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We propose two approaches to constructing deep time series probabilistic models based on a variant of RNN called an echo state network (ESN). The first is where the output layer of the ESN has stochastic disturbances and a Bayesian prior for regularization. The second employs the implicit copula of an ESN with Gaussian disturbances, which is a Gaussian copula process on the feature space. Combining this copula process with a nonparametrically estimated marginal distribution produces a distributional time series model. The resulting probabilistic forecasts are deep functions of the feature vector and marginally calibrated. In both approaches, Markov chain Monte Carlo methods are used to estimate the models and compute forecasts. The proposed models are suitable for the complex task of forecasting intraday electricity prices. Using data from the Australian market, we show that our deep time series models provide accurate short\u2010term probabilistic price forecasts, with the copula model dominating. Moreover, the models provide a flexible framework for incorporating probabilistic forecasts of electricity demand, which increases upper tail forecast accuracy from the copula model significantly.<\/jats:p>","DOI":"10.1002\/jae.2959","type":"journal-article","created":{"date-parts":[[2023,1,24]],"date-time":"2023-01-24T00:54:12Z","timestamp":1674521652000},"page":"493-511","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["Deep distributional time series models and the probabilistic forecasting of intraday electricity prices"],"prefix":"10.1002","volume":"38","author":[{"given":"Nadja","family":"Klein","sequence":"first","affiliation":[{"name":"Research Center Trustworthy Data Science and Security, Department of Statistics at TU Dortmund University Alliance Ruhr Dortmund Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael Stanley","family":"Smith","sequence":"additional","affiliation":[{"name":"Melbourne Business School University of Melbourne Melbourne Victoria Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David J.","family":"Nott","sequence":"additional","affiliation":[{"name":"Department of Statistics and Data Science National University of Singapore Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2023,3,14]]},"reference":[{"key":"e_1_2_10_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/TPWRS.2006.873409"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.eneco.2018.10.038"},{"key":"e_1_2_10_4_1","unstructured":"Australian Energy Regulator(2019).State of the energy market report: data update."},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9868.00391"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1093\/rfs\/hhaa062"},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2018.11.006"},{"key":"e_1_2_10_8_1","doi-asserted-by":"publisher","DOI":"10.5547\/01956574.37.1.dbun"},{"key":"e_1_2_10_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2019.04.024"},{"key":"e_1_2_10_10_1","doi-asserted-by":"publisher","DOI":"10.1109\/TNN.2011.2162109"},{"key":"e_1_2_10_11_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijforecast.2011.02.019"},{"key":"e_1_2_10_12_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.eneco.2015.07.014"},{"key":"e_1_2_10_13_1","doi-asserted-by":"publisher","DOI":"10.1198\/016214503000000774"},{"key":"e_1_2_10_14_1","unstructured":"Feng G. 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