Testing the Fisher Hypothesis in the G-7 Countries Using I(d) Techniques

This paper revisits the Fisher hypothesis by estimating fractional integration and cointegration models that are more general than the standard ones based on the classical I(0)/I(1) dichotomy. Two sets of results are obtained under the alternative assumptions of white noise and Bloomfield (1973) autocorrelated errors respectively. The univariate analysis suggests than the differencing parameter is higher than 1 for most series in the former case, whilst the unit root null cannot be rejected for the majority of them in the latter case. The multivariate results imply that there exists a positive relationship, linking nominal interest rates to inflation; however, there is no evidence of the full adjustment of the former to the latter required by the Fisher hypothesis.


Introduction
The long-run behaviour of interest rates is normally analysed using the so-called Fisher relationship (see Fisher, 1930) linking nominal rates and expected inflation and requiring a one-to-one adjustment of the former to the latter, in the absence of which permanent shocks to either inflation or nominal rates have permanent effects on real rates themselves, which is inconsistent with standard models of intertemporal asset pricing as well as superneutrality (to the extent that inflation is a monetary phenomenon), both requiring stationarity of real interest rates. Neoclassical models of dynamic growth also have the property that consumption growth and real rates should both be constant in the steady state.
However, the Fisher relationship is often not supported by the empirical evidence.
Many studies, going back to Mishkin (1992), find that the slope coefficient in a regression of inflation against nominal rates is significantly different from one, and the real rates exhibit a unit root. This "paradox" (see Carmichael and Stebbing, 1983) might be due to various reasons, such as not measuring accurately inflationary expectations (see, e.g. Woodward, 1992), overlooking taxation (see Darby, 1975), using short rather than long rates (see Gilbert and Yeoward, 1994), not distinguishing between short-and long-run Fisher effects (see Mishkin, 1992), and finally differences in estimation procedures; in particular, Crowder and Hoffman (1996) and Caporale and Pittis (2004) both showed that using the estimators with the best small sample properties and appropriately computed empirical critical values (rather than the asymptotical ones) produces evidence more supportive of the Fisher effect. All such studies, though, analyse the Fisher relationship on the basis of the classic dichotomy between I(0) and I(1) variables, which imposes rather restrictive assumptions on the stochastic behaviour of the variables of interest.
By contrast, in the present paper we examine the Fisher effect in the G-7 by adopting a fractional integration/cointegration framework that does not restrict the order of integration d to be 0 or 1 and allows it instead to take any real value; this is clearly a much more general specification allowing for a richer dynamic structure. In particular, consider the following model: where R t is the nominal interest rate and π t is the inflation rate (under the implicit assumption of rational expectations, i.e., e t 1 t π = π + ).
If both R t and π t are I(0) variables, standard regression methods can be applied. On the other hand, if they are I(1) a cointegration approach becomes necessary. Unlike in the classical case, we allow these variables to be I(d) with 0 < d < 1 or d > 1. First we estimate d R and d π , i.e., their respective orders of integration, then we make the series stationary I(0) by differencing them to obtain , where B stands for the backshift operator, i.e. Bz t = z t-1 , and R d and π d are the estimated orders of integration of the two variables. In other words, we test for the Fisher effect in the following model: where the null hypothesis in (2) is: and both variables are I(0) as a result of taking first differences. We also examine the cases where R d and π d are restricted to be either 1 or 0 as in the classical cointegration framework, but also test the order of integration of the errors in (2) to allow for a greater degree of generality as explained in Section 3.
The layout of the paper is the following. Section 2 briefly reviews the literature on the Fisher effect. Section 3 outlines the empirical methodology. Section 4 presents the main empirical results. Section 5 offers some concluding remarks.

The Fisher Effect: A Brief Literature Review
Early studies analysed the Fisher effect without considering stationarity issues. These include Fama (1975), Nelson and Schwert (1977) and Garbade and Wachtel (1978). Gilbert and Yeoward (1994) argued that papers using short rates are not informative about the Fisher effect unless short and long rates are strongly correlated; examples are the papers by Summers (1983) and Barsky (1987).
The following generation of studies took a cointegration approach (see, e.g., Mishkin, 1992;Evans and Lewis, 1995;Wallace and Warner, 1993). Engsted (1995) looked at the spread between the long-term (multi-period) interest rate and the one-period inflation estimating a VAR model and found considerable cross-country differences. Crowder and Wohar (1997) analysed both taxable US Treasury and tax exempt municipal bond interest rates and found a "Darby effect", i.e. evidence that rational agents require nominal rates to adjust in response to movements in "tax-adjusted" expected inflation. Crowder and Hoffman (1996) showed that the choice of estimator is crucial, an issue further investigated by Caporale and Pittis (2004), who provided extensive evidence that if the estimators with the best small sample properties are used and statistical tests are carried out with appropriate empirical critical values the data are supportive of the Fisher effect.
More recently, fractional integration and fractional cointegration techniques have been used to analyse the long memory properties of inflation and interest rates. For example, Shea (1991) investigated the consequences of long memory in interest rates for tests of the expectations hypothesis of the term structure. Phillips (1998), found stationarity but with a high degree of dependence in US interest rates. Tsay (2000) modelled interest rates as AutoRegressive Fractionally Integrated Moving Average (ARFIMA) processes and concluded that the ex-post real interest rate can be well described as a fractionally integrated or I(d) process. Further evidence of long-memory behaviour in interest rates is provided by Barkoulas and Baum (1997)

The Empirical Methodology
As mentioned before in the present study we apply fractional integration methods allowing both nominal interest rates and inflation to be I(d), where d can be a fractional value. The starting point is the estimation of the differencing parameter d in: where B stands for the backshift operator, u t is an I(0) process, and in order to allow for deterministic terms x t are assumed to be the errors in a regression model of the form: in (3) and (4)  To analyse the relationship between the two variables we run a regression of the following form: estimating simultaneously α, β and d, for different cases. In particular, we examine three cases: i) imposing a priori d R = d π = 1 in (5); ii) imposing d R = d π =0 in (5); and iii) using for d R and d π the estimated values obtained using the univariate methods, first without any restrictions and then setting d in (6) equal to 0. We assume the error terms to be in turn uncorrelated (white noise) and autocorrelated, in the latter case using the exponential spectral model of Bloomfield (1973), which performs very well in the context of fractional integration.

