As noted previously, real interest rates are non-observable and are usually proxied by the so-called post real interest rates. As is well known, however, ex-post rates include two disturbing components which can render them a misleading proxy for non-observable ex-ante real interest rates: inflation risk premia and agents' inflation expectation errors. Like ex-ante real interest rates, these two variables are also non-observable. There are reasons to think that both disturbances are probably negligible. First, inflation premia can hardly be relevant if the inflation rate is not very volatile. And second, if agents are rational when forming their inflation expectations, the expectation error should be zero on average. Although it is still an open question, this view has been recently challenged in the literature, particularly in relation to the magnitude of the inflation expectation error. Thus, a series of papers have found that, due to informational or to (monetary policy) credibility problems, inflation rates can be successfully characterised by switching-regime models à la Hamilton, not only in high-inflation countries like Argentina, Israel or Mexico (see Kaminsky and Leiderman, 1996) but also in countries whose inflation rates are lower and more stable like the US (Evans and Lewis, 1995). These switching-regime models produce inflation expectation errors which have zero-mean ex-ante but, ex-post can show a non-zero mean.
In this section the focus is on determining whether the two different methods of deriving the real interest rates yield different conclusions in testing RIP theory. In particular, we aim to investigate the question of whether real interest rates are stationary or not is sensitive to the underlying approach of deriving the rates. If there is ambiguity in identifying the stationarity of the series, this could lead to problems in selecting the methodology for conducting RIP hypothesis testing. For instance, if real rates are nonstationary, cointegration techniques would be more appropriate to test for a cointegrated relationship between two or more random walk series than a simple linear regression due to spurious regression problems as described in Granger and Newbold (1974).
Implicitly, the past literature has assumed that the method of constructing the real rate is irrelevant to the test. Therefore, real interest rates constructed differently should have similar time series properties, and the inconclusive results of RIP may come from other theoretical sources. However, if that is not the case then differences
among RIP empirical analysis may stem from deviations of the methodologies used by authors. Future investigation of issues involving use of expected real interest rates may have to be more concerned about the selection of the measuring approach.
As we found earlier, the means and the medians of the real interest rates from the two different approaches appear to be similar (as shown in Tables 2.5-2.8). Same pattern of standard deviations is also observed. The real interest rates-both short term and long term- constructed by using the year-to-year annualized inflation rate with the two methods, as shown in Figures 5-8, appear to be remarkably similar in the pattern of movements for each country. Although the derived real interest rates from the two approaches seem to follow the same pattern over the sample period, they are in fact different time series processes. As the results for ex ante rates are similar to those for ex post rates, we do not report them. We only report those for ex post rates in the next section. Overall, considering the stationarity of the series, all real rates seem to follow a random walk and this conclusion is robust to different choices of approach in constructing the series and the types of unit root tests8 used. These findings for stationarity are reported separately in the following section.
Tables 2.9 and 2.10 report the results of the mean and variance equality testing for quarterly data for all countries. We employ the analysis of variance (ANOVA) to examine whether different approaches of constructing interest rates (both short-term and long–term ones) would provide the series with equal mean. For the variance equality testing, a Brown-Forsythe test is used to evaluate the null hypothesis that the variance in all series is equal against the alternative that at least one series has a different variance.
Table 2.9: Tests for Equality of Means and Variances Real Short-Term Interest Rates from Different Approaches Country Mean Equality Test:
Test Statistics
Variance Equality Test:
Test Statistics
Australia 1.19E-06
(0.9991)
0.000184 (0.9892)
Belgium 0.016940
(0.8965)
4.47E-06 (0.9983)
Canada 3.66E-06
(0.9985)
1.05E-05 (0.9974)
France 0.025121 (0.8742)
0.001823 (0.9660)
Italy 1.89E-05
(0.9965)
3.83E-05 (0.9951)
Japan 0.019679
(0.8885)
0.005596 (0.9404)
Luxembourg 1.57E-05
(0.9968)
0.000385 (0.9844)
Netherlands 2.32E-05
(0.9962)
1.67E-05 (0.9967)
New Zealand 9.87E-06
(0.9975)
0.000101 (0.9920)
Norway 1.66E-06
(0.9990)
4.70E-05 (0.9945)
Spain 5.22E-06
(0.9982)
4.69E-06 (0.9983)
Switzerland 3.50E-06
(0.9985)
2.44E-05 (0.9961)
UK 0.044278
(0.8335)
0.002438 (0.9607)
US 0.011066
(0.9163)
0.003092 (0.9557)
Notes:
The reported test statistics are the F-statistics follow F-distribution. The parentheses display the corresponding p-values.
From the comparison of the tables 2.5 and 2.7 we find that the real short-term interest rates that we derived from the two methods with the year-to-year inflation rate yield a similar mean and median of the series. This finding is confirmed by the large probability (round 0.9) of accepting the mean equality between the real interest rates of each country, as shown in the second column of Table 2.9. The variance equality test clearly suggests that the variances of each country’s series are equal.
Table 2.10: Tests for Equality of Means and Variances Real Long-Term Interest Rates from Different Approaches Country Mean Equality Test:
Test Statistics
The reported test statistics are the F-statistics follow F-distribution. The parentheses display the corresponding p-values.
From the comparison of the tables 2.6 and 2.8 we find that the real long-term interest rates that we derived from the two methods with the year-to-year inflation rate
yield a similar mean and median of the series. This finding is also confirmed by the large probability (round 0.9) of accepting the mean equality between the real interest rates of each country, as shown in the second column of Table 2.10. The variance equality test also suggests that the variances of each country’s series are equal.
Since these two approaches in constructing the real interest rates share some similarities in regard to the descriptive statistics, mentioned in the previous sections, our next step is to test whether these rates are correlated.
Table 2.11: Correlations: Quarterly Real Short-Term Interest Rates
Ex post MA(4)
Ex post 1.000 0.999
MA(4) 1.000
Table 2.12: Correlations: Quarterly Real Long-Term Interest Rates
Ex post MA(4)
Ex post 1.000 0.999
MA(4) 1.000
Tables 2.11 and 2.12 display the correlations of the real interest rates from the two different approaches followed. It should be noted that the results presented in the above tables are coincidentally the same for all countries and this is the reason why we do not present the results for each country separately. The findings strongly indicate that, for both short-term and long-term rates, all of the real interest rate series from the two approaches are highly correlated. Specifically, the ex post is highly correlated with the MA(4) approach with the correlation coefficient of 0.99. There is generally a significant positive correlation between the real interest rate in each country and the real interest rate in the other country (US).
In summary, our findings indicate that the real interest rates obtained from different approaches yield not quite different time series processes, and they appear to have the same mean and vary across time in similar patterns over the sample period.
In the next section we test whether the stationarity of the real interest rates depends on the type of method used to construct the real rate of interest. In other words, we will test whether the stationarity of the real interest rate series are sensitive to the computations of the inflation rate.