Chapter 15: Univariate GARCH Models

Introduction

The GARCH Model

GARCH Models for a Univariate Financial Time Series

Use of PROC VARMAX to Fit a GARCH(1,1) Model

The Fitted Model

Use of PROC VARMAX to Fit an IGARCH Model

The Wage Series

Use of PROC VARMAX to Fit an AR(2)-GARCH(1,1) Model

The Conditional Variance Series

Other Forms of GARCH Models

The QGARCH Model

The TGARCH Model

The PGARCH Model

The EGARCH Model

Conclusion

Introduction

Plots of time series often show that the series include time periods with stable behavior, as well as periods with a much more interrupted structure. In financial time series, this pattern is easily interpreted as periods with stable market conditions with only minor day-to-day changes. But sometimes something more drastic happens, leading to consecutive days of larger changes. More dramatic periods can follow the presentation of the accounts, or they could be a sign of a crisis in which the market is striving to find a new equilibrium.

The same phenomena are seen for series from other sources than the economy. For example, the weather can be stable for many weeks, but in some periods the weather changes rapidly.

Such behavior violates the assumption that the residuals of a time series model should be independently distributed because the variance is not constant. The volatility is changing over the observation period. Mathematically, this means that the estimated model is insufficient. In theory, this model inadequacy means that the model estimation method is inefficient, and the reported standard deviations of the estimated parameters and the distribution of test statistics are in error.

For many purposes, this model deficit is not a serious problem if your interests are only prediction or estimation of, say, autoregressive parameters. In financial models, however, the subject of risk is of major importance. In mathematical terms, risk, in regard to many aspects, corresponds to variance as a quantitative measure of variation. GARCH models, which are the subject of this chapter, consider the variance of a time series as a time series to be modeled in its own right in order to forecast future variances. That is, the subject for predictions is future risk. Regulations of financial institutions require the institutions to report the risks in their portfolios. This subject has become very important since 2008 because of the financial crisis. The Basel III regulations for banks and Solvency II for insurance companies define minimum requirements for such risk reporting.

Even if ARIMA or VARMA models carefully treat the autocorrelation structure of the residuals, other forms of dependence are often seen. Variance clustering can be seen as a higher-order dependence than autocorrelation, which is a second-other dependence. The models presented in the next section all try to generalize models like the ARIMA models for the autocorrelation structure of the actual values of a time series to models for the variance of a time series.

The models are a bit complicated because of the many parameters and because the variance of an estimated variance is very large. These conditions lead to unstable estimation procedures in which problems occasionally occur. In some situations, it is necessary to tune the estimation process in applications of the VARMAX procedure—for example, by specification of initial values for the parameters.

The GARCH Model

The GARCH models consider the variance var(x_t) as stochastic. The series var(x_t) is considered as a time series, which is modeled by time series models of the same form as an observed series. The variance var(x_t) is, however, not directly observed like the values of the original time series x_t. But when a model for var(x_t) is formulated, it gives a statistical model for the observed time series, and the parameters of the model for var(x_t) are estimated along with the other parameters in the model.

The basic idea that the variance var(x_t) at time period t could be forecast by the conditional expectation conditioned on the previously observed values of the time series x_t−1, x_t−2 , . . . —that is, all observed values up to time t but not including time t.

In the definition, it is assumed that the mean value of x_t is zero so that var(x_t) = E[x_t²]. The conditional variance is usually denoted h_t. It can be formally defined as E[x_t²| past values] = h_t.

The GARCH(p,q) model is defined by an expression for h_t as follows:

$$h_t=\text{ω}+{\displaystyle \sum _{i=1}^q{\text{α}}_i{x}_{t-i}^2}\text{+}{\displaystyle \sum _{j=1}^p{\text{γ}}_j{h}_{t-j}^{}}$$

This formula, in some sense, mimics the definition of an ARMA(p,q) model for the time series h_t or x_t². The numbers p and q are often chosen as p = 1 and q = 1 because these values, in many cases, have proved to give a satisfactory model fit. At the same time, higher-order model orders often lead to numerical problems in the parameter estimation algorithms due to a larger number of parameters.

The parameters are subject to various restrictions in order to assure that the conditional variance is well defined as a positive number. One commonly used condition is the non-negativity condition:

ω > 0, α_i ≥ 0, γ_j ≥ 0.

