Lagrange Multiplier Tests for Fractional Difference

This paper develops Lagrange multiplier tests of ARMA( p, q ) models against fractional ARIMA( p, d, q ) alternatives. The performance of the tests is investigated for moderate‐sized samples. It is concluded that fractional difference will be difficult to detect when the orders ( p, q ) are over‐specified in an autoregressive moving‐average (ARMA) analysis. The importance of distinguishing between the mean known and mean estimated cases in fractional difference models is illustrated in the context of these tests.