Forecasting the 10-Year Treasury YieldA Supply-Side-Informed Error Correction Approach When the Phillips Curve Is Flat

Forecasts of the 10-year yield in the literature seldom improve on the forecast that the yield will not change. The hypothesis: when the Phillips curve is flat, cost pressure upstream of the consumer, read in the PPI for finished goods, carries the information about inflation that bond markets price. Three features of the design determine what the evaluation can show: every regressor is read as it stood at the forecast origin, the target is a monthly average read against two benchmarks, and the sample is split at the two dates of inflation targeting.

Has the literature placed a supply-side inflation signal in a forecasting equation for the yield, and how is a forecast of an averaged rate evaluated?

The literature takes its inflation information from consumer prices, from expected inflation, or from the factors of the yield curve; it has not, to the author's knowledge, placed a supply-side signal in a forecasting equation for the yield. The first differences of averages of a random walk are serially correlated, with a first-order autocorrelation that approaches one quarter as the number of days averaged grows, the property on which the evaluation's two benchmarks rest.

What is forecast, when is the forecast made, and what was known at that moment?

The target is the monthly average of the 10-year constant-maturity Treasury yield, and the forecast origin is the close of the final business day of a month. Two quantities known at the origin are kept apart throughout, the month's average and the yield at its close. The producer price index for finished goods, the unemployment rate, and the price index of personal consumption expenditures (PCE) enter as they stood in the vintage in force at the origin.

Figure 1, the yield and the federal funds rate

How persistent is the yield, and what does averaging over the month do to the target?

The 10-year yield and the federal funds rate are treated as persistent series, and the yield is modeled in changes. Averaging makes the target predictable from its own last change: the change in the monthly average is serially correlated at the first lag, the change in the month-end close less so. If the daily yield is a random walk, the expected value of every future monthly average is the last close, so every forecast is read against two benchmarks, the no-change forecast of the average and the month-end close.

Equation (1), the expected future average under a random walk; Table 1, unit roots and cointegration; Table 2, serial dependence under two samplings; Figure 2, autocorrelation of the monthly change

On what long-run relation does the error correction form rest, and do the data establish it?

The yield and the federal funds rate are tied by a long-run relation whose residual is the deviation the model corrects. The data do not establish the relation as a cointegrating one; the out-of-sample evaluation does not depend on it, the relation being estimated again inside every estimation window.

Equation (2), the long-run relation; Table 1, unit roots and cointegration

Which inflation signal should the equation carry, the supply-side one or the demand-side one?

On the 312 origins from August 2000, neither the coefficient on the PPI term nor that on the PCE term in its place is statistically distinguishable from zero, and the Bayesian information criterion is lower with the PCE term by less than one.

Figure 3, the yield and the two inflation rates as first published; Table 3, the inflation term under each signal

What is the full equation, how are forecasts compared, and where does the equation's fit come from?

The model takes two forms. In the equation, the change in the monthly average is regressed on its two latest changes, the change in the funds rate, the latest published changes in the PPI rate and the unemployment rate, the change in the spread of the 10-year over the 2-year yield, and the deviation from the long-run relation; the equation with the close adds the close less the month's average. In sample the fit comes from the yield's own dynamics: no other term, the PPI term and the error correction term among them, is individually distinguishable from zero.

Equation (3), the equation; Equation (4), the equation with the close; Table 4, estimates of the one-month equation; Figure 4, diagnostics

Does the estimated equation forecast, out of sample and in real time, more accurately than either benchmark?

