MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 848120 10.1155/2013/848120 848120 Research Article Maximum Likelihood Estimation of the VAR(1) Model Parameters with Missing Observations Mouriño Helena Barão Maria Isabel Xie Xuejun 1 Departamento de Estatística e Investigação Operacional Faculdade de Ciências Universidade de Lisboa Edifício C6, Piso 4, Campo Grande, 1749-016 Lisboa Portugal ul.pt 2013 22 5 2013 2013 04 01 2013 29 03 2013 08 04 2013 2013 Copyright © 2013 Helena Mouriño and Maria Isabel Barão. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Statistical analyses of data sets with missing observations have long been addressed in the literature. For instance, Morrison  deduced the maximum likelihood estimators of the parameters of the multinormal mean vector and covariance matrix for the monotonic pattern with only a single incomplete variate. The exact expectations and variances of the estimators were also deduced. Dahiya and Korwar  obtained the maximum likelihood estimators for a bivariate normal distribution with missing data. They focused on estimating the correlation coefficient as well as the difference of the two means. Following this line of research and having in mind that the majority of the empirical studies are characterised by temporal dependence between observations, we will try to generalise the previous study by introducing a bivariate time series model to describe the relationship between the processes under consideration.

The literature on missing data has expanded in the last decades focusing mainly on univariate time series models , but there is still a lack of developments in the vectorial context.

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, has missing observations. Therefore, we assume that there is also available a suitable auxiliary stochastic process, denoted by {Xt}t, which is to some extent interdependent with the main stochastic process. Additionally, the data set obtained from this process is complete. In this context, a natural question arises: is it possible to exchange information between the two data sets to increase knowledge about the process whose data set has missing observations, or should we analyse the univariate stochastic process by itself? The goal of this paper is to answer this question.

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 [3, 8, 9]. The precision of the maximum likelihood estimators is also derived.

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 by an AutoRegressive Moving Average (ARMA(2,1)) Model, which corresponds to the marginal model of the bivariate VAR(1) Model [10, 11]. The behaviour of the AutoRegressive Model of order one, AR(1), is also analysed due to its practical importance in time series modelling. Simulation studies allow us to assess the relative efficiency of the different approaches. Special attention is paid to the estimator for the mean value of the stochastic process about which information available is scarce. This is a reasonable choice given the importance of the mean function of a stochastic process in understanding the behaviour of the time series under consideration.

The paper is organised as follows. In Section 2, we review the VAR(1) Model and highlight a few statistical properties that will be used in the remaining sections. In Section 3, we establish the monotone pattern of missing data and factorise the likelihood function of the VAR(1) Model. The maximum likelihood estimators of the parameters are obtained in Section 4. Their precision is also deduced. Section 5 reports the simulation studies in evaluating different approaches to estimate the mean value of the stochastic process {Yt}t. The main conclusions are summarised in Section 6.

2. Brief Description of the VAR(1) Model

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 4.

Hereafter, the stochastic process underlying the complete data set is denoted by {Xt}t, while the other one is represented by {Yt}t. The VAR(1) Model under consideration takes the form (1)Xt=α0+α1Xt-1+ϵt,Yt=β0+β1Yt-1+β2Xt-1+ξt,t=0,±1,±2,, where ϵt and ξt are Gaussian white noise processes with zero mean and variances σϵ2 and σξ2, respectively. The structure of correlation between the error terms is different from zero only at the same date t, that is, Cov(ϵt-i,ξt-j)=σϵ  ξ, for i=j; Cov(ϵt-i,ξt-j)=0, for ij,i,j. Exchanging information between both time series might introduce some noise in the overall process. Therefore, transfer of information from the smallest series to the largest one is not allowed here.

We have to introduce the restrictions |α1|<1 and |β1|<1. They ensure not only that the underlying processes are ergodic for the respective means but also that the stochastic processes are covariance stationary (see Nunes [12, ch.3]). Hereafter, we assume that these restrictions are satisfied.

Next, we overview some relevant properties of the VAR(1) Model (1). Theoretical details can be found in Nunes [12, ch.3].

The mean values of Xt and Yt are, respectively, given by (2)E(Xt)=α01-α1,E(Yt)=α0  β2+β0(1-α1)(1-α1)(1-β1).

Concerning the covariance structure of the process Xt, (3)Cov(Xt-i,Xt-j)=σϵ2α1|i-j|1-α12,i,j.

For α1β1, the covariance of the stochastic process Yt is given by (4)Cov(Yt-i,Yt-j)=σξ2β1|i-j|1-β12  +  σϵξ  β2{β1β1|i-j|(1-β12)(1-α1β1)1β1-α1×(β1|i-j|1-β12-α1|i-j|1-α1β1)+β1β1|i-j|(1-β12)(1-α1β1)}+σϵ2  β22(β1-α1)(1-α1β1)  ×(β1β1|i-j|1-β12-α1α1|i-j|1-α12),i,j.

Considering that   α1=β1, we have (5)Cov(Yt-i,Yt-j)=β1|i-j|1-β12{(|i-j|+1+β121-β12)σξ2+2  σϵξβ2×(β11-β12+|i-j|β1)+σϵ2  β221-β12(|i-j|+1+β121-β12)(|i-j|+1+β121-β12)},fori,j.

In regard to the structure of covariance between the stochastic processes Xt and Yt, for α1β1, we have (6)Cov(Yt-i,Xt-j)=σϵξ  α1|i-j|1-α1β1+σϵ2  α1  β2α1|i-j|(1-α1β1)(1-α12),i,j.

