Which Estimate Should I Use?
The six models behind the quarterly estimates produce three different series: four give trend inflation, two give the output gap, and one of those two also gives trend output growth. Within trend inflation the four models estimate the same object under different assumptions, and they disagree materially. The current estimates and the distance between them are on the estimates page.
Each model also reports a credible interval, which describes uncertainty conditional on that model's assumptions. The two are different quantities: reporting several specifications demonstrates how far the estimate depends on assumptions, and it does not quantify model uncertainty. Where the choice of model is not clear, report more than one and say why.
| If you want | Use | The assumption you are accepting |
|---|---|---|
| The standard forecasting benchmark for inflation | Stock and Watson (2007) | The inflation gap is serially uncorrelated, both innovations carry stochastic volatility, and nothing restricts the trend |
| A trend restricted to a plausible range | Chan, Koop and Potter (2013) | Trend inflation lies between 0 and 5 percent, and the inflation gap follows a time-varying AR(1) |
| A trend estimated jointly with measured expectations | Chan, Clark and Koop (2018) | A survey measure of long-run expectations is informative about the trend, through a link that drifts over time |
| Short-run dependence held in the transitory component | Chan (2013) | The inflation gap follows an MA(1), and the trend innovation variance is constant |
| An output gap with a smooth trend and large cycles | Grant and Chan (2017, JEDC) | Shocks reach the trend only through its growth rate, whose variance the prior bounds tightly, as the Hodrick-Prescott filter implies |
| An output gap whose trend follows output closely | Grant and Chan (2017, JMCB) | The trend level takes shocks directly, around a drift that is constant between the breaks in 1973Q1 and 2007Q1 |
| Trend output growth | Grant and Chan (2017, JEDC) | As above; it is the only published trend growth series |
Trend Inflation
The four models decompose annualized quarterly PCE inflation into a trend and a transitory component, the inflation gap. They differ in what the trend is permitted to do and in what serial dependence the gap is permitted to carry. The second choice is the one most easily overlooked: Stock and Watson take the gap to be serially uncorrelated, so persistence in inflation is accommodated by moving the trend, while Chan (2013) gives the gap an MA(1), and the other two give it a time-varying AR(1).
Stock and Watson (2007)
Unobserved components with stochastic volatility in both the transitory and the permanent innovation:
y_t = tau_t + exp(h_t/2) e_t
tau_t = tau_{t-1} + exp(g_t/2) u_t
where tau_t is trend inflation, e_t and u_t are standard normal and independent across quarters, and the log-volatilities h_t and g_t are random walks. The inflation gap is exp(h_t/2) e_t, which is serially uncorrelated given its volatility path: the size of a transitory shock may change over time, and no dependence is allowed between one quarter's shock and the next. Persistence in inflation beyond a random walk therefore has one place to go, which is the trend, and that is a large part of why this trend is the most variable of the four.
This model is the standard benchmark in the inflation forecasting literature, which makes it the natural choice for comparison with published forecasting results. The sample begins in 1947Q2. Journal of Money, Credit and Banking 39(s1): 3–33.
Chan, Koop and Potter (2013)
The inflation gap follows a time-varying AR(1), and the trend is bounded:
y_t - tau_t = rho_t (y_{t-1} - tau_{t-1}) + u_t, u_t ~ N(0, exp(h_t))
tau_t = tau_{t-1} + v_t, 0 < tau_t < 5
rho_t = rho_{t-1} + w_t, 0 < rho_t < 1
where the bounds on tau_t are in annualized percent, and rho_t and the log-volatility h_t are themselves random walks. A persistent inflation gap accounts for movements that would otherwise be attributed to the trend, and the bounds restrict the trend further, which makes this the least variable of the four estimates.
The bounds and the persistent gap are the assumptions to weigh before using this series. The bounds encode a view about the monetary regime over the whole sample, including the 1970s, and a study that uses the series should state it. The sample begins in 1947Q2. Code. Journal of Business and Economic Statistics 31(1): 94–106.
