**Parameter Estimation of a Macroeconomic Model**[*]** **

E.V. Burnaev, N.N. Olenev, A.S. Starikov

*Moscow Institute
of Physics and Technology (**State** **University**),
Dorodnicyn Computing **Center** of **Russian** **Academy**
of **Sciences*

In this work a method for macroeconomic model parameter estimation based on parallel processing is proposed. For an application of this method some measure of similarity between two time-series is required and so a new wavelet based measure of similarity was elaborated. Complicated economic models with a lot of parameters can be accurately estimated by the proposed method. The use of the method is illustrated by the parameter estimation of a regional macroeconomic model.

A macroeconomic model usually contains a lot of unspecified parameters. Some variables of the macroeconomic model have initial values that are sometimes unknown and should be considered as parameters as well. In most cases mentioned parameters can't be defined on the basis of economic statistics. Moreover, even if all necessary statistics is available the quality of the data isn't always good. That's why only confidence intervals for the unknown parameters can be computed from the data.

For estimation of the unknown parameters time-series for some macro-indexes (calculated by the model) and statistical time-series for these macro-indexes should be compared by means of some measure of similarity. The unknown parameters can be determined implicitly as those parameters, which provide minimum value of the used measure of similarity. Parallel processing on a cluster of workstations or on a supercomputer enables to perform exhaustive search of the parameters within their confidence intervals (determined either from economic sense or from the available statistical data) and estimate their values for reasonable time. In order to use parallel processing the macroeconomic model should be divided into blocks so that parameter estimation of any block can be done independently of the other blocks. In this case not only real statistical data, but also data calculated from some block can be used as values for external variables of the other blocks.

Estimated values of the parameters give new knowledge about the macroeconomic system under investigation and this knowledge can force a researcher to modify the model. Thus macroeconomic model parameter estimation based on parallel processing becomes a powerful tool for mathematical modeling of economic systems.

It was mentioned that calculated and statistical time-series for some macro-indexes should be compared on the basis of some measure of similarity. As used here, two time-series are considered to be similar if they are close as functions of time (in other words, if there is a strong, possibly nonlinear dependence between two time-series). Convolution of the Theil's index and specially designed wavelet based measure of similarity was used as a characteristic of closeness between two time-series.

With linear trend appropriate rescaling with respect to ordinate axe two completely different time-series can become quite similar and the value of Euclidian distance between these time-series can decreases significantly. Such effect frequently occurs in many real situations.

Thus the similarity between two time-series should be based on certain characteristics of these time-series rather than on the raw values. These characteristics should be calculated in linear time, robust for changes in level, scale and trend of the time-series and the similarity between two time-series is simply given by the (traditional) similarity of the corresponding characteristics.

It is proposed to compute such characteristics on the basis of discrete wavelet transform. In that case these characteristics have all necessary properties. It is recommended to use Daubechies wavelet and scaling filters. If that's the case wavelet coefficients are robust for linear trend. That is if the time-series of a macro-index is simply a linear function of time, all wavelet coefficients are zero.

The described approach of exhaustive search for unknown parameters enables to find the solution, however time of calculations is rather big. A reduction of computation time can be achieved by means of special computation algorithms.

Macroeconomic model should be divided into blocks so that parameter estimation of any block can be done independently. Confidence intervals for parameters should be determined either from economic sense or from statistical estimation based on the real data.

[*] The work is in part supported by the Russian Humanitarian Scientific Foundation (Grant 06-02-91821); by the Program of State Support of Leading Scientific Schools (Grant SS-5379.2006.1); and by Presidium of the Russian Academy of Sciences, Program of Basic Research no.15.