Abstract
Missing-data problems are extremely common in practice. To achieve reliable inferential results, we need to take into account this feature of the data. Suppose that the univariate data set under analysis has missing observations. This paper examines the impact of selecting an auxiliary complete data set—whose underlying stochastic process is to some extent interdependent with the former—to improve the efficiency of the estimators for the relevant parameters of the model. The Vector AutoRegressive (VAR) Model has revealed to be an extremely useful tool in capturing the dynamics of bivariate time series. We propose maximum likelihood estimators for the parameters of the VAR(1) Model based on monotone missing data pattern. Estimators’ precision is also derived. Afterwards, we compare the bivariate modelling scheme with its univariate counterpart. More precisely, the univariate data set with missing observations will be modelled by an AutoRegressive Moving Average (ARMA(2,1)) Model. We will also analyse the behaviour of the AutoRegressive Model of order one, AR(1), due to its practical importance. We focus on the mean value of the main stochastic process. By simulation studies, we conclude that the estimator based on the VAR(1) Model is preferable to those derived from the univariate context.
Highlights
Statistical analyses of data sets with missing observations have long been addressed in the literature
This paper aims at analysing the main properties of the estimators from data generated by one of the most influential models in empirical studies, that is, the first-order Vector AutoRegressive (VAR(1)) Model, when the data set from the main stochastic process, designated by {Yt}t∈Z, has missing observations
A final point to highlight from the comparison between Figures 1 and 2 is that the increase in precision obtained by using the estimator for the mean value of {Yt}t∈Z based on the VAR(1) Model is higher when we compare its performance with the results from the AR(1) Model than when we compare the VAR(1) Model with the ARMA(2,1) Model
Summary
Statistical analyses of data sets with missing observations have long been addressed in the literature. This paper aims at analysing the main properties of the estimators from data generated by one of the most influential models in empirical studies, that is, the first-order Vector AutoRegressive (VAR(1)) Model, when the data set from the main stochastic process, designated by {Yt}t∈Z, has missing observations. In order to answer the question raised above, we must verify if the introduction of an auxiliary variable for estimating the parameters of the model increases the accuracy of the estimators To accomplish this goal, we compare the precision of the estimators just cited with those obtained from modelling the dynamics of the univariate stochastic process {Yt}t∈Z by an AutoRegressive Moving Average (ARMA(2,1)) Model, which corresponds to the marginal model of the bivariate VAR(1) Model [10, 11]. We will discard these particular cases, which means that ψ3 ≠ 0
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