## 1. Introduction

[2] The residual errors of hydrological models, which represent the combined effects of data and model errors, are usually both heteroscedastic and autocorrelated [e.g., *Sorooshian and Dracup*, 1980; *Kuczera*, 1983; *Bates and Campbell*, 2001]. Heteroscedasticity is related to larger errors being generally associated with larger rainfalls and streamflows [e.g., *Villarini and Krajewski*, 2008; *Thyer et al*., 2009]. It can be represented by directly conditioning the variance of residual errors on explanatory variables such as runoff [e.g., *Sorooshian and Dracup*, 1980; *Thyer et al*., 2009; *Pianosi and Raso*, 2012], or by applying Box-Cox and other transformations [e.g., *Kuczera*, 1983; *Bates and Campbell*, 2001; *Smith et al*., 2010]. The direct conditioning approach is of particular interest because it allows exploiting additional information through explanatory variables. Autocorrelation is related to the “memory” of hydrological models, with storage errors propagating across multiple consecutive time steps [e.g., *Kavetski et al*., 2003]. It can be represented using autoregressive (AR) models [*Kuczera*, 1983], typically under lag-1 [AR(1)] assumptions [*Schaefli et al*., 2007; *Schoups and Vrugt*, 2010].

[3] This study pursues improved probabilistic descriptions of predictive and parametric uncertainties in hydrological modeling using residual error models [e.g., *Kuczera*, 1983; *Bates and Campbell*, 2001; *Gallagher and Doherty*, 2007; *Willems*, 2009; *Smith et al*., 2010, and many others]. It focuses on the *joint* inference of heteroscedasticity, autocorrelation and hydrological parameters. Recent work in this direction includes *Schoups and Vrugt* [2010], where a heteroscedastic skewed exponential distribution was combined with an AR(1) model, and all statistical and hydrological parameters estimated jointly. In this note, we show that a seemingly straightforward combination of heteroscedasticity and autocorrelation can result in error models with poor statistical and computational properties. We then present an alternative conceptualization of the heteroscedastic AR(1) model with a notably more robust performance.

[4] The presentation is structured as follows. Section 2 derives the statistical properties of two alternative heteroscedastic AR(1) error models. Section 3 empirically compares the predictive reliability, precision, and parameter inference (including parameter interactions) for the two error models on three hydrologically distinct catchments. The note concludes with a summary of key findings and practical recommendations in section 4.