RDP 9410: An Empirical Examination of the Fisher Effect in Australia 4. Interpreting Inflation Forecasting Equations
December 1994
- Download the Paper 96KB
The conclusion from the preceding empirical analysis is that it is reasonable to assume that there is a long-run Fisher effect in Australia but not a short-run Fisher effect. This characterisation of the inflation and interest rate data along with the assumption of rational expectations can be used to provide a straightforward interpretation of when we will be likely to see estimated β coefficients substantially above zero in the inflation forecasting equation (1). As in Mishkin (1990), we can derive an expression for the coefficient β in the inflation forecasting equation (1) by writing down the standard formula for the projection coefficient β, while recognising that the covariance of the inflation forecast error with the real interest rate, rrt, equals zero given rational expectations. The resulting formula for the predicted value of β is:
where,
= σ[Et(πt)]/σ[rrt]
= the ratio of the unconditional standard deviation of the expected inflation
rate to the unconditional standard deviation of the real interest rate.
ρ = the unconditional correlation coefficient between the expected inflation rate, Et(πt), and the one-period real interest rate, rrt.
The equation above indicates that β is determined by how variable the level of expected
inflation is relative to the variability of the real interest rate (represented
by ,
the ratio of the standard deviations of Et (πt)
and rrt), as well as by the correlation of the expected
inflation rate with the real interest rate (ρ). Figure 2 shows how β varies
with
and ρ.


As we can see in Figure 2, when the variability of the level of inflation is greater
than the variability of the real interest rate, so that is above 1.0, the β coefficient will always exceed 0.5 and will increase as
increases. If inflation has a unit root and thus does not have a stationary stochastic
process, as is consistent with the empirical evidence in this paper, then its
second moment is not well defined and the standard deviation of the inflation
level will grow with the sample size. On the other hand, the existence of a
long-run Fisher effect implies that even if inflation and interest rates have
unit roots, the real interest rate has a stationary stochastic process and
will have a well defined standard deviation that does not grow with the sample
size. Hence when we are in a fairly long sample period in which inflation and
interest rates have unit roots, the existence of a long-run Fisher effect means
that
will be likely to exceed one and produce a value of β substantially above zero.
The above interpretation helps explain why we see the strong correlation between
interest rates and future inflation that we found for both pre and post 1979
sample periods in Table 1. To see this more clearly, we can calculate estimated
values of and ρ. We do this by using the procedure outlined in Mishkin (1981), where estimates
of the real rate, rrt, are obtained from fitted values
of regressions of the ex-post real rate on past inflation changes and past
interest
rates.[12]
Then the estimated expected inflation is calculated from the following definitional
relationship,
Finally estimates of σ[Et(πt)], σ[rrt], and ρ are calculated from the estimated Et(πt) and rrt.
Figure 3 shows the resulting estimates of expected inflation and real interest rates,
Et(πt) and rrt
respectively, while the estimates for each of the sample periods
are shown in Figure 2. As we can see from Figure 3, the variability of expected
inflation is greater than that of the real interest rate in the pre 1979 period,
resulting in a pre 1979
of 1.5. Also as is visible in Figure 3, expected
inflation and the real rate are highly negatively correlated with a correlation
coefficient of −0.9. As Figure 2 indicates, the predicted β using these
values of
and ρ for the pre 1979 sample period is substantially greater than one, which is
what we found in Table
4.[13]
For the post 1979 sample, expected inflation continues to be more variable
than the real rate leading to an estimate of
of 1.4, while there
is little correlation of expected inflation and real rates, ??= 0.2. The slightly
lower
and more positive ρ leads to a predicted ? which is lower in the post 1979 sample
than in the pre 1979 sample, although it is still greater than 0.5 because
is greater than one. Our analysis thus explains why the estimated β coefficient
in Table 1 falls in going from the pre 1979 to the post 1979 period.

Figure 3 also shows that estimated real interest rate had a dramatic upward shift
after 1979, a result which has also been found in other OECD
countries.[14]
This shift in the real interest rate causes the standard deviation of the real
rate to be higher for the full sample period than it is for either of the sub-periods,
thus leading to a for the whole sample of 0.9. The
less than one means
that β should drop below 0.5 and this is again what is found in Table 1. We
see that the pattern of estimated β found in Table 1 is exactly what our model
predicts.
Footnotes
The estimates described in the text were generated from OLS regressions in which
the ex-post real rate, eprrt, was regressed on it, and on
πt−1 and πt−2.
The estimated values of and ρ are robust to different
specifications of the regression equations.
[12]
This is consistent with the results of Carmichael and Stebbing (1983). [13]
See Cumby and Mishkin (1986). [14]