Out of sample and in real time, on a rolling window of 120 months, the month-end close has the lowest root mean squared error of the six procedures compared, among them an autoregression and a regression on the close alone, at each of the twelve horizons. Given the close, the model forecasts the next month's average more accurately than the no-change forecast; beyond one month neither form of the model improves on no change, and neither improves on the close at any horizon. Where the three-factor model of Diebold and Li is not distinguishable from no change at one month, the equation with the close improves on it; that improvement is not evidence of information in the model's macroeconomic terms, since the close reaches 0.752 times the no-change forecast's error with no estimation. At twelve months, where published factor models and models with macroeconomic trends report their gains, both forms of the model have errors above that of no change. Removing the PPI term lowers the error at one, three, six, and twelve months in both forms, from 1.352 to 1.340 times the close's error at one month for the equation; with the PCE rate in its place, on the origins from August 2010, the accuracy is not distinguishable from the equation's. The close keeps the lower error in each of three subperiods, under an expanding window, and on latest-vintage data; with the month-end close as the target, the equation does not improve on no change of the close. Section 8.6 states the limits.

Table 5, accuracy by horizon; Table 6, published results; Table 7, the PPI term out of sample; Table 8, subperiods; Table 9, alternative designs; Figure 5, error relative to the close; Figure 6, error over trailing windows; Figure 7, cumulative squared-error differences from the close; Figure 8, the coefficient on the PPI term in rolling windows

Does the result hold inside each of the three segments set by the dates of inflation targeting?

The Federal Open Market Committee agreed internally on an inflation objective in July 1996 and announced one in January 2012. The coefficient on the PPI term is not distinguishable from zero before July 1996, between July 1996 and January 2012, or from January 2012. The equation's slope coefficients differ across July 1996 and not across January 2012, and the interaction of the PPI term with the split at July 1996 is not distinguishable from zero. Estimated on origins from July 1996 alone and from January 2012 alone, both forms of the model have a larger error than the close. On data from July 1996, at one, three, and six months, the terms beyond the close jointly carry information beyond the regression on the close alone, and no single term is identified as its source.

Table 10, the PPI term by segment; Table 11, accuracy inside each regime

What does the model forecast from the last origin, and how are its forecast and the close weighed?

From the close of 30 September 2026, both forms of the model place the monthly average of September 2027 above the close. The equation's 95 percent interval twelve months ahead runs from 3.51 to 7.39 percent. Over the out-of-sample evaluation of Section 8, the twelve-month root mean squared errors of the two forms are 1.088 and 1.072 percentage points, against 0.856 for the close.

Table 12, the conditional forecast; Figure 9, the forecast and its intervals

Which component of the yield has moved, the real one or the inflation one?

In the fall of the nominal yield from January 2003 to its low in July 2020, and in the rise from that low to the close of the last origin, the yield on inflation-protected securities accounts for the larger part of the change and breakeven inflation for the smaller.

Figure 10, the yield, the inflation-protected yield, and breakeven inflation

What do the findings imply, and what do they not imply, for the companion papers on monetary policy rules and on inflation forecasting?

The long-run relation ties the yield to the policy rate that the companion paper on monetary policy rules studies, with a slope of 0.846. The paper's finding on the PPI term is compatible with a PPI that leads consumer inflation, under an account the paper does not test: a public signal that forecasts inflation can be priced by a bond market when it is published, and a signal already in the price does not forecast the price. The Global Supply Chain Pressure Index reached its highest value, 4.43 standard deviations above its mean, in December 2021, when the monthly average of the yield was 1.47 percent; it enters no forecast of the paper. The information the model carries beyond the close on data from July 1996 is not attributed to the long-run relation: the coefficient on the deviation from it is not statistically distinguishable from zero.

Figure 11, the yield and the supply chain pressure index

Which findings does the paper carry forward?

Given the close, the model forecasts the next month's average more accurately than the no-change forecast, with the sign of the change in 67.6 percent of months, and does not forecast it more accurately than the close at any horizon. The fit comes from the yield's own dynamics, and the coefficient on the PPI term is not distinguishable from zero in the full sample or in any of the three segments. The equation's slope coefficients differ across July 1996, one of the two dates tested. On data from that date the terms beyond the close jointly carry information beyond the regression on the close alone at one, three, and six months, and the equation that carries them has an error 1.036 times the close's at one month.