| Much data in the social sciences occurs as time series. Present in many is a trend in mean and sometimes variance. When such series are analyzed, the trend component needs to be modelled carefully, yet trend analysis is one of the least developed aspects of time series methodology. This poor development is partly attributable to the lack of a clear definition of trend and also to the difficulty of dealing with trends.;Chapter one discusses the history of time series and the recognition of trend.;Chapter two reviews past practices used to identify trend. Included are curve fitting, smoothing and moving averages, polynomial fitting, the variate difference method, differencing and regression on time.;Chapter three illustrates one danger of inappropriate detrending. When testing for the presence of cointegration, if the series in question are wrongly detrended, then the critical values of the augmented Dickey-Fuller test statistic provided by Engle and Granger are incorrect, and the power of the test will be overstated. The proper critical values are provided and it is shown that the distribution of the Dickey-Fuller test statistic when the trend is misspecified converges not to negative infinity, as usual, but to something else.;Chapter four compares two new detrending methods introduced by Watson and Granger and shows how they can be synthesized into one which outperforms the Watson model in terms of root mean square errors, and greatly increases the variety of trend shapes available. The resulting stochastic trend is modelled in an unobserved components framework and estimated using a Kalman filter.;Chapter five restricts attention to a monotonic trend model. The trend term is equal to the trend last period plus two error-correction terms, one short term to adjust for discrepancies between the two previous trend terms, yielding the smooth shape typically expected of trend, the other long term to adjust the forecasting trend to the actual value of the series. Monotonic trends are successfully estimated using nonlinear least squares for several series where linear trends are clearly inappropriate. |
