1. Stock Prediction with ARIMA
Kihwan Lee
8/23/2018

2. Exploratory Data Analysis
Data Source
Quandle
National Unemployment Rate
Apple, Google, J.P.Morgan
Foreign Exchange Rate – Korea

3. Correlation and Bollinger Band
National Unemployment Rate
Observation
Apple Stock Closing Price
-0.5
Negative correlation, somewhat noticeable
Foreign Exchange Rate with South Korea
-0.73
Negative correlation, somewhat stronger
Bollinger band
Provides a relative definition of high and low prices of a market
Low band
The price has reached a relative low value. A likely time to buy.
High band
The price has reached a relative high value. A likely time to sell.
The stock prices are well bounded by the Bollinger band
Bollinger Bands

4. What is ARIMA? p = order of Auto-Regressive model
ARMA: Auto Regression Moving Average model for the stationary data
ARIMA: Auto Regression Integrated Moving Average model – a generalization of ARMA
ARIMA model goes through differencing steps to eliminate the non-stationary part
When the data shows evidence of non-stationarity, where initial differencing steps can be applied multiple times to eliminate the non-stationarity.
 ARMA model on the differenced data
Auto Regression: y(t) = f(y(t-1)), the current value is regressed with its own lagged value
Moving Average: regression error = a linear combination of previous error terms
Integrated: data values have been replaced by the difference
p = order of Auto-Regressive model
D=order of differencing model
q = order of Moving-Average model
(p,d,q) = non-seasonal components
(P,D,Q)s = seasonal components

5. Akaike Information Criterion vs. Bayesian Information Criterion
The Akaike information criterion (AIC)
An estimator of the relative quality of statistical models for a given set of data.
AIC estimates the relative information lost by a given model: the less information a model loses, the higher the quality of that model.
The model with lowest AIC is preferred
K = number of the estimated parameters
L = maximum value of the likelihood function of the model
AIC = 2 * k = 2 ln (L)
Dickey-Fuller Test
Tests the null hypothesis that a unit root is present in an autoregressive model, meaning non-stationary
The alternative hypothesis is the data is stationary
Smaller P-value is preferred
Accepting the null hypothesis is surprising!!!!
Non-surprising is equivalent to non-stationary
Surprising is equivalent to stationary
The Bayesian information criterion (BIC) or Schwarz criterion (also SBC, SBIC)
It is independent of the prior.
It can measure the efficiency of the parameterized model in terms of predicting the data.
It penalizes the complexity of the model where complexity refers to the number of parameters in the model.
 the model with the lowest BIC is preferred.
It is based, in part, on the likelihood function
K = number of the estimated parameters
L = maximum value of the likelihood function of the model
X = observed data
N = the number of data points in x, the numbe of observations, or equivalently, the sample size
BIC = ln(n) k – 2 ln(L)
Green: Recovery with a unit root
Red: Recovery without a unit root
Quantile - Quantile Plot
45 degree reference line plotted.
Two sets of data from the same population falls approximately along this reference line.
The supplied data is plotted with a set of data from the normal distribution.
Following reference line means the supplied data follows the normal distribution closely.

6. Stock Prediction with Auto Arima
Moving
average?
Cyclical
white nose?
Good enough?
Apple stock price prediction from 2016 and 2018 based on the historical data
Auto_arima function from pyramid.arima
AIC = 1752
p = 2, q = 2, d = 1

7. Exploration on the original data

8. Exploration on the logged data

9. Looking for Seasonality…
Case A
Plot and Dickey-Fuller Test
Case A: Original data
Case B: Original data – Shift-by-1 data
Case C: Original data – Shift-by-12 data
Case D: Shift-by-1 – Shift-by-12 data
Case B
Case C
Lag
P-value
Test Statistics
Case A 14 1
3.715 > 10% CV
Case B 10
2e-5
-5.5 < 1% CV
Case C 12
0.18
-2.3 ~> 10% CV
Case D
0.071
-2.7 < 10% CV
Case D

10. Looking for Seasonality…
Case A
Plot and Dickey-Fuller Test logged data
Case A: Logged data
Case B: Logged data – Shift-by-1 data
Case C: Logged data – Shift-by-12 data
Case D: Shift-by-1 – Shift-by-12 data
Case B
Case C
Lag
P-value
Test Statistics
Case A
0.978
0.309 > 10% CV
Case B
2e-30
-17.3 < 1% CV
Case C 12
0.017
-3.25 < 5% CV
Case D
0.012
-3.36 < 5% CV
Case D

11. ARIMA Models Parameter Summary
p,d,q
P, D, Q, s
AIC
BIC
QHIC
RSS
Case 1
30, 0, 0
0, 0, 0,12
1641
1758
1688
516
Case 2
30,0,0
0,0,2,12
1664
1788
1714
517
Case 3
30,0,2
1662
1794
510
Case 11
30, 1, 3
0,1,3,12
1547
1683
1601
508
Case 12
0, 0, 10
8,4,0,12
1365
1428
1391
1027
Case 13
2, 2, 2
2,2,2,12
1628
1661
591
Case 14
0, 0, 5
25,2,0,12
753
810
776
1452
p = order of Auto-Regressive model
d = order of differencing model
q = order of Moving-Average model
(p,d,q) = non-seasonal components
(P,D,Q)s = seasonal components

12. Case 1

13. Case 1
Dynamic forecast
 Prediction 2016 ~ 2018 stock price using historical data up to the end of 2015
One-step ahead forecast
 Predicting the next step using true data
Future forecast for 2018 to 2010

14. Case 2

15. Case 2

16. Case 3

17. Case 3

18. Case 11 One-step ahead forecast
 Predicting the next step using true data
Case 11
Dynamic forecast
 Prediction using true data up to certain point
Future forecast

19. Case 12

20. Case 13

21. Case 14
Best match with random noise

22. Case 14
Shows surprisingly good match with the actual data for 2 years

23. Summary
Exploratory data analysis (EDA) performed on Apple stock price.
Demonstrated that Apple stock price variation stays within the Bollinger Bands for the relative high and low prices.
Showed periodicity in the Apple stock price history.
Demonstrated different Time-Series prediction methods using ARIMA, which were compared to the actual data.
Performed Time-Series forecast of the Apple stock price using ARIMA.
To-do
Understand the working principle of ARIMA model.
Governing equation derivation and recognize their limitations.
Apply the ARIMA model to a wider range of stock prices.
Apply correlations to different set of economic indicators, such as employment rate, inflation rate, trade deficit, and etc. and see if PCA can be applied.

24. Backup

25. Model Selection Criteria
Akaike Information Criterion and Bayesian Information Criterion
 Most commonly used criteria