3. Data

2.2.2. Variance-covariance (VACO) combination method

84 Advanced Applications for Artificial Neural Networks

Then, the combined forecast is given by <sup>b</sup>y<sup>c</sup>

all models. This guarantees that <sup>P</sup><sup>m</sup>

the weights can be calculated as

The method uses the historical achievement of the individual forecasts to compute the weights.

yj � <sup>b</sup><sup>y</sup> i j � �<sup>2</sup> " #�<sup>1</sup>

> yj � <sup>b</sup><sup>y</sup> i j

� �<sup>2</sup> " #�<sup>1</sup> (11)

<sup>t</sup> where yj is the jth actual value, <sup>b</sup>y<sup>i</sup>

� �<sup>2</sup> " #�<sup>1</sup> (12)

<sup>t</sup> of combined forecasts can be given by

<sup>t</sup> (13)

<sup>j</sup> is the jth

P T j¼1

> P T j¼1

<sup>t</sup> <sup>¼</sup> <sup>P</sup><sup>m</sup> i¼1 wiby<sup>i</sup>

forecasting value from ith individual forecasting model, and T is the total number of out-ofsample points. The weight in Eq. (11) is based on the inverse sum of squared deviation for model i as the numerator, and the denominator is the sum of these inverse contributions from

The DMSFE method weights recent forecasts more heavily than distant ones. [32] suggest that

<sup>δ</sup><sup>T</sup>�jþ<sup>1</sup> yj � <sup>b</sup><sup>y</sup>

� �<sup>2</sup> " #�<sup>1</sup>

<sup>δ</sup><sup>T</sup>�jþ<sup>1</sup> yj � <sup>y</sup>

where δ is the discount factor with 0 < δ ≤ 1, if δ ¼ 1 and then the DMSFE and VACO methods become one method, which means that the VACO is a special case of the DMSFE. Note that as

Linearity of combinations of the individual forecasts is the corner stone of linear combination method, but if the individual forecasts are based on nonlinear methods, the combinations are defined to be insufficient or if the true relationship is nonlinear. For the success of the ANN as a combination method over the linear methods, among others, see [15, 25]. Here, we use the

bc

<sup>α</sup>i, kNk,t <sup>þ</sup>X<sup>m</sup>

i¼1 βi byi

i j

> bi j

Pm i¼1

Thus, according to the VACO method, the weights determined as follows:

wi ¼

i¼1

2.2.3. Discounted mean square forecast error (DMSFE) combination method

wi ¼

mentioned above the sum of all weights is equal to one.

2.2.4. Artificial neural network (ANN) combination method

same setup used in subsection (2.1.2); the output y

where <sup>b</sup>y<sup>i</sup>

byc

<sup>t</sup> is the forecast from ith individual forecasting model.

<sup>t</sup> <sup>¼</sup> <sup>α</sup>i, <sup>0</sup> <sup>þ</sup><sup>X</sup>

K

k¼1

wi ¼ 1.

P T j¼1

> P T j¼1

Pm i¼1 For FAANN and DFM models, data are gathered that include 228 monthly time series<sup>2</sup> of which 203 are collected from South Africa, including the financial, real, nominal sectors, and confidence indices, 2 global variables, and 23 series of major trading partners and global financial markets. The AR criterion model will be used for the data which composed only the variable of interest, namely, deposit rate or share prices for gold mining or long-term interest rate. Thus, besides the national variables, the chapter uses a set of global variables such as gold and crude oil prices. Also, the data incorporate series from financial markets of major trading partners, namely, the United Kingdom, the United States, China, and Japan. For estimation data cover the period January 1992 through December 2006, while the period from January 2007 through December 2011 will be used for goodness of fit for the extracted model. For the degree of integration of all series, the augmented Dickey-Fuller (ADF) test will be used. Difference of the series is used for all nonstationary series in this study. The Schwarz information criterion (SIC) is used in selecting the appropriate lag length in such a way that no serial correlation is left in the stochastic error term. Finally, all series are standardized to have a mean of zero and a constant variance.
