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This study reports some results on the determinants of regional growth at the state level, using gross state product estimates for 1955–64 and drawing upon 28 independent variables. Theoretical hypotheses relating to the influences of demographic variables, amenities, agglomeration economies, business behaviour, labour market and policy variables are tested as well as a descriptive equation. Also, ‘good fit’ equations were derived by stepwise regression procedures. The equations had a high explanatory value. Several variables showed that the more backward states grew faster, supporting the convergence hypothesis. Differences in thespatial distribution of population and economic activity within states were closely associated with variations in growth rates, suggesting that the non-spatial models that have dominated regional growth analysis are deficient. Two instrumental variables, tourist expenditures and federal government spending, were highly significant. It is important, therefore, to incorporate the government sector in regional growth theory. On the other hand, several plausible independent variables either were insignificant or had the wrong sign: education, scientific and technical personnel, the profit rate, gross savings, income potential, unemployment and air pollution. Monocausal explanations were not supported; the regional growth process appears a much more complex and interrelated phenomenon than implied by the simple growth models. Finally, current state growth performance was not independent of past conditions, indicating that the regional growth process is to some extent historically determined.

In this paper we consider a number of estimators for the linear structural simultaneous equations model containing lagged endogenous variables and autocorrelated errors. The special case is considered in which the matrix of autocorrelation coefficients of the (vector) structural error process is diagonal. We consider the two stage least squares analogue (C2SLA) in this case, its relation to the estimators proposed earlier by Fair, the estimator obtained when the autocorrelation matrix is known, and a number of instrumental variables estimrators, as well as a modification of the method of scoring which yields an estimator that is asymptotically equivalent to the C2SLSA estimator. The asymptotic distributions of such estimators are obtained and we determine their relative asymptotic efficiencies.

The estimation method proposed here offers a way to achieve consistent estimates of simultaneous systems for which the number of observations falls short of the number of predetermined variables in the system as a whole. This method, called the iterative instrumental variables (IIV) method, can be described as iterative use of the instrumental variable interpretation of the 2SLS as given by Klein. A discussion is presented on convergence of the iterative procedure and the asymptotic properties of the IIV method. The special case of the OLS start of the iterative procedure is carefully examined. It is argued that the IIV and the 2SLS methods yield estimators with the same asymptotic distribution. An analysis of the asymptotic covariance matrix of the structural coefficients is presented and compared with that given by Basmann for the 2SLS method. Finally, we analyze the results of a test application of the IIV method to an undersized simultaneous system.

This paper deals with the two-stage least squares (2SLS) estimator of structural parameters in a system of M linear structural equations. The structural equation being estimated consists of three endogenous variables and some exogenous variables. The exact mean of the 2SLS estimator has been worked out. THE EXACT SAMPLING distribution and moments of the two-stage least squares (2SLS) estimator have been recently analyzed by Basmann [3], Kabe [7], Richardson [9], Sawa [10], Anderson and Sawa [2], Nagar and Ullah [8], and Takeuchi [12]. All these authors have considered the special case where there is only one righthand jointly dependent variable in the equation under estimation from the system of simultaneous structural equations. The exact form of the moments of the 2SLS estimator of parameters of more than one right-hand jointly dependent variable has not been worked out so far. The aim of this paper is to work out the exact mean of the 2SLS estimator in the case where there are three jointly dependent variables in the equation under estimation. The procedure requires the moment of the generalized variance and its partial derivatives with respect to noncentrality parameters. These have been given in Appendix A. The method is straight-forward and can be useful in obtaining the moments in a wide class of situations.

This article develops and examines the small sample properties of alternative estimators when the data is suspected to contain measurement error. Two types of estimators are considered: (1) a linear combination of the OLS and instrumental variable (IV) estimators and (2) a method of choosing between the OLS and IV estimators on the basis of sample information. Large sample approximations are used to derive optimal procedures that are then evaluated by Monte Carlo experiments to obtain the mean square error for small and moderate size samples.

The need to select relevant sensory and instrumental variables is discussed and illustrated by an example concerned with the textural characteristics of a gel system. Discriminant analysis is used to identify the important sensory and instrumental variables and multiple regression to verify that the instrumental variables contain the required information. The limitations of this approach and possible modifications to deal with particular problems are also discussed. The approach described has proved successful in a number of applications

A class of identification methods, proposed in [3], are based on the instrumental variable principle. This correspondence contains a continued analysis of convergence of the parameter estimates of these methods. Alternative, sufficient conditions for convergence to correct values are given. It is also shown by construction of counter-examples that the methods do not converge under general conditions.

