Nominal Interest Rates and Inflation: The U.S. Story

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.

The Relationship Between Nominal Interest Rates and Inflation: The Long Run

In the last post I discussed what I understand to be the conventional macroeconomic story about how interest rates and inflation rates behave relative to one another over short-to-intermediate time frames. While the story told doesn't capture all of the potential linkages (for example I have basically waived my hands and said the magic words with respect to role played by expectations, kinda like step 2 below),

it is, I believe, a fair representation of how most macro-economists view this relationship and it is consistent with both traditional Keynesian/Monetarist and New Keynesian theory.

Today I want to think about the long-run relationship between nominal interest rates and inflation. This topic is, I believe, less controversial since all internally consistent dynamic equilibrium models with money imply the same basic relationship between nominal interest rates and inflation. Equation (1) displays that long-run equilibrium commonly referred to as the Fisher equation. \begin{equation*} i_t = r_t + \pi^e\tag{1}\end{equation*} There are variants of (1) where we might include a risk premium, tax effects or allow the equilibrium real rate of interest to be time-varying but stationary, but all are consistent with the basic idea that in the long run, the equilibrium real interest rate should be independent of the inflation rate. This idea, independence of the real rate from inflation, does have some counter-arguments, like Mundell-Tobin effects and old-fashioned money illusion, but these effects should be short-lived so that in the long run the Fisher equation holds.

What determines trend inflation in the long run? Well there is not a whole lot of concensus on that point. Some believe that trend inflation is determined via the money supply growth rate, as in the standard quantity theory story \(MV = PY\)(Crowder, 1998). Others suggest it is a direct policy choice of the inflation target in a Taylor-type policy rule (Bernanke and Mishkin, 1997). And still others believe trend inflation is really determined by fiscal policy through the rate of deficit accumulation and the resulting seignorage requirements (Woodford, 1995). All of these "causes" of trend inflation imply that trend inflation leads or causes the trend in nominal interest rates. Causality in this sense is a little tricky since our conclusions about causality can depend on where we start the process.

Since we are interested in long-run trends and equilibria, let's start in a long-run equilibrium where output is equal to potential, unemployment is at the NAIRU and the equilibrium real rate (\(r_t\)) is constant. Note that in such a steady state the nominal interest rate is completely determined by equation (1), i.e. a constant real rate plus the rate of trend inflation. Changes that "originate" with the nominal rate will always end up back at the steady state. For example, assume the economy is in the steady state described. Now the central bank raises the nominal interest rate (whether this is through traditional open market operations or by raising the interest paid on reserves is not important) resulting in an (temporary) increase in the real rate of interest. This should lead to a fall in current spending (see previous post) and via some aggregate demand or Phillips curve relationship to lower inflation. Note that in the short-run, inflation moves in the opposite direction as the nominal interest rate.

But eventually the economy adjusts, either through intertemporal substitution effects, real wealth effects or just simple learning and modification of expectations, but eventually the central bank will reduce the nominal interest rate and inflation will return to target. The easiest way to see this is to use the simple 3-equation New Keynesian model that includes a Taylor rule \begin{equation*} i_t = r + \pi^* + \xi_y (Y_t - Y^*) + \xi_{\pi} (\pi_t - \pi^*)\tag{2}\end{equation*} an IS relationship \begin{equation*} Y_t - Y^* = -\sigma (r_t - r) + E_t(Y_{t+1} - Y^*) \tag{3}\end{equation*} and a New Keynesian Phillips curve \begin{equation*} \pi_t = \gamma E_t \pi_{t+1} + \kappa (Y_t - Y^*) \tag{4}\end{equation*} where \(i_t\) is the policy rate and \(Y^*\) and \(\pi^*\) are potential output and the inflation target.

Starting from an initial equilibrium where \(Y_t = Y^*\), \(r_t = r\) and \(\pi_t = \pi^*\), raising \(i_t\) leads to \(Y_t\) declining below potential (\(i_t \uparrow \rightarrow r_t \uparrow \rightarrow Y_t \downarrow\)) which then causes inflation to decline via (4). This situation is not consistent with long-run equilibrium. So this increase in the nominal rate is temporary or transitory. Eventually the nominal rate will fall and \(Y_t\) and \(\pi_t\) will rise to their equilibrium or targeted levels. So changes in the nominal interest rate that originate from the nominal interest rate are transitory, the effects will dissipate over time such that in the long run the nominal interest rate and inflation rate will return to their pre-shock levels.

