Forecasting

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	TUTORIAL-_OUT-OF-SAMPLE_FORECAST.xls

By now, we have learned how to prepare a data set, choose a model, calibrate its coefficients, diagnose its residuals, and compare it to other candidate models. Now we are ready to make a reliable forecast for the conditional mean, conditional volatility and confidence intervals using these models.

In-Sample vs. Out-of-Sample

When we calibrate a model, we use a given data set; so if we are to look at a model fitted value within this sample, it will referred to as an in-sample forecast.

In econometric models, the fitted value is equivalent to the in-sample one-step forecast. Aside from the model's coefficients, we used all of the information in the data set up to the point at which we compute the value for.

$r_t=F(\theta_1,\theta_2,...,\theta_N,r_0,r_1,...,r_{t-1})$

$\sigma_t=G(\beta_1,\beta_2,...,\beta_N,r_0,r_1,...,r_{t-1},\sigma_0,\sigma_1,...,\sigma_{t-1})$

Therefore, it is insightful to look at a model's fitted value to understand its forecast (lead/lag) and inherent error.

Example 1: GARCHⁱ(1,1) with GEDⁱ

The GARCH model assumes a constant conditional mean, so we will only look at the conditional volatility. NumXL has many functions that return a model's fitted values (mean or volatility) for all supported models.

GARCH in-sample volatility plot

The model's fitted volatility is consistently lower than the values calculated from EWV, and they are approximately moving in sync.

Out-of-Sample

For out of sample, we use a data set that was not used for the model calibration process. For illustration, we'll use the MSFT price sample from May 4th, 2009 to July 10th, 2009.

GARCH out-of-sample volatility forecast plot
In the graph above, we computed the out-of-sample one-step forecast for the GARCH model and plotted it with WMAⁱ and EWV.

Multi-Step Forecast

In a multi-step forecast, the forecasts from prior steps are used to compute the current forecast. Although this is mathematically correct, the confidence interval rapidly increases in width until it is no longer different from marginal distribution.

Note: If a multi-step forecast or long term forecast is required, we recommend that you resample the original data set: weekly from daily, monthly from weekly/daily, etc. CalendarXL makes this step a snap. See our documentation online.

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

The GARCH model assumes a constant (time-invariant) conditional mean, so the multi-step forecast is the same. On the other hand, the conditional volatility is time-varying, and the GARCH process attempts to revert the current value to its long-term mean.

Using GARCH_FORESD, we compute the multi-step forecast for conditional volatility. We'll start the forecast on May 4th, 2009.

GARCH out-of-sample multi-step volatility forecast

Please note that the curve is not as smooth as one may expect. This is primarily due to the use of actual dates (which span weekends) rather than step numbers. Dates are generated using DTWorkDay in CalendarXL, with the NASDAQ calendar (MSFT listed exchange) providing the raw data.