linear in 0 is equivalent to Rao's problem of variance andcovariance components. Let /i0) denote the classical likelihood function for thisproblem, and let P*(0) denote the Bayesian posterior distribution for 0 conditionalon the sample information, given diffuse prior densities For i and m (see Zellner(1971: Ch. 2)). Then .V*(0) t(0)IW(0)l'2, where W(0) is the variance matrix forthe estimator ofm, given 0. It may he shown (Rosenberg( 1973b()that the MINQUEequations bear the same relationship to maximization of t' as Anderson'siterative procedure does to maximization of .. Since and differ by a factorof order equal to the number of unknown parameters, MINQUE coincides withAnderson's procedure when no parameters are unknown, and for any vector m,the two methods coincide asymptotically asthe number ofobservations approachesinflnity. The adjustment W(0) 2 serves to achieve unbiasedness in small-sampleestimators of variance components.

Another desirable property of (f* --and therefore of MINQUE--is that it isinvariant with respect to a linear transformation of the parameters in aiicl i,whereas /' is not. For example, MINQUE and Bayesian estimation are notaffected, and Anderson's procedure and Maximum Likelihood estimation areaffected, by Cook)' and Prescott's decision to use the current parameter (which isa linear transformation of the initial parameter and of the stochastic parametershifts in the sample history) as the unknown parameter in the adaptie regressionmodel in place of the initial parameter.

Monte Carlo explorations of small-sample properties are necessary tocompare these alternative estimators. A recent study by Froelich (1973). while awelcome step n this direction, is somewhat confusing in that it identifies

it "MINQUE" with one initial guess for 0 (corresponding to homoscedastic dis-

turbances), without noting that other MINQUE estimators exist. It is importantto notethat in all but the simplest stochastic parameter models, iterative proceduressuch as Anderson's and MINQUE tire crucially dependent on the initial guess for 0.If the initial guess is good, one iteration may yield better small-sample properties

d I am indebted to Cheng Hsaio for seerat useful conjeciurcs as to the general relationshipbetween MINQIJE and maxiniuni likelihood estimation.

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thaii COiltifllEutiOfl of the iteMt ye procedure to the point i _Oe bmitial gUess is poor, the inethoJ may perform poorly.

There is a substantial applied physical scieiice literature on "ideniitjcatior''ol U in the sequential parameter model, e.g.. Astrm and FvkholT 1971). Kashyap(1970), Mehra (1970, 1972), Sage and Melsa (1971 . M ttch of the literature on thecross-section, time-series model with random time aiid in(lividtlal ellects cited inSection LII. is also interesting in that it provides a detailed exploration of' aspecial case.

All of the theory presented here is no more than a speciahizat ion of the theoryfor the general linear model, and the foundations for most of the results wereoriginally derived on that more general level. An unfortunate limitation is that allthe estimators are either optimal with rLspect to the normal distribution or elseare quadratic functions of the stochastic terms, and in neither case are the methodsrobust against stochastiedistributions with massive tails. This implies that, regard-less of the method used, the experimenter would be wise to examine the outliersin the sample, to check the robustness of the estimated parameters against dele-tion of these outliers, and, quite possibly, to delete the outliers if the results aresensitive to their presence.

V. EsrIMATI0N or THE UNKNOWN AND SrocuAsTIc Rr;REsSIoN PARAMETERSFor the general linear stochastic parameter regression model, (3. with

stochastic terms assumed normally distributed for Maximum Likelihood andBayes purposes. Rosenberg (196S, 197Th) has derived Maximum LikelihoodBayesian, and Minimum Mean Square Error Linear Unbiased Estimators(MMSLUE) conditional on 0, and Duncan and Horn (1972) and Sarris (1973)have derived MMSLUE and Bayesian estimators, respectively, under the addi-tional assumption that a proper prior distribution for the unknown vector mexists. It turns out that, conditional on 0, these three types of estimators (MLE.Bayesian, and MMSLUE) coincide, provided that careful attetition is given to theconcept of unbiasedness.

The problem with bias arises because there are several natural expectationoperators to use in discussing the bias in an estimate of a stochastic parametervector. Let bn + Bq. where A is nonsiiigular, as is usually the case. Thenthree possible expectation operators, with respect to which an estimator bn mighthe detmned as unbiased .are:

Em) Et. rn)

= I) E( . rn, b,h,)Ei I E( . rn. b,,).

The first "unconditional expectation" written as "L" by Duncan and Horn, isthe familiar expectation in a linear regression model of form (5). The secondexpectanon ;s conditional on the realized value of the parameter vector bn. withthe distribution for m being taken as conditional on b,1. The third expectation.Conditioned on both vectors, is important in discussing the behavior of b whenb differs from in. An estimator ã, will he called in - Eu ma - ] unbiasedif E[E, Emn](n) = Em[En. Emr](ai). For applications of these concepts. seeRosenberg (1972, l973a).

