Abstract
This paper presents an innovative approach to the study of regional economic dynamics within a nonlinear continuous-time econometric framework—a generalized specification of the Lotka–Volterra system of equations. This specification, which accounts for interdependent behavior of three industrial sectors and spillover effects of activities in neighboring regions, is employed in an analysis of five Italian regions between 1980 and 2003. For these regions, we report estimation results, characterize the varying systems dynamics, analyze the models’ local and global stability properties, and determine via sensitivity analyses which structural features appear to exert the greatest influence on these properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Jean Paelinck made this observation in a session on spatial econometrics at the North American Meetings of the Regional Science Association in 2005 in Las Vegas, Nevada.
We prefer to allude to the effects of distance instead of space, because space, as a metric or dimension, cannot affect anything. Neither can time. It is the intensity of some causal factor operating at a relative distance over some period of time that is effective
In employing a generalized L–V system model to study the dynamics of regional economies, we follow the suggestion of Samuelson (1971) and the examples of Dendrinos and Mullally (1985), as well as Arbia and Paelinck (2003) and Paelinck (2005). Arbia and Paelinck (2004), Paelinck (2005), and Piras et al. (2005), which is unpublished, are the only continuous-time econometric investigations of L–V system models of regional economies of which we are aware.
An anonymous referee has noted that in other literatures, generalized L–V models are referred to as ‘niche models’ or ‘interrelated logistics’ models.
The number of contiguous regions varies with the region of reference.
An anonymous referee has suggested that the nature of spillover effects between trade–partner regions should also be investigated. We agree, but time series data on interregional trade in the EU are not presently available. This referee has also suggested that human capital should be included as a state variable. We have not included this variable in the present case because of potential degrees-of-freedom problems in the system estimation, potential measurement error problems with human capital indicators, and the absence of strong hypotheses about the intersectoral and interregional effects of human capital on output and installed capacity.
Sectoral measures of real capital per employee were aggregated from annual observations on capital formation using the perpetual inventory method net of annual depreciation, whose percentages were estimated using OECD national aggregate figures. All values are expressed in 1995 current prices.
The term ’t-ratio’ simply denotes the ratio of a parameter estimate to the estimate of its asymptotic standard error, and does not imply that this ratio has a Student’s t distribution. This ratio has an asymptotic normal distribution and so in a sufficiently large sample it is significantly different from 0 at the 5% level if it lies outside the interval ±1.96 and significantly different from zero at the one per cent level if it lies outside the interval ±2.58.
Note that the required number of observations for parameter identification—and for the likelihood function to be well defined—in linear FIML estimation is the number of pre-determined variables in the model, less the number of over-identifying conditions (Sargan 1983). The considerations offered by Fisher (1966) suggest that this number should be lower in the nonlinear case. See Cramer (1986) for discussions of other conditions which may contribute to problems of parameter identification in FIML estimation.
Estimation of the eigenvalues for the eigensystems of the corresponding regional economies and the computation of the partial derivatives taken with respect to the parameter estimates were carried out with Wymer’s program CONTINEST in the WYSEA package. For a details of the calculations entailed, see Wymer (1982) or the WYSEA manual.
This is the case because in linearizing the models for the computation of the eigenvalues, the constant terms and exogenous variables vanish.
Estimates of these values were obtained for the corresponding regional economies and the computation of the partial derivatives taken with respect to the parameter estimates was carried out with Wymer’s probram APREDIC in the WYSEA package. For details of the calculation entailed, see Wymer (1997).