But the actual magnitude of the coefficients is also subject to restrictions. The GARCH(p,q) model reduces to the ARCH(q) process when p = 0. If both p = 0 and q = 0, no GARCH effects are present. Moreover, the series has a constant variance ω, and the model is homoscedastic.

The GARCH model for a time series is often applied to residuals of a time series model like the univariate ARMA models presented in the previous chapters. In these cases, it is convenient to write the residuals ε_t of the ARMA model in the following form:

$${\text{ε}}_t=e_t\sqrt{h_t}$$

where e_t are independent with a standard normal distribution. The conditional variance of the residuals, denoted h_t, is defined by the following:

$$h_t=\text{ω}+{\displaystyle \sum _{i=1}^q{\text{α}}_i{\text{ε}}_{t-i}^2}+{\displaystyle \sum _{j=1}^p{\text{γ}}_j{h}_{t-j}^{}}$$

The condition on the model parameters is as follows:

This restriction is necessary to prove that the unconditional variance and autocovariances are finite and constant over time. When the GARCH process is stationary, the unconditional variance of ε_t is computed as follows:

$$\text{Var}\left({\text{ε}}_t\right)=\frac{\text{ω}}{1-{\displaystyle \sum _{i=1}^q{\text{α}}_i}-{\displaystyle \sum _{j=1}^p{\text{γ}}_j}}$$

This expression for h_t is used by PROC AUTOREG as the default parameterization. In PROC VARMAX, the parameterizations are mainly designed in a way that is suitable for generalizations to the multivariate situation. So in PROC VARMAX, a specific option is needed to apply this parameterization. Models for multidimensional series are treated in the next chapter.

GARCH Models for a Univariate Financial Time Series

PROC VARMAX includes facilities to estimate parameters for univariate and multivariate GARCH models. In this section, a typical example analysis of a one-dimensional financial time series is presented.

Consider a series of daily observations of the quotation of a Danish bank for a period of one year 2002-2003. The time series is found in the data set QUOTES. The variable QUOTE_BANK gives the original quote while the series PERCENTAGE_CHANGE_BANK is the series of relative changes from day to day expressed as percentages. The series has 248 observations. The original series and the series of percentage changes are plotted by Program 15.1

Program 15.1: Plotting the Series

This series of relative day-to-day changes looks like independent observations that tell that the rate is a random walk. See Figure 15.1 and Figure 15.2. This point is confirmed by the autocorrelations, which are all close to zero. But lack of autocorrelation is not the same as independence when volatility clustering is present. The plots tell that volatility clustering is present in this series.

Figure 15.1: The Original Series

Figure 15.2: The Series of Daily Relative Changes

Use of PROC VARMAX to Fit a GARCH(1,1) Model

When you are using PROC VARMAX, the parameters of a GARCH(1,1) model for the day-to-day series is estimated by means of Program 15.2. In the data set, only observations for trading days exist. So when the series is considered as a series of daily observations, many observations are missing for weekends and other holidays. This is the reason that no ID statement is applied in Program 15.2. That is, PROC VARMAX does not allow for missing observations. In the data set, the variable I is the observation number that will be applied as an index variable later.

The option FORM=CCC is needed in order to use the original univariate parameterization:

$$h_t=\text{ω+}{\displaystyle \sum _{i=1}^q{\alpha }_i{x}_{t-1}^2+{\displaystyle \sum _{j=1}^p{\gamma }_j{h}_{t-j}}}$$

Otherwise, a default parameterization that is more appropriate for multivariate time series is used. This parameterization is applied in Chapter 16 on multivariate GARCH models. The actual meaning of the strange-looking option FORM=CCC is also explained in Chapter 16.

Program 15.2: Estimation a Univariate GARCH(1,1) Model

In the GARCH statement, the model is specified as GARCH(1,1) by the options P=1 and Q=1. The constant term in the series is suppressed by the option NOINT to the MODEL statement because the series of daily changes has an empirical mean of almost zero.

The estimated parameters in Output 15.1 are the most important table in the procedure output. The parameters are denoted GCHC1_1 for ω, ACH1_1_1 for α₁, and GCH1_1_1 for γ₁. The first number, 1, represents the lag, but the other two number 1s, 1_1, represent a matrix notation that is useful for multivariate GARCH models as considered in the next chapter.