The “iterative instrumental variables” (IIV) method for estimating interdependent systems, originally referred to as a symmetric counterpart to the “fix-point” (FP) method, shares its symmetry properties with Durbin's iterative method for performing the “full information maximum likelihood” (FIML) estimation. Classical interdependent systems are considered and identities may occur among the structural equations. Alternative symmetric procedures for obtaining FIML estimates are also dealt with, including the sequential maximization of the likelihood function with respect to the coefficients of one structural equation at a time. Two recent estimation methods developed by Brundy and Jorgenson (1971, Review of Economics and Statistics 53 , 207–224) as well as Dhrymes (1971, Austral. J. Statist. 13 , 168–175) can be considered the second approximation of the IIV method and Durbin's method respectively with the first approximation obtained by the “ordinary instrumental variables” (OIV) method. In practice the second approximation depends heavily on the choice of initial instrumental variables, although the asymptotic distribution is not changed by the continued iteration.

The hypothesis that public school funding is sensitive to gain or loss of residence related external benefits was tested using two stage least squares. No statistically significant relationship was detected. Arguments for providing school funding relief to declining rural areas must be defended upon equity rather than efficiency grounds.

In order to compare and evaluate identification and parameter estimation methods, three simulated linear discrete processes-a second-order oscillatory process, a second-order nonminimum phase process, and a third-order, low pass, process with delay-were identified with the following on-line methods. 1.(1) Least squares 2.(2) Generalized least squares 3.(3) Instrumental variables 4.(4) Stochastic approximation 5.(5) Correlation analysis with least squares parameter estimation 6.(6) Fourier analysis using a model with three unknown parameters. The test processes correspond to the ''multi solution test cases'' we have proposed for the 3rd IFAC-Symposium on Identification and System Parameter Estimation. The variances of parameter estimates and impulse responses are given for simulation runs using all six parameter estimation methods for two noise-to-signal ratios and three measurement time periods. The system order is assumed to be known exactly. Since the ultimate application of the parameter estimates is important in choosing an identification procedure, on example of a final goal, the design of a digital control algorthm, was chosen for these evaluations. Hence the identified models are used to optimize the parameters of a three-mode controller. Both the controller parameters and the r.m.s. error of the closed loop controlled variable are compared for the identified and the exact model.

Many time series models have autocorrelated errors. A further problem may be that some of the right-hand variables are correlated with the error. Using Monte Carlo methods we investigated several estimators of such a model. We found that a transformation to correct the autocorrelation often improved matters, but that techniques to correct the other problems such as instrumental variables were usually ineffective. In many cases ordinary least squares had the smallest mean squared error.

Summary: § 1. An outline of Theil's theory. –§ 2. An interpretation of Theil's theory – the method of instrumental variables. –§ 3. Comparison of two-rounds and limited information estimates. –§ 4. Remarks on other methods.

Summary In econometrics it is common for variables to be related together in a set of linear, multilateral and causal interdependencies. This type of system generally has properties which are unsatisfactory for application of classical regression techniques. Consequently, alternative estimation methods have been developed. This paper explores the relations between several such methods in terms of symmetric idempotents of predetermined variables and their orthogonal complements. Generalizations of two‐ and three‐stage least squares and instrumental variables are considered, including Wicken's estimator.2 The relative efficiencies of the estimators are also discussed. ∗ Some of the ideas in this paper appeared in Reference 1(a), a later version of which was presented to the World Congress of the Econometric Society, Rome, 1965.1(b) Since these papers appeared, the mathematical exposition has been improved, the results made more general and some new results (on efficiency) added. Some of the results reported...

This paper presents the results of an investigation into one of the processes by which prices could have been set in New Zealand over about the last twenty years; it does not concern itself with any processes by which wages may have been determined, as the particular price process is for a representative firm which treats its wage rate as predetermined. The firm attempts to maximise its profits, and is able to choose simultaneously its product price, its demand for labour services, and the level of real output it wishes to produce.A structural model for this representative firm is developed within a static framework in Section one. In Section two are presented the results obtained from econometric estimation of a dynamic form of the model. Aggregate data have been used both for the ordinary least squares (OLS) estimates of each of the price, labour demand, and real output equations, and for the two stage least squares (2SLS) estimates of the simultaneous equation system. Principal conclusions from the research are summarised in Section three.

