Abstract: The parameter concept in the term least squares mean is defined and given the more meaningful name population marginal mean; and its estimation is discussed.

Abstract: A solution method and an estimation method for nonlinear rational expectations models are presented in this paper. The solution method can be used in forecasting and policy applications and can handle models with serial correlation and multiple viewpoint dates. When applied to linear models, the solution method yields the same results as those obtained from currently available methods that are designed specifically for linear models. It is, however, more flexible and general than these methods. For large nonlinear models the results in this paper indicate that the method works quite well. The estimation method is based on the maximum likelihood principal. It is, as far as we know, the only method available for obtaining maximum likelihood estimates for nonlinear rational expectations models. The method has the advantage of being applicable to a wide range of models, including, as a special case, linear ,models. The method can also handle different assumptions about the expectations of the exogenous variables, something which is not true of currently available approaches to linear models.

Abstract: We consider two methods of defining a regression analog to a trimmed mean. The first was suggested by Koenker and Bassett and uses their concept of regression quantiles. Its asymptotic behavior is completely analogous to that of a trimmed mean. The second method uses residuals from a preliminary estimator. Its asymptotic behavior depends heavily on the preliminary estimate; it behaves, in general, quite differently than the estimator proposed by Koenker and Bassett, and it can be inefficient at the normal model even if the percentage of trimming is small. However, if the preliminary estimator is the average of the two regression quantiles used with Koenker and Bassett's estimator, then the first and second methods are asymptotically equivalent for symmetric error distributions.

Abstract: Traditionally, most of the effort in fitting full rank linear regression models has centered on the study of the presence, strength and form of relationships between the measured variables. As is now well known, least squares regression computations can be strongly influenced by a few cases, and a fitted model may more accurately reflect unusual features of those cases than the overall relationships between the variables. It is of interest, therefore, for an analyst to be able to find influential cases and, based on them, make decisions concerning their usefulness in a problem at hand. Based on an empirical influence function, we discuss methodologies for assessing the influence of individual or groups of cases on a regression problem. We conclude with an example using data from the Florida Area Cumulus Experiments (FACE) on cloud seeding.

Abstract: This article describes the mathematical theory underlying an interactive computer program for eliciting the hyperparameters of a subjective conjugate distribution for the multiple linear regression model with the usual normal error structure. Although the methods are heuristic, they are shown to produce hyperparameter estimates satisfying the constraints satisfied by the hyperparameters themselves. An application is given to the problem of predicting the time to fatigue failure of an asphalt-concrete road as a function of several design variables concerning the road.

Abstract: This article deals with linear models for which data have been aggregated over well-defined geographic areas. Such data may be generated by spatial processes, and these may be represented in the fo...

Abstract: SUMMARY Robust tests of general linear hypotheses in linear models are developed. These are likeli- hood ratio type tests in the same sense that M-estimates are maximum likelihood type estimates. Construction of the tests suggests a decomposition of the data into terms analogous to classical sums of squares, providing a robust analysis of variance. Asymptotic efficiency and robustness properties of the tests are the same as those of the M-estimates upon which they are based. Parameter estimation is usually only a first step in the analysis of data arising fromn a linear model. A classical least squares analysis often focuses upon the analysis of variance, which tests simultaneous hypotheses on large subsets of the parameters. Since the terms in a classical analysis of variance are quadratic forms in least squares estimates, one would expect that the sensitivity of the estimates to departures from normality should be inherited by the tests. In fact, for moderate to heavy tailed error distributions or in the presence of outliers, it appears that the classical F test does lose power. Calculations of relative efficiency for proce- dures proposed in this paper substantiate on theoretical grounds the possible inefficiency and lack of power of classical F tests. In this paper, Huber's M-estimates are adapted to hypothesis tests which can be termed likelihood ratio type tests. These procedures naturally generalize and bear a striking re- semblance to classical F tests. Robustness and efficiency properties of M-estimates apply directly to the proposed tests. Hence the case to be made for using likelihood ratio type tests rather than classical F tests is the same as that for using M-estimates in favour of least squares estimates: possible poor performance of the classical methods may be overcome with methods which perform well both when classical assumptions are met and when they are not. The proposed methods are natural, intuitive and as easily computed as M-estimates.

