Residuals Diagnosis

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	TUTORIAL-_RESIDUAL_Diagnosis.xls

We build models to help us understand underlying unknown processes, and hopefully improve our forecast accuracy. But how do we know if the model is correct or adequate. To make things even muddier, which model should we choose over several alternatives.

Early on in this tutorial, we looked at the Log-Likelihood Function (LLFⁱ) and Akaike's Information Criterion (AICⁱ) as two measures of goodness of fit. Next, we'll look at the residuals generated by each model, run the statistical tests introduced in Chapter 2, and verify each model's underlying assumptions.

Example 1: GARCHⁱ(1,1) with Normal 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, we describe the model as:

Goodness of fit

The GARCH model describes the standardized residuals ( $\{z_t\}$ ) as:

Normally distributed
Mean of zero and standard deviation of one
Statistically independent

GARCH with Normal Innovations Residual Diagnosis

Note that the GARCH model passed all tests with the exception of the Excess-Kurtosis test, which in turn caused it to also fail the Normality test.

The average of standardized residuals ( $\{z_t\}$ ) is -2.95%. This may seem high in comparison with the model's mean ( $\mu$ ), but remember that $z_t=\frac{\varepsilon_t}{\sigma_t}$ , where $\varepsilon_t$ is the actual model's residuals.

Example 2: GARCH(1,1) with GEDⁱ 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

GARCH Model with GED Distribution

By introducing the GED innovations, the new model's assumptions about the standardized residuals are:

Mean of zero and standard deviation of one
Symmetrically distributed (i.e. Skew = 0)
Independent and identically distributed (e.g. White-Noise and no ARCH Effect)

Residuals diagnosis for GARCH Model with GED Distribution #2

Summary

The GED innovations not only improve the Log-Likelihood Function for the GARCH model, they also allow us to fulfill the underlying assumptions. As a result, GARCH with GED is adequate. Is it the best model? This can only be answered in comparison with other models, but we now have the tools to help us decide.