Elaboración del manual de mantenimiento del módulo

We apply two tools in our research. The first is EViews – an econometric software that will be applied in order to derive the (OLS, BEKK & DCC) model parameters based on the in-sample data. When estimating a (bivariate BEKK) GARCH model, the software provides for the option to choose between a multivariate normal and a multivariate Student’s t-distribution of the error terms. We are therefore not confined by the assumption of joint normality.

Manual contract rolling

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The way that EViews –or any other econometric software – estimates the parameters of the GARCH models is by deploying numerical optimization algorithms (in case of EViews the Marquardt or Berndt-Hall-Hall-Hausman algorithm) to maximize some likelihood function. More specifically, the likelihood function takes as input the functions given in (10b) and (6b) & (6d) for BEKK and DCC, respectively as well as the in-sample observations. The algorithm then fits the model by virtually attempting to maximize the likelihood of the data observed in the sample by numerically optimizing the likelihood function with respect to the model parameters.

Figure 3.1 – Matif Corn and rollover cascading.

As has been explained in Section 2.2.9 conditional normality is often assumed to make those optimization calculations traceable. However, it will be established in Section 4.2 that the assumption of normality is significantly rejected for all series – at least for the unconditional case. Still, as is also shown in Section 2.2.9, even under violation of the assumption of conditional normality, optimization of the Gaussian log-likelihood function still yields quasi-maximum likelihood estimates (QMLE), which are consistent and asymptotically normal when the GARCH mean and variance functions are correctly specified.

It is found in the analysis that enforcing the assumption of a conditional t-distribution in some cases results in unstable parameters estimates – especially in the small sample context. For example, it will be shown in Section 4.2 that all return distributions are stationary. Yet, in some cases, the sum of the estimated ARCH and GARCH parameters is found to be bigger than unity, in which case the assumption of a mean-reverting volatility process would not hold.5

Moreover, in cases where estimation under the t-distribution does result in consistent estimates, they are found not to differ substantially from the QMLEs. Consequently, unless stated otherwise, all estimates will be based on conditional normality of the errors.

5 _{A mean reverting volatility process is a process, where, over time, the volatility tends to its average level and does not}

follow any trend. It is thus a direct implication of a stationary process, where the moment statistics do not vary across time. 131.25 151.25 171.25 191.25 211.25 231.25 251.25

Corn Matif EMAc1 in EUR per MT Corn Matif EMAc2 in EUR per MT

Corn Matif EMAc3 in EUR per MT Corn Matif EMAc4 in EUR per MT

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To ensure quasi-maximum likelihood estimation, we check the option ‘Heteroskedasticity consistent covariance (Bollerslev-Wooldridge)’ for the coefficient covariance.

Custom Tool

The second tool is the custom application that we have programmed in VBA Excel. For a picture of the interface assembled from screenshots, please refer to Appendix E. The program can best be described as a time series manager, which lets you query price information from databases, construct new time series from those information and plot them on a graph. We have set up structured databases for the internal replacement values and external sources including but not limited to the already mentioned Oil World and Reuters, the latter of which is also our supplier of real time FX and (commodity) exchange market price quotes.

While there are more features and functionalities, the program can roughly be split into three parts. In the first part, you choose your base products – i.e., the basic price series of any internal or external product, including FX series. You can further edit a given selection by changing its maturity, or converting its unit or currency.

Once a pool of base products has been selected, one may create linear combinations (weighted subtractions, additions, divisions, or multiplications) of those base series to create new series. Somewhat hidden in the interface, the application also lets you create and plot a volatility series from the returns of the base and newly constructed series based on variable rolling window input. Those new series may further be combined with the base series of part one to create correlation series in part three. Any combination is possible – this is true for the spread as well as for the correlation part. Part three moreover lets you specify the time interval that ought to be applied in order to calculate the returns as well as the rolling window size based on which the correlations are calculated per point in time. It is important to stress that while the graphs only plot the level prices of the series of part one and two, the volatility and correlation computations are based on the returns

of those series. Research Execution

The two models thus perfectly complement each other. While EViews provides the parameter estimates (i.e., the regression coefficient β in (4), the BEKK coefficient matrices A, B, and C in (10a), and the univariate GARCH estimates of the DCC model in (11b)) for the different models, the custom application takes those parameters as input to calculate the return series of the hedged portfolio. In case of the OLS approach, this series can be calculated right away using the built-in spread calculation feature. In the context of the time-varying models, we first of all have to combine the model parameters with the observations of the base series in order to calculate a set of series containing the second moment (i.e., (co-) variance) forecasts per point in time and per series so as to consequently construct the series of time-varying optimal hedge ratios.

In the case where the hedged portfolio includes non EUR - denoted prices, the built-in FX converter of the custom model is applied to translate those prices back to EUR. There are two possibilities to approach the FX problem in the hedging context. We could either a) conduct all the OHR calculations on the basis of the series’ base (i.e., foreign) currencies to receive some value ℎ𝑡′ and then translate

the return of the hedged portfolio back to the domestic currency, or we could b) first convert the return series to the domestic currency and then base the OHR calculation on the translated series to generate some value ℎ_𝑡′′ _{for the OHR.}

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Since the (co-) variance structure will be slightly altered after currency translation we have that, in general, ℎ_𝑡′ _{≠ ℎ}

𝑡′′. However, if we translate the currency ex post then the FX component causes

variability in hedge portfolio return, which is not accounted for in the models. Since variance minimization is what we want to achieve in the first place, it makes more sense to choose for alternative b) – and thus optimize the portfolio on the basis of the translated series, which is indeed what is also done in our thesis.

Note, that the Oil World wheat commodity spot exposure is not translated to EUR, as the price quotes refer to the US cash market. The series is rather used in order to compare the hedging effectiveness in different countries.

Since the application conveniently prints any time series (thus also the hedge portfolio return series) we can then carefully analyze the statistics of the new portfolio and judge both the conditional as well as the unconditional performance of the different hedging strategies.

EWMA and GARCH Model Initialization

Since all the time-varying methods work recursively, we somehow have to manually initialize the starting value –after all, there is no t-1 at t=0. If we have not yet collected any price data, then there is no past information on which to apply the econometric models. To make sure that the initialization phase is not characterized by excessively volatile OHR estimates, the moment estimates for t=0 are simply opted to be the sample moments.