Model Calibration

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	TUTORIAL-_MODEL_CALIBRATION.xls

Earlier, we discussed the process for selecting a tentative econometric model for the data set and defined measures for the goodness of fit. Next, we would like to find the best model parameter set for a given model.

The idea behind model calibration is relatively straight-forward; simply find a set of parameters that maximize our utility function (i.e. Log-Likelihood Function). The calibration process is in essence an optimization problem.

Excel ships with an optimization function as part of its Solver Add-in. Solver can be found under the Tools menu in Excel 2003, or in the Data tab for Excel 2007. We will use this engine for our tutorial. Please make sure it is installed and enabled.

Example 1: GARCHⁱ(1,1) with Normally Distributed Innovations

$r_t=\mu+\varepsilon_t$

$\varepsilon_t=\sigma_t\times z_t$ , $z_t\sim i.i.dN(0,1)$

$\sigma_t^2=\alpha_0+\alpha_1\times\varepsilon_{t-1}^2+\beta_1\times\sigma_{t-1}^2$

Where

$\mu$ : Conditional mean

$z_t$ : Innovations or shocks (i.i.d)

$\varepsilon_t$ : Residuals

$\sigma_t$ : Conditional volatility

In Excel, the model is defined as:

Goodness of fit

Using the Solver Add-in, we will run the optimizer to maximize the LLFⁱ value by varying the model parameters. Furthermore, we add the CHECK value as a constraint to ensure we have a stable model.

Calibration GARCH Model with Excel Solver

We select LLF in the "Set Target Cell", "Max" to indicate the type of optimization, model parameters as "Changing Cells", and CHECK to equal one in the "Subject to the Constraints" list.

Now we are ready to run the optimizer. It will start from our initial guess of parameters, look for a set of new values that improve the utility function (i.e. LLF), and finish when no further improvement can be achieved. In this case, the optimizer reports as "converged to the current solution".

Calibration GARCH Model with Excel Solver #2

Note that the marginal improvement in LLF is 5, while AICⁱ decreased further. In the next chapter, we look for other ways to examine a model by its residuals.

Example 2: GARCH(1,1) with GED Innovations

$r_t=\mu+\varepsilon_t$

$\varepsilon_t=\sigma_t\times z_t$ , $z_t\sim i.i.dGED(0,1,\nu)$

$\sigma_t^2=\alpha_0+\alpha_1\times\varepsilon_{t-1}^2+\beta_1\times\sigma_{t-1}^2$

Where

$\mu$ : Conditional mean

$z_t$ : Innovations or shocks (i.i.d)

$\varepsilon_t$ : Residuals

$\sigma_t$ : Conditional volatility

$\nu$ : Shape factor

In Excel, the new model is defined as:

GARCH with GED

Using the Solver Add-in, we follow the same process as earlier with one exception: we add the shape factor as part of "Changing Cells", allowing the optimizer to improve.

Calibration GARCH Model and GED Innovations with Excel Solver

Run the optimizer again.

Calibration GARCH Model with GED Innovations with Excel Solver #2

The improvement on LLF is marginal, but the result is higher than one with normally distributed innovation.

Note that we did not have to worry about setting the shape factors as part of the constraints (i.e. $\nu \geq 1$ ). GARCH_CHECK takes care of this for us.

Example 3: GARCH(1,1) with Student t-Distributed Innovations

$r_t=\mu+\varepsilon_t$

$\varepsilon_t=\sigma_t\times z_t$ , $z_t\sim i.i.d\sim t(0,1,\nu)$

$\sigma_t^2=\alpha_0+\alpha_1\times\varepsilon_{t-1}^2+\beta_1\times\sigma_{t-1}^2$

Where

$\mu$ : Conditional mean

$z_t$ : Innovations or shocks (i.i.d)

$\varepsilon_t$ : Residuals

$\sigma_t$ : Conditional volatility

$\nu$ : Shape factor (degrees of freedom)

In Excel, the new model is expressed as:

GARCH with Student-t

Using the Solver Add-in, we input values similar to the prior example.

Calibration GARCH Model and Student-t Innovations with Excel Solver

Run the optimizer.

Calibration GARCH Model and Student-t Innovations with Excel Solver #2

The improvement on the LLF value is marginal, hovering around the values generated with our prior example.

Summary

In this chapter, we learned how to invoke Excel's Solver Add-in and populate its dialog with our model parameters. In the examples we used, we looked primarily at the Log-Likelihood Function as a measure of goodness of fit, and we saw that GARCH with GED (or t-dist) performs better than one with normally distributed innovations.

In the next chapter, we'll take our model analysis one step further, examining standardized residuals for any bias evidence.