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.
Recommended citations
If you use the models or code below, please cite the most relevant default reference(s) from the list below. Stock and Watson (2007) is the classic starting point for the UCSV model; my papers develop extensions and testing tools. The underlying precision-based state space simulation used across these UC models build on Chan and Jeliazkov (2009).
- Underlying computation for UC / state space models (precision-based simulation): Chan and Jeliazkov (2009), Efficient Simulation and Integrated Likelihood Estimation in State Space Models, International Journal of Mathematical Modelling and Numerical Optimisation, 1, 101-120 (code)
- Bounded trend inflation and time-varying persistence (default cite for bounded trend UC models): Chan, Koop and Potter (2013), A New Model of Trend Inflation, Journal of Business and Economic Statistics, 31(1), 94-106 (code)
- Trend inflation with long-run inflation expectations (default cite when using survey expectations): Chan, Clark and Koop (2018), A New Model of Inflation, Trend Inflation, and Long-Run Inflation Expectations Journal of Money, Credit and Banking, 50(1), 5-53 (code)
- Inflation and slack (Phillips curve) with bounded trend inflation and bounded NAIRU: Chan, Koop and Potter (2016), A Bounded Model of Time Variation in Trend Inflation, NAIRU and the Phillips Curve Journal of Applied Econometrics, 31(3), 551-565 (code)
- Persistent inflation gap via moving-average errors with stochastic volatility: Chan (2013), Moving Average Stochastic Volatility Models with Application to Inflation Forecast Journal of Econometrics, 176(2), 162-172 (code)
- Testing where stochastic volatility is needed in a UC-SV specification: Chan (2018), Specification Tests for Time-Varying Parameter Models with Stochastic Volatility Econometric Reviews, 37(8), 807-823 code)
How these models are typically used
- Measurement: estimating trend inflation and uncertainty in real time and historically.
- Decomposition: separating the inflation gap (transitory component) from the trend.
- Forecasting: producing density forecasts that adapt to changing inflation volatility.
- Economic interpretation: incorporating slack (Phillips curve) or long-run expectations to sharpen trend estimates.
- Model diagnostics: testing whether stochastic volatility is needed in the trend and/or transitory component.
Univariate trend inflation models
The models below are univariate: they only require inflation data. They differ in their autocovariance structures and stochastic volatility specifications.
1. 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.
2. 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.
3. 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.
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.
1. 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.
2. 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.