Effect of autocorrelation when estimating the trend of a time series via penalized least squares with controlled smoothness

摘要

This paper studies the effect of autocorrelation on the smoothness of the trend of a univariate time series estimated by means of penalized least squares. An index of smoothness is deduced for the case of a time series represented by a signal-plus-noise model, where the noise follows an autoregressive process of order one. This index is useful for measuring the distortion of the amount of smoothness by incorporating the effect of autocorrelation. Different autocorrelation values are used to appreciate the numerical effect on smoothness for estimated trends of time series with different sample sizes. For comparative purposes, several graphs of two simulated time series are presented, where the estimated trend is compared with and without autocorrelation in the noise. Some findings are as follows, on the one hand, when the autocorrelation is negative (no matter how large) or positive but small, the estimated trend gets very close to the true trend. Even in this case, the estimation is improved by fixing the index of smoothness according to the sample size. On the other hand, when the autocorrelation is positive and large the simulated and estimated trends lie far away from the true trend. This situation is mitigated by fixing an appropriate index of smoothness for the estimated trend in accordance to the sample size at hand. Finally, an empirical example serves to illustrate the use of the smoothness index when estimating the trend of Mexico's quarterly GDP.