When α1=β1, the covariance function under study takes the form (7)Cov(Yt-i,Xt-j)=σϵ2  β2α1α1|i-j|(1-α12)2+σϵξ  α1|i-j|1-α12,i,j.

By writing out the stochastic system of (1) in matrix notation, the bivariate stochastic process Zt=[Xt        Yt] can be expressed as (8)Zt=[α0β0]+[α10β2β1][Xt-1Yt-1]+[ϵtξt]=c+Φ1Zt-1+ϵt,t, where     ϵt=[ϵt        ξt] is the 2-dimensional Gaussian white noise random vector.

Hence, at each date t,t, the conditional stochastic process Zt|Zt-1=zt-1 follows a bivariate Gaussian distribution,   Zt|Zt-1=zt-1𝒩2(μt|t-1,Ωt|t-1),  where the two-dimensional conditional mean value vector and the variance-covariance matrix are, respectively, given by (9)μt|t-1=c+Φ1Zt-1,Ωt|t-1Ω=[σϵ2σϵξσϵξσξ2].

Straightforward computations lead us to the following factoring of the probability density function of   Zt   conditional to   Zt-1=zt-1: (10)fZt|Zt-1(ztzt-1)=fXt|Zt-1(xtzt-1)×fYt|Xt,Zt-1(ytxt,zt-1).

Thus, the joint distribution of the pair Xt and Yt conditional to the values of the process at the previous date t-1, Zt-1, can be decomposed into the product of the marginal distribution of Xt|Zt-1 and the conditional distribution of Yt|Xt,Zt-1. Both densities follow univariate Gaussian probability laws: (11)XtZt-1=zt-1𝒩(α0+α1xt-1,σϵ2),for  each  date  t,t. Also,     Yt|Xt=xt,Zt-1=zt-1 follows a Gaussian distribution with (12)E(Yt|Xt=xt,Zt-1=zt-1)=β0+β1yt-1+β2xt-1+σϵξσϵ2(xt-α0-α1xt-1)=ψ0+ψ1xt+ψ2xt-1+β1yt-1, where   ψ1=(σϵξ/σϵ2) or, for interpretive purposes, ψ1=(σξ/σϵ)ρϵξ. The parameter ψ1 describes, thus, a weighted correlation between the error terms ϵt and ξt. The weight corresponds to the ratio of their standard deviations. Moreover, ψ0=β0-ψ1,ψ2=β2-ψ1α1.

The variance has the following structure: (13)Var(YtXt=xt,Zt-1=zt-1)=σξ2-σϵξ2σϵ2=σξ2(1-ρϵξ2)ψ3.

The conditional distribution of Yt|Xt,Zt-1 can be interpreted as a straight-line relationship between Yt and Xt,Xt-1, and Yt-1. Additionally, it is worth mentioning that if ρϵξ=±1 or σξ2=0, the above conditional distribution degenerates into its mean value. Henceforth, we will discard these particular cases, which means that ψ30.

3. Factoring the Likelihood Based on Monotone Missing Data Pattern

We focus here on theoretical background for factoring the likelihood function from the VAR(1) Model when there are missing values in the data. Suppose that we have the following monotone pattern of missing data: (14)x0x2xm-1xmxn-1y0y2ym-1.

That is, there are n observations available from the stochastic process {Xt}t, whereas due to some uncontrolled factors it was only possible to record m(m<n) observations from the stochastic process {Yt}t. In other words, there are n-m missing observations from Yt.

Let the observed bivariate sample of size n with missing values: (15){(x0,y0),(x1,y1),,(xm-1,ym-1),xm,,xn-1}; denote a realisation of the random process Zt=[Xt        Yt],  t, which follows a vectorial autoregressive model of order one. The likelihood function, L(θ), is given by (16)L(θ)fZ0,Z1,,Zm-1,Xm,,Xn-1×(z0,z1,,zm-1,xm,,xn-1)=fZ0(z0)t=1m-1fZt|Zt-1(ztzt-1)fXm|Zm-1(xmzm-1)×t=m+1n-1fXt|Xt-1(xtxt-1;θ)=fZ0(z0)t=1m-1fZt|Zt-1(ztzt-1)×t=mn-1fXt|Xt-1(xtxt-1), where   θ=[α0        α1        σϵ2        β0        β1        β2        σξ2        σϵξ] is the 8-dimensional vector of population parameters. To lighten notation, we assume that there is no need for conditioning the arguments of the above probability density functions on the values of the processes at date t-1. The likelihood function becomes (17)L(θ)=fZ0(z0)t=1m-1fZt|Zt-1(zt)t=mn-1fXt|Xt-1(xt).

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, Z0=[X0    Y0], as deterministic and maximising the log-likelihood function conditioned on the values X0=x0 and Y0=y0. The loss of efficiency of the estimators obtained from such a procedure is negligible when compared with the exact maximum likelihood estimators computed by iterative techniques. Even for moderate sample sizes, the first pair of observations makes a negligible contribution to the total likelihood. Hence, the exact m.l.e. and the conditional m.l.e. turn out to have the same large sample properties, Hamilton . Hereafter, we restrict the study to the conditional loglikelihood function.

