Time Series Analysis of S&P500 Index between the Years 2012 & 2022

Here is the link to the complete code snippet with results.

Table of Contents

1. Goal
2. Dataset
3. Data Pre-processing
4. Time-Series Plot
5. Analysis
   • Understanding the Time-Series Components
6. Forecasting Model Selection
   • Simple Forecasting Methods
   • Simple Exponential Functions
   • ARIMA
7. Forecast Method Evaluation Using Residual Diagnostics
8. Model Implementation
   • First Iteration
   • Second Iteration
   • Final Iteration
9. Forecasting

Goal

The project performs a preliminary analysis of the underlying time series, estimates forecasting models, and chooses the best one based on empirical evaluations. Finally, it forecasts for 6 consecutive periods in the future using the best model generated.

Dataset

The dataset contains daily closing values for the S&P500 index between the period, Dec 2012 to Dec 2022. Below is the snapshot of the same.

Fig. 1 - Snapshot of the Dataset

Data Pre-processing

Since, the dataset contains daily records with different number of days for each month, it becomes difficult to create an uniform TimeSeries. To address this problem, the average index value was calculated for each month prior to instantiating the time series object. The snapshot for the updated dataset is provided below.

Fig. 2 - Pre-processed Dataset

Time-Series Plot

With the new dataset a TS object was instantiated. This object is a vector that represents a sample of periodic data that has been uniformly spaced and sorted according to time. Fig 3. displays the time-series plot being investigated and which has been generated from the TS object.

Fig. 3 - Monthly Acerage Time-Series Plot for S&P500

Analysis

Understanding the Time-Series Components

Plotting the time-series together with its seasonal plots is a crucial first step in understanding and subsequently forecasting a time series. The charts generated are shown in the Fig 4, 5 & 6. These charts provide a better understanding of the time series' components.

Fig. 4 - Monthly Plot of S&P500 Time-Series

Fig. 5 - Seasonal Plot of S$P500 Time-Series

Following observations were made from the initial plots of the time series displayed above:

• There is a clear trend in the time-series
• The seasonal plots does not indicate any presence of seasonality.
• The time-series do not exhibit cyclical pattern.

The foregoing observations are confirmed by decomposing the time series by classical method into its constituent components. The decomposed plot is given below. Because the series shows no seasonal variation as time progresses, the seasonal component can be classified as additive, and the decomposition was performed accordingly using the additive model.

Fig. 6 - Additive Decomposition of S$P500 Time-Series

Consistent with the other plots, the auto-correlation figure below (Fig. 7) too does not demonstrate any seasonality in the time series, but it does highlight a clear upward trend.

Fig. 7 - Auto-Correlation Function (ACF) plot of S$P500 Time-Series

From the above plots, the time series can be reasonably concluded to thus have only trend, and randomcomponents within it.

Forecasting Model Selection

In time-series forecasting toolkit, there are various forecasting methods available ranging from very simple to most complicated. The methods used for the purpose of time-series analysis of the data in this project are discussed below:

Simple Forecasting Methods:

1. Average Method : This approach produces forecasts that are equal to the mean of all historical data.

2. Naive Method : The forecast is equivalent to the most recent observation in this method.

3. Seasonal Naive Method : The forecast is equivalent to the last value seen from the same season in this method.

4. Drift Method : This approach adjusts the forecast based on the change in historical data over time.

Simple Exponential Functions:

These are more advanced forecasting techniques that use weighted moving averages to improve accuracy.These are as follows:

1. Single Exponential Smoothing (ses) : This method allows forecasting of time-series without any clear trend or seasonality.

2. Double Exponential Smoothing : Also known as Holt’s linear trend method allows forecasting of time-series with a trend

3. Triple Exponential Smoothing : Commonly known as Holt’s Winter’s seasonal method, is used in forecasting time-series with seasonality.

ARIMA

It is an acronym for Auto Regressive Integrated Moving Average . This is the most complicated and advanced model of them all. It takes three parameters along with ts object to build the model. These three parametrs are:

• P: represents the effects of the previous observed values
• D: represents the trend in data and its value is the number of differencing operations required to make a time-series stationary
• Q: represents the effects of the residuals

Forecast Method Evaluation Using Residual Diagnostics:

Although RMSE and AIC are effective statistics for evaluating model performance, Residual Diagnostics is another tool for evaluating goodness of fit. It can be considered one of the most effective methods for examining time-series models. This diagnostic has been used extensively to evaluate models in this project. The successful forecasting model will exhibit some of the following significant qualities.

