Chapter 17: Multivariate VARMA-GARCH Models

Introduction

Multivariate VARMA-GARCH Models

The Wage-Price Time Series

A VARMA Model with a CCC-GARCH Model for the Residuals

A VARMA Model with a DCC-GARCH Model for the Residuals

Refinement of the Estimation Algorithm

The Final VARMA Model with DCC-GARCH Residuals

Conclusion

Introduction

In this chapter, you will see how PROC VARMAX is applied to estimate the parameters of more complicated models. These are models in which GARCH parameterizations for multivariate volatility structures are combined with VARMA models for the autocorrelation structure of multivariate time series. The theoretical specifications of the models are further developments based on a combination of the VARMA models as applied in Chapters 1–12 and multivariate GARCH models in Chapters 15 and 16.

The estimation algorithm has to be tuned in many of the examples. You use options to specify the numerical estimation and to introduce bounds and restrictions to the parameter values. Moreover, initial values and normalizations of the series might be necessary in order to reach proper results.

Multivariate VARMA-GARCH Models

In this chapter, VARMA models for multivariate time series are combined with GARCH models for the residual series. Such combined models represent both the cross-correlation and autocorrelation structures in the series by the VARMA part and the volatility structures by the GARCH part.

The VARMA part is parameterized as in Chapter 8:

$$X_t-{\phi }_1{X}_{t-1}-\mathrm{..}-{\phi }_p{X}_{t-p}=c+{\epsilon }_t-{\theta }_1{\epsilon }_{t-1}-\mathrm{..}-{\theta }_q{\epsilon }_{t-q}$$

where the coefficient matrices φ_m and θ_m are all k ×k matrices. So, generally, they each include k² parameters. This part of the model could, of course, be extended by independent variables, which would turn the model into a VARMAX model.

The GARCH part is then just a multivariate GARCH model for the multivariate residual series ε_t in the above VARMA model. This GARCH model can be any of the parameterizations, BEKK, CCC, or DCC, as introduced in Chapter 16. Moreover, the alternative parameterizations for univariate GARCH models (such as TGARCH or PARCH, as introduced in Chapter 15) could be applied.

The number of parameters in the combined VARMA-GARCH could be very large—especially when the BEKK parameterization is applied with no restrictions placed on the parameter values. The model is often over-parameterized, causing the estimation algorithm to fail.

In general statistical analyses, it is usually a pleasant working procedure to start out with a very general model that includes all possible parameters and then perhaps test out some of the parameters. But when there are numerous parameters, this “general to specific” modeling strategy is not applicable for VARMA-GARCH modeling.

Instead, it is a better idea to fit a VARMA model for the autocorrelation part of the multivariate time series. Then you can extend the model by GARCH components beginning with the parsimonious CCC parameterization. At each state of the model-fitting process, initial values for parameter values should be applied in order to refine the estimation algorithm. The initial values can be found by fitting univariate GARCH models to each component of the multivariate series or by just guessing the form from similar time series analyses.

The Wage-Price Time Series

For the wage-price series that was discussed in several previous chapters, a second-order autoregressive model was fitted with some restrictions on the parameters. In this chapter, the series DLW and DLP for the differenced log-transformed indices for wages and prices in Denmark will be applied. This AR(2) model with two parameter restrictions was estimated as shown in Program 9.6, which is presented here as Program 17.1.

Program 17.1: Fitting a VARMA Model as in Chapter 9

Among the printed output from this application, a test for ARCH effects in the residual series is found. See Output 17.1. This is a test for the ARCH parameter in a univariate GARCH(0,1) model’s being zero. This is the α₁ parameter in the univariate GARCH model, which is parameterized as follows:

$$h_t=\text{ω}+{\text{α}}_1{\epsilon }_{t-1}^2$$

In formal test theory, this test statistic is a Lagrange multiplier test, which in the present case is just the first-order estimated autocorrelation for the squared residual series ε_t.

Output 17.1: Tests for ARCH Effects in a VARMA Model

This tells you that an ARCH effect is present in the wage series but not in the price series. In order to fix this model failure, a GARCH model has to be fitted on top of the multivariate AR(2) model, which was specified in Program 17.1. The final model then has to include both the AR(2) parameters and GARCH-parameters.

A VARMA Model with a CCC-GARCH Model for the Residuals

The ARCH tests in Output 17.1 show that an ARCH effect is present only for the wage series but not for the price series. In this case, the sparsely parameterized CCC formulation for a GARCH(1,1) model is used because it allows for a GARCH model for the wage series and no GARCH model at all for the price series. The order of the GARCH model for both series is specified by the options P=1 and Q=1 in the GARCH statement. But the α₁ and γ₁ parameters in the GARCH(1,1) model are restricted to zero for the price series by the RESTRICT statement. In the code, the α₁ and γ₁ parameters are denoted ACH(1,1,1) for α₁ and the GCH(1,1,1) for γ₁; this notation is because the price series is mentioned first in the MODEL statement.