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 (17) we get (18)L(θ)=t=1m-1f(Xt,Yt)|(Xt-1,Yt-1)(xt,yt)  t=mn-1fXt|Xt-1(xt)=t=1m-1(fXt|(Xt-1,Yt-1)(xt)×fYt|Xt,(Xt-1,Yt-1)(yt))×t=mn-1fXt|Xt-1(xt)=t=1n-1fXt|Xt-1(xt)  t=1m-1fYt|Xt,(Xt-1,Yt-1)(yt).

So as to work out the analytical expressions for the unknown parameters under study, we have to decompose the entire likelihood function (18) into easily manipulated components.

For the Gaussian VAR processes, the conditional maximum likelihood estimators coincide with the least squares estimators . Therefore, we may find a solution to the problem just raised in the geometrical context. The identification of such components relies on two of the most famous theorems in the Euclidean space: the Orthogonal Decomposition Theorem and the Approximation Theorem [14, Volume I, pages 572–575]. Based on these tools it is straightforward to establish that the estimation subspaces associated with the conditional distributions Xt|Xt-1 and Yt|Xt,Xt-1,Yt-1 are, by construction, orthogonal to each other. This means that each element belonging to one of those subspaces is uncorrelated with each element that pertains to their orthogonal complement. Hence, events that happen on one subspace provide no information about events on the other subspace.

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 (18) we can, thus, decompose the conditional loglikelihood function as follows: (19)ll(θ)=logL(θ)=t=1n-1  logfXt|Xt-1(xt)+t=1m-1  logfYt|Xt,(Xt-1,Yt-1)(yt)=l1+l2.

Henceforth, l1 denotes the loglikelihood from the marginal distribution of Xt, based on the whole sampled data with dimension n, that is, x0,x1,,xn-1. The function l2 represents the loglikelihood from the conditional density of Yt|Xt,Zt-1 computed by the bivariate sample of size m: (20)(x0,y0),(x1,y1),,(xm-1,ym-1).

The components   l1 and l2   of (19) will be maximised separately in Section 4.1.

4. Maximum Likelihood Estimators for the Parameters

In Section 4.1 the m.l.e. of the parameters from the fragmentary VAR(1) Model are deduced. The precision of the estimators is examined in Section 4.2.

4.1. Analytical Expressions

Theoretical developments carried out in this section rely on solving the loglikelihood equations obtained from the factored loglikelihood given by (19). Before proceeding with theoretical matters, we introduce some relevant notation in the ensuing paragraphs.

Let X¯k(l)=(1/k)t=1kXt-l   represent the sample mean lagged l time units, l=0,1. The subscript k,k=1,,n-1, allows us to identify the number of observations that takes part in the computation of the sample mean. A similar notation is used for denoting the sample mean of the random sample Y0,,Yk, for k=1,,m-1, Y¯k(l). According to this new definition, the sample variance of each univariate random variable based on k observations and lagged l time units is denoted by (21)γ^X,k(l)=1k  t=1k(Xt-l-X¯k(l))2,γ^Y,k(l)=1k  t=1k(Yt-l-Y¯k(l))2,l=0,1.

Let   γ^X,k*(1)=(1/k)t=1k(Xt-X¯k(0))(Xt-1-X¯k(1)) describe the sample autocovariance coefficient at lag one for the stochastic process Xt, based on k observations. Its counterpart for the stochastic process Yt, γ^X,k*(1), is obtained by changing notation accordingly. The sample autocorrelation coefficient of the random process Xt at lag one is denoted by   ρ^X,k(1)=γ^X,k*(1)/γ^X,k(0)  γ^X,k(1)  . The empirical covariance between the random processes Xt and Yt lagged one time unit is represented by (22)γ^XY*(1)=1m-1  t=1m-1(Xt-X¯m-1(0))(Yt-1-Y¯m-1(1)),for  lagged  values  on  Y,γ^YX*(1)=1m-1  t=1m-1(Xt-1-X¯m-1(1))(Yt-Y¯m-1(0)),for  lagged  values  on  X.

The sample covariance coefficient of Xt and Yt computed from l time units lag for each series is given by (23)γ^XY(l)=1m-1  t=1m-1(Xt-l-X¯m-1(l))(Yt-l-Y¯m-1(l)),with    l=0,1.

Maximising the loglikelihood function l1: Using the results (11) and (19), we readily find the following m.l.e. (24)α^0=X¯n-1(0)-α^1X¯n-1(1),α^1=γ^X,n-1*(1)γ^X,n-1(1),σ^ϵ2=SSRn-1, where SSR is the respective residual sum of squares.

Maximising the loglikelihood function l2: Based on (12) and (13) we get the loglikelihood function   l2(25)l2=t=1m-1logfYt|Xt,Xt-1,Yt-1(yt)=-m-12log(2π)-m-12logψ3-12  ψ3×  t=1m-1(yt-ψ0-ψ1xt-ψ2xt-1-β1yt-1)2.

We readily find out that the m.l.e. for the parameters under study are given by (26)ψ^0=Y¯m-1(0)-ψ^1X¯m-1(0)-ψ^2X¯m-1(1)-β^1Y¯m-1(1),ψ^1=1γ^X,m-1(0){γ^XY(0)-ψ^2γ^X,m-1*(1)-β^1γ^XY*(1)},ψ^2=1(1-(ρ^X,m-1  (1))2)γ^X,m-1(1)  ×{γ^YX*(1)-γ^XY(0)γ^X,m-1*(1)γ^X,m-1(0)-β^1γ^XY(1)+  β^1γ^XY*(1)γ^X,m-1*(1)γ^X,m-1(0)},ψ^3=SSR*m-1,β^1=1(γ^Y,m-1(1)+(Y¯m-1(1))2)×{γ^Y,m-1*(1)+Y¯m-1(0)Y¯m-1(1)  -ψ^0  Y¯m-1(1)-ψ^1  γ^XY*(1)-ψ^1  X¯m-1(0)  Y¯m-1(1)-ψ^2(γ^XY(1)-X¯m-1(1)  Y¯m-1(1))}, where SSR*   denotes the corresponding residual sums of squares.

