The Fisher effect is the name given to the theory which implies that a permanent change in inflation will, in the long run, cause an equal change in the nominal interest rate. Or, equivalently, monetary shocks will have no effect on the real interest rate in the long run. This can be taken to imply a cointegrating relationship between inflation, *π*_{t}, and the interest rate, *i*_{t}, with cointegrating vector (1, –1)′. This relationship has been investigated in numerous papers for numerous countries and is often found not to hold. Beyer, Haug, and Dewalt (2011) offer a discussion of this literature and investigate whether structural breaks exist in the cointegrating relationship in a cross-country study.

In our empirical work, we look at the case of France. For this country, Beyer, Haug, and Dewalt (2011) analyze quarterly data from 1970:Q1 to 2004:Q3 and find evidence of unit roots in *π*_{t} and *i*_{t} using classical unit root tests. However, both the Johansen trace and eigenvalue tests for cointegration indicate cointegration is not present and, thus, the Fisher effect appears not to hold. They next do a classical test where the null hypothesis is that cointegration is present, but with a structural break at an unknown point in time. This test does not reject the null hypothesis and finds a break in 1981:Q4. However, when Johansen tests are done using sub-samples (before and after 1981:Q4), the trace test finds cointegration in the second sub-sample but not in the first, whereas the eigenvalue test finds cointegration in both sub-samples. We take this as an interesting case where the evidence of previous work suggests there is a great deal of model uncertainty, both about the presence of cointegration and about the break process.

Our data on French CPI inflation (quarterly inflation at an annualized rate) and the 3 month treasury bill rate runs from 1970:Q1 to 2012:Q4 and is shown in Figure 1.

Figure 1 CPI inflation (dashed line) and 3-month interest rate (solid line).

Concerning the inclusion of deterministic trends, we only put a constant in the cointegration part of the model since neither the inflation series nor the interest rate series display a trending pattern (see Franses 2001, for justification of that choice). We consider *p*=1, 2, 3 for the lag length. In each regime the cointegration relationship between the two variables follows one of the three subsequent cases. The first case (which we denote by *b*=0) assumes that inflation and the nominal interest rate are not cointegrated and the model becomes:

$$\begin{array}{c}\Delta {\pi}_{t}={\displaystyle \sum _{j=1}^{p-1}}\mathrm{(}{\gamma}_{j\mathrm{,}\text{\hspace{0.17em}}{s}_{t}}^{\pi \pi}\Delta {\pi}_{t-j}+{\gamma}_{j\mathrm{,}\text{\hspace{0.17em}}{s}_{t}}^{\pi i}\Delta {i}_{t-j}\mathrm{)}+{\epsilon}_{\pi t}\mathrm{,}\\ \Delta {i}_{t}={\displaystyle \sum _{j=1}^{p-1}}\mathrm{(}{\gamma}_{j\mathrm{,}\text{\hspace{0.17em}}{s}_{t}}^{i\pi}\Delta {\pi}_{t-j}+{\gamma}_{j\mathrm{,}\text{\hspace{0.17em}}{s}_{t}}^{ii}\Delta {i}_{t-j}\mathrm{)}+{\epsilon}_{it}\mathrm{.}\end{array}\text{\hspace{1em}(5)}$$(5)

The second case (*b*=1) assumes a cointegration rank of one but does not constrain the cointegration space:

