# Machine Learning for Quantitative Finance Applications: A Survey
## 4 Autoregressive Models  
A considerable amount of literature has examined the most promising techniques for time series forecasting. The outcomes have highlighted that these approaches could be subdivided into two separate macro-categories, which include statistical and machine-learning methods. The ARIMA model represents a generalization of the auto-regressive moving average (ARMA) model, suitable to describe non-stationary time-series. More specifically, the main benefit in using the ARIMA model is to transform a non-stationary series into a series without seasonality or trend, by applying finite differencing of data points [4]. By definition, a time series is stationary if its statistical properties are all constant over time. If a stationary series has no trend, its variations around its mean have a constant amplitude. Furthermore, the time-series autocorrelations remain constant over time. Based on these assumptions, a time series of this form can be considered as a combination of signal and noise. An ARIMA model manages the signal by separating it from the noise. After removing noise from the signal, the output of ARIMA model is the signal step-ahead to obtain forecasts. In order to select an appropriate model to improve time-series forecasting, the Box-Jenkins model [4] has been developed to determine best ARIMA model to fit to a given time-series. The Box-Jenkins model has gained momentum in forecasting time-series not only in financial sector but also to improve prediction of future values of a time-series considering electricity prices or traffic data observations [5; 6]. In order to demonstrate the effectiveness of the ARIMA model, we surveyed a set of works where the ARIMA process was applied to perform stock price forecasting. In [7] the authors investigated the problem to produce accurate prediction of stock market time series data by using ARIMA and the GARCH model. However, they indicate that a major class of works in literature used these classical approaches for solving a one-step ahead forecasting problems. In the case of a multi-step or N-step ahead prediction, the aforementioned model does not perform accurately. In particular, authors have been concerned about the decrease of accuracy and lack of maintenance of data trend or dynamics. In order to overcome these limitations, a linear hybrid model combining ARIMA and GARCH was proposed. First, they used a simple moving average filter in order to decompose the stock-market time series into two different series. One of these data streams is modeled by using ARIMA and GARCH model respectively. Finally, the outcomes from both models are then combined to perform the values predictions. Results have shown that the proposed hybrid model is suitable for multi-step ahead forecasting of high-volatile time-series. In [8] the authors have applied a statistical model based on ARIMA to the New York Stock Exchange (NYSE) and Nigeria Stock Exchange (NSE) for predicting future daily stock prices. The experimental results showed that the model has reached satisfactory results for both stock indexes. Despite preserving the data trend and obtaining good predictions in terms of accuracy, the ARIMA model has definite limitations when dealing with financial time series data which present asymmetries, irregular time intervals and high/low volatility. In fact, the main assumption of the ARIMA model regards constant variance. Considering this fact, an integrated model represents a suitable solution when dealing to data which reveal a non-complex structure. Since we have shed light to the limitation of statistical models for the analyzed time series, the next step consists in reviewing models of the machine-learning family which achieve effective results for financial data prediction. Table 1 summarizes surveyed studies in this section.