# Stock Market Crisis Forecasting Using Neural Networks with Input Factor Selection
## 2 Determination of Financial Crises  
Most current neural network approaches to financial prediction problems aim at predicting price movements, either from historical prices, or based on a set of variables (see [4]). However, future market prices are difficult to predict from the history. Moreover, price predictions are not equal to investment decisions, indicating that translating price predictions into investment solutions requires an additional manual layer (see [9] for more details). On the contrary, an accurate prediction of economic states based on the financial time series not only serves as a direct signal for the market participants, but can also be easily transmitted to investment strategies. We follow a numerical and heuristic procedure proposed in [20] to determine economic crisis states based on a stock index. This method is also used in [21]. The authors of [22] apply a different method and state that it leads to similar results as the method from [20]. The parameters which we use as thresholds are also taken from [20]. We found that the method is robust with respect to changes in the parameters of plus or minus 2%. First, we divide the time series of all daily observations of the stock index (\(B_{1},\ldots,B_{T}\)) in blocks separated by days on which the respective stock index reaches a half-year high. A half-year is defined to be 120 days or 26 weeks. This set of all days reaching half-year highs, i.e., 26 _week highs_,\[26 wh:=\{t\in\{120,\ldots,T\}:B_{t}=max\{B_{t-119},\ldots,B_{t}\}\}. \tag{1}\]  
Second, we screen each period between two elements of \(26 wh\), i.e., the interval \([26 wh_{i},26 wh_{i+1}]\), for a possible crisis. A crisis day between two _26-week-lights_ represents a minimum loss of 20% relative to the stock index at the beginning of this period. The set of _core crisis days_ in the interval \([26 wh_{i},26 wh_{i+1}]\) is correspondingly defined as:  
\[CCD_{i}=\{t\in\{26 wh_{i},\ldots,26 wh_{i+1}\}:B_{t}/B_{26 wh_{i}}\leq 0.8\}. \tag{2}\]  
There are two possible calculation results: \(CCD_{i}=\emptyset\) or \(CCD_{i}\
eq\emptyset\):  
1. If \(CCD_{i}=\emptyset\), we move on to the next interval \([26 wh_{i+1},26 wh_{i+2}]\) and repeat (2) to obtain the next possible set of _core crisis days_\(CCD_{i+1}\). 2. If \(CCD_{i}\
eq\emptyset\), we say that there is a crisis in this interval. Note that the existence of _core crisis days_ by itself does not define the crisis period. To determine the crisis period in this interval, we further define the starting and end days. We set the starting day of the crisis as the last day on which the 10% loss level relative to the stock index at \(26 wh_{i}\) is reached, i.e.,  
\[t_{Start}=1+max\;\{t\in\{26 wh_{i},\ldots,CCD_{i}[1]\}:B_{t}/B_{26 wh_{i}}>0.9\}, \tag{3}\]  
where \(CCD_{i}[1]\) refers to the first day in the \(i\)-th set of _core crisis dates_\(CCD_{i}\). For the end day of a crisis, we first denote \(t_{Low}\) as the day of the lowest index value before the next 26 week high is reached, i.e.,  
\[t_{Low}=min\;\{t\in\{26 wh_{i},\ldots,26 wh_{i+1}\}:B_{t}=min\;\{B_{26 wh_{i}}, \ldots,B_{26 wh_{i+1}}\}\}. \tag{4}\]  
In normal cases, the end day of the crisis is \(t_{Low}\), i.e., \(t_{End}=t_{Low}\). However, if a stock market upswing after the lowest index value is followed by a new downturn, we extend the crisis period if the new downturn is at least 10% below the upswing's highest value. The crisis is extended to the day of the lowest index value of the new downturn \(t_{Down}\). Two days are required for the determination of \(t_{Down}\), namely the latest possible starting day of the new 10% downturn, \(t_{DownStart}\), and consequently the latest possible end day of this downturn, \(t_{DownEnd}\).