Two decades ago, effective methods for dealing with time series models that vary with time have appeared in the statistical literature. Except in a case of marginal heteroscedasticity [1], they have never been used for official statistics. In this paper, we consider autoregressive integrated moving average (ARIMA) models with time-dependent coefficients applied to very long U.S. industrial production series. There was well an attempt to handle time-dependent integrated autoregressive (AR) models [2] but the case study was small. Here, we investigate the case of ARIMA models on the basis of [3, 4, 5]. As an illustration, we consider a big dataset of U.S. industrial production time series already used in [6]. We employ the software package Tramo in [7] to obtain linearized series and we built both ARIMA with constant coefficients (cARIMA) and ARIMA models with time-dependent coefficients (tdARIMA). In these tdARIMA models we use the simplest specification: a regression with respect to time. Surprisingly, for a large part of the series, there are statistically significant slopes, indicating that the tdARIMA models fit better the series than the cARIMA models.