Bayesian analyses covered in this thesis use a pre-sample or historical data in order to initialize the parameters or give an objective estimate of some hyperparameters. In this section, we present the method on how to estimate the essential parameters of the VECM based on a pre-sample: the long-run relations matrix Π, the lag parameters matrix Ψ and the covariance matrix Σ of the errors.
Let us assume we have a collection of data for p dierence stationary time series over a period of time length T . From that data set, we can extract a small period from time 1 until a certain time τ < T . This time-period [[1, τ]] of size τ corresponds to the time period of the pre-sample (see Figure 2.2). Then based on this pre-sample, we will obtain the parameter estimates Πb, Ψb
andΣb of the VECM model that will be used to initialize our algorithms in Chapter 3, Chapter 4 and Chapter 5 (see Algorithms 1, 4 and 8). The initial parameters will then be used sometimes to evaluate the values of certain hyperparameters (scale matrix A of the covariance matrix of the errors in the VECM in Chapters 3, 4 and 5).
The sample corresponds to the time-period [[τ + 1, T ]] (see Figure 2.2). The size of the sample is T − τ and it will be the time period on which Bayesian inference on the parameters of the VECM will be made in Chapters 3, 4 and 5. The sample and the pre-sample do not overlap and the size of the pre-sample is usually much smaller than the sample.
Figure 2.2: Arrangement of the data: Pre-sample and sample.
Based on a small pre-sample of size τ < T , we then construct and estimate the parameters of the VECM. The detailed methodology is given in Section 7.2 of Luetkepohl (2006). We obtain estimates of the parameters of the VECM (2.4): Πb,Ψb and the variance of the errorsΣb. We recall how to obtain their least squares estimates below.
We have: ∆xt= Πxt−1+ k−1 X i=1 Ψi∆xt−i+ t
where t∼ N (0, Σ). Furthermore, we assume that for each vector xt, x−k+1, · · · , x0 are available
(t = 1 is the rst time of the pre-sample). We can then build the following matrices: ∆X = [∆x1, · · · , ∆xτ] X−1 = [x0, · · · , xτ −1] ∆Z = [∆Z0, · · · , ∆Zτ −1]with ∆Zt−1= ∆xt−1 ... ∆xt−k+1 U = [1, · · · , τ]
Then, we have the following VECM for t ∈ [[1, τ]]:
∆X = ΠX−1+ Ψ∆Z + U
and we obtain the least squares estimators of Π, Ψ and Σ by:
[ bΠ, bΨ] = [∆XX−10, ∆X∆Z0] X−1X−10 X−1∆Z0 ∆ZX−10 ∆Z∆Z0 −1 (2.30) b Σ = (τ − pk)−1(∆X − bΠX−1− bΨ∆Z)(∆X − bΠX−1− bΨ∆Z)0
Chapter 3
Estimation of the cointegration rank and
the coe cients in a static model
3.1 Introduction
This chapter develops Bayesian cointegration methods for a set of time series where coin- tegration is assumed. This chapter has two aims: to estimate rst the cointegration rank by avoiding reliance upon Johansen tests and to nd the cointegrating relationships by operations based on the long-run impact matrix of the Error Correction Model (see Section 2.3). We decide to determine the cointegration rank within an MCMC procedure based on the singular values of the cointegration matrix from the Error Correction Model. Based on that rank r, we then derive rindependent cointegrating relations from the cointegration matrix. The proposed methodology is tested on simulated data sets and then illustrated with a panel data set of Eurozone economic time series consisting of net trading, long-term interest rates and the harmonized unemployment rate. Our cointegration methods will try to establish their performance, co-evolution and long- run relationships.
tion matrix, asserting that a priori we expect that the cointegration matrix has zero mean (the case of no cointegration), but there is wide uncertainty around this. We propose a new deter- mination of the cointegration rank by the number of irrelevant singular values of the estimated cointegration matrix. The estimation commences by Markov chain Monte Carlo, which provides posterior samples of the cointegration rank. Thus, we can have access to an approximation of the posterior distribution of the cointegration rank, which provides also the associated uncer- tainty around this estimation. Comparisons with Johansen's test indicate that this approach works reasonably well. The resulting cointegration relationships are derived by determining rst the cointegration rank based on the singular values of the long-run relations matrix (during the MCMC), and then decomposing the latter into two full rank matrices (after the MCMC). The proposed methodology is illustrated by considering two simulated data sets and panel data on several macroeconomic variables (net trading, long-term interest rates and unemployment rate) across four Euro zone countries (Germany, France, Italy and Spain).
The approach adopted in this chapter is somewhat associated with the embedding approach of the Error Correction Model, see Kleibergen and van Dijk (1994) and Kleibergen and Paap (2002), recalled brie y in the literature review (see Section 2.4.4). In this approach, they build an Unrestricted Error Correction model by adding a (p − r) × (p − r) parameter matrix λ to the lower rank product of matrices αβ0 of rank r. The long-run relations matrix Π of this unrestricted
ECM is then of full rank: while matrices α and β are p × r full rank matrices of rank r, it is the matrix parameter λ which controls the evaluation of the rank (see Section 2.4.4). Posterior probability distributions for the cointegration rank r can be derived by the use of Bayes factors, see Kleibergen and Paap (2002). In this chapter, the assumption of having a non-singular prior for the cointegrating matrix Π implies a prior for its singular values. We discuss brie y the prior of these singular values and how it can imply a prior for the rank (see Section 3.3.3).
The chapter is organised as follows: Section 3.2 introduces the approximation method for the determination of the cointegration rank. The Bayesian inference is in Section 3.3, which includes the operations on the long-run relationships matrix Π for the determination of the cointegrating
relationships. Section 3.4 gives the application of the methodology to the simulated data sets and to the real panel data sets of European economies. This chapter concludes with closing comments in Section 3.5.
Following standard notation seen in Chapter 2 we use the notation xt = (xit)1≤i≤p to be a
vector of economic I(1) time series represented by a p-dimensional vector autoregressive (VAR) process of lag length k = 2: an argument for this value is given in Section 3.4.3 and in Appendix B.