Modeling Financial Time Series

We are attempting to 'model' what the reality is; so that we can predict it. Statistical Modeling, in addition to being of central importance in statistical decision making, is critical in any endeavor, since essentially everything is a model of reality. As such, modeling has applications in such disparate fields as marketing, finance, and organizational behavior. Particularly compelling is econometric modeling since, unlike most disciplines (such as Normative Economics), econometrics deals only with provable facts, not with beliefs and opinions.

Time series analysis is an integral part of financial analysis. The topic is interesting and useful, with applications to the prediction of interest rates, foreign currency risk, stock market volatility, and the like. There are many varieties of econometric and multi-variate techniques. Specific examples are regression and multi-variate regression; vector auto-regressions; and co- integration regarding tests of present value models. Next section presents the underlying theory on which statistical models are predicated.

Financial Modeling: Econometric modeling is vital in finance and in financial time series analysis. Modeling is, simply put, the creation of representations of reality. It is important to be mindful that, despite the importance of the model, it is in fact only a representation of reality and not the reality itself. Accordingly, the model must adapt to reality; it is futile to attempt to adapt reality to the model. As representations, models cannot be exact. Models imply that action is only taken after careful thought and reflections This can have major consequences in the financial realm. A key element of financial planning and financial forecasting is the ability to construct models showing the interrelatedness of financial data. Models showing correlation or causation between variables can be used to improve financial decision-making. For example, one would be more concerned about the consequences on the domestic stock market of a downturn in another economy if it can be shown that there is a mathematically provable causative impact of that nation's economy and the domestic stock market. However, modeling is fraught with dangers. A model which heretofore was valid may lose validity due to changing conditions, thus becoming an inaccurate representation of reality and adversely affecting the ability of the decision-maker to make good decisions.

The examples of univariate and multivariate regression, vector autoregression, and present value cointegration illustrate the application of modeling, a vital dimension in managerial decision making, to econometrics, and specifically the study of financial time series. The provable nature of econometric models is impressive; rather than proffering solutions to financial problems based on intuition or convention, one can mathematically demonstrate that a model is or is not valid, or requires modification. It can also be seen that modeling is an iterative process, as the models must continuously change to reflect changing realities. The ability to do so has striking ramifications in the financial realm, where the ability of models to accurately predict financial time series is directly related to the ability of the individual or firm to profit from changes in financial scenarios.

Univariate and Multivariate Models: The use of regression analysis is widespread in examining financial time series. Some examples are the use of forward exchange rates as optimal predictors of future spot rates; conditional variance and the risk premium in foreign exchange markets; and stock returns and volatility. A model that has been useful for this type of application is called the GARCH-M model, which incorporates computation of the man into the GARCH (generalized autoregressive conditional heteroskedastic) model. This sounds complex and esoteric, but it only means that the serially correlated errors and the conditional variance enter the mean computation, and that the conditional variance itself depends on a vector of explanatory variables. The GARCH-M model has been further modified, a testament of finance practitioners to the necessity of adapting the model to a changing reality. For example, this model can now accommodate exponential (non-linear) functions, and is no longer constrained by non-negativity parameters.

One application of this model is the analysis of stock returns and volatility. Traditionally, the belief has been that the variance of portfolio returns is the primary risk measure for investors. However, using extensive time series data, it has been proven that the relationship between mean returns and return variance or standard deviation I weak; hence the traditional two-parameter asset pricing models appear to be inappropriate, and mathematical proof replaces convention. Since decisions premised on the original models are necessarily sub-optimal because the original premise is flawed, it is advantageous for the finance practitioner to abandon the model in favor of one with a more accurate representation of reality.

Correct specification of a model is of paramount importance, and a battery of misspecification testing criteria have been established. These include tests of normality, linearity, and homoskedasticity, and can be applied to a variety of models. A simple example which yields surprising results is the Capital Asset Pricing Model, one of the cornerstones of elementary economics. Application of the testing criterial to data concerning companies' risk premium shows significant evidence of non-linearity, non-normality and parameter non-constancy. The CAPM was found to be applicable for only three of seventeen companies that were analyzed. This does not mean, however, that the CAPM should be summarily rejected; it still has value as a pedagogic tool, and can be used as a theoretical framework. For the econometrician or financial professional, for whom the misspecification of the model can translate into suboptimal financial decisions, the CAPM should be supplanted by a better model, specifically one that reflects the time-varying nature of betas. The GARCH-M framework is one such model.