$$\begin{array}{c}\Delta {\pi}_{t}=\text{\hspace{0.05em}}{\alpha}_{{s}_{t}}^{\pi}\mathrm{(}{\beta}_{\mathrm{1,}{s}_{t}}^{\pi}+{\beta}_{\mathrm{2,}{s}_{t}}^{\pi}{\pi}_{t-1}+{\beta}_{\mathrm{3,}{s}_{t}}^{\pi}{i}_{t-1}\mathrm{)}+{\displaystyle \sum _{j=1}^{p-1}}\mathrm{(}{\gamma}_{j\mathrm{,}{s}_{t}}^{\pi \pi}\Delta {\pi}_{t-j}+{\gamma}_{j\mathrm{,}{s}_{t}}^{\pi i}\Delta {i}_{t-j}\mathrm{)}+{\epsilon}_{\pi t}\mathrm{,}\\ \Delta {i}_{t}\mathrm{=}{\alpha}_{{s}_{t}}^{i}\mathrm{(}{\beta}_{\mathrm{1,}{s}_{t}}^{i}+{\beta}_{\mathrm{2,}{s}_{t}}^{i}{\pi}_{t-1}+{\beta}_{\mathrm{3,}{s}_{t}}^{i}{i}_{t-1}\mathrm{)}+{\displaystyle \sum _{j=1}^{p-1}}\mathrm{(}{\gamma}_{j\mathrm{,}{s}_{t}}^{i\pi}\Delta {\pi}_{t-j}+{\gamma}_{j\mathrm{,}{s}_{t}}^{ii}\Delta {i}_{t-j}\mathrm{)}+{\epsilon}_{it}\mathrm{.}\end{array}\text{\hspace{1em}(6)}$$(6)

Finally, the last case (*b*=1*F*) assumes a cointegration rank of one and restricts the cointegration space according to Fisher’s hypothesis. This means that the model can be expressed as follows:^{2}

$$\begin{array}{c}\Delta {\pi}_{t}\mathrm{=}{\delta}_{\mathrm{1,}\text{\hspace{0.17em}}{s}_{t}}^{\pi}+{\delta}_{\mathrm{2,}{s}_{t}}^{\pi}\mathrm{(}{\pi}_{t-1}-{i}_{t-1}\mathrm{)}+{\displaystyle \sum _{j=1}^{p-1}}\mathrm{(}{\gamma}_{j\mathrm{,}{s}_{t}}^{\pi \pi}\Delta {\pi}_{t-j}+{\gamma}_{j\mathrm{,}{s}_{t}}^{\pi i}\Delta {i}_{t-j}\mathrm{)}+{\epsilon}_{\pi t}\mathrm{,}\\ \Delta {i}_{t}={\delta}_{\mathrm{1,}{s}_{t}}^{i}+{\delta}_{\mathrm{2,}{s}_{t}}^{i}\mathrm{(}{\pi}_{t-1}-{i}_{t-1}\mathrm{)}+{\displaystyle \sum _{j=1}^{p-1}}\mathrm{(}{\gamma}_{j\mathrm{,}{s}_{t}}^{i\pi}\Delta {\pi}_{t-j}+{\gamma}_{j\mathrm{,}{s}_{t}}^{ii}\Delta {i}_{t-j}\mathrm{)}+{\epsilon}_{it}\mathrm{.}\text{\hspace{0.05em}}\end{array}\text{\hspace{1em}(7)}$$(7)

We now turn to our empirical results that strongly favor Markov switching VECMs over structural break or constant coefficients VECMs. In fact, in a BMA exercise Markov switching models would receive virtually all of the weight. For the Markov switching case, there is never any evidence for more than two regimes. Accordingly, our empirical results focus on the Markov switching models with *M*=2. However, to illustrate the properties of our approach, we also present results for the models with structural breaks (even though there is little support for these models). For these, we do find evidence for three regimes and, accordingly, present results for structural break models with *M*=2 and *M*=3. For brevity’s sake, we do not present any results for constant coefficient VECMs (*M*=1). Trying various combinations of *b* and *p* we did not find a constant coefficient model that received considerable support.

## 4.1 Markov switching models

First, we look at results for the Markov switching case with two regimes (*M*=2). We impose an identification restriction which specifies that the variance of the interest rate equation in the first regime is bigger than the variance in the second regime.^{3} gives logarithms of marginal likelihoods for models with different cointegration relationships in the two regimes and different lag lengths. In the last line we report results for the special case where the Fisher effect holds in both regimes but only the constant in the cointegrating vector is allowed to switch.

Table 1 Markov switching case: logarithms of marginal likelihoods.