Using the results from Section 2 we get the following estimators for the original parameters: (27)β^0=ψ^0+ψ^1α^0,β^2=ψ^2+ψ^1α^1,σ^ϵξ=ψ^1σ^ϵ2,      σ^ξ2=ψ^3+σ^ϵξ2σ^ϵ2.

Thus, the analytical expressions for the estimators of the mean values, variances, and covariances of the VAR(1) Model are given by (28)μ^X=α^01-α^1,μ^Y=α^0β^2+β^0(1-α^1)(1-α^1)(1-β^1),σ^X2=σ^ϵ21-α^12,σ^Y2=σ^ξ21-β^12+2σ^ϵξβ^1β^2(1-α^1β^1)(1-β^12)+σ^ϵ2β^22(1+α^1β^1)(1-α^12)(1-β^12)(1-α^1β^1)(α1β1),σ^Y2=σ^ξ21-α^12+2σ^ϵξα^1β^2(1-α^12)2+σ^ϵ2β^221+α^12(1-α^12)3,σ^XY=σ^ϵξ1-α^1β^1+α^1β^2σ^ϵ2(1-α^1β^1)(1-α^12)(at  the  same  date  t,t).

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

4.2. Precision of the Estimators

In the section, the precision of the maximum likelihood estimators underlying equations (28) is derived. The whole analysis will be separated in three stages. First, we study the statistical properties of the vector   Θ^, where   Θ^=[Θ^1Θ^2], with Θ^1=[α^0α^1σ^ϵ2] and Θ^2=[ψ^0ψ^1ψ^2β^1ψ^3]. For notation consistency, the unknown parameter β1 is either denoted by β1 or ψ4. That is, ψ4β1. Secondly, we derive the precision of the m.l.e. of the original parameters of the VAR(1) Model (see (1)). Finally, we will focus our attention on the estimators for the mean vector and the variance-covariance matrix at lag zero of the VAR(1) model with a monotone pattern of missingness.

There are a few points worth mentioning. From Section 3 we know that there is no loss of information in maximising separately the loglikelihood functions l1 and l2 (19). As a consequence, the variance-covariance matrix associated with the whole set of estimated parameters is a block diagonal matrix. For sufficiently large sample size, the distribution of the maximum likelihood estimator is accurately approximated by the following multivariate Gaussian distribution: (29)Θ^𝒩8([Θ1Θ2],[I1-100I2-1]), where I1 and I2 denote the Fisher information matrices, respectively, from the components l1 and l2 of the loglikelihood function (see (19)). There is an asymptotic equivalence between the Fisher information matrix and the Hessian matrix (see [8, ch.2]). Moreover, as long as   Θ^Θ there is also an asymptotic equivalence between the Hessian matrix computed at the points Θ^ and Θ. Henceforth, the Fisher information matrices from (29) are estimated, respectively, by (30)I^1=-(2l1Θ1Θ1)|Θ1=Θ^1,I^2=-(2l1Θ2Θ2)|Θ2=Θ^2.

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

The variance-covariance matrix for Θ^1 takes the following form: (31)I1-1=σ^ϵ2(n-1)γ^X,n-1(1)(0)×[γ^X,n-1(1)(0)+(X¯n-1(1))2-X¯n-1(1)0-X¯n-1(1)10002σ^ϵ2  γ^X,n-1(1)(0)].

We stress that there is orthogonality between the error and the estimation subspaces underlying the loglikelihood function   l1.

Calculating the second derivatives of the loglikelihood function l2 results in the following approximate information matrix:(32)I2=1ψ^3[m-1(m-1)X¯m-1(m-1)X¯m-1(1)(m-1)Y¯m-1(1)0(m-1)X¯m-1  t=1m-1Xi2t=1m-1XtXt-1t=1m-1XtYt-10(m-1)X¯m-1(1)t=1m-1XiXi-1t=1m-1Xt-12t=1m-1Xt-1Yt-10(m-1)Y¯m-1(1)t=1m-1XiYi-1t=1m-1Xt-1Yt-1t=1m-1Yt-1200000m-12  ψ^3]  .

Once again, we mention that there is orthogonality between the error and the estimation subspaces underlying the loglikelihood function   l2. The matrix I2 can be written in a compact form: (33)I2=1ψ^3[I2100I22], where the (4×4) submatrix I21 and the scalar I22 are, respectively, defined as (34)I21=UU,I22=m-12(ψ^3)2, with (35)U=[1X1X0Y01X2X1Y11Xm-1Xm-2  Ym-2].

Using the above partition of I2 it is rather simple to compute the inverse matrix. In fact, (36)I2-1=ψ^3[I21-100I22-1], with         I21-1=(UU)-1 and I22-1=(2/(m-1))  (ψ^3)2.

Unfortunately, there is no explicit expression for the inverse matrix I21-1. As a result, there are no explicit expressions for the approximate variance-covariance of the m.l.e. for the vector of unknown parameters   Θ^2.

