User’s Guide : Multiple Equation Analysis : Vector Autoregression (VAR) Models : Views and Procs of a VAR
Views and Procs of a VAR
Residual Views
Diagnostic Views
Lag Structure
AR Roots Table/Graph
Pairwise Granger Causality Tests
Lag Exclusion Tests
Lag Length Criteria
Residual Tests
Portmanteau Autocorrelation Test
Autocorrelation LM Test
Normality Test
White Heteroskedasticity Test
Cointegration Test
Notes on System Comparability
Impulse Responses
Display Information
Standard Errors and Confidence Intervals
Impulse Definitions
Displaying Impulse Responses in EViews
Variance Decomposition
Historical Decomposition
Procs of a VAR
Make System
Estimate Structural Factorization
Once you have estimated a VAR, EViews provides various views and procs to work with the estimated VAR.
In this section, we discuss views that are specific to VARs. For other views and procedures, see the general discussion of system views in “System Estimation”.
Residual Views
You may use the entries under the Residuals and Structural Residuals menus to examine the residuals of the estimated VAR in graph or spreadsheet form, or you may examine the covariance and correlation matrix of those residuals.
The views listed under Residuals will display results using the raw residuals from the estimated VAR.
Alternately, you may display the Structural Residuals views to examine the these transformed estimated residuals. If the are the ordinary residuals, we may plot the structural residuals based on factor loadings ,
or based on weights ,
When producing results for the Structural Residuals views, you will be prompted to choose a transformation.
Note that there is a corresponding procs which will save the residuals series in the workfile (Proc/Make Residuals and Proc/Make Structural Residuals...).
Diagnostic Views
A set of diagnostic views are provided under the menus View/Lag Structure and View/Residual Tests in the VAR window. These views should help you check the appropriateness of the estimated VAR.
Lag Structure
EViews offers several views for investigating the lag structure of your equation.
AR Roots Table/Graph
Reports the inverse roots of the characteristic AR polynomial; see Lütkepohl (1991). The estimated VAR is stable (stationary) if all roots have modulus less than one and lie inside the unit circle. If the VAR is not stable, certain results (such as impulse response standard errors) are not valid. There will be roots, where is the number of endogenous variables and is the largest lag. If you estimated a VEC with cointegrating relations, roots should be equal to unity.
Pairwise Granger Causality Tests
Carries out pairwise Granger causality tests and tests whether an endogenous variable can be treated as exogenous. For each equation in the VAR, the output displays (Wald) statistics for the joint significance of each of the other lagged endogenous variables in that equation. The statistic in the last row (All) is the -statistic for joint significance of all other lagged endogenous variables in the equation.
Warning: if you have estimated a VEC, the lagged variables that are tested for exclusion are only those that are first differenced. The lagged level terms in the cointegrating equations (the error correction terms) are not tested.
Note: for VAR models with linear restrictions, EViews will only perform tests involving endogenous variable lags for which there are no restrictions.
Lag Exclusion Tests
Carries out lag exclusion tests for each lag in the VAR. For each lag, the (Wald) statistic for the joint significance of all endogenous variables at that lag is reported for each equation separately and jointly (last column).
Note: for VAR models with linear restrictions, EViews will only perform tests involving endogenous variable lags for which there are no restrictions.
Lag Length Criteria
Computes various criteria to select the lag order of an unrestricted VAR. You will be prompted to specify the maximum lag to “test” for. The table displays various information criteria for all lags up to the specified maximum. (If there are no exogenous variables in the VAR, the lag starts at 1; otherwise the lag starts at 0.) The table indicates the selected lag from each column criterion by an asterisk “*”. For columns 4–7, these are the lags with the smallest value of the criterion.
All the criteria are discussed in Lütkepohl (1991, Section 4.3). The sequential modified likelihood ratio (LR) test is carried out as follows. Starting from the maximum lag, test the hypothesis that the coefficients on lag are jointly zero using the statistics:
where is the number of parameters per equation under the alternative. Note that we employ Sims’ (1980) small sample modification which uses () rather than . We compare the modified LR statistics to the 5% critical values starting from the maximum lag, and decreasing the lag one at a time until we first get a rejection. The alternative lag order from the first rejected test is marked with an asterisk (if no test rejects, the minimum lag will be marked with an asterisk). It is worth emphasizing that even though the individual tests have size 0.05, the overall size of the test will not be 5%; see the discussion in Lütkepohl (1991, p. 125–126).
