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An earlier version of this paper was completed before Mr. Taylor joined the Fund, and it has already been accepted for publication by the Economic Journal. The author is grateful to Alan Cathcart of the Bank of England Mathematical Techniques Group, who was instrumental in providing the redemption yield data, and to Paul R. Masson for helpful comments. The usual disclaimer applies. A shortened version of this paper will be published in the May 1992 edition of the Economic Journal.
Although we shall not use time subscripts on Φ(n), nothing in this section requires the term premia to be constant—we require only the weaker assumption of stationarity.
Chow (1989) argues for the importance of allowing for alternative methods of expectations formation in the context of present value models. Taylor (1991) demonstrates the stationarity of forecast errors (when the variable being forecast is 1(1)) for a wide variety of expectations formation mechanisms.
Alternatively expressed, the test will lack power. The intuition behind Perron’s formal proof can be seen as follows. Suppose the true data-generating process is yt = α + β t + ut, where ut is stationary white noise—i.e., y is stationary about a linear trend. If we estimate the AR(1) model yt = γ + ρyt-1 + ϵt then ρ will be forced to unity, so that the AR(1) model is equivalent to yt = y0 + γt + ϵt, where
Which, in the words of Campbell and Shiller (1987), “is a stochastic singularity which we do not observe in the data.”
Asymptotic t-ratios were constructed using a White (1980) heteroscedasticity correction to the covariance matrix.
In deriving (19) we have summed the matrix geometric progression, using γn ≈ 0 for large n:
Because a constant intercept term was included in our estimated VARs, our tests in fact allow for at most constant term premia.
Although it is quite likely that this lag depth was an over-parameterization in some cases, this would have the effect of reducing the test power, so that any rejections that do occur will hold a fortiori. Moreover, we obtained qualitatively identical results for all lag lengths between four and thirteen.
Graphs of the theoretical and actual spread for the other two T-Bond maturities qualitatively similar to Figure 3 and so are not shown.
Strictly speaking, the failure of spreads to Granger cause short rate changes does not constitute a rejection of the RETS model, but a failure to confirm it.
Shiller et al. (1983) point out, however, that despite these rejections the expectations theory continually resurfaces in policy discussions: “We are reminded of the Tom and Jerry cartoons that precede feature films at movie theaters. The villain, Tom the cat, may be buried under a ton of boulders, blasted through a brick wall (leaving a cat-shaped hole), or flattened by a steamroller. Yet seconds later he is up again plotting his evil deeds” (op cit., page 175). The robustness of the theory is presumably due to lack of a sufficiently robust and well-developed alternative.
Shiller and McCulloch (1987) demonstrate that the pure ET model is based on the assumption of risk neutrality.
Approximate thirteen-week holding returns were generated using a formula analogous to (4)—see Shiller, Campbell, and Schoenholtz (1983), p. 179. Shiller et al. (1983) compare exact holding period returns with this linearization and show the approximation to be extremely close.
The additional nonlinearity induced in the model by using overlapping data is considerable. For example, the conditional standard deviation in (24) is given by the square root of:
The conditional variance of ϵt+1,
The gain in efficiency resulting from increasing the sample size thirteen-fold was thought to be important enough to outweigh these disadvantages, however.
The likelihood function was maximized using the Broyden-Fletcher-Goldfarb-Shanno positive secant update algorithm (Dennis and Schnabel (1983)). Presumably because of the high degree of nonlinearity involved this method occasionally failed (i.e., failed to converge), and the downhill simplex method of Nelder and Mead (1965) was used to restart the optimization process (see Press, et al. (1986)).
Data were also obtained from the Bank on the total nominal value of U.K. government fixed interest debt outstanding at each of the above data points, for three maturity bands: over eight and less than ten years, over ten and less than fifteen years and between fifteen and twenty years.
Masson (1978), using a structural model, reports evidence that supports the market segmentation approach, using data on Canadian government bonds.