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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.

Statistical analyses of data sets with missing observations have long been addressed in the literature. For instance, Morrison [

The literature on missing data has expanded in the last decades focusing mainly on univariate time series models [

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

Throughout this paper, we assume that the incomplete data set has a monotone missing data pattern. We follow a likelihood-based approach to estimate the parameters of the model. It is worth pointing out that, in the literature, likelihood-based estimation is largely used to manage the problem of missing data [

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

The paper is organised as follows. In Section

In this section, a few properties of the Vectorial Autoregressive Model of order one are analysed. These features will play an important role in determining the estimators for the parameters when there are missing observations, as we will see in Section

Hereafter, the stochastic process underlying the complete data set is denoted by

We have to introduce the restrictions

Next, we overview some relevant properties of the VAR

The mean values of

Concerning the covariance structure of the process

For

Considering that

In regard to the structure of covariance between the stochastic processes

When

By writing out the stochastic system of (

Hence, at each date

Straightforward computations lead us to the following factoring of the probability density function of

Thus, the joint distribution of the pair

The variance has the following structure:

The conditional distribution of

We focus here on theoretical background for factoring the likelihood function from the VAR

That is, there are

Let the observed bivariate sample of size

Two points must be emphasised: first, we emphasise that the maximum likelihood estimators (m.l.e.) for the unknown vector of parameters will be obtained by maximising the natural logarithm of the above likelihood function. Second, a worthwhile improvement in reducing the complexity of the function to maximise is to determine the conditional maximum likelihood estimators regarding the first pair of random variables,

Despite the above solutions for reducing the complexity of the problem, some difficulties still remain. The loglikelihood equations are intractable. To go over this problem we have to factorise the conditional likelihood function. From (

So as to work out the analytical expressions for the unknown parameters under study, we have to decompose the entire likelihood function (

For the Gaussian VAR processes, the conditional maximum likelihood estimators coincide with the least squares estimators [

The aforementioned arguments guarantee that the decomposition of the joint likelihood in two components can be carried out with no loss of information for the whole estimation procedure. From (

Henceforth,

The components

In Section

Theoretical developments carried out in this section rely on solving the loglikelihood equations obtained from the factored loglikelihood given by (

Let

Let

The sample covariance coefficient of

We readily find out that the m.l.e. for the parameters under study are given by

Using the results from Section

Thus, the analytical expressions for the estimators of the mean values, variances, and covariances of the VAR

These estimators will play a central role in the following sections.

In the section, the precision of the maximum likelihood estimators underlying equations (

There are a few points worth mentioning. From Section

To lighten notation, from now on we suppress the “hat” from the consistent estimators of the information matrices.

The variance-covariance matrix for

We stress that there is orthogonality between the error and the estimation subspaces underlying the loglikelihood function

Calculating the second derivatives of the loglikelihood function

Once again, we mention that there is orthogonality between the error and the estimation subspaces underlying the loglikelihood function

Using the above partition of

Unfortunately, there is no explicit expression for the inverse matrix

Now, we have to analyse the precision of the m.l.e. of the original parameters of the VAR

Recalling from Section

The parameters

The variance-covariance matrix of the m.l.e. for the vector of parameters

Writing the vector of parameters

The

For finding out the approximate variance-covariance matrix of the maximum likelihood estimators for the unknown vector of parameters

We can now deduce the approximate variance-covariance matrix of the maximum likelihood estimators for the mean vector and the variance-covariance matrix at lag zero of the VAR

According to the partition of the matrix

The

Straightforward calculations have paved the way to the desired partitioned variance-covariance matrix, called here

The matrix

The

The main point of the section is to study the variances and covariances that take part of the sub-matrix

Let

with

The

On the other hand, we can also make the following partition of the matrix

The desired variance-covariance matrix can therefore be written in the following partitioned form:

In short, the matrix defined by (

Despite the above restrictions, several investigations can be done regarding the amount of additional information obtained by making full use of the fragmentary data available. The strength of the correlation between the stochastic processes here plays a crucial role. These ideas will be developed in Section

In this section, we analyse the effects of using different strategies to estimate the mean value of the stochastic process

The m.l.e. of the mean value of the stochastic process

To avoid any confusion between the parameters coming from the bivariate and the univariate modelling strategies, from now on we denote the parameter from the VAR

The bivariate VAR

On the other hand, if we assumed that

Next, we will compare the performance of the estimators (

Following the techniques used in Section

Considering the ARMA(2,1) Model, let

In regard to the AR

Improvements in choosing the sophisticated m.l.e. for

The data were generated by the VAR

Since the correlation coefficient regulates the supply of information between the stochastic processes

We analyse the performance of the estimators based on different sample sizes,

It is worth emphasising that the estimates of the covariance terms that take part of the variances given by (

The simulation goes as follows: after each simulation run, the relative efficiency of

A word of notation: the above quantities, that is,

If

Figures

Graphical representation of

Graphical representation of

Either Figures

Further, the gain in precision by using the sophisticated estimator

A final point to highlight from the comparison between Figures

Summing up, the estimator

This article deals with the problem of missing data in an univariate sample. We have considered an auxiliary complete data set, whose underlying stochastic process is serially correlated with the former by the VAR

We have compared the performance of the estimator for

A compelling question remains unresolved. From an applied point of view, it would be extremely useful to develop estimators for the dynamics of the stochastic processes. More precisely, we would like to get estimators for the correlation and cross-correlation matrices as well as their precision when there are missing observations in one of the data sets. It was not possible to achieve this goal based on maximum likelihood principles. As we have shown in Section

This work was financed by the Portuguese Foundation for Science and Technology (FCT), Projecto Estratégico PEst-OE/MAT/UI0209/2011. The authors are also thankful for the comments of the two anonymous referees.