Empirical Findings from Yield Curve Data

Rudebusch (2002b) focuses on the R² from a prediction equation where quarterly interest rate changes, Δi_t+k = i_t+k − i_t+k−1, are regressed on lagged forward rates:

In this equation, i_t is the average federal funds rate in quarter t, and f_k,t denotes the interest rate at the end of quarter t on a Eurodollar futures contract that settles k quarters ahead.²

The reason for studying the performance of this particular prediction equation is that forward rates are believed to carry a lot of information about market expectations. For instance, if risk premia are constant (the expectations hypothesis of interest rates), then this equation should give the best possible forecast (in a mean square sense); that is, it should coincide with the mathematical expectation. In this case, f_k,t = E_ti_t+k plus a constant so that the regressor in (1) equals E_tΔi_t+k.

Equation (1) is estimated on quarterly U.S. data from 1987:Q4 to 1999:Q4, using a three-month Eurodollar futures as predictor. These futures contracts are based on the 90-day London Interbank Offered Rate, and have been traded since the mid 1980s. Table 1 reiterates the findings of Rudebusch (2002b): the R² of (1) is very low beyond the first quarter (0.56 for k = 1, 0.13 for k = 2, and 0.04 for k = 3).

These results suggest that interest rate changes are not predictable beyond the first quarter. A possible counterargument is that futures rates are bad signals of expectations because of changes in risk premia. We later study this possibility in some detail (using, among other things, survey data), and conclude that the futures data probably have the same forecasting power as market expectations.³

Role of the Dynamic Taylor Rule in a Calibrated Macro Model

The typical dynamic Taylor rule with interest rate smoothing is (constant omitted)

where i_t is a three-month T-bill rate (quarterly average),

annual (four-quarter) inflation in the GDP deflator, and y_t is the output gap (the percentage deviation of real GDP from potential GDP).⁴

The analysis by Rudebusch (2002b) focuses on the effects of a high “interest rate smoothing” parameter (ρ), which is the typical result from most estimations of the dynamic Taylor rule. It is clear that ρ has a direct (autoregressive) effect on the predictability of the interest rate, but it is also likely to have indirect effects by affecting the predictability of inflation and the output gap. To study this, we use the dynamic Taylor rule in a New Keynesian–inspired empirical model estimated by Rudebusch (2002a). The model is specified for quarterly data and includes a Phillips-type supply equation and an IS-like demand equation:

where

is annual inflation, and ε_t and η_t are cost push and demand shocks, respectively.

To parameterize the model, we choose the values for the α and β parameters and the standard deviations of the shocks ε_t and η_t estimated by Rudebusch (2002a), shown in Table 2.⁵

For different values of the interest rate smoothing parameter ρ in equation (2), we solve and simulate the model (2)–(4) 2,000 times using 80 observations. In each simulation, we estimate the prediction regression

for the forecasting horizons k = 1, 2, 3. The regressor E_tΔi_t+k is the model-consistent rational expectation of the interest rate change (calculated from the state vector of the model).

Figure 1 shows that R² of the regression (5) increases with ρ. For instance, for the two-quarter horizon (k = 2), R² = 0.21 for ρ = 0 but 0.55 for ρ = 0.8 (a common value in the empirical literature). The model results give so much higher R² than the yield curve evidence in Table 1 that something must be wrong (with either the model or the yield curve evidence).

R2 of Δit+k in model simulations, 2000 simulations of 80 quarters.

Two Simple Analytical Cases

This section uses two stylized cases to argue that smoothing of the interest level does not necessarily imply predictability of interest changes: Smoothing gives predictability of the interest rate level, not the changes.

We first consider the simplest case when inflation and the output gap in the dynamic Taylor rule (6) are white-noise processes so all predictability in the interest rate change is due to the lagged interest rate—and therefore the ρ parameter. This case is easy to analyze since i_t becomes an AR(1) process: the R² for the regression (5) is asymptotically given by (1 − ρ)/2 for the one-period horizon (see the Appendix for a derivation). This shows that the interest rate change becomes less predictable as smoothing (ρ) increases. In the limit (at ρ = 1) the interest rate level is a random walk with unpredictable changes.