Empirical Results
We start with the univariate analysis. Tables 1 and 2 display the estimates of d (along with the 95% bands corresponding to their non-rejection values using Robinson's (1994) method), for inflation and interest rates respectively. Following common practice in the unit root literature we consider the three cases of i) no deterministic terms, ii) an intercept and iii) an intercept with a linear time trend. The estimated model is: with uncorrelated and Bloomfield (1973) errors in turn. In the case of inflation rates, a time trend is required only for the US with autocorrelated errors, and no deterministic terms are required for Japan. 2 The estimated values of d under the assumption of white noise errors are much higher than 1, and evidence of a unit root (i.e., d = 1) is only obtained in the cases of Canada and Germany. With autocorrelated errors, evidence of a unit root is found in all cases except for Italy where the estimated value of d is much higher than 1.

[Insert Tables 1 and 2 about here]
Moving on to interest rates, a time trend seems to be appropriate for Japan with uncorrelated errors, and in all cases except Italy with autocorrelated ones. With white noise errors, the estimated values of d are much higher than 1 in all cases except for Japan and the US where the unit root null cannot be rejected. Under the assumption of autocorrelation, we obtain evidence of unit roots in all cases except for Japan, where the series seems to be stationary (i.e. with d < 0.5).
[Insert Table 3 about here] Table 3 summarizes the estimates of d with autocorrelated errors, a more appropriate assumption given the results of the diagnostic tests carried out on the residuals (not reported).
The unit root null is almost never rejected. The only exceptions are the inflation rate in Italy, with an order of integration much higher than 1, and interest rates in Japan, with a value significantly smaller than 1. However, in the case of Italy the confidence bands for the two variables overlap suggesting that the estimated values are not significantly different.
Tables 4, 5 and 6 show the estimated coefficients from equations (5) and (6) under three different assumptions, specifically, in Table 4 d R and d π are both set equal to 1; in Table   2 This is based on the t-values of the corresponding estimated coefficients, noting that equation (7) can be written and u t is I(0) by construction. 5, despite the evidence from Table 3, both variables are assumed to be I(0), and therefore d R = d π = 0, while in Table 6 the estimated values from Table 3 are used for d R and d π .
In Table 4, we impose d R = d π = 1 -this is a common assumption in the empirical literature that usually treats interest rates and inflation as being non-stationary I(1). Therefore, both variables in the regression model, t R and t π are expected to be I(0). In the case of Japan, given the stationarity of interest rates implied by Table 3, we also display the results with d R = 0 and d π = 1. It can be seen that with uncorrelated errors (Table 4i)

[Insert Tables 4 -6 about here]
In Table 5 we assume that d R = d π = 0, i.e. we run the regression model using the original data without taking differences. In the autocorrelation case, we cannot reject the null that the estimated errors are I(1) with the only exception of Japan, where d is found to be smaller than 1. 3 Once more, all the estimated values of β are significantly positive except in the case of Japan, though they are rather small. Finally, in Table 6 we take differences using the estimated values of d reported in Table 3. With uncorrelated errors (Table 6i) [Insert Table 7 about here] Finally, in Table 7, we compare the estimates of β with those obtained when imposing d = 0 in (6). They are generally higher than in the previous cases, especially when d R = d π = 0.
In fact, in the I(0) case, even the hypothesis β = 1 cannot be rejected for some countries (France, Germany and Italy). However, it should be noted that this specification is incorrect since the null hypothesis of d = 0 is rejected in favour of d > 0 in all the cases shown in Table   5.
Overall, the evidence based on our preferred model (Table 6ii) suggests that there exists a positive relationship between nominal interest rates and inflation, since the β coefficient is positive; however, it is statistically different from 1 (more precisely, it is smaller). Therefore, we do not find evidence of the full adjustment of nominal rates to inflation required by the Fisher hypothesis.

Conclusions
This paper revisits the Fisher hypothesis in the G7 countries using fractional integration and cointegration models that are more general than the standard ones based on the classical I(0) /I(1) dichotomy. Two sets of results are produced under the alternative assumptions of uncorrelated and autocorrelated errors. The univariate analysis suggests than the differencing parameter is higher than 1 for most series in the former case, whilst the unit root null cannot be rejected for the majority of them in the latter case. The multivariate results imply that there exists a positive relationship, linking nominal interest rates to inflation; however, the Fisher hypothesis is rejected since there is no full adjustment of the former to the latter. The  In bold the most adequate specification according to the t-values for the deterministic terms. In parenthesis the 95% confidence bands of the non-rejection values using Robinson's (1994) tests.  In bold, evidence of unit roots at the 5% level.  In bold, statistically significant positive coefficients at the 5% level. In bold, statistically significant positive coefficients at the 5% level. In bold, statistically significant positive coefficients at the 5% level. In bold, significant coefficients at the 5% level.