Proxying Inflation Forecasts With Fuller/Roy-Type Median Unbiased Near Unit Root Coefficient Estimates

The Moderate Quantity Theory of Money (McCulloch, 1980) specifies the functional form of the price adjustment equation as: pi(t) = a(m(t-1)-md(t-1)) + E(pi(t-1)) + e(t), (1) where pi(t) is inflation at time t, a is an adjustment parameter, m(t-1) and md(t-1) are real money balances and a specification for real money demand, respectively, E(pi(t-1)) is the public's expectation, as of t-1, of pi(t), and e(t) is a white noise error term. In order to implement this model, we proxy E(pi(t-1)) with forecasts that are obtained from a univariate time series model, using monthly data only available up to time t-1, over the post-war period (Jan. 1959 - May, 1999). Although equation (1) implies that the excess supply of money also affects inflation, a small value of a and a small R-squared make it plausible that this signal is too weak to be worth the public's while to try to detect, once the history of inflation is taken into account.In the early portion of our period, a unit root in inflation may be rejected, while in the later portion, it generally cannot be. Work by Andrews (1993), Andrews and Chen (1994), Fuller (1996), and Fuller and Roy (1998) has suggested that the direct modeling of a unit root or near unit root process should be done using median unbiased estimators. It is well known that the coefficient on the AR(1) term in an OLS autoregression will be biased downward as the true value of the estimator approaches one (Mariott and Pope (1954), Pantula and Fuller (1985), and Shaman and Stine (1988). To correct for this bias, these authors calculate the bias contingent on the sample size and the true AR(1) parameter. The estimated parameter is then corrected by incorporating this bias.Since the size of the root nearest unity of the U.S. annualized monthly inflation series appears to change over time, we use an expanding window. For each month, we first estimate a modified long-lag AR process using Weighted Symmetric Least Squares as in Fuller (1996), then adjust the lead coefficient along the lines he proposes. However, in order to incorporate MA terms, we then pseudo-difference the inflation series to date using the median-unbiased AR(1) coefficient, to ensure we are dealing with a stationary series. We then fit a parsimonious ARMA, as determined by the Schartz-Bayesian Criterion, and generate one-step ahead forecasts. The entire procedure is repeated each month, using only past data, and starting with data back to January 1950.The Ljung-Box Q statistic indicates that the differenced series is quasi-white noise, i.e., the inflation series has been modeled adequately. The serially uncorrelated forecasts are then used in equation (1) to test the Moderate Quantity Theory of Money and to estimate money demand.