Dynamic Generalized Structural Equation Models
Luigi Ippoliti
Department of Economics, University G. d'Annunzio, Chieti-Pescara, Italy

Abstract

This work discusses a number of conceptual issues pertaining to the study of the relationshipexisting between two groups of variables which are supposed to be spatially and temporally correlated. Since it is assumed that this relationships can be studied in a reduced latent space, we
provide an overview of the motivations for including spatial effects in a dynamic factor model,
both from a theory-driven as well as from a data-driven perspective. Considerable attention ispaid to the inferential framework necessary to carry out estimation and to the different assumptions, constraints and implications embedded in the various model specications. The discussion
combines insights from (spatial) econometrics and geostatistical literature.
Key Words: Dynamic factor models, Structural equation models, Spatio-temporal processes,

Bayesian inference, Spatial econometrics, Geostatistics.

1 Introduction
The idea of borrowing information from different but related sources can be very powerful forstatistical analysis. It proved to be very useful in the last decades where complex data structuresbegan to be tackled, as they required sophisticated modeling strategies. In this work we consider
the problem of modeling high-dimensional multivariate, spatially and temporally referenced data.
This problem has enjoyed widespread popularity in the last years and requires the denition ofgeneral and exible statistical models where the temporal and cross-sectional dependencies mustbe accommodated.
Spurred by recent advances in Geo-spatial data acquisition technologies, it is often desirable
for these data to examine the relationships existing between one or more dependent variables andsome other linked covariates. This can be achieved in a number of ways, though there might beno single approach which can be considered uniformly as being the most appropriate.
Among the different methodologies proposed in the literature, dynamic factor models (DFMs)
have grown signicantly in popularity and have been shown to be very useful for exploratory
analysis, policy analysis and forecasting in a data-rich environment. DFMs have been widely
developed in both methodological and practical issues, and have become a standard tool for increasingly high-dimensional modeling of time series. They have been extensively used in macroeconomics and nance with the core idea of explaining the common dynamic structure of themultivariate time series through a set of common (time series) factors. This is achieved by theintroduction of exible temporal correlation structures for the latent factors, previously assumed
to be independent. This renders the DFM capable of assessing the complexity of time series data.
Models along these lines include Geweke (1977), Molenaar (1985), Engle and Watson (1981),
Pe~na and Box (1987), Forni et al. (2000).
Special attention will be devoted here to the use of a dynamic factor analytic approach in
the framework of spatial statistics. It will be shown that this is not only an important area of
application but also that this area can receive several benets from this modeling approach (see,for example, Ippoliti et al., 2012; Valentini et al., 2013).
A key property of much spatio-temporal data is that observations at nearby sites and times willtend to be similar to one another. Then, factor analysis assumes that the cross dependence can becharacterized by a nite number of unobserved common factors, possibly due to common shocks
1that affect all the spatial sites, albeit with different intensities. Thus, the strong co-movement andthe high correlation among the series, amplied by the presence of spatial correlation, suggestthat both observable and unobservable factors must be at place.
In this paper, we thus approach the analysis of multivariate spatio-temporal processes fromthe perspective of recent developments of dynamic factor models. Through a fully Bayesian approach, we contribute to the recent literature by melding together dynamic factor models, spatialregression models and geostatistical techniques, in order to explain the multifactorial nature ofmany spatio-temporal data. We assume that the relationships existing between the groups ofdependent and regressor variables can be studied through a temporally dynamic and spatially descriptive model, hereafter referred to as dynamic generalized structural equation (DGSE) model.
The proposed model has an intuitive appeal and enjoys several advantages. First, our model
formulation exploits the spatio-temporal nature of the data and explicitly denes a non-separable
spatio-temporal covariance structure of the multivariate process. Second, since the data have amultivariate and multidimensional structure, in that several time series can be measured at specicspatial sites, the temporal relationships between dependent and regressor variables is modeled ina latent space. The observed processes are thus described by a potentially small set of common
dynamic latent factors with the advantage of overcoming the difculties of interpreting the relationships under study due to collinearity and low signal-to-noise ratio issues. Temporal forecastsof the variables of interest can also be obtained by only modeling the dynamics of a few commonfactors. Third, by modeling the spatial variation via spatially structured factor loadings, we entertain the possibility of identifying clusters of spatial sites that share common time series components. Through the spatial modeling of the factor loadings, spatial interpolations of the observed
variables are also straightforward. Fourth, several general structures that make use of different
covariate information, can be easily accommodated in the different levels of the hierarchy. Fifth,
the DGSE model offers a unied approach suitable to deal with variables and indicators measured at different scales and coming from different spatial sources. Hence, the model provides
a simple solution to the misalignment problems which, for example, normally occurs in healthcare research. Lastly, the model specication is not limited to normally distributed variables, but
it can be extended to handle more types of variables from an exponential family.
Details on data analysis will be given in an extended version of the present abstract in theframework of environmental and health care research.

References
Engle, R., Watson, M. (1981). A one-factor multivariate time series model of metropolitan wagerates. Journal of the American Statistical Association, 76, 774-781
Forni, M., Hallin, M., Lippi, M., Reichlin, L. (2000). The generalized dynamic factor model:identication and estimation. The Review of Economics and Statistics, 82, 540-554
Geweke, J. (1977). The dynamic factor analysis of economic time series models. In Latent Variables in Socio-Economic Models (eds. D. J. Aigner and A. S. Goldberger), pp. 365-383, North
Holland, Amsterdam
Ippoliti, L., Valentini, P., Gamerman, D. (2012). Space-Time Modelling of Coupled SpatioTemporal Environmental Variables. Journal of the Royal Statistical Society, Series C (AppliedStatistics), 61, 175-200
Molenaar, P. C. M. (1985). A dynamic factor model for the analysis of multivariate time series.
Psychometrika, 50, 181-202
Pena D., Box, G. (1987). Identifying a simplifying structure in time series. Journal of the American Statistical Association, 82, 836-843
Valentini, P., Ippoliti, L., Fontanella, L. (2013). Modeling US housing prices by spatial dynamic
structural equation models. The Annals of Applied Statistics, 7, 763-798

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