In our second simple case, we let inflation and the output gap be predictable. To focus on the direct effect of ρ, we assume that inflation and the output gap are uncorrelated with lagged interest rates and the current policy shock. This eliminates the possibility that changing ρ affects their predictability. For the one-period horizon, we then have the R² for the regression (5) (see the Appendix):

where R²_πy measures the coefficient of determination from predicting

with its rational expectation. Note that (7) simplifies to the previous expression (R² = (1 − ρ)/2) if inflation and the output gap are unpredictable. Otherwise, predictable inflation and output will help to make interest rate changes predictable—especially if the monetary policy shocks are small [which increases the ratio of variances in (7)]. However, the effect of increasing ρ is as before: A higher ρ reduces the predictability of interest rate changes.

In these examples, ρ has a negative effect on R². Although the examples are highly stylized, they suggest that the direct (pure autoregressive) effect of ρ is unlikely to explain the higher R² found in the model simulations. In the next section, we show that the basic mechanism is instead that a high ρ makes inflation and the output gap more predictable. In terms of (7), this would mean that a higher ρ also gives a higher R²_πy.

Model Results

The preceding section suggested that the direct autoregressive effect is probably not able to explain the model results. We therefore take a second look at the model to see what happens as ρ increases. In particular, we use model simulations to find the R² for prediction equations of the following type:

where x_t+k is either annual inflation

, the output gap y_t+k, or the target gap (

). The predictor, E_tx_t+k, is the model-consistent rational expectation of the dependent variable.

Figures 2A and B show the R² from (8) for the two-quarter horizon. Figure 2B shows that the predictability of inflation is high (above 0.8) for all values of ρ, but becomes very high (0.9 or higher) at typical empirical values of ρ. The same figure also shows that the predictability of the output gap increases quite rapidly as ρ increases: R² is 0.4 at ρ = 0.5 but 0.68 at ρ = 0.85.

R2 from using the optimal predictor (Etxt+k) in modelsimulations, two-quarter horizon, 2,000 simulations of 80 quarters.

As a consequence, the target interest rate (

) becomes highly predictable at high values of ρ, which carries over to the target gap—see Figure 2A. The increase in predictability is so rapid that it more than compensates for the fact that the target gap gets a lower weight (1 − ρ) in the dynamic Taylor rule (6) as ρ increases, and the unpredictable policy shock gets a relatively higher weight. This explains the high predictability of the interest rate changes in the same figure.

The reason for why a high ρ makes inflation and the output gap predictable is that monetary policy becomes fairly passive (in the short run) as the central bank smooths interest rate movements. For instance, a positive output shock is not met by a quick and large increase of the interest rate. Instead, the interest rate is increased slowly: the shock is allowed to affect the output gap. Since the output gap is partly autoregressive, this generates predictable movements. A similar story holds for the inflation shock/Phillips curve.

Compared with yield curve evidence, the model indeed implies too much predictability of interest rate changes—and this problem is larger for high values of ρ as pointed out by Rudebusch (2002b). This section has, however, shown that the effect of a high ρ is indirect, that is, it works through the predictability of inflation and the output gap. It is therefore hard to say where the problem with the model lies. It could be the dynamic Taylor rule, but it could equally well be that the Phillips curve or the aggregate demand equation are misspecified. The natural way of disentangling these effects is to bring in more direct evidence on the predictability of inflation and the output gap, which we do in Section 4. Before we do that, we take a careful look at the dynamic Taylor rule to point out some further implications for forecasting.

Forecasting Interest Rate Changes and the Target Gap

According to the dynamic Taylor rule (6), predicting interest rate changes is the same thing as predicting the target gap (

). This section outlines the forecasting implications, which are studied on survey data and a VAR model in Section 4.