The estimated value of ω is 0.34, denoted GCHC1_1 in Output 15.1. The t-statistic for the estimated ω is just 1.36, which is far from significance in a test using an approximate Student t distribution, p = .18. This could, however, not be taken as an indication that this parameter should be left out, as the parameter is assumed to be positive. It is more an indication that the approximating t-distribution is far from accurate as it allows for negative values that are outside the parameter space. The reported p-value should merely be ignored.

Note that for series with a small variance, the parameter GCH1_1 is close to zero, and it happens that the standard deviation of this estimate is printed as 0. Also, the estimated value can in some examples be printed as 0. For the present data example, this happens if the differences of the log transformed original quotes are used instead of the day-to-day changes expressed as percentages used in the present example.

Output 15.1: Estimated GARCH Parameters

The two parameters, ACH1_1_1 for α₁ and GCH1_1_1 for γ₁, are significant. The actual parameter values are discussed in the next sections. The log-likelihood of the model is −271.725.

The Fitted Model

The model fitted by Program 15.2 with the parameters seen in Output 15.1 is written as follows:

$$h_t=0.34+0.13x_{t-1}^2+0.78h_{t-1}$$

The conditional variance series is far from constant as plotted in Figure 15.3, which is generated by Program 15.3. The conditional variances h_t are written to an output data set named CONDITIONAL by the OUTHT=CONDITIONAL option to the GARCH statement. The variable name for the conditional variance is H1, where the number 1 denotes that it is the variance of the first (and only) series. In the DATA step, the time index I is defined, and the series is then plotted by PROC SGPLOT. The TEST statement in Program 15.3 is discussed in the next section.

Program 15.3: Plotting the Conditional Variances

The plot of the conditional variance, Figure 15.3, shows that the conditional variance as derived by the model is far from constant: It varies in interval 2 to 13. The vertical axis is beginning at the value 0 by the YAXIS statement in order to stress the fact that the minimal conditional variance is positive.

Figure 15.3: The Conditional Variance

In Program 15.4, the data set with the conditional variances and the original data set are merged in a combined data set called MERGED in which two standardizations are further defined. The first is the usual standardization using the empirical standard deviation 1.8855 of the series PERCENTAGE_CHANGE_BANK. The other is a standardization using the square root of the conditional variance h_t as calculated by PROC VARMAX.

Program 15.4: Plotting the Standardized Observations

On the plot, the diamonds represent the observations standardized by a constant variance. The curve shows the observations standardized by the fitted model. As shown by the plot in Figure 15.4, the model reduces many extreme observations. In terms of the GARCH model, this is due to conditional variances that are larger than observations from periods with more stable behavior.

Figure 15.4: The Standardized Observations

Use of PROC VARMAX to Fit an IGARCH Model

The condition α₁ + γ₁ = 1 for the parameters in the usual parameterization of a univariate GARCH(1,1) model turns the model into an IGARCH model. The parameters in an IGARCH model are in some sense on the boundary of the parameter space. The variance of a time series with an IGARCH model is infinite. From a practical point of view, the IGARCH model is often of interest. This is because the condition α₁ + γ₁ < 1 is not met for the estimated parameter values, and the IGARCH model in which α₁ + γ₁ = 1 then becomes a plausible alternative to a usual GARCH model.

The parameter estimates from Output 15.1 are, however, close to this limit: .13 + .78 = .91. This means that parameter values that are just a little bit larger will meet the IGARCH restriction. The TEST statement in Program 15.3 is as follows:

TEST ACH(1,1,1)+GCH(1,1,1)=1;

It gives a test for this hypothesis. The test gives acceptance has p =.22. See Output 15.2. The assumptions underlying the chi-square distribution of the test statistic are, however, not met, because the variance of the time series under the null hypothesis, the IGARCH model, is infinite.

The log-likelihood of the model is −273.527, which is smaller than the log-likelihood value for the unrestricted model in Output 15.1. A likelihood ratio test, however, does not reject the restriction.

Output 15.2: Test of a Possible IGARCH Restriction on the Parameters

The IGARCH model with this restriction on the parameter values is fitted by Program 15.5.

Program 15.5: Imposing an IGARCH Restriction on the Parameters

The resulting parameters, of course, meet the IGARCH condition by the RESTRICT statement. See Output 15.3.

Output 15.3: Estimated IGARCH Parameters

Figure 15.5 shows the estimated conditional variance, h_t, in this situation. The conditional variance behaves much the same as the conditional variance in the unrestricted case plotted in Figure 15.3. The most remarkable difference is that the peak is much larger in Figure 15.5 for the IGARCH parameterization.

