A financial conditions index (FCI) compresses a large number of financial market series – spreads, prices, volatilities, survey measures – into a single series that is meant to say whether financial conditions are tight or loose. The obvious tool for that job is a factor model. The harder question is which series should count towards the index, and by how much, given that the answer plainly changes over time: the TED spread said something quite different about the state of the financial system in 2008 than it did in 1995.
Koop and Korobilis (2014) answer it by letting the factor loadings drift. Their index comes out of a factor augmented VAR (FAVAR) in which both the loadings and the transition coefficients follow random walks, so a series can enter and leave the index as the sample proceeds. What makes the paper more than a large state space model is how it is estimated: the priors on the innovation variances are replaced by forgetting factors, which reduces the whole estimation to two passes of a Kalman filter and smoother. No posterior simulation is involved, and the entire index over 175 quarters and 18 financial series is obtained in a hundredth of a second.
This post reproduces the two exercises the paper is built around – the index itself and its real-time counterpart – using the fincond package.
The bvartools allows to perform Bayesian inference of Vector autoregressive (VAR) models, including structural VARs. This post guides through the Bayesian inference of SVAR models in R using the bvartools package.
For this illustration we generate an artificial data set with three endogenous variables, which follows the data generating process
\[y_t = A_1 y_{t - 1} + B \epsilon_t,\]
where
\[ A_1 = \begin{bmatrix} 0.3 & 0.12 & 0.69 \\ 0 & 0.3 & 0.48 \\ 0.24 & 0.24 & 0.3 \end{bmatrix} \text{, } B = \begin{bmatrix} 1 & 0 & 0 \\ -0.14 & 1 & 0 \\ -0.06 & 0.39 & 1 \end{bmatrix} \text{ and } \epsilon_t \sim N(0, I_3). \]
For some macroeconomic applications it might be interesting to see whether a set of obserable variables depends on common drivers. The estimation of such common factors can be done using so-called factor analytical models, which have the form
\[x_t = \lambda f_t + u_t,\]
where \(x_t\) is an \(M\)-dimensional vector of observable variables, \(f_t\) is an \(N \times 1\) vector of unobserved factors, \(\lambda\) is an \(M \times N\) matrix of factor loadings and \(u_t\) is an error term.
Bayesian methods have significantly gained in popularity during the last decades as computers have become more powerful and new software has been developed. Their flexibility and other advantageous features have made these methods also more popular in econometrics. This post gives a brief introduction to Bayesian VAR (BVAR) models and provides the code to set up and estimate a basic model with the bvartools package.
This post provides the code to set up and estimate a basic Bayesian vector error correction (BVEC) model with the bvartools package. The presented Gibbs sampler is based on the approach of Koop et al. (2010), who propose a prior on the cointegration space.
To illustrate the estimation process, the dataset E6 from Lütkepohl (2007) is used, which contains data on German long-term interest rates and inflation from 1972Q2 to 1998Q4.
A general drawback of vector autoregressive (VAR) models is that the number of estimated coefficients increases disproportionately with the number of lags. Therefore, fewer information per parameter is available for the estimation as the number of lags increases. In the Bayesian VAR literature one approach to mitigate this so-called curse of dimensionality is stochastic search variable selection (SSVS) as proposed by George et al. (2008). The basic idea of SSVS is to assign commonly used prior variances to parameters, which should be included in a model, and prior variances close to zero to irrelevant parameters. By that, relevant parameters are estimated in the usual way and posterior draws of irrelevant variables are close to zero so that they have no significant effect on forecasts and impulse responses. This is achieved by adding a hierarchial prior to the model, where the relevance of a variable is assessed in each step of the sampling algorithm.1