Abstract
To capture the evolving relationship between multiple economic variables, time variation in either coefficients or volatility is often incorporated into vector autoregressions (VARs). The state space representation that links the transition of possibly unobserved state variables with observed variables is a useful tool to estimate VARs with time-varying coefficients or stochastic volatility. In this paper, we discuss how to estimate VARs with time-varying coefficients or stochastic volatility using the state space representation. We focus on Bayesian estimation methods which have become popular in the literature. As an illustration of the estimation methodology, we estimate a time-varying parameter VAR with stochastic volatility with the three US macroeconomic variables including inflation, unemployment, and the long-term interest rate. Our empirical analysis suggests that the recession of 2007–2009 was driven by a particularly bad shock to the unemployment rate which increased its trend and volatility substantially. In contrast, the impacts of the recession on the trend and volatility of nominal variables such as the core PCE inflation rate and the 10-year Treasury bond yield are less noticeable.
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Notes
- 1.
If we allow for time variation in the B matrix, the model becomes a time-varying structural VAR in Primiceri (2005). And we can also incorporate the time-varying volatility of v t too to capture fluctuations in variances of innovations in trend components as in Cogley, Primiceri, and Sargent (2010).
- 2.
- 3.
Notations here closely follow those in the appendix of Cogley and Sargent (2005).
- 4.
One example of such a proposal density is \(\mathcal{N}(\mu _{it}, 0.5Q_{i})\).
- 5.
Doh (2011) estimates the same model with shorter sample data and focuses on the time-varying relationship between inflation and unemployment.
- 6.
For pre-sample data, we use total PCE inflation because core PCE inflation is not available for this period.
- 7.
For example, see Cogley, Primiceri, and Sargent (2010) and papers cited there.
- 8.
See Doh (2011) and papers discussed there.
- 9.
See Canova and Gambetti (2009) and papers cited there.
- 10.
This calculation is based on comparing the standard deviation of each variable during the relevant period.
- 11.
The drawback of this generalization of time-varying volatility is that so many volatility estimates become explosive during the mid-1970s, casting doubts on the convergence property of the model estimates. For the model without stochastic volatility for innovations in θ t , we do not observe such a convergence issue.
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The views expressed here are the opinions of the authors only and do not necessarily represent those of the Federal Reserve Bank of Kansas City or the Federal Reserve System.
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Doh, T., Connolly, M. (2013). The State Space Representation and Estimation of a Time-Varying Parameter VAR with Stochastic Volatility. In: Zeng, Y., Wu, S. (eds) State-Space Models. Statistics and Econometrics for Finance, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7789-1_6
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