Figure 15.5: The Conditional Variance of the IGARCH Model

The Wage Series

In this section, a univariate GARCH model is applied to the residuals of an autoregressive model for the wage series as considered in previous chapters. See Chapter 9 for a description of the series. An autoregressive model for series of the first differenced, logarithmically transformed wage index, the variable DLW, is estimated by Program 15.6 as in Chapter 9.

Program 15.6: Estimation of an AR(2) Model by PROC VARMAX

The application of PROC VARMAX in Program 15.6 generates all possible plots. Among these, are also a plot of the prediction errors (Figure 15.6). The prediction errors show a volatile period in the years following World War I. Moreover, two outliers in 1856/57 can also be considered a result of variance clustering, which can be modeled by GARCH models. The Q-Q plot and the residual histogram (Figure 15.7), also generated by Program 15.6, tell that the residuals are far from normal. This non-normality might be a result of a volatility clustering due to a GARCH model for the residuals of the fitted AR(2) model.

Figure 15.6: Residuals for the AR(2) Model for Wage Series

Figure 15.7: Graphical Tests for Normality for the Residuals of the AR(2) Model

These findings from the graphical output are confirmed by the tests for normality and ARCH effect. See Output 15.4. This output is printed because of the specific PRINT=(DIAGNOSE) option. Another possibility is the general PRINTALL option. The test for ARCH effects is a Lagrange multiplier test for the hypothesis that the parameter α₁ = 0 in a GARCH(0,1) model.

Output 15.4: Numerical Tests for Model Fit of the AR(2) Model

Use of PROC VARMAX to Fit an AR(2)-GARCH(1,1) Model

In Program 15.7, the AR(2) model is extended by a GARCH(1,1) model for the residuals. This order of the GARCH model is commonly applied to univariate series. In mathematical terms, the model is written as follows:

$$x_t={\text{μ+φ}}_{\text{1}}{x}_{t-1}+{\text{φ}}_2{x}_{t-2}+{\text{ε}}_t$$

where

$${\text{ε}}_t=e_t\sqrt{h_t}$$

and the conditional variance of the residuals, denoted h_t, is defined by the following:

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

Finally, the terms e_t are independent, having a standard normal distribution.

All parameters of the model—both the autoregressive parameters and the GARCH parameters—are simultaneously estimated by maximum likelihood.

Program 15.7: Including a GARCH(1,1) Model for the Residuals

The estimated parameters are given in Output 15.5. In this setup, the second-order autoregressive parameter φ₂ is insignificant: p = .78.

Output 15.5: The Estimated Parameters of an AR(2) Model with GARCH(1,1) Residuals

The Conditional Variance Series

The plot of the conditional variance h_t (Figure 15.8) is made by a program similar to Program 15.3. The conditional variance is rather constant except for two short periods for the years where the extreme observations were found in the residual plot (Figure 15.6). This curve is perhaps not what is usually seen for GARCH models. It can be taken as an indication that introducing dummy variables for the extreme observations perhaps is a more valid way to analyze the volatility of this particular series.

Figure 15.8: The Conditional Variance h_t

Other Forms of GARCH Models

PROC VARMAX supports forms of GARCH models other than the original parameterization in the previous sections. Many such extensions of the basic GARCH formulation are supported: EGARCH, TGARCH, PGARCH, and QGARCH.

PROC VARMAX is mainly dedicated to the modeling of multivariate time series. See Chapter 16. All these multivariate models in some way combine univariate GARCH models for the separate series. Also, for the other parameterizations of GARCH models, the option FORM=CCC should be used for univariate time series.

The particular parameterization of the GARCH model is found by the SUBFORM option to the GARCH statement. The value can be any of the before-mentioned abbreviations—for instance, SUBFORM=TGARCH. The extension of the generalized parameterizations in this section to the multivariate situation is considered in Chapter 16.

PROC VARMAX does not support t-distributions, which are often applied in order to model heavy-tailed residual distributions in financial data. The reader is referred to PROC AUTOREG for this possibility for univariate series.