Multivariate linear regression models apply the same theoretical framework. The principal difference is the replacement of the dependent variable by a vector. The estimation theory is essentially a multivariate extension of that developed for the univariate, and as such can be used to test models such as the stock and volatility model and the CAPM. In the case of the CAPM, the vector introduced is excess asset returns at a designated time. One application is the computation of the CAPM with time-varying covariances. Although in this example the null hypothesis that all intercepts are zero cannot be rejected, the misspecification problems of the univariate model still remain. Slope and intercept estimates also remain the same, since the same regression appears in each equation.

Vector Autoregression: General regression models assume that the dependent variable is a function of past values of itself and past and present values of the independent variable. The independent variable, then, is said to be weakly exogenous, since its stochastic structure contains no relevant information for estimating the parameters of interest. While the weak exogeneity of the independent variable allows efficient estimation of the parameters of interest without any reference to its own stochastic structure, problems in predicting the dependent variable may arise if "feedback" from the dependent to the independent variable develops over time. (When no such feedback exists, it is said that the dependent variable does not Granger-cause the independent variable.) Weak exogenetic coupled with Granger non-causality yields strong exogenetic which, unlike weak exogenetic, is directly testable. To perform the tests requires utilization of the dynamic structural equation model (DSEM) and the vector autoregressive process (VAR). The multivariate regression model is thus extended in two directions, by allowing simultaneity between the endogenous variables in the dependent variable, and explicitly considering the process generating the exogenous variables in the dependent variable, and explicitly considering the process generating the exogenous independent variables. Results of this testing are useful in determination of whether an independent variable is strictly exogenous or is predetermined. Strict exogenetic can be tested in DSEMs by expressing each endogenous variable as an infinite distributed lag of the exogenous variables. If the independent variable is strictly exogenous, attention can be limited to distributions conditional on the independent variable without loss of information, resulting in simplification of statistical inference. If the independent variable is strictly exogenous, it is also predetermined, meaning that all of its past and current values are independent of the current error term. While strict exogenetic is closely related to the concept of Granger non-causality, the two concepts are not equivalent and are not interchangeable.

It can be seen that this type of analysis is helpful in verifying the appropriateness of a model as well as proving that, in some cases, the process of statistical inference can be simplified without losing accuracy, thereby both strengthening the credibility of the model while increasing the efficiency of the modeling process. Vector autoregressions can be used to calculate other variations on causality, including instantaneous causality, linear dependence, and measures of feedback from the dependent to he independent and from the independent to the dependent variables. It is possible to proceed further with developing causality tests, but simulation studies which have been performed reach a consensus that the greatest combination of reliability and ease can be obtained by applying the procedures described.

Cointegration and Present Value Modeling: Present value models are used extensively in finance to formulate models of efficient markets. In general terms. A present value model for two variables y1 and x1, states that y1 is a linear function of the present discounted value of the expected future values of x1, where the constant term, the constant discount factor, and the coefficient of proportionality are parameters that are either know or need to be estimated. Not all financial time series are non-integrated; the presence of integrated variables affects standard regression results and procedures of inference. Variables may also be cointegrated, requiring the superimposition of cointegrating vectors on the model, and resulting in circumstances under which the concept of equilibrium loses all practical implications and spurious regressions may occur. In present value analysis, cointegration can be used to define the "theoretical spread" and to identify co-movements of variables. This is useful in constructing volatility-based tests.

One such test is stock market volatility. Assuming cointegration, second-order vector autoregressions are constructed, which show suggest that dividend changes are not only highly predictable but are Granger-caused by the spread. When the assumed value of the discount rate is increased, certain restrictions can be rejected at low significance levels. This yields results showing an even more pronounced "excess volatility" than that anticipated by the present value model. It also illustrates that the model is more appropriate in situations where the discount rate is higher. The implications of applying a cointegration approach to stock market volatility testing for financial managers are significant. Of related significance is the ability to test the expectations hypotheses of interest rate term structure.

Mean absolute error is a robust measure of error. However, one may also use the sum of errors to compare the success of each forecasting model relative to a baseline, such as a random walk model, which is usually used in financial time series modelling.

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