Cointegration in Fractional Systems with Unknown Integration Orders
研究非平稳时间序列在分数阶协整中的估计问题，允许积分阶数未知，提出频域和时域估计方法，并应用于检验购买力平价假说。
Cointegration of nonstationary time series is considered in a fractional context. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is nested, but these values have typically been assumed known. We allow one or more of them to be unknown real values, in which case Robinson and Marinucci (1997,2001) have justified least squares estimates of the cointegrating vector, as well as narrow-band frequency-domain estimates, which may be less biased. While consistent, these estimates do not always have optimal convergence rates, and they have non-standard limit distributional behaviour. We consider estimates formulated in the frequency domain, that consequently allow for a wide variety of (parametric) autocorrelation in the short memory input series, as well as time-domain estimates based on autoregressive transformation. Both can be interpreted as approximating generalized least squares and Gaussian maximum likelihood estimates. The estimates share the same limiting distribution, having mixed normal asymptotics (yielding Wald test statistics with χ 2 null limit distributions), irrespective of whether the integration orders are known or unknown, subject in the latter case to their estimation with adequate rates of convergence. The parameters describing the short memory stationary input series are √ n-consistently estimable, but the assumptions imposed on these series are much more general than ones of autoregressive moving average type. A Monte Carlo study of finite-sample performance and an empirical application to testing the PPP hypothesis are included.