Now, we have to analyse the precision of the m.l.e. of the original parameters of the VAR(1) Model, that is,   Υ=[α0α1σϵ2β0β1β2σξ2σϵξ].

Recalling from Section 2, the one-one monotone functions that relate the vector of parameters under consideration, that is, (37)    Θ2=[ψ0ψ1ψ2ψ4ψ3],    Υ2=[β0β1β2σξ2  σϵξ],areψ0=β0-α0ψ1,ψ1=σϵξσϵ2,ψ2=β2-ψ1α1ψ3=σξ2-σϵ2  ψ12,ψ4β1  .

The parameters   α0,α1, and σϵ2   remain unchanged. A key assumption in the following developments is that neither the estimates of the unknown parameters nor the true values fall on the boundary of the allowable parameter space.

The variance-covariance matrix of the m.l.e. for the vector of parameters   Υ is obtained by the first-order Taylor expansion at   Υ. We also use the chain rule for derivatives of vector fields ([for details, see [14, Volume II, pages 269–275]).

Writing the vector of parameters   Υ as a function of the vector Θ, the respective first-order partial derivatives can be joined together in the following partitioned matrix: (38)D=[D1D2D3D4], where the (3×3) submatrix D1 corresponds to the first-order partial derivatives of the vector Υ1Θ1=[α0α1σϵ2] with respect to itself, which means that D1 is nothing but the identity matrix of order 3, D1=I3. On the other hand, this statement also means that the derivatives of the parameters under consideration with respect to either ψ0,ψ1,ψ2,ψ3,   or   ψ4 are zero. In other words, the (3×5) submatrix D2 is equal to the null vector, that is, D2=0.

The (5×3) submatrix D3 and the (5×5) submatrix D4 are composed by the first-order partial derivatives of each component of the vector of parameters Υ2=[β0β1β2σξ2σϵξ] with respect to, respectively, α0,α1,σϵ2 and ψ0,ψ1,ψ2,ψ4,ψ3. Their structures are, thus, given by (39)D3=[ψ1000000ψ1000ψ1200ψ1],D4=[1α0000000100α110002ψ1σϵ20010σϵ2000]=[D41D42D43D44].

For finding out the approximate variance-covariance matrix of the maximum likelihood estimators for the unknown vector of parameters Υ, it is only necessary to pre- and postmultiply the variance-covariance matrix arising from expressions (29), (31), and (36) by, respectively, the matrix D and its transpose, D. More precisely, (40)ΣΥDI-1D=[I303×5D3D4][I1-100I2-1][I3D305×3D4]. Hence, (41)ΣΥ[I1-1I1-1D3(I1-1D3)D3I1-1D3+D4I2-1D4], with ΣΥ denoting the variance-covariance matrix of the m.l.e. for the vector of unknown parameters   Υ. A more detailed analysis of the variance-covariance matrix (41) can be found in Nunes [12, ch.3, p.91-92].

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(1) Model with a monotone pattern of missingness, represented by   Ξ=[α0        α1        σϵ2        μX        μY        σX2        σY2        σXY]. The first-order partial derivatives of the vector   Ξ with respect to the vector   Υ are placed in a matrix that is denoted by F. It takes the following form: (42)F=[F1F2F3F4].

According to the partition of the matrix D into four blocks—expression (38)—we partition the matrix F into the following blocks: the (3×3) submatrix F1 corresponds to the partial derivatives of α0,α1, and σϵ2 with respect to themselves. As a consequence, F1 is the identity matrix of order 3, that is, F1=I3. Regards to the (3×5) sub-matrix F2, its elements correspond to the partial derivatives of α0,α1, and σϵ2 with respect to β0,β1,β2,σξ2, and σϵξ. Therefore, F2=0. The partial derivatives of μX,μY,σX2,σY2, and σXY with respect to α0,α1, and σϵ2 are gathered together in the (5×3) sub-matrix F3: (43)F3=[f113  f1230f213f22300f323f3330f423f4330f523f533], where (44)f113=11-α1,f123=α0(1-α1)2,f213=β2(1-β1)(1-α1),f223=α0β2(1-β1)(1-α1)2,f323=2α1σϵ2(1-α12)2,f333=11-α12,f423=2β2(σϵξβ12(1-α12)2+σϵ2β2(β1(1-α12)+α1(1-α12β12)))×(((1-α1  β1)(1-α12))2(1-β12))-1,f433=β22(1+α1  β1)(1-α12)(1-β12)(1-α1β1),f523=σϵξβ1(1-α1β1)2+σϵ2β21+α12(1-2α1β1)(1-α1β1)2  (1-α12)2,f533=α1β2(1-α1β1)(1-α12).

The 5-dimensional square sub-matrix F4 corresponds to the partial derivatives of μX,μY,σX2,σY2, and σXY with respect to β0,β1,β2,σξ2,σϵξ: (45)F4=[0    0    0    0        0f214    f224    f234    0        00    0    0    0        00    f424f434f444    f4540    f524f5340    f554], with its nonnull elements taking the following analytical expressions: (46)f214=11-β1,f224=β0(1-β1)2+α0β2(1-α1)(1-β1)2,f234=α0(1-α1)(1-β1),f424=2σξ2β1(1-β12)2+2σϵξβ2(1+β12(1-2α1β1))(1-α1  β1)2(1-β12)2+2σϵ2β22  (α1(1-β12)+β1(1-α12β12))(1-α12)(1-β12)2  (1-α1β1)2,f434=2(1-α1β1)(1-β12){β1σϵξ+σϵ2  β2(1+α1β1)1-α12},f444=11-β12,f454=2  β1β2(1-α1β1)(1-β12),f524=(1-α12)σϵξα1+σϵ2α12β2(1-α12)(1-α1β1)2,f534=σϵ2α1(1-α1β1)(1-α12),f554=11-α1β1.

