This paper develops an asymptotic theory for residual based tests for cointegration. Attention is given to the augmented Dickey-Fuller (ADF) test and the Z(subscript alpha) and Z(subscript t) unit root tests. Two new tests are also introduced. The tests are shown to be asymptotically similar, and simple representations of their limiting distributions are given and asymptotic critical values are tabulated. The ADF and Z(subscript t) tests are asymptotically equivalent. Power properties of the test are also studied. The tests are consistent if suitably constructed, but the ADF and Z(subscript t) tests have slower rates of divergence under cointegration than the other tests. Copyright 1990 by The Econometric Society.

The theory of precautionary saving is shown in this paper to be isomorphic to the Arrow-Pratt theory of risk aversion, making possible the application of a large body of knowledge about risk aversion to precautionary saving, and more generally, to the theory of optimal choice under risk. In particular, a measure of the strength of precautionary saving motive analogous to the Arrow-Pratt measure of risk aversion is used to establish a number of new propositions about precautionary saving, and to give a new interpretation of the Oreze-Modigliani substitution effect.