Multivariate Hierarchical Bayesian space-time models in economics

Transcription

1 Multivariate Hierarchical Bayesian space-time models in economics Ioannis KAMARIANAKIS and Poulicos PRASTACOS Regional Analysis Division Institute of Applied and Computational Mathematics Foundation for Research and Technology-Hellas (FORTH) P.O. Box Heraklio GREECE Abstract: The aim of this paper is to model the Gross Regional Product of Greece at the prefecture/regional level accounting for the spatiotemporal evolution of each sector, as well a the interdependencies of the three sectors used. Hierarchical multivariate Bayesian space-time models are used to achieve more flexible methods for the analysis of data concerning the national economy. The first stage of the hierarchical model specifies a measurement error process for the data at hand, in terms of a state process. The second stage includes large-scale annual variability plus a space-time process that also accounts for sectoral interdependencies. This space-time process aims to describe the small-scale fluctuations and depends much in the notion of neighbour that is not as straightforward as it is in processes of (geo) physical type. Putting priors on previous stages parameters completes the Bayesian formulation, which is implemented in a Markov chain Monte Carlo framework. Separate multivariate space-time autoregressive models could be an alternative approach to the problem at hand. Using a hold out sample, the forecasting performance of the presented approach is compared to the performance of three separate model one for each sector of the GRP. Keywords: spatial time-serie Gibbs sampling, MCMC, spatial-temporal dependencie multivariate analysis 1. Introduction Data sets concerning economic indicators are usually organized by units of time such as month quarters or year as well a by geographical constructs such as countie regions or states. Statistical modelling of datasets that refer to processes that evolve in both space and time is a major tool for the quantification of spatial and temporal effects and the spatiotemporal interactions in the evolution of such processes. A respectable number of techniques for space-time modelling have been developed over the last decade most of them focusing on environmental and epidemiological applications. The most significant problem faced in space-time modelling is that spatial and temporal 503

2 Ionnis Kamarianakis and Poulicos Prastacos variability might be interrelated. Spatial behaviour might change at different points of time; likewise temporal cause and effects might differ at different locations in space. When modelling economic processes additional complications arise from the neighbourhood effects. Although, neighbourhood effects account for the interrelationships between adjacent location there are size/magnitude effects that can not be easily accounted for. To make this more clear consider the following example. Greece is subdivided in 51 prefecture with the Attica prefecture where Athens is located accounting for almost 0% of the national economy. It would be reasonable to expect that the Gross Regional Product (GRP) of the services sector in Heraklio, the largest prefecture in the island of Crete, will be more strongly affected by changes in the services sector of the Athens area rather than those of Rethymnon, another prefecture of Crete. Nonetheles the Athens area is geographically away but the economy is advanced, whereas Rethymnon is close by but the economy is weak. Our approach aims to take advantage of recent increases in computational speed and numerical advances (e.g. Markov chain Monte Carlo) that allow us to implement multivariate Bayesian space-time models in a hierarchical framework. Such specifications provide simple strategies for incorporating complicated space-time interactions at different stages of the model s hierarchy, and the models are feasible to implement in high dimensions. Pfeifer and and Deutch (1980 a,b) where among the first to develop space-time modelling techniques (for lattice space in the context of STARMA models. Stoffer (198) developed and applied multivariate space-time modelling techniques (STARMAX model for environmetric applications; Brown, Le and Zidek (199) adopted the Bayesian methodology to interpolate a vector-valued random response field and also demonstrated an application in environmetrics. Finally, Wikle, Berliner and Cressie (1998) made a significant contribution in environmetric space-time modelling, following the Hierarchical Bayesian approach. There are only a few cases of space-time models applied to economic none of them focusing on GRP analysis. Pfeifer and Bodily (1990) used STARMA models to describe and forecast hotel data, Gelfand et al.(1998) used Bayesian space time models with conditionally autoregressive priors for the spatial random effects in order to model residential sale whereas Pace et. al. (000) created site-specific time series and used a space-time model for the residuals in an effort to forecast more efficiently real estate prices. The objective of this paper is to develop a methodology for understanding the dynamics of the GRP of Greece by applying a Bayesian hierarchical approach that considers space, time and sectoral interrelationships. GDP growth at the country level is frequently analysed in a regression framework with labour force growth, equipment and non-equipment investment being the independent variables. De Long and Summers (1991) and Nonneman and Vanhoudt (1996) have done representative works in this context. Unfortunately, information of the aforementioned type that would almost surely have high explanatory value is not available at the prefecture level in Greece. In the following section the available datasets are introduced and the results of an explanatory analysis are presented. Then, an overview of the modelling strategy is presented followed by a discussion of the relevant model-building details and test results obtained. Finally, a short discussion on the results and their implications is done. 50

