Hello. Recently I am trying to forecast stock prices. I am still training, but I would like to output it. This time, I would like to make a prediction for the Vanguard S & P500 ETF (VOO). You can get the data at the URL below, but you need to log in to Investing.com. https://www.investing.com/etfs/vanguard-s-p-500-historical-data As a reminder, it is your responsibility to buy financial products as a result of this model.

Preparation


#Import required libraries
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import warnings
import datetime
import statsmodels.api as sm
import itertools
import matplotlib.dates as mdates

import warnings
warnings.simplefilter('ignore')

Next, read the data and prepare it.


#Data loading and indexing to_Convert with datetime
df = pd.read_csv("voo_10.csv")
df.index = pd.to_datetime(df["Date"])
df = df.sort_index()
del df["Date"]

This time, we will only look at ETF price changes in chronological order. By the way, the read data looks like this.


[out]

　　　　　    Price    Open    High    Low Vol.    Change %
Date                        
2011-01-10  116.20  115.88  116.28  115.48  124.46K -0.09%
2011-01-11  116.58  116.66  116.85  116.12  131.69K 0.33%
2011-01-12  117.62  117.32  117.72  117.10  107.31K 0.89%
2011-01-13  117.46  117.68  117.74  117.16  96.05K  -0.14%
2011-01-14  118.30  117.30  118.34  117.26  193.39K 0.72%
... ... ... ... ... ... ...
2020-12-31  343.69  341.82  344.37  341.23  3.51M   0.54%
2021-01-04  339.03  345.02  345.09  335.37  5.35M   -1.36%
2021-01-05  341.26  338.40  342.39  338.39  3.28M   0.66%
2021-01-06  343.33  339.79  346.50  339.30  4.79M   0.61%
2021-01-07  348.46  345.74  349.20  345.54  3.62M   1.49%

This Price column has a closing price. If you graph the price transition

price = pd.Series(df["Price"], dtype=float)
plt.ylabel("price, USD")
price.plot()
plt.show()

EDA The read data is the original series data, that is, the data in the observed state. Stationarity is important in the analysis of time series data, so it is necessary to confirm the presence or absence of stationarity. Stationarity means that it does not change over time. It is called a stationary process because it is assumed to have stationarity. If it is not a stationary process, it will be a completely random transition, and time series prediction will not be possible.

First, let's see how the data changes.

#Check the transition of data
fig = sm.tsa.seasonal_decompose(price, freq=80).plot()
plt.show()

The top of the image is the original series data. The second is the trend, the trend of data. It's almost rising, but sometimes it tends to go down. The third is a periodic variation, which is called seasonality. Even if it is not related to the season, it will be Seasonal if there is periodicity. Roughly speaking, prices are rising and processing periodically. Due to the large number of data, freq is also set large.

If the data is as it is, it is not a stationary process and time series analysis cannot be performed, so the difference is taken. For the time being, the floor difference is 1.


price_diff = price - price.shift()
len(price_diff)
price_diff.head()
[out]
Date
2011-01-10     NaN
2011-01-11    0.38
2011-01-12    1.04
2011-01-13   -0.16
2011-01-14    0.84
```python

 The first line is a missing value because there is no target for taking the difference.

```python

price_diff = price_diff.dropna()

Delete the lines with missing values.

The graph of the data with the difference is


plt.plot(price_diff)
plt.show()

It will be.

Since the difference is taken, the graph will repeat a certain contraction. The difference from the original series is called the difference series.

Now check the autocorrelation and partial autocorrelation of the difference series. Autocorrelation represents the correlation at two points in time. However, autocorrelation is affected between the two time points. Therefore, it is the partial autocorrelation that eliminates the influence of the period between them.

sm.graphics.tsa.plot_acf(price_diff,lags=700)
sm.graphics.tsa.plot_pacf(price_diff, lags=1095)
plt.figure(figsize=(10,8))
plt.show()

The point of time of the blue ball protruding from this blue shadow is the period when the alternative hypothesis that there is a correlation in the temporary test is adopted. The autocorrelation becomes less significant as the time points move away. Although it is a partial autocorrelation, there is a large negative correlation in a cycle of about 390 days.

Since there are some protruding parts, we will test whether it is a stationary process. We use the ADF test.

#Confirm stationarity by ADF test
check = sm.tsa.stattools.adfuller(price_diff)
print("p-value:{:.3f}".format(check[1]))

[out]
p-value:0.00000

This ADF test tests for the presence or absence of unit root processes. When the difference series calculated from the data of the non-stationary process has stationarity, it is called a unit root process. Because it is a hypothesis test Null hypothesis (hypothesis you want to deny) = not a unit root process Alternative hypothesis (hypothesis you want to affirm) = unit root process I choose either of them, but the standard is the p-value. By convention, if the p-value is less than 5%, the alternative hypothesis is adopted and the null hypothesis is denied. At this time, 5% is based on the significance level, and the severity of the test can be adjusted by the magnitude of the significance level.

Since the p-value is less than 5%, we conclude that it is a unit root process of the alternative hypothesis, that is, stationary.

Forecast

Let's move on to forecasting. Before moving on to forecasting, prepare training data and test data.

#Divide the data for training and testing
train = price_diff["2011-01-11":"2019-12-31"]
test = price_diff["2018-01-01":"2020-01-07"]

plt.plot(train)
plt.plot(test,color="r")
plt.legend(["Train", "Test"])

plt.show()

The divided ETF data looks like this. For verification purposes, both data have overlapping periods. Let's see the accuracy by the degree of overlap of this overlapping period.

Next, we will estimate the optimum parameters for tuning the model.

