TAC Method for Fitting Exponential Autoregressive Models and Others: Applications in Economy and Finance

TL;DR

This work extends the TAC framework to solve best approximations of data by exponential patterns and demonstrates its relevance to economics via the nlstac R package. It analyzes the core fit $f_k(t)=a_k e^{k t}+b_k$ and the associated error $E_ inf(k)$, provides explicit analytic forms for small datasets, and establishes a general admissibility criterion that reduces the general problem to a four-point case. The main finding is that for any dataset with at least four points, either a nontrivial exponential fit with $a k \neq 0$ exists or the best fit is a line or constant, with symmetric and limit-case behavior carefully characterized. The authors apply the method to economic problems, including a demand-curve model reparameterized as $\log_{10} Q = a e^{d C}+b$ and to exponential autoregressive time-series, using nlstac to obtain fits and assess accuracy. Overall, the paper demonstrates TAC’s robustness and its practical utility for economy and finance data analysis.

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 the economic science. We present results that completely solve the problem of the best approximation by means of exponential functions and we will be able to determine what kind of data is suitable to be fitted. Data will be approximated using TAC (implemented in the R-package nlstac), a numerical algorithm for fitting data by exponential patterns without initial guess designed by the authors. 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 TAC's utility and highlights how far this algorithm could be used.

Figures (5)

• Figure S1: In blue, the points, in black the best approximation and in green the upper and lower borders of the narrowest band that contains $(\mathfrak{t},T)$. Yes, the band has constant width.
• Figure S2: Some graphical aspects about this TAC implementation. In the numerical analisys bibliography, relative error is defined with or without sign; in this paper we will consider the latter. A spike can be seen in the small window of Figure \ref{['subfig:ex_termometro_relative_error']}. This spike should not be considered as an indicator of a poor adjustment of the curve to the data. On the contrary: the spike is due to the proximity of the data to zero and, however, the error remains bounded. This is because curve and data are close enough to control the fact that we are virtually dividing by zero
• Figure S3: Blue dots represent differences between $T$ and $\mathfrak{f}_k(\mathfrak{t})$. In 4 dots, $\mathcal{M}(T-\mathfrak{f}_k(\mathfrak{t}))$ and ${{\mathpzc{M}}}(T-\mathfrak{f}_k(\mathfrak{t}))$ are reached, and we have colored them in red. Those are the 4 points mentioned in Proposition \ref{['p_de_n_a_4']}. Please observe how the maxima and the minima are alternatively reached.
• Figure S4: Observations in blue, approximation in red.
• Figure S5: Observations in blue, approximation in red (smaller circle). Please observe how each approximation lays over the actual observation.

Theorems & Definitions (20)

• Remark 1
• Remark 2
• Proposition 1: TAC1
• Lemma 1
• Remark 3
• Lemma 2
• Remark 4
• Lemma 3
• Corollary 1
• Lemma 4