Straightforward calculations have paved the way to the desired partitioned variance-covariance matrix, called here ΣΞ, (47)ΣΞFΣΥFFDI-1DF=[ΣΞ11ΣΞ12ΣΞ21ΣΞ22], with its submatrices defined by (48)ΣΞ11=I1-1,      ΣΞ12=I1-1(F3+F4D3)=I1-1(F3+D3F4),ΣΞ21=(F3+F4D3)I1-1=(ΣF12),ΣΞ22=F3I1-1F3+F4D3I1-1F3+F3I1-1  D3F4+F4(D3  I1-1  D3+D4  I2-1  D4)F4=(F3+F4D3)I1-1(F3+F4D3)+F4D4I2-1(F4D4)=G  I1-1G+H  I2-1H.

The matrix G that has just been defined as G=F3+F4D3 corresponds to the first-order partial derivatives from the composite functions that relate μX,μY,σX2,σY2, and σXY with the vector of parameters Θ1. The elements of the matrix H=F4D4 are the first-order partial derivatives from the composite functions that relate μX,μY,σX2,σY2, and σXY with the vector of unknown parameters   Θ2.

The 3-dimensional square sub-matrix ΣΞ11 corresponds to the approximate covariance structure between the m.l.e. of the parameters α0,α1, and σϵ2. The (3×5) sub-matrix ΣΞ12 is composed of the approximate covariances between the m.l.e. that have just been cited and μX,μY,σX2,σY2, and σXY; its transpose is denoted by ΣΞ21. This is the reason why ΣΞ12, or ΣΞ21, results from the product of the variance-covariance matrix I1-1 and G. The 5-dimensional square sub-matrix ΣΞ22 is formed by the covariances between the m.l.e. for μX,μY,σX2,σY2, and σXY.

The main point of the section is to study the variances and covariances that take part of the sub-matrix ΣΞ22. Thus, it is of interest to further explore its analytical expression. The matrix G takes a cumbersome form. The most efficient way to deal with it is to consider its partition rather than the whole matrix at once.

Let (49)G=F3+F4D3=[G11G12G21G22], where the (4×2) sub-matrix G11 takes the form (50)G11=[11-α1  α0(1-α1)2ψ11-β1+β2(1-α1)(1-β1)  α0(1-α1)(1-β1)  (ψ1+β21-α1)02α1σϵ2(1-α12)20g4211],

with (51)g4211=ψ1f434+2β2(1-β12)(1-α1  β1)2(1-α12)2×(σϵξβ12  (1-α12)2+σϵ2  β2(β1(1-α12)+  α1(1-α12β12))), where f434 is defined by (45).

The 4-dimensional column vector G12, the 2-dimensional row vector G21 and the scalar G22 are, respectively, given by (52)G12=[0011-α12ψ121-β12+  β2(1-α1β1)(1-β12)  ×(2ψ1β1+β2  (1+α1β1)1-α12)],G21=[(ψ1α1+β2(1+α12  (1-2α1β1))(1-α12)(1-α1β1))01(1-α1β1)×{(ψ1α1+β2(1+α12  (1-2α1β1))(1-α12)(1-α1β1))σϵξβ11-α1β1+σϵ21-α12×(ψ1α1+β2(1+α12  (1-2α1β1))(1-α12)(1-α1β1))}],G22=11-α1β1(ψ1+α1β21-α12).

On the other hand, we can also make the following partition of the matrix H=F4D4: (53)H=[H11H12H21H22], where the sub-matrix H11 corresponds to the first order partial derivatives of the vector [μXμYσX2σY2] with respect to the vector [ψ0ψ1ψ2ψ4], whereas their derivatives with respect to the parameter ψ3 constitute the sub-matrix H12. The sub-matrix H21 is composed of the first order partial derivatives of σXY with respect to each component of the vector [ψ0ψ1ψ2ψ4]. Finally, the scalar H22=σXY/ψ3=0.

The desired variance-covariance matrix can therefore be written in the following partitioned form: (54)ΣΞ22=[Σ122Σ222Σ322Σ422], with (55)Σ122=σ^ϵ2G11(URUR)-1G11+2σ^ϵ4n-1G12G12+ψ1H11(UU)-1H11+2ψ32m-1  H12H12,Σ222=σ^ϵ2G11(URUR)-1G21+2σ^ϵ4n-1G12G22+ψ3H11(UU)-1H21,Σ322=σ^ϵ2G21(URUR)-1G11+2σ^ϵ4n-1  G22G12+ψ3H21(UU)-1H11,Σ422=σ^ϵ2G21(URUR)-1G21+2σ^ϵ4n-1  G22G22+ψ3H21(UU)-1H21, where the matrix U is defined by (35). The matrix UR takes the form (56)UR=[1X0  1X11Xm-2].