3 Multivariate Hierarchical Bayesian space-time models in economics. Description of the datasets and explanatory analysis The dataset for our study was obtained from the National Statistical Service of Greece and consists of 15 annual observations ( ). For every year and each one of the 51 prefectures in Greece there is information on the Gross Regional Product (GRP) of the three main sectors of the economy. The primary sector includes agriculture, cattle breeding, forestry and fishery activitie the secondary sector represents mining, manufacturing, construction activities and utilitie whereas the tertiary sector accounts for trading, banking, insurance, housing, transportation & communication, public administration & security, health & education and other services. Standardised per capita figures were obtained by dividing GRP by the corresponding population; logarithms were then taken in order to achieve variance homogenisation since in the early years of the study period all sectors exhibited a rather unstable behaviour. Part of the dataset is depicted in Figure 1. Only the first 1 years of observations were used for estimating the parameters of the models. The data of the last year were used for evaluating the one-step-ahead forecasting performance of the adopted modelling strategy and to compare it to the forecasts obtained from three separate multivariate space-time autoregressive models (one for each sector). Figure 1: Time series plots for the prefectures in the regions of Peloponiso West Greece and Attiki First, simple regressions for each prefecture and sector were carried out with time being the explanatory variable. The coefficients of the time variable, the slope, which depict the long-term growth behaviour, are presented in Figure. As part of the exploratory analysis the concept of β-convergence was examined, that is whether regions that start out with below average sectoral incomes tend to grow faster than regions that start with aboveaverage incomes. For an extensive discussion on β-convergence we refer to Barro and Sala- I- Martin (1995) and Funke and Strulik (1999). The scatterplots in Figure 3 constitute the traditional and simplest way of testing for β-convergence and indicate slow divergence for the primary sector and convergence for the secondary and tertiary sectors. The fit of the curves is unsatisfactory though, (there are some obvious outlier and panel data techniques should also be applied in order to confirm the above results. 505

4 Ionnis Kamarianakis and Poulicos Prastacos Figure : Magnitudes of the slopes of the regression lines fitted to each prefecture s sectoral GRP with time as the explanatory variable. For the vast majority of the case 8 % of the 3 x 51 sector-prefecture case the slope terms are statistically significant. This indicates that a model with a linear trend is appropriate for representing the temporal evolution of the area and sector specific time series. The remaining cases seem to fluctuate around a horizontal line. Additionally, for each region, the prefecture-specific time series for each sector appear to be strongly correlated; this is more pronounced in the second half of the study period. Figure 3: Scatterplots and regression lines for the detection of sectoral β-convergence. In order to proceed to the modelling stage, the neighbourhood structure of the prefectures under study had to be defined. For a prefecture in the mainland part of Greece the set of its neighbours is comprised by the geographical neighbours that are economically stronger. That i it was assumed that only the more advanced neighbours have a significant influence on a prefectures economy. For prefectures that are island parts of islands or sets of island neighbors were defined to be the geographical one where these existed, but also mainland prefectures where large harbors are located and connection through ferries has been established on a regular basis. In the modeling stage, for the sake of 506