Abstract: SUMMARY A general discussion is given of the approximate distribution of the residual sum of squares in a linear model in which a weighted analysis is made with weights estimated empirically as the reciprocals of variance estimates. Applications to testing hypotheses are made and various generalizations indicated. The initial results simplify those of James (1951, 1954). If in a regression problem the variances are not equal it is common to use the reciprocal estimated variances as weights. The residual sum of squares Q has asymptotically a x2 distribution when the degrees of freedom tend to infinity. Welch (1947, 1951) gave an approximation to the distribution of Q in the special case of the comparison of n means, by using a suitably chosen F distribution. James (1951, 1954) gave an improved approximation using the fractiles of a x2 distribution and extended the results to the general linear model. We shall show here how the results for the general linear model can be considerably simplified by using the technique due to Welch (1951), and extend the results to multivariate models and variance component models. Results will also be given on the variance of the fitted value, thereby extending the results of Jacquez, Mather & Crawford (1968) to the general linear model.

Abstract: In recent years, increasing attention has been devoted to models with a finite (usually small) number of regimes. Various strategies have been discussed in the literature to handle situations where each regime is characterized by a different value of a common parameter vector. See e.g. Barten and Bronsard (1970), Goldfeld and Quandt (1973), Poirier (1976),... . It appears however that no satisfactory treatment has yet been given to cases where the partitioning between " endogenous " and " exogenous " variables changes over time. Our objective is therefore to define a class of models with several regimes which is flexible enough to cover such situations. For convenience, we shall illustrate our argument by reference to an economy which is "controlled " by a policy maker shifting between instruments at some, possibly unknown, points of time. For tractability we shall mainly restrict our attention to a class of dynamic linear models although the concepts we introduce apply in a much broader framework. The possibility that the switching times could be endogenous to the model, such as in disequilibrium models will not be investigated here: work in progress indicates however that our approach can be extended in such directions. The paper is organized as follows: In Section 2 we shall discuss at length the issues to be faced by means of a simple example, taken from Goldfeld and Quandt (1973). In Section 3 we shall introduce the concepts which are needed for our analysis; linear dynamic models, LIML estimation and exogeneity. In Section 4, we shall discuss models with several regimes and concentrate in particular on imposing appropriate restrictions on the parameters characterizing different regimes. It will be shown that it is possible to preserve some of the operational features of LIML procedures.

Abstract: This paper develops a general procedure for performing a wide variety of model specification tests by running artificial linear regressions and then using conventional significance tests. In particular, this procedure allows us to develop non-nested hypothesis tests for any set of models which attempt to explain the same dependent variable(s), even when the error specifications of the models differ. For example, it is straightforward to test linear regression models against loglinear ones. These procedures are illustrated with an application to estimate competing models of personal savings in Canada.

Abstract: Using the concept of estimability, tests of hypotheses in multifactor fixed effects linear models are developed without resorting to the “usual assumptions.” Three types of estimable functions are defined. Each type handles unequal n's, missing cells, and any degree of confounding for any fixed effects linear model.