First, if it is possible to predict the target gap, then the dynamic Taylor rule implies that it is possible to predict interest rate changes, too. To verify that this indeed holds in the macro model discussed earlier, we compute the R² of the prediction equation (8). The first three columns of Table 3 show the results. The target gap is highly predictable in the model and the interest rate changes are somewhat less predictable because they are hit by the unpredictable policy shocks, but the difference will not be large unless the policy shocks are really volatile (discussed in some detail later).

Second, the dynamic Taylor rule implies that the best predictor of the interest rate change, E_tΔi_t+k, should predict interest rate changes and the target gap about equally well. To study this, we compute the R² from the following prediction equation:

The last three columns in Table 3 show that, in the model, E_tΔi_t+k predicts both interest rate changes and the target gap well.

Third, the dynamic Taylor rule also implies that the best predictor of interest rate changes, E_tΔi_t+k, should be close to an optimal predictor of the target gap. To study this, we compare the R² of (9) and (8). The second line in Table 3 shows that E_tΔi_t+k is indeed a very good predictor of the target gap in the model—the difference to the optimal predictor is negligible.

In Section 4, we study whether these implications are verified by survey data and a VAR model. We hope that this will reveal whether the predictability (if any) in interest rate changes is actually generated by the mechanism in the dynamic Taylor rule. Note that these implications follow directly from the dynamic Taylor rule—and do not depend on the rest of the model.

Survey Data

This section shows that survey data suggest little predictability of interest rate changes, and that the implications of the dynamic Taylor rule are not supported.

The survey data are from the Survey of Professional Forecasters (SPF), which is a quarterly survey of some 30 forecasters' views on key economic variables. The respondents, who supply anonymous answers, are professional forecasters from the business and financial community. See Croushore (1993) for further details.

It is well established, in both theory and practice, that some kind of combination (mean or median) of several different methods/forecasters typically increases the forecast precision, and that such combined forecasts are hard to beat with any method [see, for instance, Grangera and Ramanathan (1984)]. We therefore use the median of the individual forecasts of the three-month T-bill rate, quarterly GDP deflator inflation, and real GDP (GNP before 1992). From these data, we calculate the implied forecasts of the variables that enter the dynamic Taylor rule: interest rates changes, annual (four-quarter) inflation, and the output gap (assuming that potential output over the next four quarters is known).

The results for the survey data are summarized in Table 4. The first three columns show results from regressions of the same type as (8), that is, with the survey forecast of the same variable (“E_tx_t+k”) as predictor. For the interest rate changes, the results are very similar to the yield curve evidence in Table 1: some predictability at the one-quarter horizon, but none for longer horizons. If anything, the survey data suggest even less predictability than the yield curve data. It seems relatively safe to conclude that market expectations of interest rate changes have had low predictive power—in contrast to the model results.

The survey data therefore suggest that something is wrong with the model. But what? With the information we have so far, we cannot really be sure that the dynamic Taylor rule is at fault—it could equally well be that the Phillips curve and the aggregate demand curve generate too much predictability. Moreover, the dynamic Taylor rule implies that forecasting Δi_t+k is pretty much the same thing as forecasting the target gap (

)—see Section 3.3. This should hold irrespective of what the rest of the model looks like and is therefore a way of assessing the dynamic Taylor rule directly. Table 4 provides evidence on these issues.

First, the survey is pretty good at forecasting the target gap (by using the implied survey forecast of the target gap itself), but bad at forecasting interest rate changes. As mentioned earlier, this could possibly be explained by unpredictable policy shocks (or other omitted, unpredictable, variables). However, the difference in R² is so large (for instance, 0.56 versus 0.05 on the two-quarter horizon) that this would require implausibly large policy shocks. For instance, the policy shocks (or omitted variables) would have to be 10 times more volatile (in terms of the standard deviation) than in the model calibration to push the R² of the interest rate changes down to something like 0.20. However, if so much of the policy rule is driven by shocks (or omitted variables), then the rule becomes fairly uninteresting as a description of actual policy. Notice that this finding is not sensitive to the relative weights on inflation and the output gap in the dynamic Taylor rule (we use the traditional values,

) since both inflation and the output gap are quite predictable over the next few quarters.