Note: for VAR models with linear restrictions, EViews perform lag length tests using models in which the VAR linear restrictions are not applied.
Residual Tests
You may use these views to examine the properties of the residuals from your estimated VAR.
Displays the pairwise cross-correlograms (sample autocorrelations) for the estimated residuals in the VAR for the specified number of lags. The cross-correlograms can be displayed in three different formats. There are two tabular forms, one ordered by variables (Tabulate by Variable) and one ordered by lags (Tabulate by Lag). The Graph form displays a matrix of pairwise cross-correlograms. The dotted line in the graphs represent plus or minus two times the approximate asymptotic standard errors of the lagged correlations (ignoring coefficient estimation and computed as ).
Portmanteau Autocorrelation Test
Computes the multivariate Box-Pierce/Ljung-Box Q-statistics for residual serial correlation up to the specified order (see Lütkepohl, 1991, 4.4.21 & 4.4.23 for details). We report both the Q-statistics and the adjusted Q-statistics (with a small sample correction). Under the null hypothesis of no serial correlation up to lag , both statistics are approximately distributed with degrees of freedom where is the VAR lag order if there are no coefficient restrictions, and where contains adjustments for the restrictions.
The asymptotic distribution is approximate in the sense that it requires the MA coefficients to be zero for lags . Therefore, this approximation will be poor if the roots of the AR polynomial are close to one and is small. In fact, the degrees of freedom becomes negative for .
Note that adjustments are made to the computation of the p-value of these statistics when the model is a VECM (Brüggemann, Lütkepohl, and Saikkonen, 2006).
Autocorrelation LM Test
Reports the multivariate LM test statistics for residual serial correlation up to the specified order. A Breusch-Godfrey test statistic for autocorrelation at lag order is computed by running an auxiliary regression of the residuals on the original right-hand regressors and the lagged residual , where the missing first values of are filled with zeros.
See Johansen (1995, p. 22) for the formula of the LR version of this LM test. The form of this statistic employs the Edgeworth expansion correction (Edgerton and Shukur 1999). Under the null hypothesis of no serial correlation of order , the LM statistic is asymptotically distributed with degrees of freedom.
A variant of this test tests for autocorrelation for lags 1 to . The test modifies the LM statistic above by including all of the modified lagged residual regressors from . Under the null hypothesis, the LM statistic is asymptotically with degrees of freedom.
In addition to the LR version of the test, EViews computes the Rao F-test version of the LM statistic as described Edgerton and Shukur (1999) since their simulations suggest it performs best among the many variants they consider.
Normality Test
This view reports the multivariate extensions of the Jarque-Bera residual normality test, which compares the third and fourth moments of the residuals to those from the normal distribution. For the multivariate test, you must choose a factorization of the residuals that are orthogonal to each other (see “Impulse Responses” for additional discussion of the need for orthogonalization).
Let be a factorization matrix such that:
where is the demeaned residuals. Define the third and fourth moment vectors and . Then:
under the null hypothesis of normal distribution. Since each component is independent of each other, we can form a -statistic by summing squares of any of these third and fourth moments.
EViews provides you with choices for the factorization matrix :
Cholesky (Lütkepohl 1991, p. 155-158): is the inverse of the lower triangular Cholesky factor of the residual covariance matrix. The resulting test statistics depend on the ordering of the variables in the VAR.
Inverse Square Root of Residual Correlation Matrix (Doornik and Hansen 1994): where is a diagonal matrix containing the eigenvalues of the residual correlation matrix on the diagonal, is a matrix whose columns are the corresponding eigenvectors, and is a diagonal matrix containing the inverse square root of the residual variances on the diagonal. This is essentially the inverse square root of the residual correlation matrix. The test is invariant to the ordering and to the scale of the variables in the VAR. As suggested by Doornik and Hansen (1994), we perform a small sample correction to the transformed residuals before computing the statistics.
