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I basically agree with @John, let me expand: We want to model $y$ using a simple linear model, the most basic setup is $$y = c + \mathbf{X}\beta$$ with $y$ the $N$ observations, $c$ a constant, $\mathbf{X}$ the $N \times M$ matrix of regressors and $\beta$ a $M$-dimensional vector of coefficients. This model has $M$ parameters, the elements of $\beta$. ...

3

In full generality this is a very difficult question. The closest you will get to a general framework is Vapnik-Chervonenkis theory. You can read about this in Chapter 7.9 of "The elements of statistical learning" by Hastie, Tibshirani and Friedman which can be downloaded from their website . But be warned that this is a theoretical approach. Often more ...

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The following is a good way to judge the quality of fits for a model. http://en.wikipedia.org/wiki/Akaike_information_criterion

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What you are talking about is called regression using fractional polynomials and it has its merits. The canonical reference is this one: Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling by Royston and Altman (1994) From the abstract: The relationship between a response variable and one or more ...

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This book might be what you are looking for: Theory of Financial Risk and Derivative Pricing. From Statistical Physics to Risk Management by J.-P. Bouchaud and M. Potters As one reviewer from amazon wrote: Econophysics (the application of techniques developed in the physical sciences to economic, business and financial problems) has emerged as a ...

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As you suggest, in the case of non-stationary time series, the Hurst exponent is not suitable to measure the time seires persistence for the reasons you cited in the question. Particularly, when $H(q)$ is a non-linear function of q, as in the non-stationary time-series case, the time-series has to be analysed as it is a multi-fractal system (to deal this ...

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