Fractional integration and impulse responses: a bivariate application to real output in the USA and four Scandinavian countries

Abstract

This article analyzes impulse response functions in the context of vector fractionally integrated time series. We derive analytically the restrictions required to identify the structural-form system. As an illustration of the recommended procedure, we carry out an empirical application based on a bivariate system including real output in the USA and, in turn, in one of the four Scandinavian countries (Denmark, Finland, Norway, and Sweden). The empirical results appear to be sensitive, to some extent, to the specification of the stochastic process driving the disturbances, but generally a positive shock to US output has a positive effect on the Scandinavian countries, which tend to disappear in the long run.

Keywords:

• long memory
• multivariate time series
• impulse response functions

JEL Classification :

• C22

Acknowledgements

L.A. Gil-Alana gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnologia (ECO2008-03035 ECON Y FINANZAS, Spain). Useful comments of the editor and two anonymous referees are gratefully acknowledged.

Notes

An I(0) vector process is defined as a covariance stationary vector process with spectral density matrix F(λ) that is finite and positive definite.

Note that A=(a, c) ^T , (b, d) ^T is assumed to be non-singular (Section 1), so the inversion exists.

Essentially, the two procedures are equivalent. Gil-Alana 9 10 is based on the frequency domain while Nielsen 24 develops similar tests in the time domain.

The choice of 0.01 increments is arbitrary. We also conducted the tests with 0.001 increments and though the proportion of non-rejection values was clearly higher, the results were completely in line with those reported in Table 1.

Using 0.001 increments, the estimated values were (0.602, 0.657) for Denmark; (0.600, 0.633) for Finland; (0.581, 0.632) for Sweden; and (0.578, 0.664) for Norway.

Using 0.001 increments, the values were (1.313, 0.644) for Denmark; (1.221, 0.730) for Finland; (1.078, 0.973) for Sweden; and (1.171, 0.773) for Norway, and very similar results were obtained using Nielsen’s 23 likelihood approach.

Higher VAR orders were also considered and the results did not substantially differ from those based on a VAR(1) model. Though it is quite common in applied work to determine the VAR lag order based on information-based order selection criteria 13 25 34, it has not yet been theoretically justified in the context of fractional integration.

The use of other likelihood criteria like the Akaike information criterion (AIC) and Bayesian information criterion (BIC) lead essentially to the same results in favour of the white noise specification. Note, however, that these criteria may not necessarily be optimal for applications involving fractional differences, as these criteria focus on the short-term forecasting ability of the fitted model and may not give sufficient attention to the long-run properties of the autoregressive fractionally integrated moving average (ARFIMA) models 16 17.