5 Multivariate Hierarchical Bayesian space-time models in economics parsimony, at most two neighbors for each prefecture are considered. These are taken to be the ones with the most advanced economies concerning the sector under study. It is obvious from the above definitions that different neighborhood structures arise for different sectors of the GRP. A non-symmetric neighborhood pattern was adopted, that is if prefecture s 1 is considered a neighbor of s for sector then s cannot be considered a neighbor of s 1. This approach, to our knowledge, is not encountered in econometric spacetime modeling, and was proved helpful during estimation. 3. The Multivariate Hierarchical Space-Time model 3.1 Measurement error process Let Z denote the observed data and Y the process of interest. A statistical measurement error model is then specified by: Z Y, (1) 1 where θ 1 represents a collection of parameters. The sampling plan that the model represents in this case is that we observe the multivariate space-time process of interest Y, with error. The methods used for the calculation of the sectoral GRP plus the fact that the prefectures populations for the years that lie between the census ones are not observed, but expected value make it reasonable assume such a model. Hence, the dataset in our case may be represented as Z Z ( : t where, i stands for the sector under study and s and t are the spatial and the temporal indexes respectively. Conditional on Y and θ 1, the Z( are assumed independent and a Gaussian probability density is a reasonable assumption: Z () ( ~ Y (, t Thu : (, 1 i, t accounts for the measurement error variances. 3. Large and small scale variation Now by adopting a methodology similar to that of Wikle, Berliner and Cressie (1998) we L L t; a(, ( and model Y conditionally on two processes denoted by S S ( : (, and a collection of parameters θ. The large-scale temporal behavior is separated from the one at short time scales. For each sector, area and time point we have: t; a(, ( S( ( ) Y ( L t (3) 507

6 Ionnis Kamarianakis and Poulicos Prastacos L t; a(, ( is a large-scale temporal model with sector-prefecture specific parameters. As indicated previously a linear trend model may adequately describe the large-scale process. In the case of quarterly GRP data seasonal effects may be incorporated by letting L be a cosine function (see Wikle, Berliner and Cressie (1998) for an application of this kind). Thu the linear trend component is modeled as a set of lines with intercepts and slopes that vary spatially-sectorally: L t a( ( t. () where, ) ~ (, ), a i s alpha i s and, ~ beta(, ( ( i. The role of the S-process is to account for sectoral, spatial and temporal dynamics beyond those accounted for in long-term annual behavior. Roughly, we may view the contributions to Y modeled through L as each prefecture s long-term dynamic whereas S stands for the short-term neighbor and other sector effects. Its general form may be represented as: S, t c( S(, t 1 d ( S( N ), t... d ( S( N t ( 1 1 (5) f S ( j,, t f ( S ( k,, t e (, t 1( l l ), where the first term at the right hand side of the equation is an autoregressive one, the following k terms represent the small scale-fluctuations of the neighbors of prefecture s for sector i (N 1,,N l represent the neighbors of, the next two terms represent the small scale fluctuations of the remaining two sectors of the economy and e (, t is a spatially and sectorally independent white noise process. Identifiability issues arise at this stage as they do in Wikle et al (1998) and Gelfand et al. (1998) since S and ε appear only through their sum in (3). In Bayesian analysis with proper probabilities on all quantities though, such issues just require care in interpretation of the estimations (see Besag et al., 1995). For several prefectures the number of economically stronger neighbors is -6, and this causes multicollinearities. Thu for the sake of parsimony, only the two more advanced neighbors concerning the sector under study are taken into account for the prefectures that have more than two stronger neighbors. L and S are mutually independent, conditional on a set of parameters that is denoted by θ and can be written as θ =( θ L,,θ S ) or alternatively:, S L S L. (6) The above discussion may be cast into a more general framework by writing: S WS BSt et (7) t1 t1 508

7 Multivariate Hierarchical Bayesian space-time models in economics where the S t, S t+1 are vectors containing 51X3 site-sector specific element W is an exogenously constructed (51X3)X(51X3) matrix containing information on neighboring sites and covarying sector B is a vector that contains the autoregressive parameters and e t is the vector that represents the errors. W is a matrix specified by the model builders prior to analyzing the data; it is a basic mechanism for incorporating the relevant physical characteristics of the system into the model form. The ( s are zero-mean random variables which model noise. They represent the unexplained variations in the second stage, which in principle must be modeled by a AT AT covariance matrix (A accounts for the number of sector Σ for the number of prefectures and T for the length of our time serie. However, we assume that model features such as S, explain much of the sector-space-time structure of the Y proces so it might be assumed that the Y( s are conditionally independent random variables. In fact, we assume that: Y ( ~ N L t; a(, ( S(, (8) Y ( and set A,1),..., ( A, ), ( A,1),..., ( A, ) 3 Y ( 1 Y 1 Y Y 3 to account for the variance of the error process. 3.3 Priors on second stage parameters For the specification of priors for the model parameters a partition ( 1), (), (3) into hyperparameters and a conditional independence relation is assumed:, (1) () (3) 1, (9) Analogously, ( ) ( ) ( L), ( S) and the conditional independence assumption is represented as: is partitioned as ) ( L) ( ). (10) ( L S S The parameters of the explanatory variables of the L and S-processes are assumed to vary according to independent Gaussian distributions whereas the variances specified in all stages are assumed to follow independent inverse Gamma distributions. For the latter, widely dispersed noninformative priors were chosen, whereas for the former prior means and variances were estimated by regression in the exploratory phase. The exact posteriors can be derived through slight modifications of the ones presented in Wikle et al. (1998). 509