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It may he shown that the expectations E,, and E, are essentially equivalent, inhat any estimator that is ni-unbiased is also n-unbiased. However, it turns Out

that the choice of expectation does allect the minimum variance linear unbiasedestimator (MVLUE) for a stochastic parameter. Specifically. ha subscript denotesthe expectation operator with respect to which the estimator is defined, then inestimating b, MMSLUE = MMSLUErn = MVLUE, but these three estima-tors are not equal to MVLUEm. In fact, it is easily seen that ii A = I, then theMVLUE, for b, is identical to the MVLUEm for m, regardless of the stochasticvariation in Since MMSLUE = MMSLUEm. but MVLIJE MVLUEm, itis preferable to speak of a minimum mean square error property for an estimatorof a stochastic parameter rather than a minimum variance property.6 It maynow be asserted that the MLE and the Bayesian posterior means (with diffuseprior for m), for m and t, conditional on 0, coincide and are also MMSLIJE wheno is known. Since these three estiniators conditional on U do coincide, they willhe referred to hereafter as "the optimal estimators."

These estimators are not mi-unbiased. For instance, in the randomly dis-persed parameter model, the population mean estimator th is slightly mn-biasedtoward b and the individual parameter estimator b,, is mn-biased toward m, ifb,, in.

Since the three optimal estimators coincide, it is a matter of indifference bywhich criterion the estimation formulae are derived for any special case, exceptthat it is preferable to use an approach which also allows the likelihood functionand posterior distribution fof 9 to be computed. The history of the derivations ofthe optimal estimators For the special models described in Section 11 will now besurveyed briefly.

For the randomly dispersed parameter model, Rao (1965) originally derivedthe optimal estimators for the population mean parameter vector by a holisticapproach, and also the optimal estimators (predictors) lot' the individual param-eter vectors, with the population mean assumed known. More recently, Efronand Morris (1972) and Rosenberg (1973a) derived the optimal estimators for theindividual parameters with the population mean being unknown, the latter forthe genert! case.

For the sequential parameter model, Kalman (1960) and Kalman and Bucy(1961) originally derived the optimal estimators for the current parameter vector(the KalmanBucy filter). The optimal estimators for the parameter vectors in theinterior of the sample period (the smoothed estimators) and their variances werederived by Bryson and Frazier (1962), Lee (1964), Rauch (1963), Rauch. Tun. andStriebel (1965), and Meditch (1967). Other important early contributions were byBattin (1962), by Ho and Lee (1964) in expositing the Bayesian approach. and bySchweppe (1965) and Kushner (1964, 1967) in exhibiting the likelihood function.All of these works were atomistic in approach; the first holistic approach was byFagin (1964). All of the above articles assumed that there was a proper prior

'Duncan and Horn avoid this difficulty in a rather confusing way by defining a mintmum varianceestimator as thai estimalor with minimum error raria,z'.a violation of the traditton that ihe "variance"of an estimator is its own variance, not the variance of the estimation error The theorems in Duncan andHorn hold only if "minimum variance" is defined in this unusual way otherwse, the term "minimummean square error' must be subsiituted.

393

distribution for the initial parameter vector. so that no unknown paramelci \cctrwas present ThN 'starling problem or inahil it's 10 dcal ith tiiikiiot, nparameter vector was quite troublesome. As late as 967, Aok, in an otherwiseexcellent survey, proposed a solution to the starting problem that was erroneousbecause it was based on false generalized matrix inversion identities (1967

p. 80). Duncan arid Horn (1972) and Sarris (1973) fl their recent treatnietits of theproblem continue to assume a proper prior distribution for rn independent of thedata, in discussing estimation of the stochastic parameters. The first solul ions tothe starting problem were found independenil) by Fraser (1967) and Rosenberg(1967), and what is apparently the first general solution to this problem whichpermits computation of the sample likelihood IS 1)iibliSilCd in this i5SUC (Rosenber1973c, Section llBfl. The formulae for the covariances between eStifliation errors

in different periods, together with an analysis of the relationship betwceti theatomistic and holistic approaches, appear in Rosenberg (1968a, 1968h).

For the stationary stochastic parameter model. Burnett and Guthrje (1970)derived the optimal estimators conditional on the mean parameter in heinknown, and Rosenberg (1972) generalized these to the case where m was unknownWith regard to cross-section time-series models and switching regressions, thereader is referred to the articles already cited in these contexts.

VI. CoNci.usioNOn the whole, the theory of stochastic parameter regression is one of the

most exciting areas of statistical investigation. Moreover, the theory seems tohave already reached the point where it promises fruitful applications. The mostproductive applications in econometrics are likely to conic in cross-section time-series analysis, where the wealth of data offers a real opportunity to identify thepattern of parameter variation: stochastic parameter methods are especiallyneeded when estimation of the individual parameters. as distinct from the popula-tion mean alone, is of great importance. since fIxed-parameter methods arerelatively less efficient in estimating these parameters. Potential applications alsoarise whenever it is desired to analyze the process of parameter variation itself.an aspect of economic events that has been studied all too liule. To cite just oneexample, of what strength are the competitive forces that cause corporate rates ofreturn on equity to converge toward the norm for the economy. and of whatmagnitude are the stochastic shocks that allow corporations to achieve ahoe-average returns. A third promising area of application is in short-term forecasting(Rosenberg (1968. Ch. 8), Coolev (1971)), where adaptation to sequential param-eter shifts is essential.

Lnirt'rxitv of (ali/ornia, Be'rkelev

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