The QGARCH Model

In the QGARCH model, the lagged errors ε_t−i are allowed to vary around a value other than zero in the formulation of the GARCH model:

$$h_t^{}=\text{ω}+{\displaystyle \sum _{i=1}^q{\text{α}}_i{\left({\text{ε}}_{t-i}^{}-{\text{β}}_i\right)}^2}+{\displaystyle \sum _{j=1}^p{\text{γ}}_j{h}_{t-j}^{}}$$

The parameters in this model are estimated by Program 15.8 even if the underlying idea in this model is irrelevant for this particular series of daily quotations.

Program 15.8 Estimating the Parameters of a QGARCH Model

The estimated parameters (Output 15.6) clearly show that this is not necessary for the present data series. The introduced shift parameter β₁, denoted QACH1_1_1 in Output 15.6, is insignificant: p = .65. This is confirmed because the log-likelihood of the model is −271.358, which is only slightly larger than the log-likelihood value in the model without the QGARCH parameter β₁.

Output 15.6: Estimated Parameters in a QGARCH Model

When the shift parameter β₁, denoted QACH1_1_1, is restricted to zero, the model reduces to a usual GARCH(1,1) model as discussed in the previous sections.

The TGARCH Model

In the TGARCH model, the coefficient to the lagged squared error in a GARCH model is allowed to attain different values, depending on the sign of lagged error term. In the following parameterization, the indicator function is 1 if ε_t−i <  0, and 0 otherwise:

This means that for positive lagged errors, the coefficient is just the α parameter, while the coefficient for negative error terms is α + β. In Program 15.9, the parameters of this model are estimated for the series of daily relative differences for quotations of a Danish bank.

Program 15.9: Estimating the Parameters of a TGARCH Model

The estimated α₁ parameter is insignificant: p = .76. The estimated β₁ is significant, having p-value p = .02. See Output 15.7.

Output 15.7: Estimated Parameters in a TGARCH Model

The parameter α₁, which is denoted ACH1_1_1 in the output, is restricted to zero by the following statement:

RESTRICT ACH(1,1,1)=0;

With this restriction imposed in Program 15.9, the estimated parameters are as given by Output 15.8. The β₁ parameter is significant: p = .02.

Output 15.8: Estimated Parameters in a Restricted TGARCH Model

The log-likelihood value for this model is −269.938, which is larger than the value −271.725 in the original GARCH model with unrestricted parameters. The number of estimated parameters in these two models are in fact the same, so it is concluded that the TGARCH parameterization is superior for this series.

The conditional variance h_t is defined by the estimated model as follows:

This model’s structure is different from the original GARCH parameterization or the IGARCH representation. Now only negative values of the series (that is, days with a falling quotation of the bank) has an effect on future variance. This conclusion is perhaps a bit surprising, but it is found through tests. In the literature, similar conclusions are seen (for example, Engle and Ng, 1993).

The conditional variance h_t is plotted by PROC SGPLOT as in Program 15.9 with the restriction α = 0 imposed, as the series h_t is outputted to the data set named CONDITIONAL. The resulting plot, Figure 15.9, looks quite different from Figure 15.3 and 15.5. Especially the peak in Figure 15.9 is remarkably reduced by the TGARCH parameterization as compared to the previous plots.

Figure 15.9: The Conditional Variance h_t in the Restricted TGARCH Model

The PGARCH Model

The PGARCH includes an exponent λ in the model as the variance raised to the power λ is now applied in the model:

$$h_t^{\text{λ}}=\text{ω}+{{\displaystyle \sum _{i=1}^q{\text{α}}_i\left(\left|{\text{ε}}_{t-i}^{}\right|-{\text{β}}_i{\text{ε}}_{t-i}^{}\right)}}^{\text{2λ}}+{\displaystyle \sum _{j=1}^p{\text{γ}}_j{h}_{t-j}^{\text{λ}}}$$

If the parameter β_i is different from zero, then it introduces an asymmetry in the model as positive and negative values of ε_t−i give different contributions to future variances h_t.

In this PGARCH parameterization, if λ = 1 and all β_i = 0, then the PGARCH parameterization corresponds to the original GARCH parameterization. Among other possible values of the parameter λ, the value λ = .5 is especially important, because it describes the volatility as linear in the standard deviations and not in the variances. If λ = .5, then the model for the volatility, the uncertainty, is formulated in the same physical units as the original data series, which is an intuitively appealing feature of a model.

The PGARCH parameterization is applied to the series of relative changes of the quotations series in Program 15.10.

