Abstract
The current chapter outlines the second of proposed perspectives on migration forecasting, which applies the ‘from general to specific’ modelling principle in the context of nested vector autoregression (VAR) models. In this way, the impact of various theory-based interdependent variables on migration can be tested. Similarly to the previous chapter, Section 6.1 outlines the theoretical foundations of VAR modelling, while Section 6.2 illustrates the approach with empirical forecasts for emigration rates among the countries under study.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
See also Section 5.3 for more details on the construction of Minnesota priors.
- 2.
In migration studies interesting properties of the estimated VAR models have been tested by Gorbey et al. (1999, pp. 78–84), who, however, had longer time series at their disposal – about 80 quarterly observations.
- 3.
The Wishart distribution is a multivariate generalisation of the Gamma distribution (in particular, of the chi-squared distribution). Again, following the parameterisation used in the WinBUGS manual, for \(x\sim \chi ^2 (k)\), the density is \(p(x\left| k \right.) = 2^{ - k/2 } x^{k/2 - 1 } \exp ( - x/2)/\Gamma (k/2)\), where \(x > 0\) is Euler’s Gamma function. In general, for a random vector x ~ Wishart(P, k), \(p({\textbf{x}}\left| {{\textbf{P}},\;k} \right.) \propto (\det \;{\textbf{P}})^{k/2 } \left| {\textbf{x}} \right|^{(k - n - 1)/2 } \exp [ - 1/2 \cdot {\textrm{tr}}({\textbf{Px}})]\), where tr(·) is a matrix trace (a sum of the diagonal elements) and x is an n-dimensional symmetric and positive definite matrix. As in the case of τ in Chapter 5, elements of T refer to the precision of log-transformed variables, and thus to the precision with respect to the orders of magnitude for crude emigration rates.
- 4.
The prior for the Wishart distribution parameter P can be for example elicited as follows: the symmetric matrix P/k can be decomposed into a matrix product \({\textbf{P}}/k = {\textbf{D}}\;{\textbf{R}}\;{\textbf{D}}\), where \({\textbf{D}} = [d_{ij} ]_{k\,{\textrm{x}}\,k}\) denotes a very highly probable a priori diagonal matrix of standard deviations and \({\textbf{R}} = [r_{ij} ]_{k{\textrm{ x }}k}\) a very highly probable a priori symmetric matrix of Pearsonian correlations between particular variables (credit to Jacek Osiewalski). Under such transformation, in the precision classes listed in Table 7.1, the prior assumptions are thus: for very low precision: \(d_{ii} = 0.71\) and \(r_{ij} = - 0.02\) (for \(i \ne j\)); for low precision: \(d_{ii} = 0.58\) and \(r_{ij} = - 0.02\); for medium precision: \(d_{ii} = 0.32\) and \(r_{ij} = - 0.03\); and for high precision: \(d_{ii} = 0.22\) and \(r_{ij} = - 0.03\). Naturally, in all cases by definition holds: \(r_{ii} = 1\) and \(d_{ij} = 0\) for \(i \ne j\).
- 5.
Note the relationship between the chi-squared and Gamma distributions: \({\chi ^2} _{(k)} \equiv \Gamma (1/2 \cdot k,\;1/2)\), as well as the scaling property of the Gamma distribution: \(X\sim \Gamma (r,\;\mu ) \Rightarrow k \cdot X\sim \Gamma (r,\mu /k)\).
- 6.
For yearly values of forecasted median trajectories of migration determinants, see Table C.3b in Annex C.
- 7.
For detailed posterior summaries of parameters of the VAR(1) models under study, see Table C.1 in Annex C.
References
Bauwens, L., Lubrano, M., & Richard, J.-F. (1999). Bayesian inference in dynamic econometric models. Oxford, MA: Oxford University Press.
Charemza, W. W., & Deadman, D. F. (1992). New directions in econometric practice. Aldershot: Edward Elgar [(1997). Nowa ekonometria. Warszawa: PWE].
Cooley, T. F., & LeRoy, S. F. (1985). Atheoretical macroeconomics: A critique. Journal of Monetary Economics, 16(3), 283–308.
Fertig, M., & Schmidt, C. M. (2000). Aggregate-level migration studies as a tool for forecasting future migration streams (IZA Discussion paper 183). Institut zur Zukunft der Arbeit, Bonn.
Gorbey, S., James, D., & Poot, J. (1999). Population forecasting with endogenous migration: An application to trans-tasman migration. International Regional Science Review, 22(1), 69–101.
Greene, W. H. (2000). Econometric analysis. Upper Saddle River, NJ: Prentice Hall.
Hendry, D. F. (1995). Dynamic econometrics. Oxford, MA: Oxford University Press.
Jennissen, R. (2004). Macro-economic determinants of international migration in Europe. Amsterdam: Dutch University Press.
Litterman, R. B. (1979). Techniques of forecasting using vector autoregressions (Working Paper No. 115). Federal Reserve Bank of Minneapolis, Minneapolis, MN.
Osiewalski, J. (2001). Ekonometria bayesowska w zastosowaniach [Bayesian econometrics in applications]. Cracow: Cracow University of Economics.
Osiewalski, J., & Steel, M. F. J. (1996). A Bayesian analysis of exogeneity in models pooling time-series and cross-sectional data. Journal of Statistical Planning and Inference, 50(2), 187–206.
Ramos, F. F. R. (1996). VAR priors: Success or lack of a decent macroeconomic theory? Econometrics, Economics Working Paper Archive EconWPA, Internet Resource; econwpa.wustl.edu/eps/em/papers/9601/ 9601002.pdf. Accessed 31 July 2006.
Sanderson, W. (1998). Knowledge can improve forecasts: A review of selected socioeconomic population projection models. Population and Development Review, 24(Suppl.), 88–117.
Sims, C. A. (1980). Macroeconomics and Reality. Econometrica, 48(1), 1–48.
Sinn, H.-W., Flaig, G., Werding, M., Munz, S., Duell, N., & Hofmann, H. (2001). EU-Erweiterung und Arbeitskräftemigration, Wege zu einer schrittweisen Annäherung der Arbeitsmärkte. IFO-Institut für Wirtschaftsforschung, München.
Wróblewska, J. (2009). Bayesian model selection in the analysis of cointegration. Central European Journal of Economic Modelling and Econometrics, 1(1), 57–69; http://www.cejeme.eu.
Zellner, A. (1971). An introduction to Bayesian inference in econometrics. New York: Wiley.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Bijak, J. (2011). Bayesian VAR Modelling ‘from General to Specific’. In: Forecasting International Migration in Europe: A Bayesian View. The Springer Series on Demographic Methods and Population Analysis, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8897-0_6
Download citation
DOI: https://doi.org/10.1007/978-90-481-8897-0_6
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-8896-3
Online ISBN: 978-90-481-8897-0
eBook Packages: Humanities, Social Sciences and LawSocial Sciences (R0)