Abstract

This project uses advanced time series analysis techniques to forecast HDFC stock prices for the next 10 years. By leveraging historical data, the project employs ARIMA and Exponential Smoothing methods, including Holt's model, to predict stock trends and assess the accuracy of these models using standard metrics such as RMSE and MAE.
The primary goal is to provide accurate forecasts of HDFC stock prices and compare the performance of different predictive models to identify the most suitable method for long-term forecasting.

Methodology

1.1.1 Data Collection:
• The input data, HDFC.csv, contains historical stock prices for HDFC.
• The dataset is processed and analysed to ensure it is suitable for time series modelling.

1.1.2 Forecasting Models:
• ARIMA (Autoregressive Integrated Moving Average):
o Captures the linear dependencies in the time series.
o Applied after examining and differencing the data for stationarity.

• Exponential Smoothing:
o Average past data with exponentially decreasing weights.
o Provides smoother forecasts.

• Holt's Exponential Smoothing:
o Accounts for trends in the data by extending basic exponential smoothing.

1.1.3 Tools and Libraries:
The following R libraries are utilized:
• Prophet: Advanced forecasting framework.
• quantmod: For financial data modelling.
• forecast: Contains ARIMA and Exponential Smoothing functions.
• ggplot2: For visualization.
• dplyr: For data manipulation.

1.1.4 Evaluation Metrics:
• Root Mean Square Error (RMSE): Measures the average error magnitude.
• Mean Absolute Error (MAE): Represents the average error magnitude without considering its direction.

1.2 ARIMA Model (AutoRegressive Integrated Moving Average):
ARIMA is a statistical model used for time series forecasting. It combines three components: AutoRegressive (AR), Integrated (I), and Moving Average (MA) to predict future values based on past values in a time series dataset. It's widely used for modelling and forecasting time series data that shows trends or seasonal patterns.

1.2.1 Components of ARIMA:

1. AR (AutoRegressive): The relationship between an observation and a number of lagged observations (past values). It's represented as AR(p), where p is the number of lagged observations considered.
2. I (Integrated): The differencing of raw observations to make the time series stationary (i.e., constant mean and variance). This is denoted as I(d), where d is the order of differencing.
3. MA (Moving Average): The relationship between an observation and a residual error from a moving average model applied to lagged observations. It's represented as MA(q), where q is the number of lagged forecast errors considered.

1.2.2 Steps to Build an ARIMA Model:

1. Stationarity Check: Ensure the time series is stationary. If not, apply differencing (I(d)) to remove trends and seasonality.
2. Identify p and q: Use AutoCorrelation Function (ACF) and Partial AutoCorrelation Function (PACF) plots to determine the values of p (AR order) and q (MA order).
3. Fit the ARIMA model: Estimate the model parameters (p, d, q) using methods like Maximum Likelihood Estimation (MLE).
4. Model Validation: Check the residuals (errors) to ensure they resemble white noise. If they do, the model is a good fit.
5. Forecast: Use the trained ARIMA model to make predictions on future data.
   ARIMA is highly effective for non-seasonal time series data that needs differencing to achieve stationarity.

1.3 Exponential Smoothing Model:

1.3.1 Theory: Exponential Smoothing (ES) is a time series forecasting technique that assigns exponentially decreasing weights to past observations. It is based on the idea that more recent observations are more relevant for predicting future values than older ones. The model is particularly useful for forecasting data with trends and seasonality.

1.3.2 Types of Exponential Smoothing:

1. Simple Exponential Smoothing (SES): Used when there is no trend or seasonality.
2. Holt’s Linear Trend Model: Used when the data has a linear trend.
3. Holt-Winters Seasonal Method: Extends Holt's method to capture seasonality as well.

1.3.3 Steps Involved:

1. Initialize Parameters: Set initial values for L0L_0L0, T0T_0T0, and S0S_0S0.
2. Apply Formulae: Use the respective formula (SES, Holt’s, or Holt-Winters) to calculate the level, trend, and seasonality components at each time step.
3. Make Predictions: Forecast future values using the model.
4. Tune Smoothing Parameters: Use methods like cross-validation to find optimal values for α\alphaα, β\betaβ, and γ\gammaγ.
   This approach is widely used for forecasting in various domains, especially where time series data exhibits trends or seasonal variations.

Code Files:

# install required libraries

install.packages("readxl")
install.packages("ggplot2")
install.packages("dplyr")
install.packages("forecast")
install.packages("readr")
install.packages("pastecs")
install.packages("prophet")
install.packages("lubridate")
install.packages("tidyverse")
install.packages("xts")

# set working directory
setwd("C:/Users/manik/OneDrive/KARTHIK MANI/Karthik Projects/Stock Market Time Series")

# load installed library
library(ggplot2)
library(readxl)
library(dplyr)
library(forecast)
library(readr)
library(pastecs)
library(prophet)
library(lubridate)
library(tidyverse)
library(xts)

#importing data set
HDFC <- read_csv("C:/Users/manik/Downloads/HDFC.csv")

# Rows: 4977 Columns: 15                                                                                                                                      
# Column specification:
# Delimiter: ","
# chr   (2): Symbol, Series
# dbl  (12): Prev Close, Open, High, Low, Last, Close, VWAP, Volume, Turnover, Trades, Deliverable Volume, %Deliverble
# date  (1): Date