Program 15.10: Estimating the Parameters of a PGARCH Model

The estimation of the β parameter, which is denoted PACH1_1_1 in Output 15.9, gives the exact result 1 which is on the boundary of its allowed range [−1;1]. This value β = 1, however, turns this part of the parameterization into a TGARCH model where it was previously concluded that only negative terms ε_t−1 had an effect on the conditional variance h_t.

Output 15.9: Estimated Parameters in a PGARCH Model

The estimated exponent λ is larger than 1, but with a large standard deviation. So it seems that the hypotheses λ = 1 or even λ = 0 could be accepted. The log-likelihood value of this model, −269.955, is almost the same value as for the TGARCH model, which is the case of λ = 1.The estimated value of α₁ is insignificant, but the parameter α₁ is mandatory.

Figure 15.10 shows the estimated conditional variance, h_t, in this situation. This plot looks like the conditional variance in the TGARCH model plotted in Figure 15.9, but the curve for the conditional variance is somewhat smoother in periods with declining volatility.

Figure 15.10: The Variance in a Restricted PGARCH Model

The EGARCH Model

In GARCH models the time series properties of the conditional variances are modeled by models much like ARMA models. In EGARCH models the log-transformed conditional variance is modeled instead of the conditional variance itself. This log-transformation has the advantage that the conditional variance when transformed back by the exponential function is always positive, whatever the values the parameters attend. In the usual GARCH parameterizations, this positivity is secured only by complicated restrictions on the GARCH parameter values.

Moreover, the EGARCH parameterization also allows for asymmetric effects of lagged time series values such that negative error terms have a different impact than positive error terms.

The EGARCH parameterization gives a model of the following form:

$$\log \left(h_t^{}\right)=\text{ω}+{\displaystyle \sum _{i=1}^q{\text{α}}_i\left({\beta }_{\text{i}}\frac{{\text{ε}}_{t-i}^{}}{\sqrt{h_{t-i}^{}}}+\left|\frac{{\text{ε}}_{t-i}^{}}{h_{t-i}^{}}\right|-\sqrt{\frac{2}{\pi }}\right)}+{\displaystyle \sum _{j=1}^p{\text{γ}}_j\log \left(h_{t-j}^{}\right)}$$

In Program 15.11 the EGARCH model is fitted simply by the option SUBFORM=EGARCH to the GARCH statement.

Program 15.11: Generating The EGARCH Parameterization

The estimated parameter γ₁ is negative (Output 15.10), in contrast to all previous results in this chapter. Moreover, the standard deviation of the parameter β₁ is large. So according to the estimation results in Output 15.10, β₁ could take any value in a broad interval. The log likelihood value for the model fitted by Program 15.11 is −277.375.

Output 15.10: Estimated Parameters by Program 15.11

It is, however, possible that the estimation results for GARCH models are misleading because of local maxima and other numerical problems because the likelihood function for a GARCH model is very complicated.

In Program 15.12, the signs of the β₁ and γ₁ parameters are reverted by a BOUND statement. Note that the β₁ parameter is denoted EACH(1,1,1) in the coding.

Program 15.12: EGARCH Model Estimation Using a BOUND Statement

The resulting log-likelihood in this estimation is much larger (−270.451), so it seems that the estimated parameters (Output 15.11) provide a better fit to the volatility.

Output 15.11: Parameters Estimation Using a BOUND Statement

In the subsection with the TGARCH parameterization, the conclusion was that negative errors made a contribution, but positive errors had no effect at all. In the EGARCH model, this result is expressed by letting the parameter β₁ equal β₁ = −1.

If the restriction β₁ = −1 is imposed by a RESTRICT statement, the final code is given as in Program 15.13.

Program 15.13: Restriction of the EAGARCH Model to Mimic a TGARCH Model

The resulting log-likelihood is −270.459, which differs only in the third decimal place from the value in the unrestricted estimation of the β₁ parameter by Program 15.12. The parameters are given by Output 15.12. Now that the β₁ parameter is fixed, the α₁ parameter is highly significant even if the estimated value α₁ = .09 is close to 0.

Output 15.12: Estimated Parameters of the Final EGARCH Model

Conclusion

PROC VARMAX is programmed mainly for estimation in multivariate time series models. Nevertheless, PROC VARMAX includes many variations of GARCH models even for univariate time series. This is demonstrated in this chapter for a series of daily quotations of a Danish bank. One result is that the TGARCH parameterization gives the best fit to the quotation series.