The goal of this project is to develop models to forecast stock prices.
The data comes from the Nasdaq API. Specifically, we'll use equities data from the Frankfurt Stock Exhange (FSE).
- We first cast the entire data for FSE in JSON format. Then, we used a subset of that data from 2010 to 2017 as the training data, and stored that time series in a pandas dataframe. We did not want to use data before 2010 to avoid the financial crisis of 2008.
- Initially, we intended to use data after 2017 as the test data. However, we found that the data for FSE after 2017 contained a large swatch of missing values. We could have imputed those missing values, but that didn't seem like a good idea because so many of them were missing. So, we chose to evaluate our model by an in-sample forecast instead of out-of-sample forecast with the test data.
- We studied the statistical properties of the stock returns, which we derived from the stock prices by: (1) resampling the stock prices to weekly frequency, and (2) computing the percentage change of the resampled data.
- We found that the autocorrelation of the weekly return is negative (specifically -0.14), so the series is mean reversing.
- We plotted the ACF of the return to test the statistical significance of the value -0.14. We found that it is indeed significant (with 95 % confidence interval).
- Furthermore, the ACF plot reveals that - among the first 10 lags - only the lag-1 autocorrelation is statistically significant.
- The PACF plot of the returns is quite similar to the ACF plot (only the lag-1 partial autocorrelation is statistically significant), and it is negative.
- We used the augmented Dickey-Fuller test to see if the stock prices is a random walk, and we rejected the null hypothesis that it is a random walk (the p-value was 0.023, which is smaller than 0.05).
- We also used the KPSS test to see if the returns are a stationary time series, and found that we can accept the null hypothesis that it is (the p-value was 0.1, which is larger than 0.05).
- We used 5-fold cross-validation with TimeSeriesSplit (which ensures that the split into training set and validation set respects the time ordering).
We tried 3 models: the ARIMA model, the Exponential Smoothing model, and the Facebook Prophet. For the performance metric, we chose the MAPE (Mean Absolute Percentage Error). We summarize our findings below:
- Regarding the ARIMA model, we set the parameter d (in (p,d,q)) to 1, because differentiating the stock prices once yields a stationary time series. We did a hyperparameter search over the p and q parameters, and found that the choice p = 2 and q = 2 yields the smallest (cross-validated) MAPE score, which is approximately 0.15.
- Regarding the Exponential Smoothing model, we set the 'trend' parameter to be 'multiplicative', because the upward trend of the stock price doesn't look like a straight line. The MAPE score was found to be around 0.14.
- The Facebook Prophet model yielded the MAPE score of around 0.10.
We conclude that the Facebook Prophet model is the best model.