Part of my research in the last decade has been on flexible unobserved components models for trend inflation. These models are useful for measuring trend inflation (and uncertainty), decomposing inflation into trend and transitory components, and producing density forecasts that can adapt to changing volatility.
Start Here
For beginners, Section 6.1 of my teaching notes on unobserved components models provides a gentle introduction. The papers below then develop several extensions that address persistence in the inflation gap, bounded trend inflation, time-varying persistence, and additional information from slack or inflation expectations.
Current Estimates
PCE trend inflation estimates from four of these models, the UCSV model of Stock and Watson (2007), Chan (2013), Chan, Koop and Potter (2013) and Chan, Clark and Koop (2018), are re-estimated every quarter from public data and published on the estimates page, together with estimates of the output gap and trend output growth. The models page writes out each model and discusses which estimate to use, and the estimates explorer compares them side by side.
Tutorials and Code
The tutorials on unobserved components models apply the models below to one empirical question each, with MATLAB code that runs on your own data:
The code for most of these papers is archived on GitHub in statespace-toolkit, with a MATLAB library and examples. Each code page names the paper to cite, and the toolkit's CITING.md maps each method to its paper. The precision-based state space simulation used across these models builds on Chan and Jeliazkov (2009).
Univariate Trend Inflation Models
The models below are univariate: they only require inflation data. They differ in their autocovariance structures and stochastic volatility specifications.
Specification Tests for Time-Varying Parameter Models with Stochastic Volatility
Joshua Chan (2018), Econometric Reviews, 37(8): 807-823
[ Journal Version | Working Paper | Code | Toolkit ]
The unobserved components stochastic volatility (UCSV) model in Stock and Watson (2007) is the starting point for this
literature. Chan (2018) considers a reparameterization of the classic UCSV model that makes it
convenient to test whether stochastic volatility is needed in the trend component and/or the transitory component.
Code for the UCSV model is in the
code repository of my book
Bayesian Macroeconometrics, in MATLAB, Python and R (chapter10/UCSV), and
trend-cycle-toolkit re-estimates it
every quarter (ucsv_sw07).
Moving Average Stochastic Volatility Models with Application to Inflation Forecast
Joshua Chan (2013), Journal of Econometrics, 176(2): 162-172
[ Journal Version | Working Paper | Code | Toolkit ]
In the original UCSV model, the cyclical component is assumed to have no persistence. This can be restrictive, as the inflation
gap is often autocorrelated. Chan (2013) introduces unobserved components models with stochastic
volatility and moving-average errors. These models allow a persistent transitory component and can improve inflation forecasts
relative to the original UCSV.
A New Model of Trend Inflation
Joshua Chan, Gary Koop and Simon Potter (2013), Journal of Business and Economic Statistics, 31(1): 94-106
[ Journal Version | Working Paper | Code | Toolkit ]
In many inflation targeting economies, trend inflation plausibly evolves within a narrow band (related to the inflation target),
rather than following an unconstrained random walk. Motivated by this observation,
Chan, Koop and Potter (2013) introduce a bounded trend inflation model with two features:
(i) trend inflation is constrained to lie in an interval; and (ii) the persistence of the transitory component can vary over time.
The Stochastic Volatility in Mean Model with Time-Varying Parameters: An Application to Inflation Modeling
Joshua Chan (2017), Journal of Business and Economic Statistics, 35(1): 17-28
[ Journal Version | Working Paper | Code | Toolkit ]
Chan (2017) extends the stochastic volatility in mean model to time-varying parameters, so that the volatility of inflation
enters its conditional mean with a time-varying coefficient. For US, UK and German inflation, the coefficient on the
volatility varies substantially over time, and the model forecasts better than standard benchmarks.
Stochastic Volatility Models with ARMA Innovations: An Application to G7 Inflation Forecasts
Bo Zhang, Joshua Chan and Jamie Cross (2020), International Journal of Forecasting, 36(4): 1318-1328
[ Journal Version | Working Paper ]
Zhang, Chan and Cross (2020) introduce stochastic volatility models with ARMA errors, whose conditional mean can take a
state space form or a dynamic regression form, and estimate them with a precision-based posterior simulator. In point and
density forecasts of inflation for the G7 economies, the new models are competitive with standard benchmarks.
Multivariate Trend Inflation Models
Additional information can help to estimate trend inflation more precisely. The multivariate models below incorporate other sources of information related to inflation.
A Bounded Model of Time Variation in Trend Inflation, NAIRU and the Phillips Curve
Joshua Chan, Gary Koop and Simon Potter (2016), Journal of Applied Econometrics, 31(3): 551-565
[ Journal Version | Working Paper | Code | Toolkit ]
The Phillips curve relates the inflation gap to measures of economic slack. Chan, Koop and
Potter (2016) propose a bivariate unobserved components model for inflation and unemployment with two features: (i) both trend
inflation and the NAIRU evolve within bounds; and (ii) the Phillips curve slope and inflation persistence are allowed to vary over time.
A New Model of Inflation, Trend Inflation, and Long-Run Inflation Expectations
Joshua Chan, Todd Clark and Gary Koop (2018), Journal of Money, Credit and Banking, 50(1): 5-53
[ Journal Version | Working Paper | Code | Toolkit ]
Short-horizon inflation forecasts from professional forecasters are generally accurate and informative, but longer-horizon forecasts
are more mixed. This raises the question of whether long-horizon expectations can still refine trend inflation estimates.
Chan, Clark and Koop (2018) introduce a model of trend inflation that uses inflation data and long-run
inflation expectations, while also allowing time-varying persistence in the transitory component and stochastic volatility.
Measuring Inflation Expectations Uncertainty Using High-Frequency Data
Joshua Chan and Yong Song (2018), Journal of Money, Credit and Banking, 50(6): 1139-1166
[ Journal Version | Working Paper | Code | Toolkit ]
Chan and Song (2018) augment an unobserved components model of inflation with daily break-even inflation, a noisy and
possibly biased measure of inflation expectations from financial markets. The result is a model-based measure of inflation
expectations uncertainty, a direct gauge of how well anchored the expectations are. With US data, this uncertainty changes
significantly during the Great Recession.