1. IN THIS NOTE we consider the estimation of linear models when the coefficients of the structural form are not the same for all observations for which the model is postulated to be valid. An example of such a model is given in [3], where some structural relations have a piecewise linear form. Another example is the water melon market model of Suits [2] where there are two alternative harvest supply schedules. Also discussed here is the case where for one part of the sample period one or more variables are endogenously determined while for another part they are exogenous, for instance, the wage rate or the rate of exchange. Such a change in the nature of the model can also be interpreted as a change in the coefficients of the structural form. It is assumed throughout that it is known a priori for what observations each specification holds. 2. A shift in the value of the coefficients of a predetermined variable does not cause special problems. If, say, only one shift occurs, one defines two predetermined variables to replace the original one. The vector of observations for the first of these consists of the observations on the original variable with the exception of those observations for which the second value is supposed to hold. These latter observations are replaced by zero's. The vector of observations on the second variable is simply the difference between the vector of observations for the original variable and the one for the first variable. 3. Next, consider the case where there is a shift in the value of one or more structural coefficients associated with an endogenous variable. Let the model in structural form be (1) Yt = y'B + x'C + ur where yt is an M-element vector of endogenous or jointly dependent variables, xt an L-element vector of predetermined variables, while ut is the M-element vector of structural disturbances. The matrix B is the M x M matrix of coefficients associated with the endogenous variables and C is the L x M matrix of coefficients associated with the predetermined variables. It is assumed that one or more elements of B take for some observations a different value than for others. The superscripts a and b are used to distinguish between the two situations. The following partitioning of the sets of T observations in two subsets of T7 and Tb observations, respectively, are introduced

The factor analysis model is rewritten as a system of linear structural relations with errors in variables. The method of instrumental variables is applied to this revised form of the model to obtain estimates of the factor loading matrix. The relation between this method and interbattery analysis, proportional profile analysis, and canonical factor analysis is pointed out. In addition, an estimation procedure based on replicated sampling different from proportional profile analysis is given.

A statistical technique for the simultaneous study of response to several variables has been used in the estimation of “optimum„ operating conditions for the element Calcium in atomic absorption spectroscopy. The method is based on “response surface„ exploration, originally due to G.E.P. Box. This technique has several advantages over the traditional method of varying “one factor at a time.„ Direct estimates of the magnitude of the effects of changes in each instrumental variable are obtained, together with estimates of the interactions between two or more variables. The method leads to the efficient estimation of “optimum„ operating conditions, i.e., those levels of the instrumental variables at which percent transmission is a minimum and also the estimation of alternative levels of the several variables at which transmission can be maintained at a minimum. The results showed that, for the estimation of Ca, the four variables air pressure, air flow, gas flow, and “height above the blue cone„ are compensating in the sense that departure from minimum transmission due to change in one variable can be compensated by a suitable change in the other variables. In the absence of a theoretical solution to the problem of estimating “optimum„ operating conditions, it would seem that this method is the best available at the present time.

Improving personal income in low income rural communities is of widespread interest in many areas of the United States, Canada and in other parts of the world. This paper suggests a useful method of analyzing inter-area income differences and identifying factors that are useful in developing alternative policies to raise low per capita income levels as wetl as in delineating areas for public investment. While the results of the study are of special relevance to the Omro area of the United States, the mythology and the general reacts have much wider implications.

This paper presents the derivation and evaluation of an asymptotically unbiased statistical estimator for the parameters in the vector difference equation canonical description of a linear multi variable system disturbed by correlated plant and measurement noise processes having rational spectral densities. The sequential estimation algorithm utilizes instrumental variables generated by a recursive filter incorporating the current parameter estimate in a novel manner. A spectral factorization technique for the identification of the noise process is discussed. Numerical results comparing the estimation algorithm with other methods are reported for scalar-output and multivariable-output cases.

