An aspect of the wald test for linear restrictions in the seemingly unrelated regressions model

This paper shows that, in a SUR context, there is an exact relationship between the Wald statistic for testing linear restrictions and the ‘F statistic’ based upon an estimate of the covariance matrix that adjusts for degrees of freedom.     

An adjustment to the likelihood ratio statistic when testing hypotheses in the multivariate linear model using large samples: A note

Leamer's (1978) criterion for testing restrictions in the single equation linear model in which the level of significance decreases with sample size is generalized to the case of common, within equation linear restrictions in the multivariate linear model.     

A linear demand system within a seemingly unrelated time series equations framework

A seemingly unrelated time series equations framework for the linear almost ideal (AID) demand system is considered. The framework is applied to a consumer demand system covering nine non-durable commodities. Within a specification where the static linear AID system is augmented by latent variables representing stochastic trends and seasonality, demand homogeneity is tested; both in each equation and in the system as a whole. Income and own-price elasticities are calculated under homogeneity restrictions. Although the homogeneous model is formally rejected by statistical tests, it performs well with respect to interpretability, parameter stability and forecasting.     

A system of demand equations for Austria

This paper presents results of parameter estimations of a small system of demand equations for Austria. The functional form of the equations follows the log-linear specification well known as the “Rotterdam”-System. Using annual data from 1954 to 1977 we estimate the absolute price version for a rather aggregated system consisting of four sectors of consumption expenditures. Aitken estimation with and without linear restrictions is the adopted estimation method. Tests for the validity of the general linear restrictions axe performed employing the usual criteria. Relations among the test statistics are discussed. Taking into accountBeaton's [1972] argument of the appropriate use of likelihood ratio tests we present results also after iterating on the restricted error-covariance matrix. The question of negative semidefiniteness of the matrix of price coefficients is examined by inspection of its characteristic roots and the calculation of their approximated asymptotic covariance matrix. Finally, our results are confronted with such of other comparable studies.     

Normalization in cointegrated time series systems

Abstract .  A method for normalizing cointegrating vectors is proposed for cointegrated time series systems containing multiple cointegrating vectors, a method requiring that an identity matrix appear in the normalized cointegrating matrix with unit coefficients attached to the endogenous or choice variables. The preferred method causes the normalized cointegrating matrix and the adjustment matrix to be consistent with the implications of static and dynamic economic theory. Alternative normalizations generate cointegrating and adjustment matrices that do not match up well with economic theory and do not reveal the testable restrictions implied by static economic theory.     

Polynomial approximation of distributed lags and linear restrictions: A Bayesian approach

In this paper, the polynomial approximation of distributed lags is investigated within the framework of linear restrictions in linear regression models. In the first part, the polynomial approximation is analysed assuming well known the truncation point and the degree of the polynomial. The polynomial approximation is shown to involve linear restrictions on regression coefficients; two equivalent representations of these restrictions are used to clarify relationships between previous works byAlmon and byShiller. The difficulties related to the treatment of exact restrictions in a Bayesian framework are then tackled in the present context and alternative procedures are presented. In the second part, the analysis is extended to the case of unknown truncation point and/or unknown degree of the polynomial. This leads to consider mixed prior distributions as for the problem of choosing among different models. The paper ends by investigating the sensitivity of a particular set of data w.r.t. changes in the truncation point, in the degreee of the polynomial and in the prior tightness of the polynomial approximation.     

Maximum likelihood estimation of linear expenditure systems with serially correlated errors: An application

The present paper uses Canadian data for the period 1947–1972 and three commodity groups to examine the empirical importance of restrictions imposed by autocorrelated disturbances on the static linear expenditure system, LES. For comparison a simple habit persistence model is also estimated. Results of applications of likelihood ratio tests indicate that autocorrelation is present in the data, that a simple habit persistence hypothesis on the structure is implied and that the restrictions imposed by the form of the utility function and maximization problem are inconsistent with the data whether or not adjusted for the autocorrelation in errors. Despite these differences, the estimated price and income elasticities remain fairly constant across the various specifications of the LES that were considered.     

Asymptotic distribution of least squares estimator and a test statistic in linear regression models

This paper presents a fairly general and sufficiently weak condition on the observation matrix for the asymptotic normality of the least squares estimator of coefficient vector in a linear regression model. The asymptotic distribution of a statistic for testing the nullity of coefficient vector is also considered.     

General instrumental variables estimation under stochastic linear restrictions

This letter extends the Theil-Goldberger ‘mixed’ regression estimator, for models subject to stochastic linear restrictions, to the case of stochastic regressors. A general instrumental variables ‘mixed’ estimator is discussed. The asymptotic distribution of the estimator is obtained, and an asymptotic test of the compatibility of the sample and prior information is presented.     

Tests in contingency tables as regression tests

Applied researchers often use tests based on contingency tables, especially in preliminary data analysis and diagnostic testing. We show that many such tests may be alternatively implemented by testing for coefficient restrictions in linear regression systems.     

A note on the compatibility of a system of difference equations and a time-independent linear equation

Solutions to systems of difference equations subject to linear constraints are studied. Restrictions on the coefficients necessary to maintain a compatible system are developed. Applications to the dynamics of expenditure shares and to lagged adjustment models with adding up constraints are given.     

On least squares estimation with a particular linear function of the dependent variable

We consider the standard linear regression model where the endogenous variable y is substituted by Ty, T being a symmetric, idempotent matrix. Comparing the mean square error (MSE) matrices we show that a ‘naive’ LS-procedure may work better than a competing estimator usually proposed in the literature and may even perform better than the LS-estimator based on untransformed data. We derive necessary and sufficient conditions for MSE-dominance and outline some ideas for testing.     

