On testing weak separability

A new framework for testing for the existence of a consistent aggregator for a subset of inputs in a production function is developed in terms of the variable profit function. In contrast to the Berndt–Christensen framework, in which the parametric restrictions required to attain weak separability also impose unwanted restrictions on the form of the aggregator, the aggregator function has a flexible functional form. Consequently this procedure should permit a less restrictive test of separability or aggregation. Application of the procedure to the data for U.S. manufacturing assuming a production function involving two labour inputs (blue- and white-collar workers) and two capital inputs (structures and equipment) leads to the conclusion that there does not exist a consistent aggregator for labor whereas there is some mild support for the existence of a consistent aggregator for capital.     

Separability test for the electricity supply industry

This paper examines the vertical integration issue of the electricity industry. This industry is typically vertically integrated and heavily regulated. The paper investigates the potential technological efficiency loss due to the vertical disintegration using the translog production function. Three separability hypotheses using the Allen-Uzawa elasticity of substitution are tested to examine the feasibility of various degrees of disintegration of the current industry structure. All three separability hypotheses are rejected, and the cost of disintegration is estimated. Further, estimates indicate that the electricity industry appears to have exhausted economies of scale. Unlike other deregulated industries, disintegration of the electricity industry will entail technological efficiency loss.     

On testing separability restrictions with flexible functional forms

In this paper, we examine the implications of imposing separability on the translog and three other flexible forms. Our results imply that the Berndt-Christensen ‘nonlinear’ test for weak separability tests not only for weak separability, but also imposes a restrictive structure on the macro and micro functions for all currently known ‘flexible’ functional forms. For example, testing for weak separability using the translog as an exact form is in fact equivalent to testing for a hybrid of strong (additive) separability and homothetic weak separability with Cobb-Douglas aggregator functions. Our results show that these ‘flexible’ functional forms are ‘separability-inflexible’. That is, they are not capable of providing a second-order approximation to an arbitrary weakly separable function in any neighbourhood of a given point.     

A nonparametric test of weak separability and consumer preferences

Two stochastic nonparametric procedures are developed to evaluate the significance of violations of weak separability. When the data have measurement error, we show that the necessary and sufficient weak separability conditions of Varian [Varian, H., 1983. Nonparametric tests of consumer behavior. Review of Economic Studies 50, 99–110] must also satisfy the Afriat inequalities. The tests detect weak separability with high probability for weakly separable data. In addition, the procedures correctly reject weak separability for both nonseparable and random utility simulated data sets. The tests also fail to reject weak separability for a monetary and consumption data set which suggests that measurement error may be the source of the observed violations.     

Unbiased determination of production technologies

To determine whether an industry exhibits constant returns to scale, whether the production function is homothetic, or whether inputs are separable, a common approach is to specify a cost function, estimate its parameters using data such as prices and quantities of inputs, and then test the parametric restrictions corresponding to constant returns, a homothetic technology, or separability. Statistically, such inferences are valid if the true cost function is a member of the parametric class considered, otherwise the inference is biased. That is, the true rejection probability is not necessarily adequately approximated by the nominal size of the statistical test. The use of fixed parameter flexible functional forms such as the Translog, the generalized Leontief, or the Box-Cox will not alleviate this problem.The Fourier flexible form differs fundamentally from other flexible forms in that it has a variable number of parameters and a known bound, depending on the number of parameters, on the error, as measured by the Sobolev norm, of approximation to an arbitrary cost function. Thus it is possible to construct statistical tests for constant returns, a homothetic technology, or separability which are asymptotically size α by letting the number of parameters of the Fourier flexible form depend on sample size. That is, the true rejection probability converges to the nominal size of the test as sample size tends to infinity. The rate of convergence depends on the smoothness of the true cost function; the more times is differentiable the true cost function, the faster the convergence.The method is illustrated using the data on aggregate U.S. manufacturing of Berndt and Wood (1975, 1979) and Berndt and Khaled (1979).     

The industrial and commercial demand for electricity under time-of-use pricing

In this paper we have developed and estimated the demand for electricity by an industrial (commercial) firm subject to time-of-use (TOU) pricing of electric power. In the application we use a quadratic production function and directly incorporate into the production process the restrictions that some inputs cannot vary over the day. We show that the TOU structure implies a unique set of parameter restrictions across the demand functions for inputs.     