This study has two main purposes: the estimation of certain demand relations, and the assessment of the usefulness of Canadian consumer expenditure surveys as a source for the derivation of such estimates. Results from five surveys of Canadian urban family expenditures in the post-war period have been published, and the returns from a sixth are now being processed. These reports have received relatively little attention from economists, probably because of the limited scope of most of the surveys. Their principal objective was to provide appropriate commodity weights (or a check on existing weights) for the Consumer Price Index (CPI); the sample interviewed was mostly restricted to the CPI target income group and often to only a few cities. The surveys for 1947–48 and 1959, however, do not contain any family-size or income restrictions. The former sampled the non-farm population and the latter those living in cities with populations of 15,000 and over. For these two surveys, expenditures on major commodity groups were cross-classified by income and family-size. It is only for these two surveys that Engel curves can be fitted. Estimates of income and family-size parameters were obtained for this study by using two methods of estimation—least squares and instrumental variables. The results obtained from the latter method provide, under certain assumptions, a measure of the extent of the downward bias in the least squares estimates of the parameters caused by transitory income. In addition, estimates of the income elasticity of consumption are obtained on the basis of different methods of grouping the spending units surveyed, in an attempt to test the hypothesis that this value is unity.

Using cross‐section data, the demand elasticities for basic living materials were estimated by least‐squares and instrumental variables. Income and own‐ and cross‐price elasticities were derived from time‐series data by ordinary least‐squares and combined technique. It is interesting to compare the estimated parameters using different estimating methods. The order of magnitude of the income elasticities contained few surprises. Most of the price elasticities carried correct signs, but the majority of them were insignificant. Income is much more important than prices in explaining Japanese consumption. Most of the estimates are similar to those obtained by others. Expenditure patterns between the Japanese farm households and the U.S. farm households can be compared.

Pseudo-instrumental variables, defined generally as functions of the independent or predictor variables, are utilized in the estimation of straight-line relations when both variables are subject to error. For such relations it is known that some form of prior information is necessary for purposes of estimation. As such, we assume that the independent variables are only partially in error and write the instrumental variable as a function of the correct portion of the independent variable. This technique leads to consistent slope and intercept estimates and may have broader applications than the Wald-Bartlett estimation techniques.

An automatic computational scheme is presented for the determination of the order of a linear system. The method is developed for systems capable of being modeled by a linear constant coefficient differential equation of finite order. Use is made of the Poisson filtered excitation and response signals to obtain parameter estimates. The order is obtained by employing a statistical measure of the consistency of the parameter estimates for successively increasing assumed system order. The instrumental variable method is incorporated to allow the system order to be determined in the presence of increased amounts of measurement noise. Demonstration of the performance of the method is provided by the results of several computer simulation runs.

Friedman (1956, p. 4) has indicated that the demand for money by "the ultimate wealth-owning units in [a] society can be made formally identical with that of the demand for a consumption service." As a particular consumer good, then, the demand for money can be considered as a special case of the permanent-income hypothesis (PIH) for consumption goods in general. The main requirement on the PIH in extending it to individual consumer goods is that a unitary elasticity of demand is no longer required (although, in the case of money, this would be an interesting hypothesis to test). From the PIH framework the demand for money as a function of income can be accordingly described by the following three equations

