For purposes of maximum likelihood estimation, we show how to compute the Gaussian likelihood function when the data are generated by a higher-order continuous-time vector ARMAX model and are observed as stocks and flows at different frequencies. Continuous-time ARMAX models are analogous to discrete-time autoregressive moving-average models with distributed-lag exogenous variables. Stocks are variables observed at points in time and flows are variables observed as integrals over sampling intervals. We derive the implied state-space model of the discrete-time data and show how to use it to compute the Gaussian likelihood function with Kalman-filtering, prediction-error, decomposition of the data.

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. The paper illustrates and evaluates a Kalman filtering method for forecasting German real GDP at monthly intervals. German real GDP is produced at quarterly intervals but analysts and decision makers often want monthly GDP forecasts. Quarterly GDP could be regressed on monthly indicators, which would pick up monthly feedbacks from the indicators to GDP, but would not pick up implicit monthly feedbacks from GDP onto itself or the indicators. An efficient forecasting model which aims to incorporate all significant correlations in monthlyquarterly data should include all significant monthly feedbacks. We do this with estimated VAR(2) models of quarterly GDP and up to three monthly indicator variables, estimated using a Kalman-filtering-based maximum-likelihood estimation method. Following the method, we estimate monthly and quarterly VAR(2) models of quarterly GDP, monthly industrial production, and monthly, current and expected, business conditions. The business conditions variables are produced by the Ifo Institute from its own surveys. We use early insample data to estimate models and later out-of-sample data to produce and evaluate forecasts. The monthly maximum-likelihood-estimated models produce monthly GDP forecasts. The Kalman filter is used to compute the likelihood in estimation and to produce forecasts. Generally, the monthly German GDP forecasts from 3 to 24 months ahead are competitive with quarterly German GDP forecasts for the same time-span ahead, produced using the same method and the same data in purely quarterly form. However, the present mixed-frequency method produces monthly GDP forecasts for the first two months of a quarter ahead which are more accurate than one-quarter-ahead GDP forecasts based on the purely-quarterly data. Moreover, quarterly models based on purely-quarterly data generally cannot be transformed into monthly models which produce equally accurate intra-quarterly monthly forecasts. Terms of use: Documents inJEL classification: E37, C32.

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