#Make a copy of the data set
HDFC_COPY= data.frame(HDFC)
HDFC_COPY1 = data.frame(HDFC_COPY)

# exploring data set
# view the first 5 rows of the data set using head ()
head(HDFC)
head(HDFC,10)

# view the last 5 rows of the data set using tail ()
tail(HDFC)
tail(HDFC,15)

# view dimension of the data set using dim ()
dim(HDFC)

# view the statistical vectors using summary ()
summary(HDFC)

# to know more about column and its name using variable_name ()
variable.names(HDFC)

# to know data types of the column using str ()
str(HDFC)

# to view number of missing data in data set using is.na ()
is.na(HDFC)
colSums(is.na(HDFC))

# to visualize the data of a particular column by '$' symbol and column name 
head(HDFC$Turnover, 20)

# to know more briefly about descriptive statistic and its calculated values using stat.desc ()
stat.desc(HDFC)

# Remove missing values
HDFC <- na.omit(HDFC)

# line plot of the trending close price
ggplot(HDFC, aes(x = Date, y = Close)) +
  geom_line(color = "blue") +
  labs(title = "HDFC Stock Price Over Time", x = "Date", y = "Close Price")

# ARIMA MODEL building...
class(HDFC)

# Assuming HDFC has a Date column and Close column
HDFCtime <- ts(HDFC$Close, 
               start = c(as.numeric(format(min(HDFC$Date), "%Y")), as.numeric(format(min(HDFC$Date), "%j"))), 
               end = c(as.numeric(format(max(HDFC$Date), "%Y")), as.numeric(format(max(HDFC$Date), "%j"))), 
               frequency = 365.25)
class(HDFCtime)

library(tseries)
plot(HDFCtime)

# Auto Correlation BEFORE ARIMA model 
acf(HDFCtime)
pacf(HDFCtime)
adf.test(HDFCtime)

#Augmented Dickey-Fuller Test
#data:  HDFCtime
#Dickey-Fuller = -2.1372, Lag order = 14, p-value = 0.5202
#alternative hypothesis: stationary

# Auto Arima Function
HDFCmodel = auto.arima(HDFCtime, ic="aic", trace = TRUE)
#ARIMA(0,1,0)                                : 31417.37
#Best model: ARIMA(0,1,0) 
HDFCmodel

# Auto Correlation AFTER ARIMA model 
acf(ts(HDFCmodel$residuals))
pacf(ts(HDFCmodel$residuals))

# forecast
Myforecast_HDFC = forecast(HDFCmodel, level = c(95), h = 10*365.25)
Myforecast_HDFC

plot(Myforecast_HDFC)

# validate the forecast
Box.test(Myforecast_HDFC$residuals, lag = 15, type = "Box-Pierce")
Box.test(Myforecast_HDFC$residuals, lag = 10, type = "Box-Pierce")
Box.test(Myforecast_HDFC$residuals, lag = 5, type = "Box-Pierce")
Box.test(Myforecast_HDFC$residuals, lag = 20, type = "Box-Pierce")

# EXPONENTIAL SMOOTHING MODEL BUILDING...
HDFC_ES_model = ses(HDFCtime, alpha = 0.2, h = 100)
HDFC_ES_model

plot(HDFC_ES_model, main = "Forecast of the Exponential Smoothing")

#USING HOLT EXPONENTIAL SMOOTHING
HDFC_HOLT = holt(HDFCtime, h = 100)
HDFC_HOLT
plot(HDFC_HOLT, main  = "Forecast of Exponential Smoothing using HOLT method")

# Facebook's Prophet model Building...
#HDFC_DF = data.frame(ds = index(HDFC_COPY$Date), y = as.numeric(HDFC_COPY$Close))
#HDFC_prophet = prophet(HDFC_DF)
#future = make_future_dataframe(HDFC_prophet, period = 30)
#forecast_prophet = predict(HDFC_prophet, future)
#plot(HDFC_prophet, forecast_prophet)

# CALCULATING THE ACCURACY FOR THE ABOVE RESULTED MODELS....
accuracy(Myforecast_HDFC)
#                ME    RMSE      MAE         MPE     MAPE       MASE       ACF1
#Training set 0.1903026 36.2822 14.36358 -0.03355592 1.324054 0.03537981 0.01778181

#Summary of Ideal Accuracy Ranges for ARIMA:
#  RMSE: Lower is better, ideally < 5% of actual values.
#  MAE: Lower is better, ideally < 5% of actual values.
#  MAPE: Lower is better, ideally < 5% (or less than 10% for good models).
#  AIC/BIC: Lower is better; use to compare different models.
#  ACF of Residuals: Should be close to 0 (no significant autocorrelation).

accuracy(HDFC_ES_model)
#                  ME     RMSE      MAE        MPE    MAPE       MASE      ACF1
#Training set 0.9015601 59.58318 23.98849 -0.1686734 2.30609 0.05908751 0.7930406
accuracy(HDFC_HOLT)
#                  ME     RMSE      MAE        MPE     MAPE       MASE       ACF1
#Training set 0.7432281 36.30499 14.39091 0.03704185 1.327261 0.03544713 0.01803578

Results

Model Accuracy:
Model RMSE MAE
ARIMA 36.2822 14.36358
Exponential Smoothing 59.58318 23.98849
Holt's Method 36.30499 14.39091