Inverse Square Root of Residual Covariance Matrix (Urzua 1997): where is the diagonal matrix containing the eigenvalues of the residual covariance matrix on the diagonal and is a matrix whose columns are the corresponding eigenvectors. This test has a specific alternative, which is the quartic exponential distribution. According to Urzua, this is the “most likely” alternative to the multivariate normal with finite fourth moments since it can approximate the multivariate Pearson family “as close as needed.” As recommended by Urzua, we make a small sample correction to the transformed residuals before computing the statistics. This small sample correction differs from the one used by Doornik and Hansen (1994); see Urzua (1997, Section D).
Factorization from Identified (Structural) VAR: where , are estimated from the structural VAR model. This option is available only if you have estimated the factorization matrices and using the structural VAR (see (here)).
EViews reports test statistics for each orthogonal component (labeled RESID1, RESID2, and so on) and for the joint test. For individual components, the estimated skewness and kurtosis are reported in the first two columns together with the p-values from the distribution (in square brackets). The Jarque-Bera column reports:
with p-values from the distribution. Note: in contrast to the Jarque-Bera statistic computed in the series view, this statistic is not computed using a degrees of freedom correction.
For the joint tests, we will generally report:
If, however, you choose Urzua’s (1997) test, will not only use the sum of squares of the “pure” third and fourth moments but will also include the sum of squares of all cross third and fourth moments. In this case, is asymptotically distributed as a with degrees of freedom.
White Heteroskedasticity Test
These tests are the extension of White’s (1980) test to systems of equations as discussed by Kelejian (1982) and Doornik (1995). The test regression is run by regressing each cross product of the residuals on the cross products of the regressors and testing the joint significance of the regression. The No Cross Terms option uses only the levels and squares of the original regressors, while the With Cross Terms option includes all non-redundant cross-products of the original regressors in the test equation. The test regression always includes a constant term as a regressor.
The first part of the output displays the joint significance of the regressors excluding the constant term for each test regression. You may think of each test regression as testing the constancy of each element in the residual covariance matrix separately. Under the null of no heteroskedasticity or (no misspecification), the non-constant regressors should not be jointly significant.
The last line of the output table shows the LM chi-square statistics for the joint significance of all regressors in the system of test equations (see Doornik, 1995, for details). The system LM statistic is distributed as a with degrees-of-freedom , where is the number of cross-products of the residuals in the system and is the number of the common set of right-hand side variables in the test regression.
Cointegration Test
This view performs the Johansen cointegration test for the variables in your VAR. Note that Johansen cointegration tests may also be performed from a Group object.
To perform the test, select Views/Cointegration Test... from an estimated VAR object window using. In the latter case, the test dialog will be pre-filled with information from the VAR (or VEC) specification:
Enter the lag specification and exogenous variables specifications in the appropriate edit fields, and specify the deterministic trend specifications using the dropdown. The Summarize all deterministic case assumptions checkbox may be used to compute the test statistic for all of the deterministic trend cases.
See “Johansen Cointegration Test” for a description of the basic test methodology and test settings. See also “The VECM Specification” and “Estimating VEC Models in EViews” for a discussion of VEC specifications.
Bear in mind that the Lag intervals lag specification for is for the error correction form of the VEC model, not the levels of a VAR. If performing the test on a specification in levels as in an ordinary VAR, the lag structure will differ in the test equation.
Similarly, note that exogenous variables in a VAR will be placed in the Short-run (outside cointegrating equation) edit field, which may not be the desired test specification in the error correction form of the VEC.
Notes on System Comparability
Many of the diagnostic tests given above may be computed “manually” by estimating the VAR using a system object and selecting View/Wald Coefficient Tests... We caution you that the results from the system may not match those from the VAR diagnostic views for various reasons:
The system object will, in general, use the maximum possible observations for each equation in the system. By contrast, VAR objects force a balanced sample across equations in cases where there are missing values.
The estimates of the weighting matrix used in system estimation do not contain a degrees of freedom correction (the residual sums-of-squares are divided by rather than by ), while the VAR estimates do perform this adjustment. Even though estimated using comparable specifications and yielding identifiable coefficients, the test statistics from system SUR and the VARs will show small (asymptotically insignificant) differences.
Impulse Responses
A shock to the i-th variable not only directly affects the i-th variable but is also transmitted to all of the other endogenous variables through the dynamic (lag) structure of the VAR. An impulse response function traces the effect of a one-time shock to one of the innovations on current and future values of the endogenous variables.
If the innovations are contemporaneously uncorrelated, interpretation of the impulse response is straightforward. The i-th innovation is simply a shock to the i-th endogenous variable .