The code is given in Program 17.2. The second-order autoregressive model for the VARMA model is stated by the option P=2 in the MODEL statement, as in Chapter 9. The GARCH model is specified in the GARCH model as discussed in Chapters 15 and 16. All restrictions to the VARMA model used in Program 17.1 and the further restrictions to the parameters of the GARCH model are stated in the same RESTRICT statement in Program 17.2.

Program 17.2: Fitting a Combined VARMA-GARCH Model

The estimated parameters are shown in Output 17.2. The model includes a constant correlation .44, denoted CCC1_2 in Output 17.2. Moreover, the GARCH parameters for the wage series are significant even if the significance of γ₁ is borderline, p = .034. This parameter is denoted GCH1_2_2 in Output 17.2.

Output 17.2: Estimated Parameters of a Combined VARMA-GARCH Model

A VARMA Model with a DCC-GARCH Model for the Residuals

Even if, according to the tests in Output 17.1, no ARCH effect seems to be present for the price series, the correlation between the two series could be volatile. This means that a DCC model could be appropriate. The DCC model is fitted by simply changing the FORM option in the GARCH statement to FORM=DCC. The estimated parameters for the GARCH part of the model are given in Output 17.3.

Output 17.3: Estimated Parameters in a DCC Parameterization

In Output 17.3, the estimates for both the DCCA and DCCB parameters are estimated to values equal to or close to zero, and the standard deviation is reported as zero. These extra parameters α and β give weight to the time-varying components in the volatile correlation, q_ijt, in the parameterization:

$$q_{ijt}=\left(1-\text{α}-\text{β}\right)s_{ij}+\text{α}\frac{{\text{ε}}_{i\left(t-1\right)}^{}}{\sqrt{h_{ii\left(t-1\right)}}}\frac{{\text{ε}}_{j\left(t-1\right)}^{}}{\sqrt{h_{jj\left(t-1\right)}}}+\text{β}{q}_{ij\left(t-1\right)}$$

If these parameters are both zero, then the DCC parameterization turns into a CCC parameterization.

Refinement of the Estimation Algorithm

Reported parameter estimates and standard deviations equal to 0 are seen in other GARCH examples as well, and they are usually an indication of failure in the parameter estimation. To study this further, include an NLOPTIONS statement in Program 17.3 with the option PALL (short for printall) so that details of the optimization are printed. Otherwise, the code for estimation the DCC model is exactly as in Program 17.2.

Program 17.3: The NLOPTONS Statement

One of these tables is Output 17.4.

Output 17.4: Optimization Results from Program 17.3

Output 17.4 shows that something went wrong. The iteration algorithm has stopped because of a criterion that determines that the function value, which is, generally speaking, the log-likelihood function value, cannot be improved. But the normalized gradient is very large (equal to 622), as opposed to the optimal value of zero. Moreover, a warning message is written to the SAS log because this maximum gradient value exceeds 10⁻³.

NOTE: Convergence criterion (FCONV=2.220446E-16) satisfied.

NOTE: At least one element of the (projected) gradient is greater than

1e-3.

The situation is that the estimation algorithm has been stopped by a criterion using function values because the log-likelihood in numeric terms is flat around the printed solution. But this solution is far from optimal under other criteria.

One possible remedy is to adjust the series by multiplying it by a constant. In Program 17.4, the series DLP and DLW are both multiplied by 100 in the DATA step. Afterward, these new series are applied in an application of PROC VARMAX.

The new option in the NLOPTIONS statement GCONV=1E-14 is used because the iteration process is otherwise stopped by a criterion on the non-normalized gradient. Without the option GCONV=1E-14, the following notes are printed to the SAS log:

NOTE: Convergence criterion (GCONV=1E-8) satisfied.

NOTE: At least one element of the (projected) gradient is greater than

1e-3

These notes mean that the problem of a large normalized gradient is not yet mended.

The GCONV option in Program 17.4 is used to strengthen this gradient criterion to 10⁻¹⁴ instead of the default 10⁻⁸.

Program 17.4: Scaling the Series and Introducing Extra Options to Refine the Estimation Algorithm

The table of parameter estimates now shows that both the α and β, the DCA and DCB parameters, in Output 17.5, are significant. The parameters GCHC1_1 and GCHC_2 are, of course, now much larger compared to Output 17.3 because both the series are multiplied by 100. This scaling by 100 increases the variance by a scaling factor of 10000. Moreover, the GARCH parameter γ₁ (denoted GCH1_2_2) has now become insignificant.

Output 17.5: Estimated Parameters from Program 17.4

The tests for the restrictions (Output 17.6) show that the second restriction, which is the restriction on the AR2_2_1 parameter, is rejected. This rejection means that the off-diagonal element in the second-order autoregressive parameter should be included in the model. When this restriction is removed, the restriction on the AR2_1_1 parameter is, however, also rejected. So it is concluded that all parameters in the autoregressive model should be included when GARCH effects for the wage series are present in the model.