INTRODUCTION In recent years, the dynamic panel data literature has begun to focus on panels in which the number of cross-sectional observations (N) and the number of time-series observations (T) are both large. The availability of data with greater frequency is certainly a key contributor to this shift. Some cross-national and cross-state data sets, for example, are now large enough in T such that each nation (or state) can be estimated separately. See Blackburne and Frank, (2007) for further details. The asymptotics of large N, large T dynamic panels are quite different from the asymptotics of traditional large N, small T dynamic panels. Small T panel estimation usually relies on fixed or random effects estimators, or a combination of fixed effects estimators and instrumental variable estimators, such as the Arellano and Bond, (1991) GMM estimator. These methods require pooling individual groups and allowing only the intercepts to differ across the groups. One of the central findings from the large N, large T literature, however, is that the assumption of homogeneity of slope parameters is often inappropriate. This point has been made by Pesaran and Smith (1995); lm et al. (2003), Pesaran et al; (1997, 1999), Phillips and Moon, (2000) (1). With the increase in time observations inherent in large N, large T dynamic panels, nonstationarity is also a concern. Recent papers by Pesaran et al. (1997, 1999) offer two important new techniques to estimate nonstationary dynamic panels in which the parameters are heterogeneous across groups: the mean-group and pooled mean-group estimators. The mean-group estimator (MG) (see Pesaran and Smith, 1995) relies on estimating TV time series regressions and averaging the coefficients, while the pooled mean-group estimator (PMG) (see Pesaran et al., 1997, 1999) relies on a combination of pooling and averaging of coefficients. In recent empirical research, the MG and PMG estimators have been applied in a variety of settings. Freeman, (2000), for example, uses the estimators to evaluate state-level alcohol consumption over the period 1961 to 1995. Martinez-Zarzoso and Bengochea-Morancho, (2004) employ them in an estimation of an environmental Kuznets curve in a panel of 22 OECD nations over a period 1975 to 1998. Frank, (2005) uses the MG and PMG estimators to evaluate the long-term impact of income inequality on economic growth in a panel of U.S. states over the period 1945 to 2001. This paper applies the MG and PMG estimators to a panel of OECD nations for the years 1970-2004. We present a simple dynamic model of oil consumption as a function of income and prices. As in previous studies, we allow demand to respond asymmetrically to price shocks. Specifically, this paper has three goals: * test the degree of heterogeneity in oil consumption among the OECD nations * test the asymmetric response of oil consumption with respect to price * estimate precise price and income elasticities for OECD oil consumption This paper proceeds as follows. Section 2 discusses the methods involved, including price decomposition and alternative dynamic panel estimators. Section 3 briefly describes the data. Section 4 presents the results and Section 5 concludes. METHODOLOGY Demand Asymmetries Following the recent work of Gately and Huntington, (2002), this paper allows for asymmetric price response in oil demand. Models that assume price symmetry when, in fact, it does not exist introduce model misspecification and downwardly bias income elasticity estimates. Accordingly, we decompose the world price of oil (in logs), Pr, into three components: [P.sub.max,t] = max([P.sub.t], [P.sub.t-1]) (1) [P.sub.rec,t] = [T.summation over (t=1)] max(0, ([P.sub.t] - [P.sub.t-1]) - ([P.sub.max,t] - [P.sub.max,t-1])) (2) [P.sub.cut,t] = [T.summation over (t=1)] min(0, ([P.sub.t] - [P. …

ECONOMIC problems present themselves sometimes as problems of individual firms and households, sometimes as problems of national concern. At levels which are intermediate with respect to geographical bounds, a unique set of economic problems arise. These are the problems of communities, areas, and regions. The theory used to explain the economic behavior of firms and households is tailored to meet the needs of firm managers and household heads. Techniques such as budgeting, functional analysis, and linear programming relate the theory to data available through descriptive profit statements and balance sheets. The theory of the firm provides a manager with information on which to base decisions about instrumental variables

This paper explores the possibility of using the instrumental variable method to estimate the parameters of linear time-invariant discrete-time systems. The existence of optimal estimates is established, methods for their approximate computation are given, and an on-line identification scheme based on recursive computation is proposed. Experimental results are included.

A method is developed for expressing a dependent variable that is subject to systematic regional variations (trend) in terms of a set of independent geological variables. Trend surfaces are fitted to the independent variables, and the dependent variable is regressed on both the fitted trend values and the original observations for the independent variables. The method is applied to evaluate the areal distribution of gold occurrences in the Greenbelt of western Quebec as a function of a set of lithological variables.

In this paper we study a model that minimizes the sum of production, employment smoothing, and inventory costs subject to a schedule of known demand requirements over a finite time horizon. The three instrumental variables are work force producing at regular-time, work force producing on overtime, and the total work force. Overtime is limited to be not more than a fixed multiple of regular time. The idle portion of the work force and the levels of inventory are resultant variables. We postulate the following shape characteristics for the cost functions production costs are convex-like, smoothing costs are V-shaped, and holding costs are increasing, both the production and holding cost functions need not be stationary. In this paper, we provide upper and lower bounds on the cumulative regular-time plus overtime work force for any sequence of demand requirements. We also give the form of an optimal policy when demands are monotone (either increasing or decreasing). Finally, we derive the asymptotic behavior of optimal policies when demands are monotone and the planning horizon becomes arbitrarily long. All of these results, which convey information about the numerical values of optimal policies, given specific demands and an initial level of inventory, depend only on the shape characteristics of the cost functions. Algorithmic techniques are discussed elsewhere [Lippman, S. A., A. J. Rolfe, H. M. Wagner, J. S. C. Yuan. Algorithms for optimal production scheduling and employment smoothing. Opns Res. To appear.], [Yuan, J. S. C. 1967. Algorithms and multi-product model in production scheduling and employment smoothing. Technical Report 22 (NSF GS-552), Stanford University, August.].

