Abstract
There are a couple of purposes in this paper: to study a problem of approximation with exponential functions and to show its relevance for economic science. The solution of the first problem is as conclusive as it can be: working with the max-norm, we determine which datasets have best approximation by means of exponentials of the form f(t)=b+aexp(kt), we give a necessary and sufficient condition for some a,b,k∈R to be the coefficients that give the best approximation, and we give a best approximation by means of limits of exponentials when the dataset cannot be best approximated by an exponential. For the usual case, we have also been able to approximate the coefficients of the best approximation. As for the second purpose, we show how to approximate the coefficients of exponential models in economic science (this is only applying the R-package nlstac) and also the use of exponential autoregressive models, another well-established model in economic science, by utilizing the same tools: a numerical algorithm for fitting exponential patterns without initial guess designed by the authors and implemented in nlstac. We check one more time the robustness of this algorithm by successfully applying it to two very distant areas of economy: demand curves and nonlinear time series. This shows the utility of TAC (Spanish for CT scan) and highlights to what extent this algorithm can be useful.
Highlights
This paper is designed to cover a couple of major objectives
We show that the coefficients given by nlstac give a realistic approximation of such datasets
We will deal with approximations by means of exponential functions, and we will measure the error with the max-norm
Summary
This paper is designed to cover a couple of major objectives. Broadly, the first one is to solve one of the remaining issues in [1]. In these sections, we will deal with approximations by means of exponential functions, and we will measure the error with the max-norm. If some data (t, T) are to be fitted with a decreasing function and we are measuring the error with the max-norm, the maximum value in T must be attained before the minimum. 29 or [8]) we know that the line that best approximates any dataset behaves this way, the opposite way or as described in Remark 1 Please observe that this Theorem would not apply so to approximations with general degree polynomials.
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