The Role Of Initial Values In Conditional Sum-Of-Squares Estimation Of Nonstationary Fractional Time Series Models

In this paper, we analyze the influence of observed and unobserved initial values on the bias of the conditional maximum likelihood or conditional sum-of-squares (CSS, or least squares) estimator of the fractional parameter, d, in a nonstationary fractional time series model. The CSS estimator is popular in empirical work due, at least in part, to its simplicity and its feasibility, even in very complicated nonstationary models. We consider a process, X t , for which data exist from some point in time, which we call –N 0 + 1, but we only start observing it at a later time, t = 1. The parameter (d, μ, σ 2) is estimated by CSS based on the model ${\rm{\Delta }}_0^d \left( {X_t - \mu } \right) = \varepsilon _t ,t = N + 1, \ldots ,N + T$ , conditional on X 1,..., X N . We derive an expression for the second-order bias of $\hat d$ as a function of the initial values, X t , t = – N 0 + 1,..., N, and we investigate the effect on the bias of setting aside the first N observations as initial values. We compare $\hat d$ with an estimator, $\hat d_c $ , derived similarly but by choosing μ = C. We find, both theoretically and using a data set on voting behavior, that in many cases, the estimation of the parameter μ picks up the effect of the initial values even for the choice N = 0. If N 0 = 0, we show that the second-order bias can be completely eliminated by a simple bias correction. If, on the other hand, N 0 > 0, it can only be partly eliminated because the second-order bias term due to the initial values can only be diminished by increasing N.