Limit theory for nonlinear functions of ‘weakly nonstationary’ linear processes [joint with I. Kasparis]


James Duffy

Date & Time

21 March 2017




UCL Economics Department
Drayton House,
30 Gordon Street,

We develop limit theory for standardised sums of non-linear transformations of (arrays of) linear processes that lie at the ‘boundary’ demarcating stationarity from non-stationarity; processes which we term ‘weakly nonstationary’. This includes, as leading examples, autoregressive processes with roots drifting slowly towards unity (‘moderate deviations from unity’ in the terminology of Phillips & Magdalinos, 2007, JoE), and fractionally integrated processes with d = 0.5.

On the one hand, these processes are sufficiently dependent to be non-stationary, and thus resist the application of limit theorems appropriate to stationary linear processes. On the other hand, their dependence is also sufficiently weak to prevent the appeal to convergence results that rely on the finite-dimensional convergence of standardised processes to tight limits (e.g. Pötscher, 2004, ET).

The present work aims to fill this significant lacuna in the existing theory. As discussed in Duffy (2015) [] one application of these results is to proving the uniform validity of standard testing procedures in a nonparametric predictive regression, in the sense that the size of these tests may be controlled, asymptotically, uniformly in the parameter governing the degree of the persistence of the regressor.