Working With Systems—433

is the unconditional variance of the residuals. By default, the smoothing parameter, l is set to 0.7. However, you have the option to choose from a number of weights from 0.1 to 1, in increments of 0.1. Notice that if the parameter is set to 1 the initial value is simply the unconditional variance, i.e. backcasting is not performed.

•EViews will report the robust standard errors when the Bollerslev-Wooldridge SE box is checked.

For basic specifications, ARCH analytic derivatives are available, and are employed by default. For a more complex model, either in the means or conditional variance, numerical or a combination of numerical and analytics are used. Analytic derivatives are generally, but not always, faster than numeric.

In addition, the Options tab allows you to set a number of options for estimation, including convergence criterion, maximum number of iterations, and derivative calculation settings. See “Setting Estimation Options” on page 751 for related discussion.

Estimation Output

The system estimation output contains parameter estimates, standard errors, and t-statistics (or z-statistics for maximum likelihood estimations), for each of the coefficients in the system. Additionally, EViews reports the determinant of the residual covariance matrix, and, for ARCH and FIML estimates, the maximized likelihood values, Akaike and Schwarz criteria. For ARCH estimations, the mean equation coefficients are separated from the variance coefficient section.

In addition, EViews reports a set of summary statistics for each equation. The R2 statistic, Durbin-Watson statistic, standard error of the regression, sum-of-squared residuals, etc., are computed for each equation using the standard definitions, based on the residuals from the system estimation procedure.

In ARCH estimations, the raw coefficients of the variance equation do not necessarily give a clear understanding of the variance equations in many specifications. An extended coefficient view is supplied at the end of the output table to provide an enhanced view of the coefficient values involved.

You may access most of these results using regression statistics functions. See Chapter 18, page 16 for a discussion of the use of these functions, and Chapter 1. “Object View and Procedure Reference,” on page 2 of the Command and Programming Reference for a full listing of the available functions for systems.

Working With Systems

After obtaining estimates, the system object provides a number of tools for examining the equation results, and performing inference and specification testing.

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System Views

•The System Specification view displays the specification window for the system. The specification window may also be displayed by pressing Spec on the toolbar.

•Representations provides you with the estimation command, the estimated equations and the substituted coefficient counterpart. For ARCH estimation this view also includes additional variance and covariance specification in matrix formation as well as single equation with and without substituted coefficients.

•The Estimation Output view displays the coefficient estimates and summary statistics for the system. You may also access this view by pressing Stats on the system toolbar.

•Residuals/Graphs displays a separate graph of the residuals from each equation in the system.

•Residuals/Correlation Matrix computes the contemporaneous correlation matrix for the residuals of each equation.

•Residuals/Covariance Matrix computes the contemporaneous covariance matrix for the residuals. See also the function @residcov in “System” on page 559 of the Command and Programming Reference.

•Gradients and Derivatives provides views which describe the gradients of the objective function and the information about the computation of any derivatives of the regression functions. Details on these views are provided in Appendix C. “Gradients and Derivatives,” on page 763.

•Conditional Covariance… gives you the option to generate conditional covariances, variances, correlations or standard deviations for systems estimated using ARCH methods.

•Coefficient Covariance Matrix allows you to examine the estimated covariance matrix.

•Coefficient Tests allows you to display confidence ellipses or to perform hypothesis tests for restrictions on the coefficients. These views are discussed in greater depth in “Confidence Intervals and Confidence Ellipses” on page 140 and “Wald Test (Coefficient Restrictions)” on page 146.

•A number of Residual Diagnostics are supported, including Correlograms, Portmanteau Autocorrelation Test, and Normality Test. For most estimation methods, the Correlogram and Portmanteau views employ raw residuals, while Normality tests are based on standardized residuals. For ARCH estimation, the user has the added option of using a number of standardized residuals to calculate Correlogram and Portmanteau tests. The available standardization methods include Cholesky, Inverse Square Root of Residual Correlation, or Inverse Square Root of Residual Covariance. See “Residual Tests” on page 464 for details on these tests and factorization methods.

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•Make Endogenous Group creates an untitled group object containing the endogenous variables.

•Make Loglikelihoods (for system ARCH) creates a series containing the log likelihood contribution.

•Make Conditional Covariance (for system ARCH) allows you to generate estimates of the conditional variances, covariances, or correlations for the specified set of dependent variables. (EViews automatically places all of the dependent variables in the Variable field. You have the option to modify this field to include only the variable of interest.)

If you select Group under Format, EViews will save the data in series. The Base name edit box indicates the base name to be used when generating series data. For the conditional variance series, the naming convention will be the specified base name plus terms of the form “_01”, “_02”. For covariances or correlations, the naming convention will use the base name plus “_01_02”, “_01_03”, etc., where the additional text indicates the covariance/correlation between member 1 and 2, member 1 and 3, etc.

If Matrix is selected then whatever is in the Matrix name field will be generated for what is in the Date (or Presample if it is checked) edit field.