Restrictions on variables

Accounting identities impose exact restrictions on the endogenous variables of econometric models. Such restrictions are usually met by choosing a closing entry or by building an allocation model. Selecting a closing entry may be difficult or arbitrary, while allocation models admit little flexibility in the choice of explanatory variables and lagged adjustment schemes. This paper studies a third solution which in a sense lies inbetween: freely chosen equations for all variables are adjusted additively such that the restrictions will hold. The adjustment involves some new parameters which can be estimated simultaneously with the original parameters using Maximum Likelihood techniques.An application is provided for a financial model of the Dutch private sector. Our approach here proves superior to any choice of closing entry in the system.     

On Some Characteristics of Gaussian Covariance Functions

The concepts of isotropy/anisotropy and separability/non‐separability of a covariance function are strictly related. If a covariance function is separable, it cannot be isotropic or geometrically anisotropic, except for the Gaussian covariance function, which is the only model both separable and isotropic. In this paper, some interesting results concerning the Gaussian covariance model and its properties related to isotropy and separability are given, and moreover, some examples are provided. Finally, a discussion on asymmetric models, with Gaussian marginals, is furnished and the strictly positive definiteness condition is discussed.     

Restricted forecasting with VAR models: An analysis of a test for joint compatibility between restrictions and forecasts

A restricted forecasting compatibility test for Vector Autoregressive Error Correction models is analyzed in this work. It is shown that a variance–covariance matrix associated with the restrictions can be used to cancel out model dynamics and interactions between restrictions. This allows us to interpret the joint compatibility test as a composition of the corresponding single restriction compatibility tests. These tests are useful for appreciating the contribution of each and every restriction to the joint compatibility between the whole set of restrictions and the unrestricted forecasts. An estimated process adjustment for the test is derived and the resulting feasible joint compatibility test turns out to have better performance than the original one. An empirical illustration of the usefulness of the proposed test makes use of Mexican macroeconomic data and the targets proposed by the Mexican Government for the year 2003.     

Conditional Demands and Marginal Tax Reform

This paper examines Irish demand patterns using conditional demand functions. This overcomes the problems faced by traditional demand analysis which neglects the influence of labour supply and thus assumes weak separability. The conditional approach allows for more exact tests of weak separability using more flexible functional forms than is possible when estimating an unconditional commodity demand–labour supply model. The impact of the conditioned demand responses and the relaxation of weak separability on measures of marginal tax reform is examined.     

Commodity and asset demands with and without quantity constraints in the labour market

This paper reformulates the consumer's decision-making problem in a temporary framework with and without quantity constraints in the labour market in such a way as to define a complete demand system which includes consumption, labour and saving. Then, based on American and Canadian data (1948–1980), this extended complete demand system is estimated using a priori theoretical properties. Quantity constraints in the labour market cannot be rejected, and marginal propensities to consume out of labour income are very different from the marginal propensities out of non-labour income. Temporal separability is rejected and Slutsky conditions are preserved.     

The existence of a real value-added function in the Canadian manufacturing sector

Sims (1969), Gordon (1969), and Arrow (1972) have demonstrated that the capability of using real value-added to measure the output of a sector when material inputs are employed in the production process rests upon primary factor inputs being weakly separable from material inputs. In Canada, the double deflation technique is used by Statistics Canada to measure the real value-added output of an industry; this technique assumes strong separability. This paper tests both of these separability hypotheses for the Canadian manufacturing sector using data from 1950 to 1972 and a translogarithmic production function. Our tests lead us to reject both the strong and weak separability hypotheses. Furthermore, our estimates of the Allen partial elasticity of substitution of capital for labor are radically different from the results of previous Canadian studies.     

Separability and separability flexibility of the CRESH function

The CRESH functional form has many desirable properties. It is globally regular for a wide range of parameter values, it includes the CES as a special case, and it has stable patterns of relative substitutability. However, while the CRESH form is defined to be implicitly additively separable, theordinary separability structure of the CRESH form has never been investigated. In this article, that deficiency is corrected by describing the separability structure of CRESH functions. In particular, it is shown that the CRESH form is separability flexible in the important three input case. This and its other properties may make the CRESH function the form of choice among existing three input production functions.The editor for this paper was Michael Denny.     

Aspects of second‐order analysis of structured inhomogeneous spatio‐temporal point processes

Statistical methodology for spatio‐temporal point processes is in its infancy. We consider second‐order analysis based on pair correlation functions and K‐functions for general inhomogeneous spatio‐temporal point processes and for inhomogeneous spatio‐temporal Cox processes. Assuming spatio‐temporal separability of the intensity function, we clarify different meanings of second‐order spatio‐temporal separability. One is second‐order spatio‐temporal independence and relates to log‐Gaussian Cox processes with an additive covariance structure of the underlying spatio‐temporal Gaussian process. Another concerns shot‐noise Cox processes with a separable spatio‐temporal covariance density. We propose diagnostic procedures for checking hypotheses of second‐order spatio‐temporal separability, which we apply on simulated and real data.     

Estimating restricted structural change models

This paper considers issues related to multiple structural changes, occurring at unknown dates, in the linear regression model when restrictions are imposed on the parameters. This includes, for example, the important special case where different nonadjacent regimes are the same. The estimates are constructed as global minimizers of the restricted sum of squared residuals and we provide an extension of the algorithm discussed in Bai and Perron [2003b, Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1–22] to efficiently compute them. We show that the estimates of the break dates have the same asymptotic properties with or without the restrictions imposed; that is, in large samples, there is no efficiency gain from imposing valid restrictions as far as the estimates of the break dates are concerned. Of course, efficiency gains occur for the other parameters of the model. Simulations show that in small samples, all parameters are more efficiently estimated using the restrictions. We also consider tests of the null hypothesis of no structural change. These are also more powerful when the restrictions are imposed. A Gauss code for all the procedures discussed in this paper is available from the authors.     

Market Values of Environmental Amenities: A Latent Variable Approach

This study presents a latent variable framework to provide consistent and efficient estimates of market values of amenities. A model for property values of residential housing using different indicators for neighborhood quality and property value is estimated using data from the U.S. American Housing Survey. The estimated effect of neighborhood quality on property values is positive and more significant compared to the estimates obtained by ordinary least squares and instrumental variable methods. Variances of errors of measurement and variances of the latent structures are shown to be positive and significant without imposing nonnegativity restrictions.     

CONTINUOUS PREFERENCE ORDERINGS REPRESENTABLE BY UTILITY FUNCTIONS

This paper surveys the conditions under which it is possible to represent a continuous preference ordering using utility functions. We start with a historical perspective on the notions of utility and preferences, continue by defining the mathematical concepts employed in this literature, and then list several key contributions to the topic of representability. These contributions concern both the preference orderings and the spaces where they are defined. For any continuous preference ordering, we show the need for separability and the sufficiency of connectedness and separability, or second countability, of the space where it is defined. We emphasize the need for separability by showing that in any nonseparable metric space, there are continuous preference orderings without utility representation. However, by reinforcing connectedness, we show that countably boundedness of the preference ordering is a necessary and sufficient condition for the existence of a (continuous) utility representation. Finally, we discuss the special case of strictly monotonic preferences.     

A two-stage procedure for partially identified models

This paper studies a two-stage procedure for estimating partially identified models, based on Chernozhukov, Hong, and Tamer’s (2007) theory of set estimation and inference. We consider the case where a sub-vector of parameters or their identified set can be estimated separately from the rest, possibly subject to a priori restrictions. Our procedure constructs the second-stage set estimator and confidence set by taking appropriate level sets of a criterion function, using a first-stage estimator to impose restrictions on the parameter of interest. We give conditions under which the two-stage set estimator is a set-valued random element that is measurable in an appropriate sense. We also establish the consistency of the two-stage set estimator.     

Expenditure allocation across nondurables,services, durables and savings: An empirical study of separability in the long run

The separability of categories of consumption expenditures and savings are empirically examined in the context of an error-correction model of aggregate quarterly expenditure allocations in Canada. Some evidence is found for separability of consumption allocations from savings in both the long run structure and short run dynamics of the model lending some support to the use of the assumption in modelling consumption behaviour.