The problems of seasonal adjustment and other forms of prior adjustment have seldom been integrated into a general framework of estimation. A well-known result, due to Frisch and Waugh, has been used to demonstrate how linear seasonal influences might be treated in the context of the general linear statistical model. In this paper, the author establishes a proposition that one form of prior adjustment is consistent with two different estimating techniques that are in common use.

The use of modern control theory in various dynamic economic models has raised a number of interesting issues in the theory of economic policy and its operational applications to problems of economic growth, stabilisation and development planning Two of these seem to be of great importance: econometric estimation viewed as a part of the decision-making process by a policymaker and the operational linkages between a consistency model without any explicit optimisation criterion and an optimisation model with an explicit objective function defined in a programming framework. In order to compare and evaluate alternative economic policies defined within a dynamic econometric model, these two problems become most relevant and they have to be resolved in some manner. As an example of the first type of problem one may refer to the use of the Brookings quarterly econometric model of the U.S. economy by Fromm and Taubman [1] for evaluation of the relative desirability of a set of monetary and fiscal policy actions. They noted that the method of optimum growth defined in a Ramsay-type framework of maximisation of a utility functional over a horizon is not applicable to cyclical paths; moreover it ignores the disutility of the time path of variances of the arguments (e.g. consumption, etc.) in the utility function.

This paper discusses the concept of strong instrumental variables and strong instrumental matrix sequences for the estimation of the transfer-function parameters of discrete-time, time-invariant models of linear systems. It is shown that strong instrumental variable estimators are strongly consistent and a sufficient condition for the estimator to be asymtotically unbiased is given. Moreover, it is shown that with a persistently exciting signal of appropriate order for an input, "virtually" any discrete-time, time-invariant, linear system model of appropriate order can be used to generate strong instrumental variables.

In this chapter we shall consider alternative distribution-free estimators, that is, estimators whose derivation does not depend on explicit specification of the form of the distribution of the error terms of the system. In particular, we shall consider indirect least squares and instrumental variables estimators, and in the context of the former we shall discuss, in somewhat greater detail than previously, the identification problem. Finally, we shall examine the simplifications that accrue to the estimation problem when the econometric model under consideration is recursive.

The field of identification and process-parameter estimation has developed rapidly during the past decade. In this survey paper the state-of-the-art/science is presented in a systematic way. Attention is paid to general properties and to classification of identification problems. Model structures are discussed; their choice hinges on the purpose of the identification and on the available a priori knowledge. For the identification of models that are linear in the parameters, the survey explains the least squares method and several of its variants which may solve the problem of correlated residuals, viz. repeated and generalized least squares, maximum likelihood method, instrumental variable method, tally principle. Recently the non-linear situation, the on-line and the real-time identification have witnessed extensive developments. These are also reported. There are 230 references listed, mostly to recent contributions. In appendices a resume is given of parameter estimation principles and a more detailed exposition of an example of least squares estimation.

A new approach to the choice of econometric estimators, called small-sigma asymptotics, is introduced and applied to the choice of k-class estimators of the parameters of a single equation in a system of linear simultaneous stochastic equations. I find that when the degree of overidentification is no more than six, the two stage least squares estimator uniformly dominates the limited information maximum likelihood estimator in a certain sense. The small sigma method can be used on many problems in statistics and econometrics. THE STUDY OF simultaneous equation econometric models has led to many alternative estimators to ordinary least squares: single equation limited information maximum likelihood, and two stage least squares, for example. The behavior of these estimators has been difficult to describe, however, and it has been difficult to choose among these estimators. The work described in this paper explores this problem for the case in which lagged dependent variables are not permitted. To be most useful for normative purposes, a description must be detailed enough to give a good approximation and expose differences between estimators, and yet be simple enough to strengthen intuition and yield easily described comparisons. Since detail and simplicity are in conflict, approaches may differ in this respect. This paper introduces a new approach, based on asymptotic series in a scalar multiple, a, of the variance of the disturbance in the model. As a -+ 0 the regression function is an increasingly good description of the random variables generated. Intuitively this is suggested by Gauss' "Theory of Errors" the errors were never intended to be so large as to swamp the regression function. One important approach used in the past is large sample asymptotic theory. This reveals a persistent bias in ordinary least squares, and a large sample asymptotic equivalence between two stage least squares and single equation limited information maximum likelihood. Additionally, Nagar [13] found the 11T term in the large sample asymptotic bias and the 1/T and I/T2 terms of the moment matrix of two stage least squares. Economists have been uneasy, however, about application of large sample theory to samples which may not be "large" in the relevant sense. Additionally large sample asymptotic results often depend on an assumption about the asymptotic behavior of the moment matrix of exogenous variables which is difficult to justify.

