In my last two posts I have been discussing what I understand to be the relationship between nominal interest rates and inflation. It is complicated by the fact that the relationship depends on the behavior of other variables, like real output, and expectations of future values, and the dynamics underlying all of these relationships are not well understood. In this post I want to provide some empirical evidence that I think is consistent with the discussion so far.
The starting point is the Fisher equation, \begin{equation*} i_t = r_t + \pi_{t+1}\tag{1}\end{equation*} where I have replaced expected inflation with actual ex-post inflation. The graph below displays the inflation rate (measured as the year-over-year percent change in the PCE deflator less food and energy) and the 3-month Treasury bill interest rate for the U.S. over the period of January 1960 to December 2008 (for now I don't want the post-crisis data to mess up the discussion, but I'll bring it back in to the analysis eventually). The graph looks to be broadly consistent with the idea that trend inflation and nominal interest rates are positively related over long spans.
Equation (1) can be interpreted in several different ways. For example, it can be a way to simply define the real interest rate as the nominal rate less the inflation rate. Or we could assume that the real interest rate is constant or at least independent of inflation and that gives the long-run super-neutrality condition that nominal rates move in proportion to inflation, what is often called (by me at least) the Fisher effect. It is this second interpretation that is of interest to me. Why should we believe that the real interest rate is constant? We shouldn't. It's not. But in a world in which things are changing (a stochastic environment) the relevant concept isn't really a constant real rate, but rather a stationary real interest rate. A stationary variable is simply one that has a constant mean or average value such that any deviations from that equilibrium value revert back to the mean over time. When thinking about stationarity think mean reversion. Using the data in the above figure, the ex-post real interest rate is plotted in the graph below.
Clearly the real interest rate has not been constant over this period in U.S. history. But it does appear to have a constant mean (at a value of approximately 1.6%). And when we subject the series to statistical analysis (unit root tests) the results confirm that the series is stationary. Not constant, but a constant mean to which the series eventually reverts over time. There are also theoretical reasons why the real interest rate should be stationary. For example, the real interest rate is a measure of the marginal product of capital (mpk) and balanced growth restrictions imply the mpk should be stationary. So when the real interest rate is stationary nominal interest rates move one-for-one with trend inflation in the long run.
Testing if inflation causes the nominal interest rate in the long run can be done using the real interest rate graphed above. The idea is that the real interest rate represents the equilibrium error in the long-run relationship between the nominal rate and inflation. That is, when the real rate is above its average (equilibrium) value either the inflation rate must rise or the nominal interest rate must fall, or some combination of the two, to restore the equilibrium. If there really is a long-run equilibrium between nominal interest rates and inflation then at least one or the other must respond to past values of the real interest rate. We can test this simple idea using a simple OLS regression model as in equation (2),
\begin{equation*} \Delta \pi_t = \beta_0 + \beta_1 r_{t-1} + \sum_{j = 1}^k \gamma_j \Delta \pi_{t-j} + \sum_{j = 1}^k \delta_j \Delta i_{t-j} + \varepsilon_{\pi,t}\tag{2}\end{equation*}where the hypothesis of interest is \(\beta_1 = 0\). This would mean that the inflation rate does not change in response to a deviation of the real interest rate from its long-run equilibrium value last period. If inflation does not adjust in order to correct errors (deviations) in the equilibrium real interest rate the nominal rate must.
Why is this relevant to long-run causality? Let's assume \(r_t = 0\) in the long-run equilibrium, so that \(i=\pi\) in the long run. If \(r_t > 0\) either \(i_t\) is too high (relative to the equilibrium value) or \(\pi_t\) is too low. But if \(\beta_1 = 0\) the inflation rate does not change to restore the equilibrium. So it must be that the nominal interest rate falls to restore the equilibrium. But why was \(r_t > 0\) in the first place? If \(\pi_t\) is too low, i.e. inflation independently fell leading to \(r_t > 0\), and if \(\beta_1 = 0\) in (2), then the nominal rate must fall to the new lower inflation rate. On the other hand if it was a rise in \(i_t\) that created the \(r_t > 0\) disequilibrium, assuming \(\beta_1 = 0\) in (2), \(i_t\) must again fall to restore equilibrium, this time back to its original value. So we see that when \(\beta_1 = 0\) in (2) changes in the long-run trend inflation are permanent requiring the nominal rate to adjust to restore equilibrium while changes in the nominal interest rate are temporary, requiring the nominal rate to move back to it's original value. It is the trend inflation rate which pulls the nominal rate along so that in the long run inflation causes the nominal interest rate.
Using the data graphed above to estimate equation (2) I find that the hypothesis that \(\beta_1 = 0\) cannot be rejected (see this paper for details). So at least up until 2008, it is the case that inflation causes the nominal interest rate in the long run. The policy implication is that raising or lowering the nominal interest rate will not impact trend inflation.
But the situation is very different after 2008. That will be the subject of my next entry.
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