Innovations, however, are usually correlated, and may be viewed as having a common component which cannot be associated with a specific variable. In order to interpret the impulses, it is common to apply a transformation to the innovations so that they become uncorrelated:
where is a diagonal covariance matrix. As explained below, EViews provides several options for the choice of which produce a diagonal matrix.
To compute and display impulse responses, click on View/Impulse Response.... on the toolbar of an estimated VAR object to display the two tab dialog interface:
There are sections for Display information and Response standard errors on the first tab, and settings for Impulse Definition on the second tab.
Display Information
In the Display information section you will define the Impulses and Responses you initially wish to display, specify the Horizon length over which you wish to compute those impulse responses, and indicate whether to compute cumulative responses using the Accumulate responses checkbox.
Note that the Impulses and Responses edit fields are for specifying the initial results to display. By default, these fields are filled out with the names of all of the variables in the VAR, but you will later be able to modify the display interactively. Note also that the impulse occurs in period 1. The default horizon length of 10 displays the period in which the impulse occurs (period 1) and the nine periods that follow.
Standard Errors and Confidence Intervals
The Method dropdown in the Response standard errors group offers several methods for computing IR standard errors (SEs) and confidence intervals (CIs). The following choices are available:
None—no SEs and no CIs are produced.
Analytic (asymptotic)—SEs based on the response asymptotic distribution (Lütkepohl, 1990); CIs computed based on SEs.
Monte Carlo—SEs derived by simulating impulses by drawing innovations from the standard normal distribution; CIs computed based on SEs.
Bootstrap: Standard percentile—SEs and (traditional) CIs derived using the standard percentile bootstrap ( “Standard Percentile Confidence Interval”).
Bootstrap: Hall’s percentile—SEs and (traditional) CIs derived using Hall’s percentile bootstrap ( “Hall’s Percentile Confidence Interval”).
Bootstrap: Hall’s studentized—SEs and (traditional) CIs derived using Hall’s studentized bootstrap ( “Hall’s Studentized Confidence Interval”).
Bootstrap: Kilian’s studentized—SEs and (traditional) CIs derived using Kilian’s unbiased confidence intervals (1998) ( “Killian’s Unbiased Confidence Interval”).
Coverage for the CIs may be specified in the Confidence intervals levels edit field. You may provide one or more values in the form of a space delimited list.
For Monte Carlo and bootstrap methods, the number of replications is specified in the Response standard error options group. For instance, when performing the MC procedure, replications are specified in the Monte Carlo replication edit field. Alternatively, when doing a bootstrap procedure, the first (possibly the only) stage bootstrap replications are specified in the Single bootstrap replications edit field. Should a bootstrap procedure require a nested bootstrap loop, the second stage bootstrap replications are set in the Double bootstrap replications edit field. Should the use of a full double bootstrap be too computationally expensive, fast bootstrap iteration can be achieved by checking the Fast double bootstrap approximation checkbox.
By default, EViews uses shading to display CI bands. To use lines instead, check the box next to Display intervals using lines.
Impulse Definitions
The Impulse Definition tab provides the following options for specifying the impulses:
Residual—One Unit sets the impulses to one unit of the residuals. This option ignores the units of measurement and the correlations in the VAR residuals so that no transformation is performed. The responses from this option are the MA coefficients of the infinite order Wold representation of the VAR.
Residual—One Std. Dev. sets the impulses to one standard deviation of the residuals. This option ignores the correlations in the VAR residuals.
Cholesky uses the inverse of the Cholesky factor of the residual covariance matrix to orthogonalize the impulses. This option imposes an ordering of the variables in the VAR and attributes all of the effect of any common component to the variable that comes first in the VAR system. Note that responses can change dramatically if you change the ordering of the variables. You may specify a different VAR ordering by reordering the variables in the Cholesky Ordering edit box.
The (d.f. adjustment) option makes a small sample degrees of freedom correction when estimating the residual covariance matrix used to derive the Cholesky factor. The (i,j)-th element of the residual covariance matrix with degrees of freedom correction is computed as where is the number of parameters per equation in the VAR. The (no d.f. adjustment) option estimates the (i,j)-th element of the residual covariance matrix as .
