THEORETICAL ASPECTS CONCERNING THE USE OF THE STATISTICAL- ECONOMETRIC INSTRUMENTS THE ANALYSIS OF THE FINANCIAL ASSETS

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1 THEORETICAL ASPECTS CONCERNING THE USE OF THE STATISTICAL- ECONOMETRIC INSTRUMENTS THE ANALYSIS OF THE FINANCIAL ASSETS Professor Constantin ANGHELACHE PhD. Academy of Economic Studies, Bucharest, Artifex University of Bucharest Lecturer Mădălina Gabriela ANGHEL PhD. Artifex University of Bucharest Abstract The econometric modelling of the fi nancial variables aims to obtain models meant to forecast to the best their future values, taking into account the inertial character of the progress of the analysed processes as well as the relatively predictable character of their evolution in response to certain deviations from the observed past. The econometric regression models or those based on the use of the chronologic series allow us to do prognoses on the ground of the observations subject of the analysis. Although requiring a volume of work quit signifi cant, the regression models allow the identifi cation of certain functional dependences between the various components of the capital market which secures a real possibility to forecast the phenomena subject of the analysis over a time horizon well established. Keywords: fi nancial asset, regression model, ARMA, ARCH, GARCH 1. Introduction The portfolios management is representing a complex activity based on a series of scientific models through which it is possible to permanently evaluate of the financial performances of the transacted instruments as well as the risks to which the investors are exposed. The portfolio administration is representing the totality of the methods and models through which an optimum combination of the assets is achieved, based on the correlation profitableness risk. In these conditions, the decisions being taken in the activity of the portfolio management is grounded on the portfolio yield and on the corresponding risk. This activity requires a permanent development view that the modifications arisen on the variables determinant for the yield and risk of the component instruments of the portfolio will imply modifications to be done in the portfolio structure. The construction of the portfolio of financial instruments aims to diversify the investments being done with the final result of diminishing the total risk relating to them, its structure depending on the investor s behaviour as to the risk and the capital he is in the position to invest within a certain interval of time. 2. Literature review The econometric regression model and the actual possibilities to use it in the economic activities have been permanently analysed during the previous century, in 44

2 this respect a series of works of reference being available as references in this domain, such as those signed by Graybill (1961), Sprent (1969), Theil (1971), Chow (1978) or Belsley, Kuh and Welsch (1980). During the last years, these models of analysis have been completed and adapted to the specific requirements of the modern economy. In this respect, to remark to works signed by Guijarati (2004), Bardsen (2005), Wooldrige (2006). Dougherty (2007), Andrei and Boubonnais (2008), Anghelache (2013). The models of the type ARCH (Auto Regressive Conditional Heteroskedasticity) have been created by Engle (1982), who is a Nobel Prize laureate (2003) for economy, for the activities carries on in the field of the econometric modelling of the financial time series. Further on, the models type ARCH have been adapted and developed by Bollerslev (during the period ). In the frame of the ARCH models, the hypothesis of the homoscedasticity of the stock exchange profitableness is given up. Starting from this theoretical basis, the literature of speciality submitted during the last years, a series of adaptations of the ARCH model to the various situations specific to the capital market. In this respect there are the works signed Audrino (2005), Zumbach (2008) to be reminded. The models of this category evaluated from the classical model ARCH to developments such as GARCH, ARMA-GARCH or ARMA-M. The introduction of the GARCH model in the econometric analyses performed on the capital market is the result of the activities carried on, basically, by Engle and Bollerslev (2008). This model also has been applied subsequently on certain specific situations of the activities of the capital markets, the outcomes of these researches being synthetized in works of speciality such as the one signed by Barone Adesi, Engle and Mancini (2008), Audrino and Trojani (2007), Tang, Chiu and Xu (2003). 3. The main elements concerning the statistical econometric instruments applied in the analysis of the financial instruments The econometric models are based either on the regression equations, such as the models through which the functional dependences between the various components of the capital market are established, or the analysis of the time series, for instance the models which allow the evaluation of the evolution of the capital market, in general, or of certain components of this one, on the basis of the observations recorded in respect of the levels recorded by the corresponding indicators during the previous periods of time. The simple regression model (the unifactorial regression model) is defined through a mathematical relation built up on the basis of the economic theory which assumes that the economic phenomenon effect is the result of the cumulated action of two categories of factors, as follows: a main factor, determinant and all the other factors which can be considered as non-essential, with alleatory or constant action, invariable, on the economic phenomenon effect. From mathematical point of view, the simple regression model implies a direct relation between two variables, out of which one is considered independent (explicative) and the second one is dependent Such an econometric model is transcribed, in mathematical form, as follows: Revista Română de Statistică - Supliment nr. 9 /

