WHEN MONEY AND PRICES ARE INTEGRATED of order two, I(2), and shocks to money demand or velocity are stationary, then the Cagan (1956) monetary model of hyperinflation has the implication that real money balances cointegrate, in the sense of Engle and Granger (1987), with the rate of inflation. As a result, one can estimate the interesting parameter in the model, the semielasticity of the demand for real balances w.r.t. expected inflation, super-consistently in a cointegrating regression without having to specify the exact expectations formation process. In addition, simultaneity or omitted variables bias vanishes asymptotically. In a recent paper in this journal, Taylor (1991) makes use of these insights in a reexamination of some of the classic interwar European hyperinflations, and although the results are mixed, he generally finds support for the Cagan model with stationary velocity shocks.
In the present paper I show, in the next section, that by making the stricter assumptions of rational expectations and no bubbles, an additional cointegrating relationship can be derived from the Cagan model, namely that real money balances cointegrate with money growth. I then show how, under these assumptions, the Cagan specification imposes testable restrictions on a VAR model that involves this additional cointegrating relationship. As opposed to the VAR model derived by Taylor (1991), my model entails interesting cross-equation parameter restrictions, and can be used to evaluate the economic significance of a possible statistical rejection of the restrictions, along the lines suggested by Campbell and Shiller (1987). In section 2 I use these methods on data from the German 1920-23 hyperinflation. Some concluding remarks are given in section 3.
The Cagan model under rational expectations and instantaneous clearing in the money market is given as [Mathematical Expression Omitted] where [m.sub.t] and [p.sub.t] are natural logarithms of the money stock and the price level, respectively, and [alpha] and [Beta] are parameters to be estimated. [E.sub.t] is the conditional expectations operator and [u.sub.t] is a stochastic variable representing velocity and/or demand shocks.
Equation (1) can be rewritten into the following regression equation: [Mathematical Expression Omitted] where [[epsilon].sub.t+1] = [p.sub.t+1] - [E.sub.t][p.sub.t+1] is the serially uncorrelated rational expectations error at time t + 1. When both inflation and real balances are I(1)-processes, (2) is a cointegrating regression, and stationarity of [u.sub.t] then implies cointegration between [[Delta]p.sub.t+1] and [m.sub.t] - [p.sub.t], in which case [beta] is estimated super-consistently (Stock 1987).(1)
Imposing the non-bubble transversality condition [Mathematical Expression Omitted], (1) can be solved recursively forward to give the following present value relation: [Mathematical Expression Omitted] Multiplying (3) with - 1, adding [m.sub.t] on both sides and rearranging terms, gives [Mathematical Expression Omitted] which shows that in case of no velocity shocks, real balances predict future money supply changes in the Cagan model. If [m.sub.t] - [p.sub.t] rises, it is an indication of expectations of future decline in money growth, which leads to lower future inflation and therefore higher demand for real balances today. This is a general implication of present value models. In the permanent income model of consumption, for instance, savings predict future labor income changes (Campbell 1987), and in the expectations theory of the term structure the spread between long and short interest rates predicts future interest rate changes (Campbell and Shiller 1987).
Campbell and Shiller (1987) have devised an appealing method to test such implications of present value models when the underlying time series are integrated of order one. Under hyperinflation, however, …