Comments on "Econometric Analysis of Fisher's Equation" by Peter C. B. Phillips

American Journal of Economics and Sociology, The,  Jan, 2005  by John Rust

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Like stocks, bonds have uncertain payout streams and can be traded in secondary markets, so bondholders have the option of making capital gains (or losses) and are not necessarily committed to a "buy and hold" strategy. Thus, there is no reason why "asset bubbles" or any of the other explanations for the excess volatility of stock returns do not also apply to bond returns. Another important factor affecting nominal interest rates that is ignored in the simple Fisher equation is the impact of large players in the bond market such as the government and central banks. Even when these large players are not directly intervening in the Treasury Bill market, they can have huge impact on bond prices via simple "announcement effects" in an environment with market bubbles or when private traders have fickle expectations. Through various interventions the Federal Reserve can affect both the level and volatility of nominal interest rates.

The role of these additional factors as determinants of the nominal interest rate is evident from the plots of time series for the 90-day Treasury Bill rate and the three-month inflation rate in Figures 14 and 15 of Phillips's paper. For example, there is a suspiciously long period from 1942 to nearly 1948 where nominal interest rates remained virtually unchanged at a very low level of less than 1%. Of course, this was during World War II when the U.S. government and the Federal Reserve appealed to patriotism and moral suasion to "jawbone" investors into buying Treasury Bills at par for abnormally low nominal interest rates. For most of this period wartime price controls were also in effect (so inflation was essentially zero), but the simple Fisher equation would be hard-pressed to explain why real interest rates should have remained constant over this period as the wartime production effort was initially geared up and then wound down.

It seems clear that we need to account for these additional neglected factors to obtain an adequate statistical model of nominal interest rates. One way to do this is to include some sort of residual or additional factor in the Fisher equation. For example, we might account for the missing factors through a time varying "market risk premium," [p.sub.t], yielding a modified "three-factor" version of the Fisher equation:

(10) [i.sub.t] = [r.sup.e.sub.t] + [[pi].sup.e.sub.t] + [[rho].sub.t]

Fisher clearly recognized that risk premiums must be accounted for in a fuller account of the determination of nominal interest rates: "We see, then, that the element of risk introduces disturbances into those determining conditions which were expressed in previous chapters as explaining the rate of interest" ([1930] 1971: 222); "instead of a series of equalities which we found to hold true in the vacuum case, so to speak, where risk was absent, we have only a tendency toward equalities, interfered with by the limitations of the loan market, and which, therefore, result in series of inequalities. Rates of interest, rates of preference, and rates of return over cost are only ideally, not really, equal" ([1930] 1971: 226). However, once we acknowledge the presence of risk premiums, we essentially have a decomposition of nominal interest rates in terms of at least three latent variables. This compounds the identification problem, since with three factors it is even less clear whether any movement in nominal interest rates should be attributed to a change in the real rate of interest, expected inflation, or the risk premium [p.sub.t].