Keywords: poverty measurement inequality measurement Gini index inequality measures decomposition latent variables financial risk statistics for financial markets

Latent variables and financial markets

The starting point is given by the concept of latent variables, which consents to study the main characteristics of financial markets, like risk, by referring to well-known statistical methodologies, like factor analysis.

The reference economic model for the interpretation and the measure of financial asset returns is given by the Arbitrage Pricing Theory (APT) of Stephen Ross.

A Monte Carlo experiment, in which the factor structure of the simulated data is known a priori, has been conducted in order to measure the distance between the “true” number of factors and the indications given by the alternative methodologies. The relative results suggest that, in the presence of a one-factor structure, Akaike information criterion and the likelihood ratio test overestimate the number of factors, while the Schwarz information criterion indicates the true value. However, as the number of factors increases, the Akaike information criterion seems to be the most precise method to evaluate the dimension of the factor structure.

An alternative method to the factor structure identification is given by reduced rank regression, which allows to analyse the relation between latent factors and macroeconomic variables. A number of factor which is strongly lower than the number of factors suggested by factor analysis, is suggested: in the Italian stock market, only 2-3 factors are relevant to the explanation of returns variability.

Furthermore, by using a state-space representation and the Kalman filter are used, dynamic factors in the Italian stock market are not detected.

In the following part of the research a new information criterion is suggested. As classic information criteria, the new criterion is based on two components: the first is given by the maximised log likelihood, which gives a measure of the empirical fit of the model, while the second, which is given by the product between the number of parameters and a coefficient alpha, represents a loss for an increase of the number of parameters, accordingly to the parsimony principle. In the new suggested criterion, the alpha coefficient is neither constant, like in the Akaike information criterion, nor merely a function of the number of observations, like in Schwarz information criterion and in the Hannan and Quinn information criterion, but it is a function of all the available information, i.e. in the case of the factor model of the number of observable variables and the number of observation.

Within the framework of the Italian stock market, by referring to Monte Carlo experiments an increasing relation between the number of observations and alpha, and a decreasing relation between the number of observable variables and alpha are detected. In order to formalise these relations, two different approaches are suggested. Firstly, the main reference is given by artificial neuronal networks, which are able to approximate unknown non linear relations between sets of variables. The obtained results is an improvement with respect to classic information criteria. Furthermore, a linear specification, where alpha is given as a function of the number of observation and the number of assets is proposed. Also in this case it is proved that the new criteria is more efficient and precise with respect to classical methods. Moreover, for the Italian stock market during the period 1990-1994 the presence of 2-3 factors is detected, while likelihood ratio test finds 3-4 and Akaike information criterion finds 3-5.

In the last part of the research, it is shown that one of the strongest hypothesis of the factor model, i.e. the cross-sectional independence of the error terms, is inadequate to measure real phenomena. Consequently, it is proposed to analyse financial data by referring to the approximate factor model, which generalises the classic factor model by relaxing the hypothesis of independence of the error terms. In this framework, classical methods of factor dimension estimation are not valid and hence it is necessary to refer to alternative strategies, which allow estimation and testing of the model. For the Italian stock market, only two factors are relevant in order to explain the returns variability.

Although of great importance, risk measurement does not exhaust the topic of unobservability in financial data analysis. A further important and still unresolved question regards portfolio choice, where the investment decision is observed, but the decision process remains unobserved. The suggestion is to deal with the unobservability problem in portfolio choice through a latent class analysis approach. The goal of this methodology is to identify latent classes representing groups of investors subject to different constraints on their assets. Such latent classes are determined through the relationship among a set of observed categorical variables representing socio-demographic and economic characteristics of the investors and one manifest variable measuring the effective portfolio choice.

This methodology is applied to international investment decisions in order to shed some light on the equity home bias puzzle. The basic idea is to classify the population of investors who do not effectively hold foreign equities into two sub-groups: one made of investors who are not precluded from the investment in foreign assets (potential international investors with potentially unexplained home bias), the other of investors who are actually prevented from any investment in foreign assets (home bias explained by unobserved constraints).