Generalized Impulses as described by Pesaran and Shin (1998) constructs an orthogonal set of innovations that does not depend on the VAR ordering. The generalized impulse responses from an innovation to the j-th variable are derived by applying a variable specific Cholesky factor computed with the j-th variable at the top of the Cholesky ordering.
Structural Decomposition uses the orthogonal transformation estimated from the structural factorization matrices. This approach is not available unless you have estimated the structural factorization matrices as explained in “Structural (Identified) VARs”.
User Specified allows you to specify your own impulses. Create a matrix (or vector) that contains the impulses and type the name of that matrix in the edit box. If the VAR has endogenous variables, the impulse matrix must have rows and 1 or columns, where each column is an impulse vector.
For example, say you have a variable VAR and wish to apply simultaneously a positive one unit shock to the first variable and a negative one unit shock to the second variable. Then you will create a vector containing the values 1, -1, and 0. Using commands, you can enter:
vector(3) shock
shock.fill 1,-1,0
and type the name of the vector SHOCK in the edit box.
Displaying Impulse Responses in EViews
To display the impulse responses, for now click on OK to accept the default settings and display the impulse responses. Here, EViews displays 9 impulse response graphs including 95% analytic asymptotic confidence bands:
Users may wish to display multiple CI bands. To show 95% and 70% CI bands, for example, enter 0.95 0.7 in the Confidence intervals levels edit field. Doing so yields the following:
Note that the impulse response output is an interactive spool object with the spool tree listing two collapsible categories (Variables and Output Views) and selectable entries on the left of the window. A category is said to be collapsible when its contents can be collapsed or expanded by clicking on, respectively, the minus or plus box to the left of the category name.
The Output Views node may be collapsed or expanded to hide or show the three entries which control the type of output displayed:
Table — if selected, will display a table of the impulse response values and standard errors in the right pane.
Multiple Graphs — if selected, will plot impulse response graphs with one for each requested impulse-response combination.
Combined Graphs — if selected, will plot separate graphs for each requested impulse, with all of the requested responses.
Selecting one of these entries will display the corresponding contents in the right pane of the output window. For example, clicking on Table changes the display to show numeric values for the impulse responses and standard errors:
Clicking on the Combined graphs entry yields:
showing the impulse responses in graphs grouped by response variables.
The Variables node may be collapsed or expanded to hide or show the three node entries: Impulses, Responses, and if appropriate, Cholesky ordering.
The Impulses and Responses nodes contain entries for the variables in the VAR with which you may interact to change the display behavior.
The Cholesky ordering node is informational and does not allow for interaction with the entries in the node.
You may interact with entries in the Impulses and Responses nodes by:
Double clicking on (toggling) each variable to show or hide the impulses or responses of the variable in the output views.
Drag the entries within the node to change the order of variables in the output views.
Entries with “*” by their name indicate that they will be displayed in the output views. Clicking on a “*” entry will remove it from the output view. Clicking on an entry without a “*” will add it to the output view. Here, we see that all impulses and responses will be displayed.
If we double click on the Impulses entries for IP and TB3 to remove the “*”, and double click on Multiple graphs, the view will change to show only responses to the M1 impulses:
We may click-and-drag the TB3 entry in the Responses node to rearrange the order of the graphs
Variance Decomposition
While impulse response functions trace the effects of a shock to one endogenous variable on to the other variables in the VAR, variance decomposition separates the variation in an endogenous variable into the component shocks to the VAR. Thus, the variance decomposition provides information about the relative importance of each random innovation in affecting the variables in the VAR.
To obtain the variance decomposition, select View/Variance Decomposition... from the VAR object toolbar. You should provide the same information as for impulse responses above.
Note that since non-orthogonal factorization will yield decompositions that do not satisfy an adding up property, your choice of factorization is limited to the Cholesky Decomposition and the Structural Decomposition, the latter if available.
The table format displays a separate variance decomposition for each endogenous variable. The second column, labeled “S.E.”, contains the forecast error of the variable at the given forecast horizon. The source of this forecast error is the variation in the current and future values of the innovations to each endogenous variable in the VAR. The remaining columns give the percentage of the forecast variance due to each innovation, with each row adding up to 100.
As with the impulse responses, the variance decomposition based on the Cholesky factor can change dramatically if you alter the ordering of the variables in the VAR. For example, the first period decomposition for the first variable in the VAR ordering is completely due to its own innovation.