Abstract: One of the problems most frequently encountered by the applied econometrician is the choice between logarithmic and linear regression models. Economic theory is rarely of great help although there are cases where one or other specification is clearly inappropriate; for example, in demand analysis constant elasticity specifications are inconsistent with the budget constraint. Nor are standard statistical tests very useful; R2 statistics are not commensurable between models with dependent variables in levels and in logarithms and the comparison of likelihoods has no firm basis in statistical inference. In this paper, we develop a practical text based upon Cox's ((1961) and (1962)) procedure for testing separate families of hypotheses; the work is thus an extension of earlier econometric applications of Cox's test to single equation linear regressions in Pesaran (1974) and to many equation non-linear regression in Pesaran and Deaton (1978). The test we develop here is applicable to two competing single-equation models, one of which explains the level of a variable up to an additive error, the other of which explains its logarithm, again up to an additive error. Hence, in terms of the levels of the variables, we are testing for multiplicative versus additive errors, and it is this which differentiates this paper from the earlier work in which an additive error was always assumed. We shall also allow, as in the earlier papers, the deterministic parts of the regressions to be linear or non-linear and to have the same or different independent variables; it is thus possible to test for functional form and specification in a very general way. Section 1 of the paper defines the problem and derives the test statistics. The formulae allow the calculation of two statistics, No and N1 say, the first of which is asymptotically distributed as N(0, 1) if the logarithmic specification is correct, the second, for all practical purposes, as N(0, 1) if the linear model is true. Section 2 discusses problems associated with the calculation of the statistics and shows how they can be surmounted. Section 3 presents the results of Monte-Carlo experiments designed to evaluate the potential of the test in practice. We investigate, in particular, the shape of the actual distributions of No and N1 in samples of sizes 20, 40 and 80 as well as comparing the performance of the Cox procedure with that of the likelihood ratio test, as proposed by Sargan (1964). Finally, we offer some evidence of the ability of the procedure to detect total misspecification when neither of the hypotheses is true. Section 4 contains a summary and conclusions. The general issues of statistical inference raised by the use of the Cox procedure in econometrics as well as alternative testing procedures have already been widely discussed, see Pesaran and Deaton (1978), Quandt (1974) and Amemiya (1976). In this case, however, there exists one very obvious alternative procedure. This is to specify the model,

Abstract: In this paper, the following problems associated with genetic evaluation of categorical traits by linear models are discussed: (1) Scores are arbitrarily assigned to response categories. (2) Mixed model solutions do not incorporate the restriction in the estimation space that the sum of response probabilities must total 1 across categories (3) The variance in the observed scale is not constant and depends on the genotypic value of the candidates for selection. (4) The additive genetic variance in the observed scale depends on the mean incidence of the character in the subpopulations considered in the model. (5) Nonadditive genetic variation is present in the observed scale. (6) Linear relationships fail outside a restricted range of the data. (7) Ranking optimality of best linear predictors is lacking when the conditional expectation of the predictand given the data is not linear. A method of sire evaluation of dichotomies based on a log-linear model is introduced, its properties are discussed and examples are presented.

Abstract: SUMMARY Some simple cases of optimal Bayes designs are investigated for linear models with prior information represented in hierarchical form.

Abstract: The parameter values for this model are specific to the human eye movement systems; however, the form of the model is applicable to other neurological motor control systems. The muscle length-tension diagram was modeled with an ideal spring. The muscle force-velocity relationship was linearized in a manner that produced a linear model. Initial parameter estimates were based on physiological data, human when possible. Then a function minimization program was used to fine tune model parameters. These parameter values were compared to the original physiological data to ensure that they were within the range of variability of the data. The antagonist dashpot value was selected to minimize the mean squared error between human and model responses; the value produced suggested a unique simplified representation for the original physiological data. The parameter estimation routine was applied to make the model match atypical human eye movements; these simulations suggested that glissades in normals are caused by pulsewidth, not pulse height errors.

Abstract: We say that a regression model is partially nonlinear in its parameter vector θ if some components of θ “appear linearly” while others “appear nonlinearly” in the form of the model. A theorem shows that a D-optimal discrete design for such a model is independent of the values of the linear parameters.

Abstract: This paper reviews models for the occurrence of outliers in data from the linear model. The Bayesian analyses are all closely similar in form, but differ in the way they treat suspected outliers. The models are compared on Darwin's data and one of them is used on data from a 25 factorial experiment. The question on how many outliers are present involves comparison of models with different number of parameters. A solution using proper priors on all parameters is given. On two trial datasets it is found insensitive to choice of priors on all except the parameters representing the amount of contamination in the outliers. Here, choice of even a slightly wrong prior can be very misleading. Moreover, it is difficult to choose an appropriate prior when contaminations can be both positive or negative

Abstract: Linear, normal, and beta statistical models are fit to 2,450 overlapped sprinkler irrigation patterns. The deviations of these three models from the observed water distribution are compared. Of the three, the beta distribution provides the best fit but is presently least practical to use. For uniformity coefficients above about 0.65, the normal model generally fits observed sprinkler distributions better than does the linear model. At lower uniformities, iuniformities, the opposite is true.

