The Fisher hypothesis says that nominal interest rates move one for one with expected inflation, so that the real interest rate is stationary. In a vector error correction model (VECM) of a short-term interest rate \(r_t\) and inflation \(\pi_t\) it implies a single cointegrating relationship with the vector \((1, -1)'\). Tests of it on long samples give mixed results, and one reason may be that the relationship is not the same throughout: a hypothesis that holds in some decades and fails in others is rejected by a model that insists on one answer for all of them.
Koop, León-González and Strachan (2011) develop a VECM whose cointegration space is allowed to change over time, and use it on exactly this question. The difficulty they solve is not computational but conceptual. The natural extension of a time varying parameter VAR – letting the cointegrating vectors follow a random walk – does not work, because only the space spanned by the vectors is identified, and a random walk on a normalised vector says nothing sensible about how that space moves. Their state equation instead implies that the space in one period is centred on the space of the period before, and that its marginal prior is uniform in every period.
This post reproduces the two results of the application with bvartools: the path of the cointegrating coefficient in the paper’s preferred model (Fig. 1) and the log predictive likelihoods of the model variants (Tables 1 and 2).
Basically stationarity means that a time series has a constant mean and constant variance over time. Although not particularly important for the estimation of parameters of econometric models these features are essential for the calculation of reliable test statistics and, hence, can have a significant impact on model selection.
To illustrate this concept, let’s look at quarterly data on disposable income in billion DM from 1960 to 1982, which is data set E1 from Luetkepohl (2007).
One of the prerequisits for the estimation of a vector autoregressive (VAR) model is that the analysed time series are stationary. However, economic theory suggests that there exist equilibrium relations between economic variables in their levels, which can render these variables stationary without taking differences. This is called cointegration. Since knowing the size of such relationships can improve the results of an analysis, it would be desireable to have an econometric model, which is able to capture them. So-called vector error correction models (VECMs) belong to this class of models. The following text presents the basic concept of VECMs and guides through the estimation of such a model in R.
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.