An Extended Yule-walker Method for Estimating a Vector Autoregressive Model with Mixed-frequency Data

Zadrozny (1990) proposed and illustrated a nonlinear Kalman-filtering (KF) method for estimating a vector autoregressive moving-average (VARMA) model with mixed-frequency and partly temporallyaggregated data. The present paper proposes an optimal three-step linear instrumental variable method for estimating a VAR model with mixed-frequency data. The method compensates for missing data arising from mixed frequencies by using restrictions implied by extended Yule-Walker (XYW) equations, beyond the usual minimum YW equations for estimating a VAR model with fully-observed single-frequency data. The theory of generalized method of moments is used to derive an asymptotically efficient XYW estimator, determine its asymptotic distribution, and provide asymptotic tests of overidentifying restrictions. The KF method can simultaneously handle missing data, temporal aggregation, measurement errors, reporting delays, and revisions, but performs poorly or not at all on large models with many parameters. The XYW method can handle any pattern of missing data (subject to parameters being identified), but it is not yet clear how it might handle the other mentioned data problems. Having the computational complexity of generalized least squares, the XYW method can handle much larger models. The XYW method is illustrated and compared to the KF method using real and simulated macroeconomic monthly-quarterly data. The large number of available macroeconomic and financial time series at observation frequencies ranging from annual to "tick by tick" offer wide possibilities for applications of the XYW method. The analysis and conclusions of the paper represent the authors' views and not necessarily those of the Bureau of Labor Statistics. Stefan Mittnik provided valuable comments at several stages. Forthcoming 1998 in Advances in Econometrics: Messy Data, Missing Observations, Outliers, and Mixed-Frequency Data, Vol. 13, edited by T.B. Fomby and R.C. Hill, JAI Press, Greenwich, CT. 2