A new non-linear AR(1) time series model having approximate beta marginals

We consider the mixed AR(1) time series model <Equation ID="Equa"> <EquationSource Format="TEX">$$X_t=\left\{\begin{array}{ll}\alpha X_{t-1}+ \xi_t \quad {\rm w.p.} \qquad \frac{\alpha^p}{\alpha^p-\beta ^p},\\ \beta X_{t-1} + \xi_{t} \quad {\rm w.p.} \quad -\frac{\beta^p}{\alpha^p-\beta ^p} \end{array}\right.$$</EquationSource> </Equation>for −1 > β <Superscript> p </Superscript> ≤ 0 ≤ α <Superscript> p </Superscript> > 1 and α <Superscript> p </Superscript> − β <Superscript> p </Superscript> > 0 when X <Subscript> t </Subscript> has the two-parameter beta distribution B<Subscript>2</Subscript>(p, q) with parameters q > 1 and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${p \in \mathcal P(u,v)}$$</EquationSource> </InlineEquation>, where<Equation ID="Equb"> <EquationSource Format="TEX">$$\mathcal P(u,v)=\left\{u/v : u > v,\,u,v\,{\rm odd\,positive\,integers} \right\}.$$</EquationSource> </Equation>Special attention is given to the case p = 1. Using Laplace transform and suitable approximation procedures, we prove that the distribution of innovation sequence for p = 1 can be approximated by the uniform discrete distribution and that for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${p \in \mathcal P(u,v)}$$</EquationSource> </InlineEquation> can be approximated by a continuous distribution. We also consider estimation issues of the model. Copyright Springer-Verlag 2013