446—Chapter 31. System Estimation

Covariance specification: Constant Conditional Correlation

GARCH(i) = M(i) + A1(i)*RESID(i)(-1)^2 + B1(i)*GARCH(i)(-1)

COV(i,j) = R(i,j)*@SQRT(GARCH(i)*GARCH(j))

Transformed Variance Coefficients

Is this model better than the previous model? While the log likelihood value is lower, it also has fewer coefficients. We may compare the two system by looking at model selection criteria. The Akaike, Schwarz and Hannan-Quinn all show lower information criteria values for the VECH model than the CCC specification, suggesting that the time-varying Diagonal VECH specification may be preferred.

Technical Discussion

While the discussion to follow is expressed in terms of a balanced system of linear equations, the analysis carries forward in a straightforward way to unbalanced systems containing nonlinear equations.

Denote a system of m equations in stacked form as:

Technical Discussion—447

Under the standard assumptions, the residual variance matrix from this stacked system is given by:

Other residual structures are of interest. First, the errors may be heteroskedastic across the m equations. Second, they may be heteroskedastic and contemporaneously correlated. We can characterize both of these cases by defining the M ¥ M matrix of contemporaneous

correlations, S , where the (i,j)-th element of S is given by jij = E(eitejt) for all t . If the errors are contemporaneously uncorrelated, then, jij = 0 for i π j , and we can write:

More generally, if the errors are heteroskedastic and contemporaneously correlated:

Lastly, at the most general level, there may be heteroskedasticity, contemporaneous correlation, and autocorrelation of the residuals. The general variance matrix of the residuals may be written:

where Sij is an autocorrelation matrix for the i-th and j-th equations.

Ordinary Least Squares

The OLS estimator of the estimated variance matrix of the parameters is valid under the assumption that V = S ƒ IT . The estimator for b is given by,

Weighted Least Squares

The weighted least squares estimator is given by:

residual variance estimator:

448—Chapter 31. System Estimation

where the inner product is taken over the non-missing common elements of i and j . The max function in Equation (31.17) is designed to handle the case of unbalanced data by down-weighting the covariance terms. Provided the missing values are asymptotically negligible, this yields a consistent estimator of the variance elements. Note also that there is no adjustment for degrees of freedom.

When specifying your estimation specification, you are given a choice of which coefficients to use in computing the sij . If you choose not to iterate the weights, the OLS coefficient estimates will be used to estimate the variances. If you choose to iterate the weights, the current parameter estimates (which may be based on the previously computed weights) are used in computing the sij . This latter procedure may be iterated until the weights and coefficients converge.

The estimator for the coefficient variance matrix is:

The weighted least squares estimator is efficient, and the variance estimator consistent, under the assumption that there is heteroskedasticity, but no serial or contemporaneous correlation in the residuals.

It is worth pointing out that if there are no cross-equation restrictions on the parameters of the model, weighted LS on the entire system yields estimates that are identical to those obtained by equation-by-equation LS. Consider the following simple model:

y1 = X1b1 + e1

(31.19)

y2 = X2b2 + e2

If b1 and b2 are unrestricted, the WLS estimator given in Equation (31.18) yields:

The expression on the right is equivalent to equation-by-equation OLS. Note, however, that even without cross-equation restrictions, the standard errors are not the same in the two cases.

Seemingly Unrelated Regression (SUR)

SUR is appropriate when all the right-hand side regressors X are assumed to be exogenous, and the errors are heteroskedastic and contemporaneously correlated so that the error variance matrix is given by V = S ƒ IT . Zellner’s SUR estimator of b takes the form:

If you include AR terms in equation j , EViews transforms the model (see “Estimating AR Models” on page 89) and estimates the following equation:

where ej is assumed to be serially independent, but possibly correlated contemporaneously across equations. At the beginning of the first iteration, we estimate the equation by nonlinear LS and use the estimates to compute the residuals eˆ . We then construct an estimate of S using sij = (eˆi¢eˆj) § max(Ti, Tj) and perform nonlinear GLS to complete one iteration of the estimation procedure. These iterations may be repeated until the coefficients and weights converge.

Two-Stage Least Squares (TSLS) and Weighted TSLS

TSLS is a single equation estimation method that is appropriate when some of the variables in X are endogenous. Write the j-th equation of the system as,

the matrix of endogenous variables and X is the matrix of exogenous variables; Yj is the matrix of endogenous variables not including yj .

where Zˆ j = (Yˆ j, Xj). The residuals from an equation using these coefficients are used for form weights.