8 Ionnis Kamarianakis and Poulicos Prastacos. Bayesian Estimation The posterior distributions of model parameters and one-step-ahead forecasts are derived by implementing the Gibbs sampling approach, which is very frequently adopted in Statistics literature since the work of Geman and Geman (198). The Gaussian distributions and the conjugate priors presented in section 3 of this paper imply their straightforward derivation. With respect implementation issue it must be noted that at a preliminary stage simulations with different starting values were run and the Gelman and Rubin (199) convergence criterion was examined. Convergence was suggested for all parameters after the first 500 iterations. Thu we run a 0000-iteration chain and posterior means of all quantities were estimated by the corresponding sample means (neglecting the burn-in period). All MCMC samples had autocorrelations that dropped to insignificance after the first lags. The Gibbs sampling results proved to be close to our prior beliefs for all identifiable parameters and the rate of convergence was satisfactory. Care is needed though, for the interpretation of Y, S (the variances of the first two stage. From a classical perspective these variance components are not identifiable in the model and one not be very confident about their posterior results. However, sensitivity analysis with different priors showed that the remaining parameters are not sensitive to these changes. 5. The forecast test As mentioned earlier, the observations of the last year of the study period were used as a hold out sample so as to compare the forecasting performance of the presented approach to that of three separate models (one for each sector). It should be pointed out that this process was carried out just for illustrative purposes. We are not suggesting that either of the two modeling approaches is necessarily best suited for forecasting the GRP of the prefecture since several important explanatory variables could be used to create a forecasting system that could possibly be superior to the approaches considered here. Additionally, the rather limited size of the available dataset does not justify much confidence on the comparison. The alternative models belong to the family of multivariate space-time autoregressive models where the current state of a process y t may be presented in terms of its previous states y t-1, y t-,,, y t-q and a set of covariates c t and their previous states c t-1, c t-,, c t-m. The length and denseness of our time-series however, do not encourage us to implement time lagging of high order. The models used with are of the general form: y it 1 j0 t j D (11) ij y t j B z i it w it 510

9 Multivariate Hierarchical Bayesian space-time models in economics where y it in our case are vectors containing the site specific observations for sector i at time t, φ t-j are parameter z it represents a set of (sector specific) covariate B i is the appropriate (sector specific) coefficient matrix and w it is a white noise process. D i0 is a square matrix that contains the neighborhood structure identified separately for each sector. It has a dimension equal to the number of areas and the elements are zeros or ones. One is used to denote a pair of neighbor while all other elements are zero. D i1 is a sparse matrix that contains only one nonzero element in each line. That element accounts for the autoregressive coefficient of the area-sector under study. Finally, the reader should notice the existence of the instantaneous spatial component (j=0), which is not present in STARMA or STARMAX approaches of other authors (for a discussion on the usefulness of the instantaneous spatial interaction we refer to Cressie (1993) p.50). That component does not create problems during the estimation of the model parameters since a hierarchical neighborhood structure has been adopted. For each sector, the set of possible covariates included the time and the observations on the other two sectors. The neighborhood structure for each sector remained the same as before. Conditional least squares estimation indicated that time should be used as the only covariate in all sectors models; moreover, time lagging proved to be unnecessary for all cases. Thu a multivariate spatial model was the result. Some of the results obtained are shown in Table 1. A detailed pairwise comparison of forecast errors showed that the Multivariate Hierarchical Bayesian approach outperforms the alternatives in 58 % of the cases for the primary sector, 1% for the secondary and 5% for the tertiary sector. In fact, for 81 out of 153 forecasted values the sum of squared forecast errors across the 3 sectors was smaller than the alternative approach. A related paired t-test was used to test whether the average of the 153 differences between the sum (across prefectures and sector of squared forecast errors for the two approaches was significant of zero; its level of significance (as expected) indicated that the differences in forecasting performance for the two models are not insignificant. Figure : Observed and predicted values produced by the multivariate spatial model. 511