Second, it could actually be argued that “ E_tΔi_t+k” (the forecast of the interest rate change) does predict interest rate changes and the target gap equally well. That is, equally poorly: The R² is very low (with the possible exception of interest rate changes at the one-quarter horizon).

Third, the interest rate forecasts are far from being an optimal predictor of the target gap. For instance, at the two-quarter horizon, using “E_tx_t+k” as the predictor gives an R² of 0.56, but using “ E_tΔi_t+k” as the predictor gives an R² of 0.01.

The results in Table 4 are very hard to reconcile with a dynamic Taylor rule. It seems that the predictability of actual interest rate changes is more or less unrelated to the predictability of what the interest rate changes should be according to the dynamic Taylor rule. This is not too surprising since the data in Figure 3A showed large and persistent differences between the interest rate change and the target gap.

The reason for why the model generates too much predictability of interest rate changes is not that it generates too much predictability in inflation and the output gap, because survey data strongly suggest that these variables are quite predictable—see Table 4. The reason is instead that the model uses a dynamic Taylor rule.

Couldn't the model be saved by assuming a much lower interest rate smoothing parameter, ρ? Figure 2 shows that a much lower value of ρ would change the interest rate predictability in the right direction (although insufficiently so), but at the price of making the predictability in the output gap (and therefore the target gap) much too low. In these experiments, we have assumed unpredictable policy shocks. In contrast, Rudebusch (2002b) suggests setting ρ = 0 and letting the policy shocks be autocorrelated. He shows that this brings down the predictability of interest rate changes in the model, while preserving good econometric properties of estimated policy rules. To study this, we simulate the model with the suggested policy rule ((2) but with ρ = 0 and ζ_t+1 = 0.85ζ_t + v_t+1 where Std(v_t) = 0.4). The results (not shown) indicate that the autocorrelated shocks make relatively little difference: The overall picture is the same as with just a low ρ discussed earlier.

VAR Model

This section repeats the steps in the survey data section, but with forecasts from a VAR model instead.

Table 5 shows the R² from forecasts made by a VAR(3) model of interest rates, inflation, and the output gap. The VAR model is estimated on first differences and then transformed to fit our data definitions (quarterly interest rate changes, annual inflation, and output gap). To approximate the information set available to the futures market and the survey respondents, the VAR is estimated on a moving data window of 40 quarters. For instance, the forecasts made in 1987:Q4 use a VAR model estimated on 1978:Q1–1987:Q4, while the forecasts made in 1988:Q1 use a VAR model estimated on 1978:Q2–1988:Q1, and so forth. The VAR results confirm most of the findings from survey data: Interest rate changes are not predictable, it is possible to forecast the target gap but not interest rate changes, and the forecast of interest rate changes is far from an optimal predictor of the target gap.

It may seem surprising that the VAR results are so different from the model simulations. After all, the model (2)–(4) is a dynamic system with a few lags, and so, its reduced form is essentially a low-order VAR system. It is then a bit strange that an estimated VAR produces so much less predictability than an estimated/calibrated model (especially for interest rate changes). There may be several explanations for this, but we believe that the tendency for OLS to overfit in small samples plays an important role.

To illustrate this, we reestimated the VAR on a fixed sample (1987:Q4–1999:Q4). The results are shown in Table 6. The most striking result is that the interest rate changes appear to be much more predictable than in the VAR estimated on a moving data window (see Table 5). Some experimenting with the VAR system (estimating some equations on a fixed sample and other equations on a moving data window) suggests that most of the difference is due to the interest rate equation.

The intuition for this result is that the VAR is estimated with OLS, which always tries to maximize the in-sample R² of the one-step-ahead forecasts. With many coefficients to pick, OLS typically can create a good fit in small samples. Since the macro model (2)–(4) is estimated in a fairly similar way, we believe that part of the high predictability in the model is caused by similar in-sample overfitting—perhaps mostly in the estimated dynamic Taylor rule.