I HAVE DISCOVERED an error in the computations of regression (3) and the regressions in Table 1 in my paper in Brookings Papers on Economic Activity (2:1973), on pages 452-53 and 460. With the exception of the lagged unemployment rates in those regressions, the instrumental variables in Ot were all inadvertently lagged four more quarters than is reported in the paper. For example, mti_ in equation (3) is actually mt_5. The correct calculations are reported here in regression (3) and Table 1. The F-statistic for regression (3) is now 4.5, which exceeds the value of 2.503 reported in my paper, and so is even more significant statistically. Thus, the test continues to point toward rejection of the natural rate hypothesis. The corrected results for Table 1 now detect neither a short-run nor a long-run Phillips curve using the log of the GNP deflator, p, the coefficient on the systematic part of Ap having t-statistics close to zero. The t-statistic on the systematic part of wage inflation in regression (5.2) is now 1.28, and fails to support rejection of the natural rate hypothesis. However, the coefficient on the "random" part of w in (5.2) is now larger than before both in absolute value and in statistical significance, so that the amended results for w are more favorable to the hypothesis of a tradeoff between unemployment and the surprise component of wage inflation.

The approach to discrete system identification described in a recent paper by Mehra is shown to be one example of a whole class of instrumental variable (IV) solutions. A recursive version of this IV solution is presented and an alternative, statistically more efficient, approximate maximum likelihood procedure is outlined.

RegWuTest performs a Wu (or Durbin-Wu-Hausman) specification test on a regression just estimated by instrumental variables. Because it works off the last regression, there are no parameters. Wu(1973), "Alternative tests of independence between stochastic regressors and disturbances", Econometrica vol 42, 529-546.(This abstract was borrowed from another version of this item.)

has recently been discussed by Goldberger [7], Zellner [19]. We observe Yi, Y2,, = 1,.. , N, but not the physically meaningful variable y*. Unobservable variables models of various types have recently been the subject of empirical study (see e.g., Aigner [1], Goldberger [8], Griliches [10], Griliches and Mason [11]). For (1) and (2) various estimators of the parameter ao have been suggested. First, there is the factor analysis approach of making very strong assumptions about the covariance structures of Y) and the residuals uIn and u2n. Second, one can use (2) to substitute for y* in (1), giving an equation that can be consistently estimated if one has access to observationis on an instrumental variable that is correlated with Y2n' but not with uln, aOu2n. Such estimates are unlikely to be fully efficient. Thus Zellner [19] suggested an asymptotically efficient least squares (LS) method that relies on ull and U2,, being uncorrelated and y* being modelled as2

(1) D = (y _ y*)2 + 9g( G*)2 where Y is national income; G is the instrumental variable, government spending; Y* and G* are planned values of Y and G; and g is the weight attached to squared deviations from planned values of government spending. Wood2 has criticized the use of the squared deviation of Y in equation (1) on the grounds that it gives equal weight to undershooting and overshooting planned income; while he believes that "higher real income always increases economic welfare, given deviations of realized from desired government expenditures." 8 Wood has proposed, therefore, that the social welfare function should be defined by (2) U= Y-g(G-G*)2 instead of equation (1). Wood appears to have assumed tnat the relation between income and welfare is a monotonic one. This writer would question that assumption. It need not be the case that an increase in income is always desirable especially if the increased income is connected with decreased leisure.4 Even economists may sometimes wish that they could have more time to think instead of more income. In addition, it is the contention of this writer that Wood's criticism loses much of its significance for economic planning problems when equations like (1) and (2) are considered not just as representations of "aggregate social welfare" but as products of attempts to solve actual planning problems and to develop practicable decision rules. It should be noted that most planning problems are highly complex. Planners often do not have substantial information about the parameter values and the nature of the relations among the system variables.5 Given the present state of planning techniques

