Quantifying chaos: a tale of two maps

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This paper presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.

The relative importance sector and regional factors in Italy

The benefits of sector and regional diversification have been well documented in the literature but have not previously been investigated in Italy. In addition, previous studies have used geographically defined regions, rather than economically functional areas, when performing the analysis even though most would argue that it is the economic structure of the area that will lead to differences in demand and hence property performance. This study therefore uses economically defined regions of Italy to test the relative benefits of regional diversification versus sector diversification within the Italian real estate portfolio. To examine this issue we use constrained cross-section regressions the on the sector and regional affiliation of 14 cities in Italy to extract the “pure” return effects of the different factors using annual data over the period 1989 to 2003. In contrast, to previous studies we find that regional factors effects in Italy have a much greater influence on property returns than sector-specific effects, which is probably a direct result of using the extremely diverse economic regions of Italy rather than arbitrary geographically locations. Be that as it may, the results strongly suggest that that diversification across the regions of Italy used here is likely to offer larger risk reduction benefits than a sector diversification strategy within a region. In other words, fund managers in Italy must monitor the regional composition of their portfolios more closely than its sector allocation. Additionally, the results supports that contemporary position that ‘regional areas’ based on economic function, provide greater diversification benefits rather than areas defined by geographical location.