The methodology proposed is applied to the 2002 wave of the Survey of Household Income and Wealth of the Bank of Italy. The results show that 90 percent of households who do not hold foreign assets, are completely prevented from investing in foreign assets. They represent 89 percent of the whole population. Of the remaning 11 percent of households without restrictions precluding investment in foreign assets, 10 percent do not invest in foreign equities. These results imply that the equity home bias is completely explained for 89 percent of households, while it could be unexplained for 11 percent of households.

It is important to stress that, in order to correctly measure equity home bias, it is necessary to detect investors who are precluded from operating on foreign markets (89% of total households). With respect to the existing literature, where investors are considered as a single group and the equity home bias is measured by referring to all households, a strategic distinction is introduced.

The results also provide estimates of the effects of social, demographic, economic and financial characteristics on the probability of being an international investor. Globally it is possible to define the household profile consistent with the higher probability to be an international investor: between 40 and 60 years old, male, with a large houshold, resident in the northern Italy, and with a university degree.

An evaluation of the equity home bias based on the distinction between non-international and potential international investors is also proposed: considering this distinction gives powerful insights of the equity home bias. On the basis of traditional procedures home bias reaches a level of 84.5 per cent against the 73.5 per cent of our approach, thus indicating the relevant role of unobserved constraints.

Measurement and analysis of inequality

a) The measurement of inequality between performed by resorting to the Gini index decomposition allows to avoid the major difficulties related to the traditional income inequality measures decompositions and, at the same time, allows to correctly evaluate and compare the effects of different factors on total inequality. When a population is divided in two subgroups, Gini inequality index enables to obtain an immediate and straightforward measure of the contribution to total inequality attributable to the differences between non overlapping subgroups: this contribution is given by the difference between the population share and the income share of the poorest subgroup. Furthermore, also for the case of two overlapping subgroups it is derived a simplified measure of both transvariation and inequality between. Finally, by resorting to a simulation study, it is demonstrated how the difference between the population share and the income share of the poorest subgroup still represents the main source of the inequality between also for the case of overlapping subgroups.

b) Perception of poverty changes greatly in the last decades, leading to a wide theoretical debate, which states that income and wealth provide insufficient information on poverty condition. The adoption of a more general and multidimensional definition of poverty requires to adequate methodological tools for the measurement of poverty, actually generally still obtained on the basis of income only.

With respect to the traditional framework, a fuzzy multidimensional approach is proposed, wich offers fuzzy set poverty ratios for: (i) each household; (ii) the population of households; and (iii) the population of households by attribute. These ratios accurately represent the state of poverty, social exclusion and deprivation of the poor, and clearly identify the causes of poverty by order of importance.

From the information provided by the European Community Household Panel a set of 7 composite indicators for 12 European countries is obtained. Among these indicators the main factors of poverty are identified in the education and the activity of the reference person and in the dimension of the household residence. It is quite interesting to observe the great stability of poverty structure among European countries, which share the same problems in the field of social exclusion. Only for Spain an high source of poverty is detectable in heating and bath facilities of the household residence, while in the Netherlands and Ireland the indicator related to household structure and activity of the reference person seems to be a relevant element in poverty condition.

By identifying the poverty structure, the multidimensional approach can be extremely useful in order to implement socio-economic actions to reduce poverty diffusion: on the basis of the previous results, these actions should be addressed to reform educational system and labour market and to improve housing conditions.

A comparison between traditional unidimensional approach to the measurement of poverty and the new multidimensional approach is also performed: the results of a rank correlation analysis allow to demonstrate that the two approaches define two different sets of poor households.

Therefore any socio-economic policy to reduce poverty developed on the basis of income information is likely to not achieve its proposed goals, being addressed to socioeconomic units which are, in effect, non-poor. Only in the framework of the multidimensional approach it is possible to correctly individuate the set of the poor and to formulate actions able to reduce poverty.