A simple method for detecting fractional cointegration relation: an application to Finnish data

The cointegration technique, based originally on nonstationary component, reduced 'Dimension Reduction' to some linear combination that is stationary. The obtained dimension has unit minimum and s - 1 maximum linear combination setting, where s stands for number of components in the system. In this paper a new reduction concept is provided that explains fractional cointegrated system. This reduction is located on specific angle and defined as the cointegration direction. The reduction 'Level Reduction' exists with smaller level of fractional orders, i.e. when -0.5 < dz < 0.5, where dz represent the fractional differences of the linear combination series, while the original series orders are in ≥ 0.5 level and the system is fractionally cointegrated. This, of course, can be considered as a new way to detect fractional cointegration relation among variables of the same order of differencing in the frequency domain. A concept Periodogram Roughness (PR) is used to estimate and test the cointegration direction. The description of the direction property will be presented. A test statistic is presented that is based on the difference between the maximum and minimum of the PR. Using Monte Carlo simulations, it is found that the properties of the test perform satisfactorily regarding both size and power. Fractional differencing of linear combination series is estimated using Periodogram Roughness. This approach is illustrated using quarterly data on the government spending and revenue in Finland during the period of 1960 to 1997.