Abstract: ALTHOUGH MACROECONOMETRIC MODELS are widely used to analyze the effects of alternative government actions on the economy, estimates of the uncertainty of these effects are rarely, if ever, presented. This is, of course, not surprising, since most macroeconometric models are nonlinear. Unlike for linear models, formulas for the asymptotic variances of impact and dynamic multipliers are not known for nonlinear models.2 It is possible, however, to estimate these variances for nonlinear models by stochastic simulation, and the purpose of this paper is to discuss the method by which this can be done. The method is discussed in Section 2, and results of applying the method to eight policy experiments for the model in Fair [7, 10] are presented in Section 3.3 Given the obvious importance of knowing how much confidence to place on the results of any particular policy experiment in a model, it is hoped that this study will stimulate others to obtain uncertainty estimates for their models similar to those presented in Section 3. 2. THE METHOD The method can be applied to a model that is nonlinear in both variables and coefficients. Let G denote the total number of equations in the model, M the number of stochastic equations, and N the total number of predetermined (both exogenous and lagged endogenous) variables. Assume (for expositional convenience only) that the model is quarterly, and let the ith equation of the model for quarter t be written: (1) 4ki(yit, * * * , YGt, Zlt, * * *, ZNt, 1i) = Eit (i = . ), where the Yit are the endogenous variables, the zit are the predetermined variables, fli is the vector of unknown coefficients in equation i, and sit is the error term corresponding to equation i. For identities, sit is zero for all t. Also, let fl denote the vector of all the unknown coefficients in the model, and let Et denote 1 The research described in this paper was financed by grant SOC77-03274 from the National

Abstract: We consider the general linear model Y = Aβ + e in the Bayesian framework and examine the implications of the statement that the posterior expectation of β, given Y, is a linear function of Y. Under various conditions on the model, it is shown that this linear posterior expectation implies that both β and e are normally distributed. For most of the practical situations in which linear models are used, only normal distributions have linear posterior expectations.

Abstract: : This report discusses tests of hypotheses for regression parameters in the power transformation model. In this model, a simple test consists of estimating the correct scale and then performing the usual linear model F-test in this estimated scale. We explore situations in which this test has the correct level asymptotically as well as better power than Wald's test or the likelihood ratio test.

Abstract: SUMMARY Spectral analysis techniques are employed to analyze the dynamic response of a six-axle locomotive on tangent track to vertical and lateral random track irregularities. The locomotive is represented by a thirty-nine (39) degrees of freedom model. A linear model is employed by considering small displacements, linear suspension elements and a linear theory for the wheel-rail interaction. Power spectral densities of displacements, velocities and accelerations and the statistical average frequencies of the system are obtained for each degree of freedom. Comparison of the calculated dominating frequencies with existing experimental values shows good agreement. The technique of spectral analysis is an effective tool for model validation, and for the determination of rail vehicle response to track irregularities. The probability functions for the response can be used as a measure for the ride quality of rail vehicles and for the study of fatigue damage of components. § Numbers in brackets designate refer...

Abstract: In experimental statistics the usual method of estimating treatment effects is to introduce arbitrary linear restrictions among the treatment effects in order to obtain solutions of the normal equations. A comparatively recent approach is to dispense with the linear restriction and use a generalized inverse solution to the normal equations. The present note is an attempt to bring these two methods closer together and show their correspondence; namely, for each given linear restriction, there exists a corresponding generalized inverse that yields the same solution for the treatment effects as the linear restriction does.

Abstract: This paper describes the application of a recursive least squares algorithm to identify a laboratory turbogenerator system. The algorithm was initially tested in a computer simulation, which indicated that identified low-order models can represent the small-signal dynamics of a non-linear system more accurately than the corresponding linearized analytical models. This was confirmed by laboratory tests, which clearly showed the difficulty of representing the system by analytical models, and the substantial improvement obtained by identification The availability of accurate low-order linear models is particularly important in the design of controllers, to facilitate the design procedures and reduce the complexity of the system. Controllers based on identified models of the plant were designed and tested over a wide range of operating conditions, and found to give very good performance.