Example

As an illustration of the process of estimating a system of equations in EViews, we estimate a translog cost function using data from Berndt and Wood (1975) as presented in Greene (1997). The data are provided in “G_cost.WF1”. The translog cost function has four factors with three equations of the form:

parameters to be estimated. Note that there are cross equation coefficient restrictions that ensure symmetry of the cross partial derivatives.

We first estimate this system without imposing the cross equation restrictions and test whether the symmetry restrictions hold. Create a system by clicking Object/New Object.../ System in the main toolbar or type system in the command window. Press the Name button and type in the name “SYS_UR” to name the system.

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and click OK. The test result:

Wald Test:

System: SYS_UR

Null Hypothesis: C(3)=C(6), C(4)=C(10), C(8)=C(11)

Restrictions are linear in coefficients.

fails to reject the symmetry restrictions. To estimate the system imposing the symmetry restrictions, copy the object using Object/Copy Object, click View/System Specification and modify the system.

We have named the system SYS_TLOG. Note that to impose symmetry in the translog specification, we have restricted the coefficients on the cross-price terms to be the same (we have also renumbered the 9 remaining coefficients so that they are consecutive). The restrictions are imposed by using

the same coefficients in each equation. For example, the coefficient on the log(P_L/P_M) term in the C_K equation, C(3), is the same as the coefficient on the log(P_K/P_M) term in the C_L equation.

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In this example, we elect to produce ordinary residuals. EViews opens an untitled group window containing the residuals of each equation in the system. To compute descriptive statistics for each residual in the group, select

View/Descriptive Stats/Common Sample from the group window toolbar.

The Jarque-Bera statistic rejects the hypothesis of normal distribution for the second equation but not for the other equations.

The estimated coefficients of the trans-

log cost function may be used to construct estimates of the elasticity of substitution between factors of production. For example, the elasticity of substitution between capital and labor is given by 1+c(3)/(C_K*C_L). Note that the elasticity of substitution is not a constant, and depends on the values of C_K and C_L. To create a series containing the elasticities computed for each observation, select Quick/Generate Series…, and enter:

es_kl = 1 + sys_tlog.c(3)/(c_k*c_l)

To plot the series of elasticity of substitution between capital and labor for each observation, double click on the series name ES_KL in the workfile and select View/Graph/Line & Symbol:

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System: SYS01

Estimation Method: ARCH Maximum Likelihood (Marquardt)

Covariance specification: Diagonal VECH

Date: 08/13/09 Time: 10:40

Sample: 12/31/1979 12/25/2000

Included observations: 1096

Total system (balanced) observations 3288

Presample covariance: backcast (parameter =0.7)

Convergence achieved after 127 iterations

The variance coefficients are displayed in their own section. Coefficients C(4) to C(9) are the coefficients for the constant matrix, C(10) to C(15) are the coefficients for the ARCH term, and C(16) through C(21) are the coefficients for the GARCH term.

Note that the number of variance coefficients in an ARCH model can be very large. Even in this small 3-variable system, 18 parameters are estimated, making interpretation somewhat difficult. To aid you in interpreting the results, EViews provides a covariance specification section at the bottom of the estimation output that re-labels and transforms coefficients:

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A single equation representation of the variance-covariance equation may be viewed by clicking on View/Representations and scrolling down to the

Variance and Covariance Equations section.

The GARCH equations are the conditional variance equations while the COV equations are the conditional covariance

equations. For example GARCH1 is the conditional variance of Japanese yen. COV1_2 is the conditional covariance between the Japanese Yen and the Swiss Franc.

Before proceeding we name the system SYS01 by clicking on the Name button and accepting the default name.

A graph of the conditional variance can be generated using View/Conditional Covariance…. An extensive list of options is available, including Covariance, Correlation, Variance, and Standard Deviation. Data may also be displayed in graph, matrix or time series list format. Here is the correlation view:

The correlation looks to be time varying, which is a general characteristic of this model.

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Another possibility is to model the covariance matrix using the CCC specification, which imposes a constant correlation over time. We proceed by creating a new system with specification identical to the one above. We'll select Constant Conditional Correlation this time as the Model type for estimation and leave the remaining settings as they are. The basic results:

System: UNTITLED

Estimation Method: ARCH Maximum Likelihood (Marquardt)

Covariance specification: Constant Conditional Correlation

Date: 08/13/09 Time: 10:51

Sample: 12/31/1979 12/25/2000

Included observations: 1096

Total system (balanced) observations 3288

Presample covariance: backcast (parameter =0.7)

Convergence achieved after 44 iterations

Note that this specification has only 12 free parameters in the variance equation, as compared with 18 in the previous model. The extended variance section represents the variance equation as,

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

while the model for the covariance equation is:

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

The lower portion of the output shows that the correlations, R(1, 2), R(1, 3), and R(2, 3) are 0.5713, -0.4032, and -0.6773, respectively: