Testing for rational bubbles in a co-explosive vector autoregression

We derive the parameter restrictions that a standard equity market model implies for a bivariate vector autoregression for stock prices and dividends, and we show how to test these restrictions using likelihood ratio tests.  The restrictions, which imply that stock returns are unpredictable, are derived both for a model without bubbles and for a model with a rational bubble.  In both cases we show how the restrictions can be tested through standard chi-squared inference.  The analysis for the no-bubble case is done within the traditional Johansen model for I(1) variables, while the bubble model is analysed using a co-explosive framework.  The methodology is illustrated using US stock prices and dividends for the period 1872-2000.