On the static and dynamic costs of trade restrictions for small developing countries

We analyze the costs of trade restrictions for a small developing economy (LDC). Intermediate goods invented elsewhere are only introduced on the LDC market if it is profitable to do so. The LDC economy evolves to a balanced growth path in which income, welfare, and the share of available goods increase if trade restrictions fall. The adjustment path is asymmetric: an increase in trade restrictions leads to a slow-down of economic growth, while a decrease may lead to a rapid catch-up process. The dynamic costs of trade restrictions are in general substantially larger than the static costs.     

Linear demand functions in theory and practice

Integrability conditions for an incomplete system of linear demand functions are considered. The parameter restrictions consistent with integrability are identified, and the structure of the conditional preference map is obtained. It is found that the conditional preferences for a set of linear demand functions are either quadratic or Leontief from a translated origin. Welfare analysis with linear demand models is also considered.     

Maximum likelihood estimation of sum-constrained linear models with insufficient observations

A maximum likelihood procedure for estimating sum-constrained linear models is presented, which seems to provide a good balance between excessive observational requirements on the one hand and an unduly restrictive specification of the contemporaneous covariance matrix on the other.     

Transformations for an exact goodness-of-fit test of structural change in the linear regression model

This paper considers testing for structural change of unknown form in the linear regression model as a problem of testing for goodness-of-fit. Transformations of recursive (or other LUS) residuals that reduce the problem to one of testing independently distributed uniform variables are presented. Exact empirical distribution function tests can then be applied without having to estimate unknown Parameters. The tests are illustrated by their application to a money demand model.     

An intertemporal system of dynamic consumer demand functions

This paper constructs and estimates a system of dynamic consumer demand equations under the assumption of rational expectations about anticipated human wealth. The traditional one-period budget constraint is replaced by the lifetime anticipated wealth constraint. Lagged dependent variables are rationalized by an adjustment cost argument. In the model presented, both the dynamic adjustment coefficients as well as the parameters characterizing the underlying long-run preferences are identified and can be estimated. We find that a weak version of the REH cannot be rejected on the data. In common with most empirical studies on demand behaviour we have to reject the symmetry restrictions.     

A turnpike property of optimal programs for a class of simple linear models of production

Summary. This paper establishes a ‘turnpike theorem’ for a closed linear model of production with a primitive input requirement matrix. Optimal programs of resource allocation have a ‘turnpike property’ if the growth factor of every sector in the economy converges, in the long run, to a common value. The usefulness of such a theorem is due to the fact that the input requirement matrix for an economy with a large number of goods may be primitive (some power of the matrix is strictly positive). Received: April 19, 1998; revised version: July 15, 1998     

Effects of changes in outputs and in prices on the economic system: an input–output analysis using the spectral theory of nonnegative matrices

The (physical) output adjustment model and the price adjustment model are presented. By the two models we quantitatively analyze the influences of alterations of one sectoral (physical) gross output and of one sectoral price on another sectoral (physical) gross output and on another sectoral price, respectively. Hence, a basic nature of the Ghosh inverse and a fundamental character of the monetary Leontief inverse are obtained. The proposition that a matrix of intermediate output (input) coefficients alters if the vector of output (price) adjustment coefficients is nontrivial holds, if and only if this matrix is C-irreducible. It is impossible that (i) the adaptation of output system causes all sectoral final output rates (or input multipliers) either to rise or to fall collectively, or (ii) an adjustment of price system causes all sectoral value-added rates (or output multipliers) either to increase or to decrease jointly. However, maybe (i) a change of output system enables some sectoral final output rates (or input multipliers) to rise (fall) and all others to be constant, and (ii) an alteration of price system enables some sectoral value-added rates (or output multipliers) to increase (decrease) and all others to be fixed, whose necessary and sufficient condition is that the matrix of intermediate output (or input) coefficients has at least one non-final (or non-initial) class. The proposition that the vector of final output rates (or input multipliers) changes if the vector of output adjustment coefficients is nontrivial is true, if and only if the matrix of intermediate output coefficients has only one final class. The proposition that the vector of value-added rates (or output multipliers) alters if the vector of price adjustment coefficients is nontrivial holds, if and only if the matrix of intermediate input coefficients has only one initial class. The necessary and sufficient conditions and the matching economic explanations for possibility and uniqueness of the economic adjustment that enables (i) all sectors to have a uniform final output rate (or input multiplier), and (ii) all sectors to have the same value-added rate (or output multiplier) are respectively given. I would like to thank an anonymous referee for helpful comments and suggestions.     

Investigating the US consumer credit determinants using linear and non-linear cointegration techniques

This paper has investigated the determinants of total consumer credit for the USA over the period 1968:Q1 to 2011:Q3. Using Breitung's (2001) non-parametric rank tests, we find the existence of linear cointegrating relationships in the consumer credit models. Enders and Siklos' (2001) threshold adjustment tests revealed that non-linearity is present slightly (with a statistical significance of 10% level) in the consumer credit model with a short-term interest rate (federal funds rate), while there exists a linear and symmetric cointegrating relationship in the models with medium (3 years) and long (10 years) term interest rates. Application of the linear cointegrating techniques (fully modified OLS, canonical cointegrating regression and general to specific) show that consumer credit responds more significantly to the medium and long-term interest rates than the short-term interest rate. We use these results to assess the popular belief that abnormality in the consumer credit set the stage for the 2007–08 crisis and severe recession.