Time Series Modeling

	Attachment	Size
	TUTORIAL-_INNOVATIONS_MODEL.xls

Let's summarize what we have learned about the time series:

Small or insignificant mean
- Sample Mean Test
Symmetry of distribution
- Skewness Test
Heavy tails on both sides of the distribution
- Excess-Kurtosis Test
- Normality Test
Slow moving conditional mean
- White-Noise Test
- WMAⁱ
Time-varying conditional volatility
- ARCHⁱ Effect Test
- EWVⁱ
Significant but small serial correlation
- ACFⁱ & ACF Correlogram
- PACFⁱ & PACF Correlogram

Given the information we gathered so far, we are tempted to propose a GARCHⁱ, GARCH-Mⁱ or EGARCHⁱ model for this series.

Goodness of Fit Measure

To evaluate the models consistently, we need to define two important measures for goodness of fit: the Log-Likelihood Function (LLFⁱ) and the Akaike-Information Criterion (AICⁱ).

Log-Likelihood Function (LLF)

The Likelihood Function is defined as the probability of generating an observed data sample for a given model. In other words, assuming we have the
$\left \{ x_0,x_1,...,x_{T-1},x_T \right. \}$
data set and a hypothetical model that defines the conditional probability density function in terms of $x_t$ as:

$f(x_t\mid \left \{ x_0,x_1,...,x_{t-1} \right. \},\left \{ \theta_1,\theta_2,...,\theta_N \right. \} )$

Where $\{ \theta_1,\theta_2,...,\theta_N \}$ are the model parameters, then the Likelihood Function is defined as:

$L^*=\prod_{t=0}^{T}f(x_t\mid \{x_0,x_1,...,x_{t-1}\},\{\theta_1,\theta_2,...,\theta_N\})$

Using a logarithmic transform, the Log-Likelihood Function is expressed as:

$ln(L^*)=\sum_{t=0}^{T}ln(f(x_t\mid \{x_0,x_1,...,x_{t-1}\},\{\theta_1,\theta_2,...,\theta_N\}))$

Please note that we are using the probability density function value as if it were equivalent to the probability of the occurrence of the value. This is not perfectly correct; the probability is defined as the area under the density function. But if we replace each individual observation with a small interval (e.g. $x_t\in \left [ x_t- \Delta/2<br /> ,x_t +\Delta/2 \right ]$ ), then we can simply multiply the Likelihood Function with a constant (or add a constant to the Log-Likelihood Function). Either way, the Likelihood Function can still work as a relative measure of goodness of fit.

Akaike Information Criterion (AIC)

As with any other modeling technique, the higher the model order, the greater the fit is. This is exactly what the AIC is defined to address; it penalizes higher order models.

In general, the AIC is defined as the Log-Likelihood Function minus a penalty for the model order. For instance, the AIC formula for the GARCH(p,q) model is defined as:

$AIC=2\times (p+q+1)-ln(L*)$

Note that the AIC has the opposite sign of the Log-Likelihood Function.

Example 1: GARCH Model

Let's propose a GARCH model for MSFT log daily returns with normally distributed innovations or shocks.

$r_t=\mu+\varepsilon_t$

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

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

Where

$\mu$ : Conditional mean

$z_t$ : Innovations or shocks (i.i.d, independent and identical distributed random variables)

$\varepsilon_t$ : Residuals

$\sigma_t$ : Conditional volatility

In Excel, we describe the model as:

Goodness of fit

The initial choice of a parameter model is important. We suggest using the values you computed during the earlier analysis, making sure that the GARCH_CHECK function returns a value of one ("1").

Innovations Distribution

In our analysis of financial time series, we run into situations where GARCH/EGARCH can't capture the entire excess kurtosis in a series, so we then look for an innovation process with inherent leptokurtic distribution.

NumXL supports two commonly used processes: Student t-Distribution and Generalized Error Distribution (GEDⁱ). Both have one parameter, the shape factor ( $\nu$ ), that define their final distributions.

Please note that the shape factor for the Student t-Distribution must be greater than four, while for the GED it must be greater than one.

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

$r_t=\mu+\varepsilon_t$

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

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

Where

$\mu$ : Conditional mean

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

$\varepsilon_t$ : Residuals

$\nu$ : Shape factor

$\sigma_t$ : Conditional volatility

In Excel, the new model is expressed as:

GARCH with GED

General notes about the GED:

The maximum excess kurtosis for a GED distribution is 3.0 ( $\nu=1$ )
For $\nu=2$ , the GED is normally distributed
As $\nu\gg 2$ , the GED approaches uniform distribution and its excess kurtosis is negative

What does this mean to us?

The GED is used to capture small-to-moderate excess kurtosis. For other cases, consider the Student t-Distribution.
Using the GED does not lock us into leptokurtic innovation. As we optimize the model, the shape factor can yield a normally distributed innovation.

Example 3: GARCH(1,1) with Student t-Distribution Innovation

$r_t=\mu+\varepsilon_t$

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

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

Where

$\mu$ : Conditional mean

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

$\varepsilon_t$ : Residuals

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

$\sigma_t$ : Conditional volatility

In Excel, the new model is expressed as:

GARCH with GED

Please note that these parameters are not optimal. We should think of them as our initial guess.

Using the NumXL Toolbar

You can use the NumXL toolbar to construct a model in a few steps. For instance, to construct an EGARCH(1,1) model with Student t-Distribution innovations, do the following:

Using the NumXL toolbar (or Excel 2003 NumXL menu), select GARCH.
The GARCH model dialog box pops up. Fill in the location of your data, series time order, "EGARCH" for model, output options and location for the table and graphs to be generated in your worksheet.
Upon completion, the model parameters, goodness of fit calculation and residuals diagnosis analysis are generated (along with their formulas) in the selected output range.