Evaluating Multivariate GARCH Models in the Nordic Electricity Markets

ABSTRACT

This article considers a variety of specification tests for multivariate GARCH models that are used for dynamic hedging in electricity markets. The test statistics include the robust conditional moments tests for sign-size bias along with the recently introduced copula tests for an appropriate dependence structure. We consider this effort worthwhile, since quite often the tests of multivariate GARCH models are omitted and the models become selected ad hoc depending on the results they generate. Hedging performance comparisons, in terms of unconditional and conditional ex-post variance portfolio reduction, are conducted.

Keywords:

• Conditional moments
• Hedging performance
• Multivariate GARCH

Mathematics Subject Classification:

• 62H12
• 62H15
• 60G35

Notes

Panel A reports summary statistics and unit root tests of the logarithmic spot and futures prices for the full sample period between January 1996 and November 2002. ADF and KPSS denote the Augmented Dickey–Fuller test and Kwiatkowski–Phillips–Schmidt–Shin test, respectively. The trend is included in the tests. ADF tests the null hypothesis of a unit root, whereas KPSS tests the null of stationarity. The critical values for ADF are −4.0 and −3.5 at 1% and 5% levels. The corresponding critical values for KPSS are 0.216 and 0.146 for 1% and 5% levels. Q(k) is the Ljung–Box test for autocorrelation. 99% critical value for Jarque–Bera is 9.21. Panel B reports the Johansen cointegration test for spots and futures. For the Johansen's trace test and the eigenvalue test, the first row tests the null of no cointegration and the second row tests the null of one cointegrating vector. The lag orders used in the unit root and cointegration tests were selected using a sequence of Lagrange ratio tests for VARs of different orders. Italic-bold denotes rejection at 1% significance level.

The table reports the estimates of the ECM(2) for the mean in daily spot and futures prices (the first 1,400 observations: between January 1996 and August 2001). The futures strategy is based on buying contracts with three weeks left to maturity and rolling over one week prior to expiration. Q(k) denotes the Ljung–Box test for the first k lags. The model order was selected using a series of Lagrange ratio tests. Q ²(k) is the Ljung–Box test for the squared residuals. 99% critical value for Jarque–Bera normality test is 9.21. Italic-bold denotes rejection at 1% significance level.

The table gives diagnostic tests for the univariate GARCH specifications obtained from CCORR. The ARCH and GARCH tests denote the LM-test proposed by Lundbergh and Teräsvirta (1998). The sign and size bias tests of standardized residuals are based on Engle and Ng (1993). QGARCH and LSTGARCH denote the direct tests against these alternatives as proposed by Hagerud (1997). The parameters constancy tests denote the Lundbergh and Teräsvirta (1998) tests against ANST-GARCH.

K-S T-dist, K-S GED-dist, and K-S Normal-dist statistics represent the Kolmogorov–Smirnov tests the null that the errors are from the given distribution. The N(0, 1) distributed copula statistics are based on the work of Chen et al. (2004).

The table presents the maximum likelihood parameter estimates for the constant correlation model. The lower part of the table gives standardized residual statistics. Log L denotes the loglikelihood value and denotes the Ljung–Box test for the remaining serial correlation in the squared residuals. 99% critical value for Ljung–Box is 16.8 and for the Lilliefors normality test 0.0276. The DCC-MGARCH test statistic is χ² distributed with nlags+1 degrees of freedom. The pval denotes the probability that the correlation is constant.

AIC = −2 Log L + 2K, where K is the number of estimated parameters and Log L is the log-likelihood function.

SIC = −2 Log L + K log(T), where T = 1517 is the sample size and K is the number of estimated parameters and Log L is the log-likelihood function

This table gives summary statistics for the variance, covariance, and hedge ratio series estimated from the different MGARCH models discussed in the article. All models have been applied to the same dataset between January 1996 and August 2001. ϵ_1t is the spot return innovation and ϵ_2t is the innovation to the futures return. h _11t , h _22t , and h _12t denote the estimated spot variance, futures variance, and their covariance, respectively.

This table gives correlations between the variance, covariance, and hedge ratio series estimated from the different MGARCH models discussed in the article. All models have been applied to the same dataset between January 1996 and August 2001.

Panel A gives the unconditional daily portfolio return variance and percentage variance reduction in the insample and out-of-sample compared to the non hedged spot position. The insample period covers the first 1,400 observations between January 1996 and August 2001, the outsample period covers the last 299 observations between August 2001 and November 2002. Panel B gives the daily portfolio return conditional variance (average value over the test period) and percentage reduction of this average conditional variance compared to the non hedged spot position. The conditional variance tests assume that the true return and variance processes are generated by one of the GARCH models under consideration.