References
Arbia G, Paelinck JHP (2003) Economic convergence or divergence? Modeling the interregional dynamics of EU regions, 1985–1999. J Geogr Syst 5:291–314
Barro RJ, Sala-I-Martin X (1995) Economic growth. McGraw-Hill, New York
Carlberg M (1980) A Leontief model of interregional economic growth. J Reg Sci 20:30–40
Cramer JS (1986) Econometric applications of maximum likelihood methods. Cambridge University Press, Cambridge
Dendrinos DS, Mullally H (1985) Studies in the mathematical ecology of cities. Oxford University Press, Oxford
Dendrinos DS, Sonis M (1986) Variational principles and conservation conditions in Volterra’s ecology and in urban relative dynamics. J Reg Sci 26:359–377
Donaghy K (2000) General stability analysis of a non-linear dynamic model. In: Reggiani A (eds) Spatial economic science. Springer, Berlin, pp 243–257
Donaghy K, Dall’erba S (2004) Structural and spatial aspects of regional inequality in Spain. Paper presented at the North American Meetings of the Regional Science Association International, November, Seattle Washington
Fingleton B (2001) Equilibrium and economic growth: spatial econometric models and simulations. J Reg Sci 41:117–147
Fisher FM (1966) The identification problem in econometrics. McGraw-Hill, New York
Frisch R (1933) Editorial. Econometrica 1:2
Gandolfo G (1980) Economic dynamics: methods and models, 2nd Revised edn. North-Holland Publishing Company, Amsterdam
Hahn W (1963) Theory and applications of Lyapunov’s direct method. Prentice-Hall, Englewood Cliffs
La Salle J, Lefschetz S (1961) Stability analysis by Lyapunov’s direct methods. Academic, New York
Lotka A (1932) The Growth of mixed populations: Two species competing for a common food supply. In: Scudo F, Ziegler J (eds) The golden age of theoretical ecology: 1923–1940. Lecture Notes in Biomathematics, vol 22. Springer, Heidelberg, (1978) 274–286
Medio A (1992) Chaotic dynamics: theory and applications to economics. Cambridge University Press, Cambridge
Paelinck JHP (1992) De l’économètrie spatiale aux nouvelles dynamiques spatialisées. In: Derycke PH (ed) Espace et dynamiques territoriales. Economica, Paris, pp 137–154
Paelinck JHP (2005) Lotka–Volterra models revisited: A mixed specification with endogenous generated SDLS-variables, and a finite automaton version. Paper presented to the North American Meetings of the Regional Science Association International, November, Las Vegas
Piras G, Donaghy K, Arbia G (2005) Continuous-time regional convergence: A comparison of various estimation procedures. Paper presented to the North American Meetings of the Regional Science Association International, November, Las Vegas
Puga D (2001) European regional policies in light of recent location theories. CEPR Discussion Paper 2767
Samuelson P (1971) Generalized predator–prey oscillations in ecological and economic equilibrium. Proc Natl Acad Sci 68:980–983
Sargan JD (1983) Identification of models with autoregressive errors. In: Karlin S, Amemya T, Goodman L (eds) Studies in econometric time series and multivariate statistics. Academic, New York, pp 169–205
Sydsaeter K, Strøm A, Berck P (1999) Economists’ mathematical manual, 3rd edn. Springer, Berlin
Volterra V (1926) Variations and fluctuations in the numbers of coexisting animal species. In: Scudo F, Ziegler J (eds) The golden age of theoretical ecology: 1923–1940. Lecture Notes in Biomathematics, vol 22. Springer, Berlin (1978), pp 11–17
Wymer CR (1972) Econometric estimation of stochastic differential equation systems. Econometrica 40:565–577
Wymer CR (1982) Sensitivity analysis of economic policy. Presented at workshop on the application of sensitivity analysis to the evaluation of financial policies, World Bank, Washington. In: Gandolfo G, Marzano F (eds) (1987) Essays in memory of Vittorio Marrama. Giuffrè, Milano, pp 953–965
Wymer CR (1993) Estimation of nonlinear continuous-time models from discrete data. In: Phillips PCB (ed) Models, methods, and applications of econometrics. Basil Blackwell, Cambridge, Mass, pp 91–114
Wymer CR (1997) Structural non-linear continuous-time models in econometrics. Macroecon Dyn 1:518–548
Wymer CR (2004) WYSEA (Wymer System Estimation and Analysis) software
Acknowledgments
We are grateful to Jean Paelinck and two anonymous referees for their constructive comments, criticisms, and suggestions for improvement. The usual disclaimers apply.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A
Table 5.
Appendix B
Table 6.
Appendix C
Table 7.
Rights and permissions
About this article
Cite this article
Piras, G., Donaghy, K.P. & Arbia, G. Nonlinear regional economic dynamics: continuous-time specification, estimation and stability analysis. J Geograph Syst 9, 311–344 (2007). https://doi.org/10.1007/s10109-007-0049-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10109-007-0049-x
Keywords
- Regional dynamics
- Continuous-time econometrics
- Generalized Lotka–Volterra system
- Local and global stability analysis