Chan, Clark and Koop (2018)
Inflation and a survey-based measure of long-run inflation expectations are modeled jointly. The survey enters a second measurement equation whose intercept and loading on the trend are time-varying, so the link between the two is estimated and allowed to drift. The survey measure is PTR from the Federal Reserve Board's FRB/US model. The inflation gap follows a time-varying AR(1) with stochastic volatility, so persistence in inflation is again accommodated outside the trend.
The second observable is itself informative about the trend, which is why this credible interval is much narrower than the others. The series is not independent evidence that expectations are anchored: the link between expectations and the trend is imposed by the specification, and what the model estimates is its strength.
The sample begins in 1960Q2 and ends where PTR ends, which is often a quarter behind the other models. Code. Journal of Money, Credit and Banking 50(1): 5–53.
Chan (2013)
Unobserved components with an MA(1) transitory component and stochastic volatility:
y_t = tau_t + u_t + psi u_{t-1}, u_t ~ N(0, exp(h_t)), |psi| < 1
tau_t = tau_{t-1} + v_t, v_t ~ N(0, sigma_tau^2)
where the log-volatility h_t is a stationary AR(1). The inflation gap is u_t + psi u_{t-1}, so one quarter's gap carries part of the previous quarter's shock. This is the assumption that separates the model from Stock and Watson, whose gap is serially uncorrelated: the persistence that model can accommodate only by moving the trend is accommodated here within the transitory component, which leaves the estimated trend much smoother. A second difference works in the same direction, since the trend innovation variance is constant here.
Use this model where short-run dependence in inflation is better treated as transitory than as movement in the trend. The sample begins in 1947Q2. Code. Journal of Econometrics 176(2): 162–172.
The Output Gap
Both models decompose 100 log real GDP into a trend and a cycle, both estimate a correlation between the trend and cycle innovations, and both allow the cycle to be serially correlated. They differ in the trend equation, and that difference drives the difference between their gaps.
| Model | Trend | What the innovation hits | Prior bound on its variance |
|---|---|---|---|
| Grant and Chan (2017, JEDC) | second-order Markov, tau_t - tau_{t-1} = tau_{t-1} - tau_{t-2} + eta_t |
the growth rate | 0.01 |
| Grant and Chan (2017, JMCB) | random walk with regime drift, tau_t - tau_{t-1} = mu_s(t) + eta_t |
the level | 3 |
In the second model the level of the trend takes each shock directly, and the prior admits a large innovation variance, so the trend follows a sharp movement in output and leaves little of it in the cycle. In the first only the growth rate takes the shock, under a prior that bounds its variance at 0.01, so the trend turns slowly and a sharp movement stays in the cycle. In 2020Q2 the two gaps are roughly five times apart, and the same mechanism separates them over the whole sample.
The question to ask is how potential output behaves over the sample being studied. An analysis in which potential output is smooth, and recessions are large deviations from it, matches the JEDC model. An analysis in which potential output itself moves with the economy, so that recessions leave smaller gaps, matches the JMCB model. The break dates belong to the second assumption: they are set at 1973Q1 and 2007Q1 and are not estimated. Both models are uncertain about the present to a degree worth reporting, and their credible intervals at the end of the sample are wide. JEDC code, JMCB code.
Trend Output Growth
Annualized trend growth, 4 (tau_t - tau_{t-1}), is published from the JEDC model and computed from the same draws as its output gap. The JMCB model implies a trend growth rate as well. Its drift is a step function with three levels, while its realized trend growth is not, since that carries the trend innovation. Only the first series is published.
Reading the Published Series
Every value is smoothed with the whole sample. The estimate reported for 1998Q3 uses data through the end of the vintage, and it differs from the estimate that would have been produced in 1998. These are not real-time series. Every release is frozen on GitHub, so a vintage records what was reported at the time.
The ends of the sample are the least precisely estimated part. A smoother conditions on data on both sides of a date, and at the last observation there is data on one side only.
Estimates for past quarters change between releases, for three reasons: the source data were revised, the sample grew, or the sampler is stochastic. Each release publishes the total change and the part of it attributable to the longer sample.
Models end at different quarters, since Chan, Clark and Koop (2018) depends on the FRB/US package. Read each model's last observation off the date column, which the table on the estimates page also reports.
This page describes the models. For the numbers, the downloads and how to cite them, see the estimates page; for the code, the guide in the repository.