Knowledge that Transforms

The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples. If each element of a vector of time series x first achieves stationarity after differencing, but a linear combination a'x is already stationary, the time series x are said to be co-integrated with co-integrating vector a. There may be several such co-integrating vectors so that a becomes a matrix. Interpreting a'x,= 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance. The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representations for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application. In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.

[Andersen (1970) considered the problem of inference on random effects linear models from binary response panel data. He showed that inference is possible if the disturbances for each panel member are known to be white noise with logistic distribution and if the observed explanatory variables vary over time. A conditional maximum likelihood estimator consistently estimates the model parameters up to scale. The present paper shows that inference remains possible if the disturbances for each panel member are known only to be time-stationary with unbounded support and if the explanatory variables vary enough over time. A conditional version of the maximum score estimator (Manski, 1975, 1985) consistently estimates the model parameters up to scale.]

This paper describes a simple method of calculating a heteroskedasticity and autocorrelation consistent covariance matrix that is positive semi-definite by construction. It also establishes consistency of the estimated covariance matrix under fairly general conditions.

For every range of admissible incomes, the authors characterize the class of Engel curves with the property that if an economy has, first, a price independent distribution of income and, second, preferences which are identical across consumers and generate Engel curves in the class, then the corresponding aggregate demand function satisfies the Weak Axiom of Revealed Preference. This class is defined by two simple conditions. The no-torsion condition says that, in the relevant range of income, the Engel curve is contained in a plane through the origin. The uniform-curvature condition says that, in addition, the Engel curve is either convex or concave to the origin.

This paper extends the simple errors-in-variable bound to the setting of systems of equations. Both diagonal and nondiagonal measurement error covariance matrices are considered. In the nondiagonal case, the analogue of the simple errors-in-variable interval of estimates is an ellipsoid with diagonal equal to the line segment connecting the direct least squares with a two-stage least squares estimate. For the diagonal case, the set of estimates under some conditions must lie within the convex hull of 2k points.

The problem of the existence of a competitive equilibrium in models with hidden knowledge and self-knowledge has been discussed previously by M. Rothschild and J. E. Stiglitz_(1976), C. A. Wilson_(1977), and J. G. Riley_(1979). Recent analyses of such models by I. Cho and D. Kreps_(1986) and Riley argue for a particular outcome - the Pareto-dominant separating, zero-profit one. The authors prove the existence of such an outcome under very general conditions and, generalizing the reactive equilibrium concept introduced by Riley, they prove this outcome is the unique reactive equilibrium.

This paper considers estimation and testing using location measures for regression m odels that are based on an asymmetric least-squares criterion functio n. These estimators have properties that are analogous to regression quantiles, but are easier to calculate, as are the corresponding test statistics. Asymmetric least-squares tests of homoskedasticity and s ymmetry compare quite favorably with other tests of these hypotheses in terms of asymptotic relative efficiency. Consequently, asymmetric least-squares estimation provides a convenient and relatively efficie nt method of characterizing the conditional distributi on of a dependent variable given some regressors.

The local power of test statistics is analyzed by considering sequences of data-generating processes (DGPs) that approach the null hypothesis without necessarily satisfying the alternative. The three classical test statistics-LR, Wald, and LM-are shown to tend asymptot ically to the same random variable under all such sequences. The powe r of these statistics depends on the null, the alternative, and the sequence of DGPs in a geometrically intuitive way. This implies that, for any statistic that is asymptotically chi-squared under the null, there exists an "implicit alternative hypothesis" against which that statistic will have highest power.