In short, the matrix defined by (54) corresponds to the approximate variance-covariance matrix of the m.l.e. for the mean vector and variance-covariance matrix at lag zero for the VAR(1) Model with missing data. We cannot write down explicit expressions for those variances and covariances. The limitation arises from the inability to invert the matrix product   UU   in analytical terms (see (36)). Hence, its inverse can only be accomplished by numerical techniques using the observed sampled data. This point will be pursued further in Section 5.

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 5.

5. Simulation Studies

In this section, we analyse the effects of using different strategies to estimate the mean value of the stochastic process {Yt,t}, denoted by μY. More precisely, the bivariate modelling scheme and its univariate counterparts are compared. Simulation studies are carried out to evaluate the relative efficiency of the estimators with interest.

The m.l.e. of the mean value of the stochastic process {Yt,t} based on the VAR(1) Model is obtained by the second equation of the system of (28). We need to compare this estimator to those obtained by considering the univariate stochastic process {Yt,t} itself. More precisely, having in mind that we are handling a bivariate VAR(1) Model, the corresponding marginal model is the ARMA(2,1) [10, 11]. On the other hand, the AR(1) Model is one of the most popular models due to its practical importance in time series modelling. Therefore, the behaviour of the AR(1) Model will be also evaluated. In short, we will compare the performance of the VAR(1) Model with both the ARMA(2,1) and the AR(1) Models.

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(1) Model by μVAR, whereas those from the ARMA(2,1) and the AR(1) Models are represented by μARMA and μAR, respectively.

The bivariate VAR(1) Model is described by the system of (1). Thus, the univariate stochastic process {Yt,t} follows an ARMA(2,1) Model, and the m.l.e. of the mean value are given by (57)μ^ARMA=β^01-α^1(1-β^1)-β^1.

On the other hand, if we assumed that {Yt,t} followed an AR(1) Model, the m.l.e. of the mean value would be given by (58)μ^AR=β^01-β^1.

Next, we will compare the performance of the estimators (57) and (58) with the m.l.e. based on the VAR(1) Model (second equation of the system (28)). It is important to stress that the strategy behind the AR(1) Model has not taken into account the relationship between the stochastic processes {Xt,t} and {Yt,t}. This feature will certainly introduce an additional noise in the overall estimation procedure.

Following the techniques used in Section 4.2 for determining the precision of the estimators under consideration, here we also have used the first-order Taylor expansion at the mean value μY for computing the estimate of the variance of μY.

Considering the ARMA(2,1) Model, let   θ=[β0β1α1]   be the vector of the unknown parameters. Then, (59)Var(μ^ARMA)i=13(μ^ARMAθiθi=θ^i)2Var(θ^i)+2i=13j=i+13μ^ARMAθi|θi=θ^i×μ^ARMAθj|θj=θ^jCov(θ^i,θ^j).

In regard to the AR(1) Model, μ^AR is given by (58) and (60)Var(μ^AR)2β^0(1-β^1)3Cov(β^0,β^1)+Var(β^0)(1-β^1)2+β^02Var(β^1)(1-β^1)4.

Improvements in choosing the sophisticated m.l.e. for μY based on the VAR(1) Model rather than considering its univariate counterparts are next discussed. Simulation studies are carried out to evaluate the relative efficiency of the estimators under consideration.

The data were generated by the VAR(1) Model (system of (1)). In order to make comparisons on the same basis, a few assumptions to the parameters of the VAR(1) Model are made. We consider that μX=μY=0. These restrictions have no influence on the results because they are equivalent to   α0=β0=0, that is, the constant terms of the VAR(1) Model are equal to zero (system of (1)). Additionally, we introduce the restriction σϵ2=σξ2=1.

Since the correlation coefficient regulates the supply of information between the stochastic processes {Xt}t and {Yt}t, particular emphasis is given to this parameter. Using the grid of points ρϵξ=0.1,0.5,0.75,0.9, the Gain index is computed. We stress that the value ρϵξ=1 is not allowable in this context (see Section 2 for the details).

We analyse the performance of the estimators based on different sample sizes, n=50,100,250, and 500. The simulations reported next are based on different percentages of missing observations referred to the dimension of the sampled data from the auxiliary random process {Xt}t. Simulation runs for each combination of the parameters are based on 1000 replicates.

It is worth emphasising that the estimates of the covariance terms that take part of the variances given by (59) and (60) were computed by the R package tseries .

The simulation goes as follows: after each simulation run, the relative efficiency of μ^VAR with respect to each estimator μ^ARMA and μ^AR is quantified by the Gain index, GI1 and GI2, respectively, expressed as percentage: (61)GI1=Var(μ^ARMA)-Var(μ^VAR)Var(μ^ARMA)×100%,GI2=Var(μ^AR)-Var(μ^VAR)Var(μ^AR)×100%.

A word of notation: the above quantities, that is, GI1 and GI2, were computed from the estimates of the corresponding variances. To lighten the notation, we skipped the conventional nomenclature used to represent the estimates.

If GI1>0, then μ^VAR is more precise than μ^ARMA. Otherwise, μ^VAR loses precision, and μ^ARMA becomes a better estimator for the mean value of {Yt}t. A similar reasoning applies to the comparison between μ^VAR and μ^AR.

Figures 1 and 2 display the main results from the simulation studies. The estimators μ^VAR and μ^ARMA are compared in Figure 1, whereas Figure 2 exhibits the comparison between μ^VAR and μ^AR. For each combination of the parameters of the model, we represent graphically the gain indexes as functions of the percentage of missing data in the sampled data from the stochastic process {Yt}t.

