This paper applies techniques of Quantile Data Analysis to non-parametrically analyze time series functions such as the sample spectral density, sample correlations and sample partial correlations. The aim is to identify the memory type of an observed time series, and thus to identify parametric time domain models that fit an observed time series. Time series models are usually tested for adequacy by testing if their residuals are white noise. It is proposed that an additional criterion of fit for a parametric model is that it has the non-parametrically estimated memory characteristics. An important diagnostic of memory is the index δ of regular variation of a spectral density; estimators are proposed for δ. Interpretations of the new quantile criteria are developed through cataloging their values for representative time series. The model identification procedures proposed are illustrated by analysis of long memory series simulated by Granger and Joyeux, and the airline model of Box and Jenkins.