Abstract
The current chapter focuses on two aspects of evaluating the forecasts of migration flows, presented earlier in Chapters 5, 6 and 7. Firstly, Section 8.1 deals with the sensitivity of the results to changes in the assumed prior distributions and in the information they carry. The focus of the discussion is on the precision parameters of the assumed random processes. Secondly, in Section 8.2, selected Bayesian forecasts are compared with their frequentist counterparts in terms of errors ex ante and ex post, in the latter case computed for forecasts for 2000–2007 calculated on the basis of series truncated in 1999.
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Notes
- 1.
Poirier and Tobias (2004, p. 12) observed that any statistical model consists of two inherently subjective components: ‘(i) a window (likelihood) for viewing observables in the world, and (ii) a prior reflecting a professional position of interest’. They also noted that, even if there is an agreement among the researchers with respect to the likelihood function, there are high chances of ‘agreement to disagree’ about prior distributions, especially in complex models with multi-dimensional parameter space (idem, credits to Jacek Osiewalski for drawing my attention to this).
- 2.
The parametric family of distributions F is conjugate with respect to the sample distribution \(p(x\left| \ \theta \right.)\), if and only if both the prior p(θ) and posterior \(p(\theta \left| \ x \right.)\) belong to F (e.g., Męczarski, 1998, pp. 21–22). Examples of such families are: Normal distributions for Normal samples (for with regard to the mean μ), Beta distributions for binomial samples (for the probability of success p), Gamma distributions for Poisson samples (with respect to λ), etc. (idem).
- 3.
The non-orthodox Bayesian character of empirical methods should be borne in mind, cf. http://Chapter 2.
- 4.
As indicated by Bernardo and Smith (2000, pp. 314, 361), under asymptotic Normality conditions, h(θ) is Fisher’s information about θ (or Fisher’s information matrix H(θ) for multivariate θ, when \(\pi ({\boldsymbol{\uptheta}}) \propto {\textrm{det[}}{\textbf{H}}({\boldsymbol{\uptheta}}){\textrm{]}}^{0{\textrm{.5}}}\)). For its formal derivation, see a part devoted to likelihood-based estimation in Section 8.2.
- 5.
For comparison, more detailed results for the longer-sample case are available in Bijak (2008b).
- 6.
MAPE and its non-percentage counterpart, the mean absolute error (MAE) were used in previous chapters to evaluate the ex-post forecast errors for 2005–2007.
- 7.
For a demographic application of a non-parametric bootstrap, consisting in resampling from the same set of observations, see e.g. Keilman and Pham (2006).
- 8.
In general, for a more detailed critical treatment of various inferential aspects the traditional sampling-theory approach made from the Bayesian perspective, see Bernardo and Smith (2000, pp. 443–488).
- 9.
The whole R environment is available from the R Development Core Team (2008), R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna; http://www.r-project.org (accessed on 10 June 2007). In computations, the built-in functions arima and predict.arima from the stats package of R 2.4.1 were used. For more information on the applications of R to Bayesian computations, see Albert (2007), Lynch (2007), and http://Chapter 9 of this book.
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Bijak, J. (2011). Evaluation of Presented Forecasts of European Migration. 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_8
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