Temporal Aggregation of Garch Processes

• Published in: Econometrica Vol. 61; no. 4; pp. 909 - 927
• Main Authors: Drost, Feike C., Nijman, Theo E.
• Format: Journal Article
• Language: English
• Published: Malden, MA Econometric Society 01.07.1993; Blackwell; George Banta Pub. Co. for the Econometric Society; Blackwell Publishing Ltd
• Subjects: Aggregates; Aggregation; Autoregressive moving average; Consistent estimators; Econometrics; Economic models; Economic theory; Exact sciences and technology; Exchange rates; Foreign exchange rates; High frequencies; Inference from stochastic processes; time series analysis; Kurtosis; Low frequencies; Mathematics; Parametric models; Probability and statistics; Sciences and techniques of general use; Statistical variance; Statistics; Stochastic models; Studies; Time series; Variables; Variance analysis; ARMA model; Parameter estimation; Time series; GARCH model; High frequency; Low frequency; Exchange rate; Heteroscedasticity
• ISSN: 0012-9682; 1468-0262
• DOI: 10.2307/2951767

We derive low frequency, say weekly, models implied by high frequency, say daily, ARMA models with symmetric GARCH errors. Both stock and flow variable cases are considered. We show that low frequency models exhibit conditional heteroskedasticity of the GARCH form as well. The parameters in the conditional variance equation of the low frequency model depend upon mean, variance, and kurtosis parameters of the corresponding high frequency model. Moreover, strongly consistent estimators of the parameters in the high frequency model can be derived from low frequency data in many interesting cases. The common assumption in applications that rescaled innovations are independent is disputable, since it depends upon the available data frequency.