10 Ionnis Kamarianakis and Poulicos Prastacos Bayesian Model Multivariate spatial model Prefecture (Sector) Neighbors 95%Conf. Interval Forecast Observed data (199) Forecast 95%Conf. Interval Hraklio(p) None Hraklio( Attiki Hraklio( Attiki Lasithi(p) Hraklio Lasithi( Hraklio Lasithi( Hraklio Rethymno(p) Chania Rethymno( Chania Hraklio Rethymno( Chania Hraklio Chania(p) None Chania( Attiki Chania( Attiki Lakonia(p) None Lakonia( Arkadia Lakonia( Messinia Arkadia Messinia(p) Ilia Lakonia Messinia( Arkadia Messinia( None Arkadia(p) Lakonia Messinia Arkadia( Korinthos Hlia Arkadia( Argolida Messinia Table 1: Representative results for several prefectures obtained from the estimation of the two models Model Sector Primary Secondary Tertiary Bayesian M-STAR Table : Average forecast errors for the holdout sample 6. Conclusions This paper has illustrated the Multivariate Hierarchical Bayesian space-time approach in the setting of econometrics. The adopted modeling strategy is useful in both predicting future observations and understanding the interrelations of different prefectures and economic sectors. There are several built-in advantages of this approach over univariate or multivariate (non space-time) modeling procedures. First, this model comprises all the amount of information available. It also has a physical interpretation and its forecasting 51

11 Multivariate Hierarchical Bayesian space-time models in economics ability is competitive to the one that alternative space-time methodologies achieve. Our approach though, is not the most general and many extensions and modifications for the large and small-scale fluctuations could be explored since the model was tailored for the problem at hand. References [1] Barro, R.J., Sala-I-Martin, X., (1995). Economic Growth. New York/London. [] Besag, J., Green, P., Higdon, D. and Mengersen, K. (1995) Bayesian computation and stochastic systems. Statistical Science, 10, [3] Brown, P.J., Le, N.D. and Zidek, J.V. (199) Multivariate spatial interpolation and exposure to air pollutants. The Canadian Journal of Statistic, [] Cressie, N.A.C. (1993). Statistics for Spatial Data. Wiley, New York. [5] De Long, P.J., Summer L.H., (1991). Equipment investment and economic growth. Quarterly Journal of Economics 106, [6] Funke, M. and Strulik H. (1999). Regional growth in West Germany: convergence or divergence? Economic Modelling, 16, [7] Gelfand, A.E., Ghosh S.K., Knight, J.R. and Sirman C.F. (1998). Spatiotemporal modelling of residential Sales data. Journal of Business and Economics Statistic 16, [8] Gelman, S. and Rubin, D.B. (199). Inference from iterative simulation using multiple sequences. Statistical Science, 7, [9] Geman, S. and Geman, D. (198). Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, [10] Nonneman, W., Vanhoudt, P., (1996). A further augmentation of the Solow model and the empirics of the economic growth for OECD countries. Quarterly Journal of Economics 111, [11] Pace, K. R., Barry, R., Gilley, W. and Sirman C.F. (000) A method for spatial-temporal forecasting with an application to real estate prices. International Journal of Forecasting, 16, 9-6. [1] Pfeifer, P.E., and Bodily, S.E. (1990). A test of space-time ARMA modelling and forecasting of hotel data. Journal of Forecasting, 9, [13] Pfeifer, P.E. and Deutsch, S.J. (1980a). Identification and interpretation of first order space-time ARMA models. Technometric, [1] Pfeifer, P.E. and Deutsch, S.J. (1980b). A three stage iterative procedure for space time modelling. Technometric, [15] Stoffer, D.S. (198). Estimation and Identification of space-time ARMAX models in the presence of missing data. Journal of the American Statistical Association, 81, [16] Wikle, C.K., Berliner L.M. and Cressie, N.A.C. (1998) Hierarchical Bayesian space-time models. Environmental and Ecological Statistic 5,