Output 17.6: Tests for the Restriction in Program 17.4

The Final VARMA Model with DCC-GARCH Residuals

The estimated value of the parameter α₁, the ACH1_2_2 parameter, is 0.91. See Output 17.5. According to the reported standard deviation, this estimated value is close to 1.0. This indicates that the volatility in the wage series is almost persistent. Persistent volatility corresponds to the value α₁ = 1.0, which is in fact outside the parameter space for the usual GARCH models. The value α₁ = 1.0 is the situation covered by the IGARCH model. See Chapter 15.

The IGARCH model for the wage series is estimated using the DCC parameterization for the volatile correlation between the wage and price series as estimated by Program 17.5. In Program 17.5, a BOUND statement is applied in case the estimated value turns out to be smaller than the boundary value 1.0.

In Program 17.5, the AR(2) model for the bivariate series is fitted without any restrictions to the parameters for the reasons discussed above. The GARCH parameter γ₁ is set to zero by removing the option P=1 so that only the α₁ parameter in the GARCH model is included in the model by the option Q=1. In this situation, the GARCH model is suppressed for the price series by restricting the ACH(1,1,1) parameter in the RESTRICT statement because the GCH(1,1,1) parameter is excluded as p = 0 in the GARCH model.

Program 17.5: The Final VARMA-GARCH Model Using DCC Parameterization and an IGARCH Model

Output 17.7 gives all estimated parameter values as calculated by Program 17.5. All parameters in the model are significant. Output 17.8 gives test results for the individual restrictions, showing that all restrictions are accepted. This means that the model is acceptable in these aspects. Some indication of lack of fit of this model is found in the cross-correlation and autocorrelation structures for the residual series, where a significant autocorrelation is found for the wage series for lag 2 and for the price series at lag 4. Moreover, a significant cross-correlation is present at lag 4. As in Chapter 9, these indications of misfit by the model are ignored because it takes too many parameters to mend these problems.

Output 17.7: Estimated Parameters in Program 17.5

Output 17.8: Tests for Restrictions in Program 17.5

The model is written in details as follows:

$$x_{1t}=1.25+0.33x_{1t-1}\text{+0}\text{.28}{x}_{2t-1}+0.16x_{1t-2}-\text{0}\text{.30}{x}_{\text{2}t-2}\text{+}{\epsilon }_{1t}^{}$$

$$x_{\text{2}t}=2.21+0.24x_{1t-1}\text{+0}\text{.66}{x}_{\text{2}t-1}+0.17x_{\text{1}t-2}-\text{0}\text{.27}{x}_{\text{2}t-2}\text{+}{\epsilon }_{2t}^{}$$

For the variance of the residuals (ε_1t, ε_2t), the model is a constant for the variance of ε_1t :

$$h_{1t}=34.03$$

And the variance of ε_2t is an IGARCH:

$$h_{\text{2}t}=7.25+{\epsilon }_{2(t-1)}^2$$

Note that the constants 34.03 and 7.25 are for the differenced log-transformed series multiplied by 100. See Program 17.5. For the original, differenced, log-transformed series the numbers should be .0034 and .0007.

The covariance is volatile, according to the DCC parameterization:

$$q_{12t}=\left(1-0.16-0.65\right)\times 0.52+0.16\frac{{\text{ε}}_{1\left(t-1\right)}^{}}{\sqrt{h_{11\left(t-1\right)}}}\frac{{\text{ε}}_{2\left(t-1\right)}^{}}{\sqrt{h_{2\left(t-1\right)}}}+0.65q_{12\left(t-1\right)}$$

The variance series for ε_2t, the variable H2_2, and the correlation between ε_1t and ε_2t are plotted by Program 17.6. In Program 17.6, the correlation is defined using the three variance series h_1t, h_2t, and h_12t, which are defined by the GARCH equations. These equations are stored in the output data set CONDITIONAL by means of the option OUTHT=CONDITIONAL in Program 17.5. The constant variance 7.25 for ε_1t is stored as the variable H1_1, and the covariance is the variable H1_2.

In the new data set CONDITIONAL defined in Program 17.6, the variance series are shifted by 2 because of the lag 2 in the AR(2) part of the model. The DATA step in Program 17.6 also deletes the first two observations in the data set CONDITIONAL for which the conditional variance is undefined.

Program 17.6: Plotting the Variance for the Wage Series and the Correlation from Program 17.5

The plot of the conditional variance function (Figure 17.1) for the wage series has very large variances for two periods of, say 10 years. But the variance is rather stable around the constant variance parameter GCHC2_2 = 7.25 for most of the years.

Figure 17.1: The Conditional Variance Series of the Residual Wage Series

The correlation (Figure 17.2) has a similar behavior. The correlation is largest, up to 0.8, in periods with the largest variances for the wage series. But for most days in the sample period, the correlation is close to the value .45 as estimated by the CCC parameterization. See Output 17.2.

Figure 17.2: The Conditional Correlation Series of the Two Residual Series

Conclusion

You can apply combined VARMA models and multivariate GARCH models by using PROC VARMAX. This application of a model combination is demonstrated in this chapter for the bivariate series of yearly prices and wages in Denmark for more than 100 years. It must be stressed, however, that the models are involved, so some extra options are required in order to achieve acceptable results.