3 Y = f(x) + ε Y = the resulting variable (for the phenomenon effect); X = the explicative variable, or factorial, (for the elements determining the occurrence of the phenomenon effect); ε = the residual variable. The residual variable is normally distributed, having a mean zero and a constant dispersion and it is included in the model because in the real economy a linear functional dependence between two variables is not always met but one of probabilistic type; the data series are jeopardized by errors of measurement with influence on the estimation of the two parameters; the data series are established through observations on certain samples. From the category of the unifactorial regression models, the largest utilisation in the economic analyses is held by the linear model. In the case of this model, the relation between the resulting variable (Y) and the causal variable (X) is described through a function of the form: yi b a x i, i y i the resulting characteristic (explained), x i the factorial characteristic (explicative); ε i the residual variable. In practice, the forecast on the evolution of the profitableness of the financial assets can be achieved with the help of the econometric linear models of the time series type (ARMA Auto Regressive Moving Average). In this case, the subsequent profitableness of a financial instrument or of a portfolio of financial instruments can be estimated on the basis of the values recorded by this indicator during the previous periods of time. In the heteroskedastic models the modelling of the dispersion associated to the profitableness is attempted according to certain rules specific to the evolution. From a statistical mathematical point of view, the autoregressive processes with mobile average - Auto Regressive Moving Average ARMA can be defined as follows: - The process (X t ) t is a stationary process. and - X t Φ 1 X t-1 Φ 2 X t-2... Φ n X t-n = Z t + θ 1 Z t-1 + θ 2 Z t θ m Z t-m (Z t ) t white noise (the residual term) which is a series of independent variables of mean zero and constant dispersion σ 2. In practice, most of the times, the independence condition is hard to accomplish and is replaced by the condition of noncorrelation of the errors of the econometric model. Φ i and θ i coefficients. From the theoretical point of view, in every model of the type autoregressive with mobile average (ARMA) there are two distinct components which can be identified, namely: the auto regressive component defined by the term from the left side of the previous equality (X t Φ 1 X t-1 Φ 2 X t-2... Φ n X t-n ) and the component of mobile average defined by the term of the right side of the previous equality (Z t + θ 1 Z t-1 + θ 2 Z t θ m Z t-m ). 46

4 Engle s (the creator of the models of the type Auto Regressive Conditional Heteroskedasticity ARCH)) speech at the moment of the Nobel Prize awarding became one of the most significant source of information concerning the models ARCH and GARCH. This speech has been afterwards synthetized in the frame of the article titled Risk and volatility: econometric model and financial practice, initially published in the year 2004 in the frame of American Economic Review and re-published, in an updated form, during the year The model is submitted bellow: r t, = series of yields with t = 1, 2,..., T; I t = available information at the moment t, formed by the past and present values of this series; E(r t I t-1 ) = the anticipated average yield for the moment t depending on the available information at the moment (t-1), I t-1 : σ 2 (r t I t-1 ) = the yields dispersion, estimated on the account of the available information at the moment (t-1); c = constant; a = significance factor grated to the errors of forecasting related to the previous periods; q = number of the previous periods for which it is considered that the errors have n influence on the dispersion. In the frame of the ARCH model, the estimated dispersion, depending on the available information presently, is a function of the squares of the forecasting errors from the past. The model GARCH - Generalised AutoRegressive Conditional Heteroskedasticity- is representing a generalization of the previous model. According to this model, the dispersion depends on two components, respectively it is a linear combination of the squares of the forecasting errors (the auto regressive term) and of the dispersions conditioned by the previously available information, stated out in the past (the mobile average term). The model is represented by the following relation: r t = ε t Revista Română de Statistică - Supliment nr. 9 /

5 In the frame of the ARCH models the hypothesis of the homoscedasticity of the stock exchange profitableness is given up. Under these circumstances, it is allowed that the volatility is modelled in a scholastic manner, taking into account its variations over the time and, consequently, trying to implement a superior degree of accuracy for the estimation of the future profitableness of an asset. The ARCH model is considering that the estimated dispersion presently, depending on the available information presently, is a function of the squares of the forecasting errors in the past. According to the GARCH model, the dispersion depends on two components, respectively it is a linear combination of the squares of the forecasting errors (the auto regressive term) and on the dispersions depending on the previous available information, stated out in the past (the mobile average term). 4. Conclusions Once the econometrics notions appeared, specific models have been developed meant to secure the reflection of the specific realities of the financial monetary markets in the form of well defined statistical mathematical relations. These models have been steadily developed, the major goal of the researchers of the domain consisting of adapting and improving the econometric models in accordance with the specific dynamics of the capital market. With the support of the econometric models, the dependences between the main elements defining the capital market as well as the evolution in time of the main of its indicators are analysed. References 1. Andrei, T.; Bourbonais, R. (2008). Econometrics, Economica Publishing House, Bucharest 2. Anghel, M.G. (2014). Econometric model used in the capital market analysis, Theoretical and Applied Economics, Volume XXI, No Anghel, M.G. (2013). Models of management and analysis of the portfolios, Economic Publishing House, Bucharest 4. Anghel, M.G. (2011). Analysis of the evolution of a financial assets on the basis ARCH model (Auto Regressive Conditional Heteroskedasticity), Romanian Statistical Review Supplement, Trim III 5. Anghelache, C., Anghel, M.G., Manole A. (2015). Economic, fi nancial-banking and informatics modeling, Artifex Publishing House, Bucharest 6. Anghelache, C., Anghel, M.G. (2014). Using the regression model for the portfolios analysis and management, Theoretical and Applied Economics, Volume XXI, No.4 7. Anghelache, C. (2013). Elements of theoretical econometrics, Artifex Publishing House, Bucharest 8. Barone Adesi, G.; Engle, R.; Mancini, L. (2008). A GARCH Option pricing model with filtered historical simulation, The Review of Financial Studies, Vol. 21, Issue 3, pg Benjamin, C.; Herrard, N.; Houée-Bigot, M.; Tavéra, C. (2012). Forecasting with an Econometric Model, Springer 10. Bollerslev, T. (2008). Glossary to ARCH (GARCH), Duke Department of Economics Research Paper No Dougherty, C. (2008). Introduction to econometrics. Fourth edition, Oxford University Press 12. Dragotă, V. (coordinator) (2009). Portfolio management securities second edition, Economic Publishing House, Bucharest 13. Engle, R.F. (2004/2007). Risk and Volatility: Econometric Models and Financial Practice, The American Economic Review 14. Engle, R.F. (2001). GARCH 101: An introduction to the use of ARCH/GARCH models in applied econometrics, NYU Working Paper no. FIN Wooldrige, J. (2006). Introductory econometrics. A modern approach 2 edition, MIT Press 48