where ut is a random disturbance with zero mean. Koyck pointed out that subtracting Xyt-i from (1) produced the equation yt = A'+xyt_i+ bxt+ut' (2) where A' = A(1 -X) and ut' = ut -Xut1. Thus instead of being forced to deal with the model in its distributed lag form (1), which involves the seemingly intractable task of estimating a relation with an infinite number of explanatory variables from a finite amount of data, we can estimate the parameters of the autoregressive form of the model given in equation (2). However, this apparent simplification is purchased only at a cost, for consistent estimation of relation (2) requires that we face several estimation problems associated with equations in which lagged dependent variables appear as explanatory variables. While ordinary least squares estimates of the parameters of (2) are consistent provided that the disturbances ut' are serially independent and follow a distribution which satisfies the assumptions of the central limit theorem, even in this case a small sample bias exists. If the disturbances are serially dependent, an asymptotic bias exists.' Moreover, the transformation from (1) to (2) has changed both the variance and the serial correlations of the disturbances. Hence, if the disturbances in (1) are serially independent, those in (2) are necessarily autocorrelated, which means that applying least squares to equation (2) yields inconsistent estimates of the parameters. In addition to ordinary least squares (OLS), several techniques for estimating such distributed lag relations are available. Generally these techniques have been recommended on the basis of their desirable asymptotic properties. However, for economists, who are forced to work in a world where data are scarce, asymptotic properties are frequently of little relevance. What is more often required is knowledge of the properties of the estimators in small samples. Unfortunately, it has proved difficult to investigate these properties analytically. In the absence of such results, sampling or "Monte Carlo" experiments provide an alternative, if less elegant, source of information. Accordingly, this paper presents the results of a "Monte Carlo" study of several lag estimators under conditions in which the disturbances ut' of relation (2) are serially correlated. In addition to ordinary least squares, the following five methods were studied. 1) Two Stage Regression (TSLS): This is an application of Leviatan's instrumental variable approach. Leviatan [12] has suggested that xti1 be used as an instrument for yt-i in estimating relation (2). In order to increase the efficiency of the technique, we employed a linear combination of lagged x's as the instrument. The linear combination was determined by first estimating the equation

T HE present study applies the two-stage least squares principle to a nonlinear least squares estimation method; the nonlinear least squares method is based on Marquardt's maximum neighborhood method. The method is applied to the CES production functions of the Canadian manufacturing industries. Section II explains the application of the nonlinear least squares method to the CES production functions; in section III the estimated results are presented; section IV gives some qualifications to the results obtained in the present study.

This paper reports on a study of the timing of the economic impact of government defense procurement. By assuming that the letting of new orders signals the beginning and shipments signal the end of the impact of defense procurement, this research investigates the effects of changes in product mix and capacity utilization on the duration of such impacts. The effects of changes in product mix and capacity utilization on inventory accumulation are also investigated. RECENT DISCUSSIONS on econometric model building have pointed out that the development of a realistic model of the government sector is a prerequisite to the effective use of econometric models in evaluating the impact of government operations on the economy [7]. There are two aspects to this problem: (i) An econometric model should include appropriate instrumental variables-variables that can be controlled by policy makers [11], and (ii) the model should properly capture the impact of the government actions [2, 3,5,13]. This paper presents an explanatory study related to the second aspect of the above problem. Since defense procurement accounts for nearly ten per cent of GNP, a question which naturally arises is: Can defense procurement be manipulated by the government to help stabilize economic activity or to offset cyclical fluctuations? It is to be expected that the timing of defense procurement is determined primarily by noneconomic considerations. In peace time, however, a certain degree of flexibility is presumed to exist in the scheduling of defense procurement. For this reason, defense procurement is a useful instrumental variable for an econometric model of the government sector. In considering defense procurement as an instrumental variable, the following question is raised: What stage in the defense procurement process is most important from the viewpoint of measuring its impact on economic activity? In their studies of inventory investment, Lovell and Suits [10, 12] emphasized that the Department of Defense orders have an immediate impact upon inventories in advance of its expenditures. In a substantial number of other econometric models, however, the impact of defense procurement was measured at the expenditure stage [4,6,8,9].