# Stock Market Crisis Forecasting Using Neural Networks with Input Factor Selection
## 5 Application of the Model Selection Procedure
Based on our criterion to choose the largest gap provided that at least six factors are included in the model, we choose \(n^{*}_{LG}=9\), i.e., the last variable which is included in the final model is the change in the federal government debt (GFDEBTN Change). Thus, our final model consists of nine input factors. Performing a principal component analysis, we discover that the first nine principal components explain 89.7% of the total variance in the data set. This gives us an indication that choosing nine variables enables us to capture the main characteristics of the data set well. The chosen input factors reflect various aspects of the financial market and the economy. The volatility (Volatility), the exchange rate (EURODOLLAR) and the federal funds rate (FEDFUNDS) capture different areas of the financial market, namely the stock market, international influences and the central bank policy. The product manufacturing index (PMI) reflects the current situation of companies, while the consumer sentiment index (UMSCENT), the personal savings rate (PSAVERT) and the housing starts (HOUST) incorporate the situation of the households. The unemployment rate (UNRATE) can be interpreted as a factor, in which information on consumers and companies is included. Finally, the change in the government debt (GFDEBTN Change) reflects governmental action. For Model 2, we observe that the first two gaps are given by 2.12 and 1.28 (see Table 3). While the largest gap occurs after the first variable, we decide to cut the factors in this model after the second variable which is added (UMSCENT) for two reasons: first, such a large gap does not occur later anymore, second, the model has eight input factors in total, which makes it comparable to the model selected from Table 2, which has nine input factors. Later, we use Model 2 primarily as a benchmark for Model 1. This means, we add two input factors in Model 2--the unemployment rate (UNRATE) and the consumer sentiment index (UMSCENT). While the model from [15] consists largely of financial market indicators, these two additional factors add information on consumer sentiment (UMSCENT) and the unemployment rate which includes information on consumers and companies. In the following, we assess the quality of the out-of-sample predictions for these models. Model 1 represents our suggested input factor selection process. Model 2 serves as a benchmark as it represents a model which is largely chosen by another input factor selection process and is only extended by our approach. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Variable** & **Average Ranking** & **Gap to Next Factor** \\ \hline Volatility & 1.1 & 4 \\ PMI & 4.4 & 1.05 \\ EURODOLLAR & 4.6 & 1 \\ UMCSENT & 4.6 & 1.3 \\ UNRATE & 5.8 & 1.31 \\ FEDFUNDS & 7.6 & 1.07 \\ PSAVERT & 8.1 & 1.05 \\ HOUST & 8.5 & 1.01 \\ GFDEBTN Change & 8.6 & 1.3 \\ BAAMinusAAA & 11.0 & 1.08 \\ M3 Change & 11.9 & 1.06 \\ GSI0 MinusFEDFUNDS & 12.6 & 1.09 \\ TCU & 13.7 & 1.03 \\ PPIACO Change & 14.1 & 1.04 \\ TOTALSL Change & 14.7 & 1.05 \\ WTISPLC Change & 15.5 & 1.02 \\ PAYEMS Change & 15.8 & 1.13 \\ CPIAUCSL Change & 17.8 & 1 \\ W823 RCI Change & 17.8 & 1.01 \\ GPD Change & 18.0 & 1.08 \\ TRESEGUSM025N Change & 19.4 & 1.04 \\ INDPRO Change & 20.2 & 1 \\ OECD CLI & 20.2 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Selection with initial \(S=\varnothing\) (Model 1). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Variable** & **Average Ranking** & **Gap to Next Factor** \\ \hline BAAMinusAAA & - & - \\ FEDFUNDS & - & - \\ GSS0 MinusFEDFUNDS & - & - \\ GS10 MinusFEDFUNDS & - & - \\ OECD CLI & - & - \\ Voidtility & - & - \\ UNRATE & 1.7 & 2.12 \\ UMCSENT & 3.6 & 1.28 \\ EURODOLLAR & 4.6 & 1.04 \\ PSAVERT & 4.8 & 1.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input factors from [15] in \(S\) initially (Model 2).