The Asymptotic Efficiency of Cointegration Estimators Under Temporal Aggregation

Abstract

This paper examines the effects of temporal aggregation on the asymptotic variances of estimators in cointegrated systems+ Two important findings are obtained+ First, estimators based on flow data alone are more efficient than when the data are all stocks or a mixture of stocks and flows+ Second, estimators based on flow data are as efficient as when the data are recorded continuously+ A method of improving efficiency with stock variables is also proposed, and an empirical illustration of the method is provided in the context of long-run money demand regressions+ I thank Roy Bailey, Rex Bergstrom, Roderick McCrorie, a co-editor, and two anonymous referees for helpful com-ments+ I also thank Katsumi Shimotsu for help with some data issues+ None of these individuals are implicated, however, in any possible shortcomings of this paper+ The financial support provided by the ESRC under grant R000221818

The long-run determinants of fertility: one century of demographic change 1900–1999

“…(20 time series observations per country). An estimator of cointegrating vectors constructed with temporally aggregated data is consistent (see, e.g., Granger 1990) and asymptotically as efficient as the estimator based on disaggregated time series, when, as in our case, all the variables in the cointegrating vector are flows (see, e.g., Chambers 2003). Chambers (2001), for example, demonstrates that estimates from cointegrating regressions are remarkable stable across frequencies, which is consistent with the well-known finding that the power of unit root and cointegration tests depends far more on the time span than on the number of observations (see, e.g, Shiller and Perron 1985;Hakkio and Rush 1991;Lahiri and Mamingi 1995).…”

Frequency domain estimation of temporally aggregated Gaussian cointegrated systems

“…(3) of Chambers (2003). Writing this as ðDI m þ JAÞyðtÞ dt ¼ wðtÞ dt and assuming that wðtÞ may be represented as a continuous time VAR(q) process of the form CðDÞwðtÞ dt ¼ dbðtÞ, with CðzÞ ¼ z q I m þ P qÀ1 j¼0 C j z j , we obtain the system CðDÞðDI m þ JAÞyðtÞ dt ¼ dbðtÞ which may be written as…”

“…3 Note that, when l ¼ 0, we define Mð0; y 1 Þ using the limits lim l!0 gðilÞ ¼ 1 and lim l!0 hðilÞ ¼ 1=2; see Lemma A.5 of Chambers (2003). approximation to (6) based on the discrete Fourier transform (dFt) of (4):…”

Frequency Domain Gaussian Estimation of Temporally Aggregated Cointegrated Systems

“…Combining (1) and (2) we may write dy(t) = −JAy(t)dt + w(t)dt, where the matrices J and A are defined in Lemma 1 and w(t) depends on u(t); see equation (3) of Chambers (2003). Writing this as (DI m + JA)y(t)dt = w(t)dt and assuming that w(t) may be represented as a continuous time VAR(q) process of the form Ψ(D)w(t)dt = db(t), with Ψ(z) = z q I m + q−1 j=0 Ψ j z j , we obtain the system Ψ(D)(DI m + JA)y(t)dt = db(t) which may be written…”