Factorization based on structural orthogonalization is available only if you have estimated the structural factorization matrices as explained in “Structural (Identified) VARs”. Note that the forecast standard errors should be identical to those from the Cholesky factorization if the structural VAR is just identified. For over-identified structural VARs, the forecast standard errors may differ in order to maintain the adding up property.
As with the output for impulse responses ( “Impulse Responses”), the resulting output is interactive. You may click on any of the Output Views node entries to change the display type. Here we see a Stacked graphs representation of the decomposition.
Similarly, you may click on the Decompositions node under Variables and toggle on the variable names to show or hide the display of the decomposition of specific variables.
Historical Decomposition
An alternative method of innovation accounting is to decompose the observed series into the components corresponding to each structural shock. Burbridge and Harrison (1985) propose transforming observed residuals to structural residuals, and then for each observation beyond some point in the estimation sample, computing the contribution of the different accumulated structural shocks to each observed variable.
To obtain the historical decomposition, select View/Historical Decomposition... from the VAR object toolbar. You should provide the same information as for impulse responses and variance decomposition above.
In addition, you may use the Include baseline in decomposition to choose whether to include the base projection in the decomposition, or whether to plot only the stochastic accumulations and total.
Lastly, you may use the edit fields to specify an optional Start and End period to the decomposition. Both periods must be within the range of the original estimation period. If you leave the fields blank, EViews will perform the decomposition from the start to the end of the full estimation period.
Procs of a VAR
Most of the procedures available for a VAR are common to those available for a system object (see “System Procs”). Here, we discuss only those procedures that are unique to the VAR object.
You may produce forecasts directly from an estimated VAR object by clicking on the Forecast button or by selecting Proc/Forecast. EViews will display the forecast dialog:
Most of the dialog should be familiar from the standard equation forecast dialog. There are, however, a few minor differences.
First, the fields in which you enter the forecast name and optional S.E. series names now refer to the character suffix which you will use to form output series names. By default, as depicted here, EViews will append the letter “f” to the end of the original series names to form the output series names. If necessary, the original name will be converted into a valid EViews series name.
Second, if you choose to compute standard errors of the forecast, EViews will obtain those values via simulation. You will be prompted for the number of Simulation repetitions, and the “% failed reps before halting” the simulation setting.
Lastly, in addition to a Forecast evaluation, you are given a choice of whether to display the output graphs as Individual graphs, as Multiple graphs, or both.
Clicking on OK instructs EViews to perform the forecast and, if appropriate to display the output:
In this case, the output consists of a spool containing the forecast evaluation of the series in the VAR, along with individual graphs of the forecasts and the corresponding actuals series.
Make System
This proc creates a system object that contains an equivalent VAR specification. If you want to estimate a non-standard VAR, you may use this proc as a quick way to specify a VAR in a system object which you can then modify to meet your needs. For example, while the VAR object requires each equation to have the same lag structure, you may want to relax this restriction. To estimate a VAR with unbalanced lag structure, use the Proc/Make System procedure to create a VAR system with a balanced lag structure and edit the system specification to meet the desired lag specification.
The By Variable option creates a system whose specification (and coefficient number) is ordered by variables. Use this option if you want to edit the specification to exclude lags of a specific variable from some of the equations. The By Lag option creates a system whose specification (and coefficient number) is ordered by lags. Use this option if you want to edit the specification to exclude certain lags from some of the equations.
For vector error correction (VEC) models, treating the coefficients of the cointegrating vector as additional unknown coefficients will make the resulting system unidentified. In this case, EViews will create a system object where the coefficients for the cointegrating vectors are fixed at the estimated values from the VEC. If you want to estimate the coefficients of the cointegrating vector in the system, you may edit the specification, but you should make certain that the resulting system is identified.
You should also note that while the standard VAR can be estimated efficiently by equation-by-equation OLS, this is generally not the case for the modified specification. You may wish to use one of the system-wide estimation methods (e.g., SUR) when estimating non-standard VARs using the system object.
Estimate Structural Factorization
This procedure is used to estimate the factorization matrices for a structural (or identified) VAR. The details for this procedure are provided in “Structural (Identified) VARs” below. You must first estimate the structural factorization matrices using this proc in order to use the structural options in impulse responses and variance decompositions.