#A function that automatically calculates the optimum parameters of Sarimax
def selectparameter(DATA,s):
    p = d = q = range(0, 2)
    pdq = list(itertools.product(p, d, q))
    seasonal_pdq = [(x[0], x[1], x[2], s) for x in list(itertools.product(p, d, q))]
    parameters = []
    BICs = np.array([])
    for param in pdq:
        for param_seasonal in seasonal_pdq:
            try:
                mod = sm.tsa.statespace.SARIMAX(DATA,
                                            order=param,
                                            seasonal_order=param_seasonal)
                results = mod.fit()
                parameters.append([param, param_seasonal, results.bic])
                BICs = np.append(BICs,results.bic)
            except:
                continue
    return parameters[np.argmin(BICs)]
best_params = selectparameter(price, 12)

#Let the model train and predict
sarimax = sm.tsa.statespace.SARIMAX(train,order=best_params[0],
                                    seasonal_order=best_params[0]).fit()
pred = sarimax.predict("2018-02-01", "2019-11-29")

The SARIMA model also requires parameter tuning, but it is difficult to try them one by one, so we will automate it. Use the Bayesian Information Criterion for tuning to avoid error in learning results and overfitting. SARIMA is a model that uses ARIMA after considering periodic fluctuations.

best_params[0], best_params[1]
[(1, 1, 0), (0, 0, 0, 12)]

The optimal parameters are as above.
Since it takes too much time due to the PC specifications, the range of parameters to be given is set to 0 or 1.

There is also a function that automatically estimates the order of SARIMAX, but only the autocorrelation degree and moving average can be estimated.
Two criteria, AIC and BIC, can be used for estimation, but here we will create a model estimated based on Akaike's information criterion.
Manually enter the parts that cannot be estimated automatically.

```python
# Estimate parameters with AIC
# Manual specification of guidance, seasonal_order, etc.
def sarimax_aic(data, sp, sd, sq, s):
    param = sm.tsa.arma_order_select_ic(data, ic="aic")
    model = sm.tsa.statespace.SARIMAX(data, order=(param["aic_min_order"][0], 1, param["aic_min_order"][1]),
                                    seasonal_order=(sp, sd, sq, s)).fit()
    return model, param

# Let the model estimated by AIC perform training and prediction
aic_model, aic_param = sarimax_aic(price_diff, 0 ,0 ,0, 12)
pred_aic = aic_model.predict("2018-02-01", "2019-11-29")

The optimum parameters estimated by the two criteria are 'aic_min_order': (4, 2) is.

Visualize the predictions to verify the accuracy of the prepared model.

# Visualization
plt.figure(figsize=(8,8))
plt.plot(test, color="r")
plt.plot(pred, color="c")
plt.plot(pred_aic, color="y")
plt.legend(["Test", "Prediction", "AIC Prediction"], loc="lower right")
plt.tight_layout()
plt.show()

Regarding model prediction, we compare the period predicted from the test data and training data left as the correct answer for accuracy verification. As mentioned above, we are comparing by the overlapping period. The model tuned with BIC traces the trend of price changes, but there is a gap. I also wanted to tune the AIC under the same conditions, but it takes too long, so I use the same values for the manual part as the BIC model. However, this one was not very accurate. The parameter of the part representing the cycle was set to 12 months, but the result was the same when I set it to 13 with reference to the correlogram of partial autocorrelation. By the way, the result is the same even if the period of the given data is shortened. And the error of prediction

# Make the model prediction error a correlogram
sm.graphics.tsa.plot_acf(sarimax.resid, lags=365*2)
sm.graphics.tsa.plot_pacf(sarimax.resid, lags=365*2)
plt.show()

Since the autocorrelation and periodicity between the errors remain, it means that the prediction cannot be made well ...

It was a very disappointing result, but finally I will make the prediction I wanted to make this time.

Finally, since we want to know the future price transition, let the model predict the entire period of the data with the estimated parameters.

# Predict the range outside the period given to the model
sarimax = sm.tsa.statespace.SARIMAX(price_diff,order=(1, 1, 0),
                                    seasonal_order=(0, 0, 0, 12)).fit()
forecast =  sarimax.forecast(steps=10)

# For convenience, create a subplot of one graph
fig = plt.figure(figsize=(7,7))
ax = fig.add_subplot(1,1,1)

# Set the date of the predicted period on the x-axis and draw the predicted value on the y-axis
fig = plt.figure(figsize=(7,7))
ax = fig.add_subplot(1,1,1)
days = mdates.DayLocator() 
daysFmt = mdates.DateFormatter('%m-%d')
x = pd.date_range(price_diff.index[-1], periods=len(forecast), freq="B")
y = forecast
ax.plot(x,y)
ax.set_xlabel("Date")
ax.set_ylabel("USD")

plt.tight_layout()

plt.show()

forecast()By using, you can predict after the end of the period used for learning. Specify 10 destinations in step, and since it is date and time data, predict the period up to 10 business days ahead. The horizontal axis sets the date for the predicted future period. Predicted price on the vertical axis(US dollar unit)It is described in. Since the predicted value is a difference, it is a difference. As a result, it seems that the current peak is at the peak and then goes down and level off. However, the accuracy of the model itself is not high, so it cannot be used for trading.

#Summary How to tune the model(Especially the calculation time)Was the biggest challenge. seasonal_order is a parameter of the effect on seasonal seasonal fluctuations, and I wanted to adjust it as well, but I gave up because it would take too long to set it to 1 or more. I wanted to expand the range of specifications given to the parameters, but for the same reason I gave up ... I also had a problem understanding the mathematical part.

`Written by Professor Okimoto Measurement time series analysis of economic and finance data'

bought. I'm just starting to read it, but I recommend it because it clearly describes what you need for time series analysis.

Thank you for watching until the end.

I tried to predict the price of ETF

Preparation

Forecast