by Bob Burkhardt
14.386
Converted to HTML July 31, 1998.
1 Introduction (July 31, 1998)
This is a short paper I did as a graduate student in economics at M.I.T. for an econometrics class I was taking. A version in PDF format is available. The equations in that version use more standard symbols.
2 Introduction (Spring, 1977)
In a paper forthcoming in the American Economic Review,1 Charles Nelson and William Schwert (henceforth N-S) provide an alternate test of hypotheses tested in a 1975 AER article by Eugene Fama.2 Their main purpose in putting forward this new test is that they see it as more powerful than the tests Fama used. The purpose of this paper is to review Fama's work and the flaws which have been seen in it and to analyze N-S's handling of the testing of Fama's hypotheses.
3 Fama's Test
The purpose of Fama's 1975 article was to test whether market expectations of inflation (or equivalently market expectations of the real rate) are formed efficiently in the Treasury Bill market. Fama sees this as being a necessary condition for efficiency in the Treasury Bill market. To accomplish this test, he tests a joint hypothesis. The first part of this joint hypothesis is that the ex ante real rate implied by inflationary expectations is constant. The second part is that inflationary expectations are efficient, i.e. that the market expectations of inflation are in some sense the best predictors of actual inflation.
These hypotheses are tested in several ways. First, autocorrelation coefficients of the Treasury Bill rate are examined. They are seen not to be significant as implied by Fama's joint hypothesis. Next the Treasury Bill rate is regressed on inflation. Its coefficient is indistinguishable from unity as implied by the hypothesis. The battery of tests is concluded by seeing whether there is more information about future inflation in lagged inflation rates than in the nominal interest rate. Again as implied by the hypothesis, such information appeared not to be present. Fama concluded by accepting the joint hypothesis.
4 Tobin-Begg Criticism of Fama's Tests
The predominate criticism of Fama's tests is that they aren't very powerful. One type of criticism along these lines is the Begg-Tobin3 errors-in-variables analysis. The basic outline of that criticism is that if there is a random component of the Treasury Bill rate which is orthogonal to expected inflation, the biased least-squares estimator may show the Treasury Bill rate moving one-for-one with the inflation rate even if it doesn't move one-for-one with expected inflation. The analysis leading to this conclusion is as follows:
where Rt is the Treasury Bill rate, pte and pt are expected and actual inflation respectively, and et and nt are error terms which are uncorrelated with each other and the other time series.
If et were excluded, and a1 = 1, these equations constitute, for the present purposes, Fama's joint hypothesis. Thus:
By assumption, et is positively correlated with Rt and thus the coefficient of Rt estimated by least-squares will be downward biased. So even if a1 < 1, the hypothesis a1 = 1 could be accepted even asymptotically. Thus, under these conditions, the regression analysis may not detect departures of the Treasury Bill market from Fama's joint hypothesis.
5 Nelson and Schwert's Criticism of Fama's Tests
N-S's criticism of Fama's tests of his joint hypothesis focus more on the first and third of Fama's tests outlined above. Again the criticism is that these tests are not very powerful. According to N-S, the first test is not very powerful since the large errors in inflationary expectations dominate the autocorrelation statistics and thus the statistics reflect the autoregressive properties of the inflation expectation errors and not those of the ex ante real rate. The third test is criticized because only a weak competitor is offered to take explanatory power away from the Treasury Bill rate. Specifically, they criticize Fama for only examining lagged inflation in his search for time series which have information about future inflation which the Treasury Bill rate doesn't possess.
N-S's basic contribution is to redo Fama's third test in a more satisfactory manner. They do this by using the methods of Box-Jenkins4 to construct a better predictor of future inflation than just lagged inflation. Their first step was to formulate and fit a time-series model to the inflation time series. The forecast provided by this series was then compared to the Treasury Bill rate to see if it possessed additional information about future inflation. Unlike lagged inflation, the N-S construction appeared to possess a lot of additional explanation of future inflation. They follow this test up by one which estimates the time series model of inflation as a variable in the regression of pt on Rt. This is done to avoid the criticism that, in estimating their time series model outside the final regression model, they gave their inflation predictions information the market wouldn't have. This test confirms their previous result. Thus, they rightly reject Fama's joint hypothesis.
However, they then conclude that the ex ante real Treasury Bill rate is not constant. They justify this conclusion by citing Fama's 1970 survey article's5 strong evidence that the stock market is efficient. Since the Treasury Bill market and stock market are somewhat related, they understandably interpret this as indicating the Treasury Bill market is also efficient. Following Fama's logic, they conclude price expectations in that market must be efficient. Thus Fama's hypothesis that inflation expectations are efficiently formed is accepted on this basis and his model of the ex ante real rate rejected.
6 Criticism of Nelson and Schwert
The latter conclusion runs afoul of Tobin's6 criticism of Fama that there is no a priori reason to expect inflationary expectations to be formed efficiently in the Treasury Bill market. In Fama's survey of the tests of stock-market pricing efficiency, one is constantly aware of the arbitrage opportunities which would open up if the stock market were inefficient in its pricing. Inefficiency is practically equivalent to the presence of arbitrage opportunities. However, neither Fama nor N-S outline arbitrage opportunities which would give an advantage to an investor with efficient expectations in a market with inefficient expectations. Such an opportunity would exist say if there were an instrument of Treasury Bill maturity which offered a return indexed in terms of consumer prices. In this case, the ex ante real rate would be observable and speculators with efficient expectations could reap arbitrage gains in the Treasury Bill market if there were inefficient expectations of inflation and thus incorrect assessment of the ex ante real return offered by Treasury Bills there. But no such instrument exists, and Tobin can see no close market substitute for such an instrument. Thus, efficient pricing in the Treasury Bill market, here interpreted as the absence of arbitrage opportunities, need not imply that expectations of inflation formed in that market are efficient. The basic import of this line of criticism is that tests of efficient market properties of the Treasury Bill market, or for that matter any of its properties, are uninformative if they rely on assumptions or tests of the efficiency of inflation prediction in that market.
7 Conclusion
Thus it can be concluded that N-S's tests are informative in that they reject Fama's joint hypothesis. However, it must also be concluded that neither N-S's nor Fama's line of research provides any evidence on the properties of the ex ante real Treasury Bill rate or the efficiency of the Treasury Bill market.