A number of recent studies address the problem of assessing the contributions of market fundamentals and rational bubbles to stock-price fluctuations-see, for example, Olivier Blanchard and Mark Watson, 1982; Robert Flood, Robert Hodrick, and Paul Kaplan, 1986; and Kenneth West, 1986, 1987. A rational bubble reflects a self-confirming belief that an asset's price depends on a variable (or a combination of variables) that is intrinsically irrelevant-that is, not part of market fundamentals-or on truly relevant variables in a way that involves parameters that are not part of market fundamentals. A basic difficulty involved in testing for the existence of rational bubbles, pointed out by Flood and Peter Garber, 1980, and emphasized by James Hamilton and Charles Whiteman, 1985, is that the contribution of hypothetical rational bubbles to asset prices would not be directly distinguishable from the contribution to market fundamentals of variables that the researcher cannot observe. For example, as Hamilton, 1986, shows, a researcher who is unable to observe or to infer changes in the expectations of market participants, especially if they involve the probable future occurrence of relevant events that are infrequent and discrete, might falsely conclude that rational bubbles exist. In the present context, the probabilities that investors attach to possibilities for future tax treatment of dividend income could act like such an unobservable variable. Diba and Grossman, 1984, and Hamilton and Whiteman, 1985, propose an empirical strategy based on stationarity tests for obtaining evidence against the existence of explosive rational bubbles without precluding the possible effect of unobservable variables on market fundamentals. The present paper implements such tests for explosive rational bubbles in stock prices using a model that assumes a constant discount rate, but that allows unobservable variables to affect market fundamentals and also allows different valuations of expected capital gains and expected dividends. If the first differences of the unobservable variables and the first differences of dividends are stationary (in the mean) and if rational bubbles do not exist, then the model implies that first differences of stock prices are stationary. The model also implies, using an argument adapted from John Campbell and Robert Shiller, 1987, that, if the levels of the unobservable variables and the first differences of dividends are stationary, and if rational bubbles do not exist, then stock prices and dividends are cointegrated of order (1,1). These theoretical results do not imply that the finding that first differences of stock prices are nonstationary, or that stock prices and dividends are not cointegrated, would establish the existence of rational bubbles. A finding that stock prices and dividends are not cointegrated could result from the nonstationarity of the unobservable variables in market fundamentals, and a finding that stock-price changes are nonstationary could result from the nonstationarity of changes in these unobservable variables. Such findings also could arise from the inappropriateness of the implicit assumption that dividends are generated by an ARIMA process. The converse inference, however, is possible. That is, evidence that first differences of stock prices have a stationary mean and/or evidence that stock prices are cointegrated with dividends would be evidence against *Research Department, Federal Reserve Bank of Philadelphia, Philadelphia, PA 19106, and Department of Economics, Brown University, Providence, RI 02912, respectively. The views expressed are solely those of the authors and do not necessarily represent the views of the Federal Reserve Bank of Philadelphia or of the Federal Reserve System. We thank John Campbell, Robert Shiller, and anonymous referees for helpful comments on earlier versions of this paper.
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