We find that the model with the highest marginal likelihood has a lag length of two and specifies that both regimes are cointegrated but the Fisher effect restriction only holds in the first regime. For this “best model” the top panel in Figure 2 plots the posterior probability that the regime where the Fisher effect holds occurs. It can be seen that this probability is very high during the 1970s and in the beginning of the 1980s. After that the probability is very low for much of the time. If this were the full story, then we would expect a structural break model to work well, with a break occurring around 1983. The timing of the break is similar to that reported in Beyer, Haug, and Dewalt (2011). However, there are three, relatively short, time periods where the Fisher effect seems to hold again (in the mid-1990s and at the end of the sample). This kind of behavior is more consistent with a Markov switching process than a structural break model and this is why Markov switching models perform so well in our analysis.

Figure 2 Markov-switching case: results for the “best model.”

(A) Posterior probability that the Fisher effect holds. (B) Posterior median and 16% and 84% posterior quantiles of $${\tilde{\beta}}_{3}.$$

The same conclusion can be drawn from the bottom panel in Figure 2. Here, the cointegration space is normalized to be a vector $$\mathrm{(}{\tilde{\beta}}_{1}\mathrm{,}\text{\hspace{0.17em}}\mathrm{1,}\text{\hspace{0.17em}}{\tilde{\beta}}_{3}\mathrm{}\mathrm{)}\mathrm{.}$$ In this normalization, $${\tilde{\beta}}_{1}$$ is the normalized intercept and the Fisher hypothesis tells us that $${\tilde{\beta}}_{3}$$ should be –1. The posterior median of $${\tilde{\beta}}_{3}$$ and its 16% and 84% posterior quantiles are drawn. As expected, the posterior median of $${\tilde{\beta}}_{3}$$ is equal to –1 at the same times that the top panel says there is a high probability that the Fisher effect holds.

So far, we have presented results for the single model with highest marginal likelihood. However, there are many other models whose marginal likelihoods are only slightly smaller than that of the “best model.” For example, the second best model is the one where the first regime is cointegrated and the Fisher effect holds but with no cointegration in the second regime. Faced with such model uncertainty, the researcher may wish to do BMA. Figure 3 gives the results of a BMA exercise. It plots the posterior probabilities of the three cointegration cases at each point in time, averaged across all the models in . The story told by Figure 3 is similar to that in Figure 2. Up until 1983, in the mid-1990s and at the end of the sample the Fisher effect is supported. But elsewhere it is not. Furthermore, in the periods where the Fisher effect is not supported, there is great uncertainty over whether cointegration occurs or not.

Figure 3 Markov-switching case: posterior probabilities of the three cointegration cases averaged over all models.

## 4.2 Structural break models

Now we discuss results for the structural break case with two and three regimes (*M*=2, 3). gives the logarithms of marginal likelihoods for the different models. The special cases where the Fisher effect holds in all regimes but only the constant in the cointegrating vector is allowed to switch are marked by asterisks again.

Table 2 Structural break case: logarithms of marginal likelihoods.

As discussed previously, the logarithms of marginal likelihoods are much lower than for the Markov switching models and we include these structural break models for illustrative purposes only.

The “best model” with two regimes has a lag length of two. Both regimes are cointegrated with the Fisher effect holding in the first regime. The “best model” with three regimes also has a lag length of two. Again, all regimes are cointegrated and the Fisher effect only holds in the first regime. However, in the close second best model the first regime is cointegrated and the Fisher effects holds, the second regime is not cointegrated and the third regime is cointegrated again but the Fisher effect does not hold. Note that conventional models of cointegration with structural breaks could not handle such a case where the cointegrating space switches between cointegrating ranks as well as switching between restricted and unrestricted cointegrating spaces. This illustration shows that such cases are empirically relevant, highlighting the importance of a modelling approach which allows for such possibilities.

Figure 4 plots the posterior probabilities for each regime to occur for these two “best models.” It can be seen that the structural break models are trying (poorly) to approximate the Markov switching properties of Figure 2.

Figure 4 Structural break case: posterior probabilities of regimes.

(A) Two regimes, solid line: Pr(s=1), dashed line: Pr(s=2). (B) Three regimes, solid line: Pr(s=1), dashed line: Pr(s=2), dotted line: Pr(s=3).

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