Graphical representation of GI1. The data were obtained from a VAR(1) Model, with α0=β0=0, α1=0.6, β1=0.7, and β2=0.8.

ρ ϵ ξ = 0.1

ρ ϵ ξ = 0.5

ρ ϵ ξ = 0.75

ρ ϵ ξ = 0.9

Graphical representation of GI2. The data were obtained from a VAR(1) Model, with α0=β0=0,α1=0.6,β1=0.7, and β2=0.8.

ρϵξ=0.1

ρ ϵ ξ = 0.5

ρ ϵ ξ = 0.75

ρ ϵ ξ = 0.9

Either Figures 1 or 2 shows that the plot of the gain index against the percentage of missing data in the sample from the stochastic process {Yt}t behaves roughly as a linear function, regardless of the combination of the parameters. In outline, the more the percentage of missing values in the sampled data is, the more precise is the estimator μ^VAR when compared with the univariate context, that is, μ^ARMA or μ^AR (see Figures 1 and 2).

Further, the gain in precision by using the sophisticated estimator μ^VAR rather than μ^ARMA or μ^AR increases as the strength of the linear relationship between the processes {Xt}t and {Yt}t (described by the correlation coefficient) rises from ρ=0.1 to ρ=0.9. This statement is true for both the ARMA(2,1) and AR(1) modelling schemes (see Figures 1 and 2).

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 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. This feature emphasises the idea that has already been raised that the ARMA(2,1) Model describes more accurately the dynamics of the stochastic process {Yt}t than the AR(1) Model does. In short, it seems that the AR(1) Model is not a good approach in this context because it incorporates a noise term related to the simulation scheme that we cannot control.

Summing up, the estimator μ^VAR is preferable to those explored in the univariate context, that is, either μ^ARMA or μ^AR.

6. Conclusions

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(1) Model structure. We have proposed maximum likelihood estimators for the relevant parameters of the model based on a monotone missing data pattern. The precision of the estimators has also been derived. Special attention has been given to the estimator for the mean value of the stochastic process whose sampled data has missing values, μY.

We have compared the performance of the estimator for μY based on the VAR(1) Model with a monotone pattern of missing data with those obtained from both the ARMA(2,1) Model and the AR(1) Model. By simulation studies, we have showed that the estimator derived in this article based on the VAR(1) Model performs better than those derived from the univariate context. It is essential to emphasise that, even numerically, it was quite difficult to compute the precision of the later estimators as we have shown in Section 4.2.

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 4.2, we have only developed estimators for the covariance matrix at lag zero. In future research, we will try to solve this problem in the framework of Kalman filter.

Acknowledgments

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.

Morrison D. F. Expectations and variances of maximum likelihood estimates of the multivariate normal distribution parameters with missing data Journal of the American Statistical Association 1971 66 335 602 604 Dahiya R. C. Korwar R. M. Maximum likelihood estimates for a bivariate normal distribution with missing data The Annals of Statistics 1980 8 3 687 692 MR568732 10.1214/aos/1176345020 ZBL0435.62032 Gómez V. Maravall A. Estimation, prediction, and interpolation for nonstationary series with the Kalman filter Journal of the American Statistical Association 1994 89 426 611 624 MR1294087 ZBL0806.62076 Jones R. H. Maximum likelihood fitting of ARMA models to time series with missing observations Technometrics 1980 22 3 389 395 10.2307/1268324 MR585635 ZBL0451.62069 Kohn R. Ansley C. F. Estimation, prediction, and interpolation for ARIMA models with missing data Journal of the American Statistical Association 1986 81 395 751 761 MR860509 10.1080/01621459.1986.10478332 ZBL0607.62106 Pourahmadi M. Estimation and interpolation of missing values of a stationary time series Journal of Time Series Analysis 1989 10 2 149 169 10.1111/j.1467-9892.1989.tb00021.x MR1005894 ZBL0686.62067 Gómez V. Maravall A. Peña D. Missing observations in ARIMA models: skipping approach versus additive outlier approach Journal of Econometrics 1999 88 2 341 363 10.1016/S0304-4076(98)00036-0 MR1666907 ZBL1054.62621 Little R. J. A. Rubin D. B. Statistical Analysis with Missing Data 1987 New York, NY, USA John Wiley & Sons MR890519 Sparks R. SUR models applied to an environmental situation with missing data and censored values Journal of Applied Mathematics and Decision Sciences 2004 8 1 15 32 10.1207/s15327612jamd0801_2 MR2042167 ZBL1055.62136 Heij C. De Boer P. Franses P. H. Kloek T. van Dijk H. K. Econometric Methods with Applications in Business and Economics 2004 New York, NY, USA Oxford University Press Tsay R. S. Analysis of Financial Time Series 2010 3rd Hoboken, NJ, USA John Wiley & Sons 10.1002/9780470644560 MR2778591 Nunes M. H. Dynamics relating phytoplankton abundance with upwelling events. An approach to the problem of missing data in the gaussian context [Ph.D. thesis] 2006 Lisbon, Portugal University of Lisbon Hamilton J. D. Time Series Analysis 1994 Princeton, NJ, USA Princeton University Press MR1278033 Apostol T. M. Calculus 1969 2nd Singapore John Wiley & Sons MR0248290 Trapletti A. Hornik K. tseries: Time series analysis and computational finance R package version 0.10-25, 2011, http://CRAN.R-project.org/package=tseries