## Historical Data Decomposition (a.k.a Observables Decomposition)

**Historical data decomposition**for linear state-space models is the "inverse" of historical shock decomposition and is a very useful tool for structural Dynamic Stochastic General Equilibrium (DSGE) models.

Linear DSGE models, or other structural models, are usually expressed in a state-space form, with the measurement equations, \(Y_{t} = Z X_{t} + H\varepsilon_{t}\), and the transition equations: \(X_{t} = T X_{t-1} + R\varepsilon_{t}\). The Kalman filter is then used to estimate the historical structural shocks and other unobserved model variables, given the observed data, the "observables". It is very common to feed back the historical shock estimates \(\varepsilon_{t}\) through the model to explain the contributions of various structural shocks to the observed data, to interpret the history through the lens of the model using the

**historical shock decomposition,**i.e. \(Y = g(\varepsilon)\).

The

**historical data decomposition**, or

**decomposition into observables**, reverses the question! Given the estimates of unobserved variables \(X_t\) and shocks \(\varepsilon_t\), how are the observables being used to generate the estimate of the unobserved states? It finds out \(\widehat{\varepsilon} = f(Y^{obs})\) and \(\widehat{X} = f(Y^{obs})\)

For instance, in a semi-structural output gap model, or in a DSGE model, how does the observed GDP, inflation, or interest rate contribute to the estimate of the

**output gap**, or the potential output? What variables, and at what frequencies, the model uses to identify the productivity shock? Also, observables decomposition enable interpretation of revised real-time estimates and interpret the new data! Many simple modifications can be used for understanding the model, e.g. chain the two to get \(Y = \sum_{i}^{ne}g(f_{i}(Y^{obs} ))\) or explain shocks of one model in terms of shocks of the other model, etc.

To our best knowledge, observable decomposition (historical data decomposition), has been developed and regularly used for the Czech National Bank's (CNB) core forecasting model "g3", see

**Andrle et al. (2009)**. The motive was to understand better the identified shocks... The approach is routinely used to gain insight into IMF's output-gap models.

**PAPERS:**

******I****mplementing the New Structural Model of the Czech National Bank**,**Czech National Bank WP 2009/2**[pdf] (joint with T. Hledik, O. Kamenik and J. Vlcek). Describes the model and some novel analytical methods of analysis for structural forecasting models. This was the first time we described the "observables decomposition" developed for the CNB's "g3" projection model.

**Understanding DSGE filters in forecasting and policy analysis**,**DYNARE WP No. 16/2012**[pdf], updated in**IMF WP/13/98**[pdf]. Presented at 8th DYNARE conference. (Useless to academia(?), useful for policy and forecasting...). Note, there is way more formulas than needed for basic usage but helpful for understanding of the DSGE filters in time- and frequency domain.

**What Is in Your Output Gap? Unified Framework & Decomposition into Observables**, 2013,**IMF WP/13/105**, [pdf] This paper presents a unifying framework for output gap estimation -- theory of linear filters. It also explains how to decompose the output gap measures, even using DSGE models, as a function of observed data inputs, e.g. inflation, output or unemployment.

**SOFTWARE/CODE:**

**Simple and flexible example in IRIS Tbx [file] -**this simple code shows an implementation that exploits the linearity of the problem and allows very flexible treatment of groups of variables, sub-periods, and missing observations. Easy to implement with ANY software that implements the Kalman filter!

**IRIS Toolbox**-- Mirek Benes agreed to implement observable decomposition into the IRIS toolbox. The implementation also relies on the linearity of the problem and handles missing observations easily...

**YADA -- the ECB's DSGE toolbox YADA**by Anders Warne has also implemented the observable decompositions. Anders used the explicit derivation through the implied filter weights, as derived by Harvey and Koopmans (2003).

**Some other applications of the method...**

- Melolinna, M. and M. Toth (2016): Output gaps, inflation and financial cycles in the United Kingdom, [pdf], Bank of England Staff WP No. 585, 2016
- Hledik, T. and J. Vlcek (2018): Quantifying the Natural Rate of Interest in a Small Open Economy – The Czech Case, [pdf] Czech National Bank WP No. 7/2018
- ...

**The views expressed herein are those of the author and should not be attributed to the International Monetary Fund, its Executive Board, or its management.**