So how could we get a permanent increase in the nominal rate? We would need a higher target inflation rate. If the central bank raises its inflation target, it will need to lower the policy rate initially, so that in (2) \(\pi^* = \pi_t\) initially but now \(\pi^* > \pi_t\). As before the lower nominal rate lowers the real rate temporarily which stimulates economic activity through the various effects already described. The Phillips relationship in (4) then tells us that as output rises above potential actual inflation begins to rise and will continue to do so until it equals the (now higher) inflation target. But as both output and inflation increase, the policy rate does also. The permanently higher inflation rate (and inflation target, remember trend inflation is a policy choice here) causes a permanently higher nominal interest rate. Trend inflation causes the nominal interest rate in the long run.

So where are we? In the short run nominal interest rates cause inflation. Raising (lowering) nominal interest rates leads to lower (higher) inflation temporarily. As the economy adjusts over time inflation will eventually return to trend and when inflation expectations are well anchored that will be the central banks target rate of inflation, i.e. the expected inflation in (1) will be the same as the target inflation in (2). But in order to change the nominal policy rate permanently there must first be a change in the trend (target) inflation rate, so that in the long run inflation causes the nominal interest rate.

The Relationship Between Nominal Interest Rates and Inflation: The Short Run

The conventional view of monetary policy highlights the short- and intermediate-run relationship between nominal interest rates and inflation. Namely that they are inversely related. This negative or inverse relationship is related to the "liquidity effect". The liquidity effect refers to the mechanism through which decreases (increases) in the policy rate are achieved by monetary injections (contractions) via open market purchases (sales), i.e. adding (subtracting) liquidity moves the interest rate in the opposite direction. The resulting decline (increase) in real interest rates, a consequence of inflation inertia, lead to greater (lesser) economic activity by altering consumption, through the intertemporal substitution effect, and investment, through a cost of capital effect. Eventually the increase (decline) in economic activity results in higher (lower) inflation via the Phillips curve relationship. So the conventional view of monetary policy is that decreases in the policy rate lead to greater economic activity and higher inflation while increases should eventually lead to lower activity and lower inflation.

This relationship is highlighted in Taylor-type monetary policy rules such as equation (1), \begin{equation*} i_t = r + \pi^* + \xi_y (Y_t - Y^*) + \xi_{\pi} (\pi_t - \pi^*)\tag{1}\end{equation*} where \(i_t\) is the policy rate that responds positively when output, \(Y_t\), or inflation, \(\pi_t\), are above their targeted values, \(Y^*\) and \(\pi^*\), respectively, and negatively when either are below target. When combined with a Phillips curve like equation (2), \begin{equation*} \pi_t = \zeta E_t \pi_{t+1} -\eta (U_t - U^*) \tag{2}\end{equation*} where \(U_t\) is the unemployment rate and \(U^*\) is the natural or equilibrium rate of unemployment, or a New Keynesian Phillips curve relationship like (3), \begin{equation*} \pi_t = \gamma E_t \pi_{t+1} + \kappa (Y_t - Y^*) \tag{3}\end{equation*} and some type of IS relationship like \begin{equation*} Y_t - Y^* = -\sigma (r_t - r^*) \tag{4}\end{equation*} we get the conventional monetary policy result. When output is below target (or equivalently unemployment is above the natural rate) we can trace the conventional monetary policy causal chain as something like \(i_t \downarrow \rightarrow r_t \downarrow \rightarrow Y_t \uparrow \text{and/or}\; U_t \downarrow \rightarrow \pi_t \uparrow\). The implication is that in the short-run the nominal interest rate causes or leads inflation.

The Neo-Fisher Hypothesis

John Cochrane and Stephen Williamson have been blogging about a pretty radical idea for monetary policy at the zero lower bound (ZLB). They suggest that in order to increase inflation the Federal Reserve should raise its target interest rate and the rate it pays on reserves. The mechanism by which this would lead to higher inflation is not particularly well articulated (understood?) but is based on the long-run super-neutrality condition known as the Fisher relation. In the long run nominal interest rates and inflation must move proportionally to maintain the constant long-run equilibrium real rate of interest. Over the next few posts I intend to examine this idea empirically using some simple techniques.