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ADF test AIC SIC allow for fractional autoregressive alternatives Canada and Japan Carl Walsh cointegrating regression cointegration approach cointegration test CPIs Critical values cumulative impulse responses deviations from PPP Diebold and Rudebusch Donald Wittman economic empirical results Engle and Granger find cointegration fractional alternatives fractional cointegration analysis fractional differencing fractionally integrated equilibrium fractionally integrated models fractionally integrated process Geweke and Porter-Hudak GPH estimate GPH spectral regression GPH test homogeneity condition hypergeometric function impulse response analysis integrated equilibrium errors Italy lag polynomial long-run equilibrium relationship long-run PPP maximum likelihood mean reversion behavior Michael Hutchison model specification Monte Carlo method moving average notion of cointegration notion of fractional null hypothesis persistence in deviations Porter-Hudak 1983 PPP hypothesis PPP relationship PPP shocks price indices Purchasing Power Parity Rudebusch 1989 sample short-run Sowell spectral density Table test for cointegration test for long-run testing for fractional unit root tests United Kingdom Vector Autoregression WPIs zero