The increment ratio statistic

We introduce a new statistic written as a sum of certain ratios of second-order increments of partial sums process S n = ∑ t = 1 n X t of observations, which we call the increment ratio (IR) statistic. The IR statistic can be used for testing nonparametric hypotheses for d -integrated ( - 1 2 < d < 3 2 ) behavior of time series X t , including short memory ( d = 0 ), (stationary) long-memory ( 0 < d < 1 2 ) and unit roots ( d = 1 ). If S n behaves asymptotically as an (integrated) fractional Brownian motion with parameter H = d + 1 2 , the IR statistic converges to a monotone function Λ ( d ) of d ∈ ( - 1 2 , 3 2 ) as both the sample size N and the window parameter m increase so that N / m → ∞ . For Gaussian observations X t , we obtain a rate of decay of the bias E IR - Λ ( d ) and a central limit theorem ( N / m ) 1 / 2 ( IR - E IR ) → N ( 0 , σ 2 ( d ) ) , in the region - 1 2 < d < 5 4 . Graphs of the functions Λ ( d ) and σ ( d ) are included. A simulation study shows that the IR test for short memory ( d = 0 ) against stationary long-memory alternatives ( 0 < d < 1 2 ) has good size and power properties and is robust against changes in mean, slowly varying trends and nonstationarities. We apply this statistic to sequences of squares of returns on financial assets and obtain a nuanced picture of the presence of long-memory in asset price volatility.