Tying Maximum Likelihood Estimation for Dependent Data

This study proposes a tying maximum likelihood estimation (TMLE) method to improve the estimation performance of parametric models for dependent data where some time series are long while others are shorter. The TMLE achieves this by flexibly tying some parameters of the long time series to those of the short ones, facilitating the transfer of valuable information to improve parameter estimation accuracy for the short series. We derive the asymptotic properties of the TMLE and its finite-sample risk bound under a tuning parameter that determines the strength of the tying. The theoretical analysis shows that the TMLE not only substantially outperforms the standard MLE under the correct tying but also can retain this strength under a local misspecification setting. We propose a feasible bootstrapping procedure for selecting the tuning parameter to reduce the finite-sample risk, with supporting finite-sample theory guiding implementation. Extensive simulations demonstrate that the TMLE exhibits superior performance compared to alternative methods. An empirical application based on a multivariate GARCH-type model for the volatility dynamics of Facebook and Twitter stock returns shows improved predictive performance attributable to tying parameters, consistent with the view that these two stocks have similar exposure to common risk factors and related volatility dynamics.