Weighted TSLS applies the weights in the second stage so that:

where the elements of the variance matrix are estimated in the usual fashion using the residuals from unweighted TSLS.

450—Chapter 31. System Estimation

If you choose to iterate the weights, X is estimated at each step using the current values of the coefficients and residuals.

Three-Stage Least Squares (3SLS)

Since TSLS is a single equation estimator that does not take account of the covariances between residuals, it is not, in general, fully efficient. 3SLS is a system method that estimates all of the coefficients of the model, then forms weights and reestimates the model using the estimated weighting matrix. It should be viewed as the endogenous variable analogue to the SUR estimator described above.

The first two stages of 3SLS are the same as in TSLS. In the third stage, we apply feasible generalized least squares (FGLS) to the equations in the system in a manner analogous to the SUR estimator.

SUR uses the OLS residuals to obtain a consistent estimate of the cross-equation covariance matrix S . This covariance estimator is not, however, consistent if any of the right-hand side variables are endogenous. 3SLS uses the 2SLS residuals to obtain a consistent estimate of

S .

In the balanced case, we may write the equation as,

If you choose to iterate the weights, the current coefficients and residuals will be used to

estimate ˆ .

S

Generalized Method of Moments (GMM)

The basic idea underlying GMM is simple and intuitive. We have a set of theoretical moment conditions that the parameters of interest v should satisfy. We denote these moment conditions as:

The method of moments estimator is defined by replacing the moment condition (31.30) by its sample analog:

However, condition (31.31) will not be satisfied for any v when there are more restrictions m than there are parameters v . To allow for such overidentification, the GMM estimator is defined by minimizing the following criterion function:

452—Chapter 31. System Estimation

where:

You also need to specify the kernel function k and the bandwidth q .

Kernel Options

The kernel function is used to weight the covariances so that ˆ is ensured to be positive k Q

semi-definite. EViews provides two choices for the kernel, Bartlett and quadratic spectral (QS). The Bartlett kernel is given by:

while the quadratic spectral (QS) kernel is given by:

where x = j § q . The QS has a faster rate of convergence than the Bartlett and is smooth and not truncated (Andrews 1991). Note that even though the QS kernel is not truncated, it still depends on the bandwidth q (which need not be an integer).

Bandwidth Selection

The bandwidth q determines how the weights given by the kernel change with the lags in the estimation of Q . Newey-West fixed bandwidth is based solely on the number of observations in the sample and is given by:

where int( ) denotes the integer part of the argument.

EViews also provides two “automatic”, or data dependent bandwidth selection methods that are based on the autocorrelations in the data. Both methods select the bandwidth according to the rule:

The two methods, Andrews and Variable-Newey-West, differ in how they estimate aˆ (1) and aˆ (2).

Andrews (1991) is a parametric method that assumes the sample moments follow an AR(1) process. We first fit an AR(1) to each sample moment (31.33) and estimate the autocorrela-

454—Chapter 31. System Estimation

Note that while Andrews and Monahan (1992) adjust the VAR estimates to avoid singularity when the moments are near unit root processes, EViews does not perform this eigenvalue adjustment.

Multivariate ARCH

ARCH estimation uses maximum likelihood to jointly estimate the parameters of the mean and the variance equations.

Assuming multivariate normality, the log likelihood contributions for GARCH models are given by:

Given a specification for the mean equation and a distributional assumption, all that we require is a specification for the conditional covariance matrix. We consider, in turn, each of the three basic specifications: Diagonal VECH, Constant Conditional Correlation (CCC), and Diagonal BEKK.

Diagonal VECH

Bollerslev, et. al (1988) introduce a restricted version of the general multivariate VECH model of the conditional covariance with the following formulation:

where the coefficient matrices A , B , and Q are N ¥ N symmetric matrices, and the operator “•” is the element by element (Hadamard) product. The coefficient matrices may be parametrized in several ways. The most general way is to allow the parameters in the matrices to vary without any restrictions, i.e. parameterize them as indefinite matrices. In that case the model may be written in single equation format as:

where, for instance, (Ht )ij is the i-th row and j-th column of matrix Ht .