• The residuals should have no autocorrelation.
• The residuals should have a mean zero
• The residuals should have constant variance
• The residuals should be normally distributed

The Ljung-Box test is a comparable and more precise test for determining autocorrelation and was used to present the same in the model summary (Table 1).

Model Implementation

The iterations in search of the optimal time-series forecasting model are listed below.

Iteration 1 :

Simple forecasting approaches were utilized to construct benchmark models in the first iteration. All the four techniques were chosen in this iteration. The results of the forecast are displayed in a composite chart in the Fig 2. Drift was found to be the most effective of the four strategies based on the RMSE value presented in the model summary (Table 1) below. As a result, this model was used for more advanced evaluation using residual diagnostics.
Below are the plots (Fig 8) associated with this evaluation.

Fig. 8 - Residual Diagnostic Plots for Drift model from Iteration 1

Methods	RMSE	Box Ljung (P Value)	AIC
Average	875.47	2.2e-16	--
Naive	107.76	0.3526	--
Seasonal Naive	475.83	2.2e-16	--
Drift	105.67	0.3004	--
SES	107.318	0.2469	1717.834
ETS (AAN)	105.554	0.1758	1717.822
ETS (ZZZ)	105.313	0.3653	1665.839
Holt's	105.554	0.1758	1717.822
Holt's Winters	101.274	0.0005	1731.805
ARIMA (0,1,1)	106.315	0.4851	1465.5
ARIMA (auto)	105.231	0.3429	1463.02

Table. 1 - Model Summary

Iteration 2 :

Because the previous iteration failed to produce satisfactory forecasting models, this iteration employed more complex algorithms. These algorithms are known as simple exponential smoothing functions. They are based on weighted moving averages. Some of the algorithms in this set are unsuitable for application because of the lack of seasonality and the existence of trend in the time-series, but they were nevertheless used here for learning and exploratory reasons. The forecasting results are shown in the composite chart in the Fig 9. Based on the RMSE values as displayed in the model summary (Table 1), Holt's Winters model was selected for further analysis to establish the goodness of the fit.
(Fig 10) Similar to the last best model, the auto-correlation spikes for the lags in the ACF chart of the residuals for this model was statistically significant as well, indicating room for further improvement.

Fig. 10 - Residual Diagnostic Plots for Holts Winter's model from Iteration 2

Iteration 3 :

The RMSE value of the last best model from the previous iteration improved significantly, although residual diagnostics presented unsatisfactory results. As a result, more sophisticated algorithms were applied in the final iteration. This approach, known as ARIMA, is used to fit stationary time series. Stationary time-series are defined as those with constant variation and mean 0. The following steps were taken to correctly apply ARIMA on the given time-series:

1. To evaluate whether the time-series under investigation is stationary, appropriate pre-built R function (ndiffs) were used. The examination revealed that the series was not stationary, thus first order differencing (diff) was advised to transform it into the same. Hence D is 1.

2. To identify values for P and Q, Auto-Correlated Function (ACF) and Partial Auto-Correlated Function (PACF) charts were generated.

3. Following an examination of the ACF and PACF plots, P was judged to be 0 and Q to be 1.

4. Using the chosen values of P, D, and Q, an ARIMA model was created and evaluated. Because the results were unsatisfactory, several more similar models were created by tweaking the P, D, and Q parameters slightly.

5. Finally, due to the disappointing results of the previous ARIMA models, a new model was constructed using the auto.arima function to round out the report. Based on AIC, this function returns the optimal model. This model performed the best during the evaluation and can be used for forecasting. The residual diagnostics for this final model is displayed in Fig 11. below

Fig. 11 - Residual Diagnostic Plots of final ARIMA model

(Fig 11) Apart from a few spikes, the residuals in the ACF plot appear to have no autocorrelation. The spikes can be attributed to chance occurrence. Furthermore, the residuals appear to have a mean of 0 and a distribution that is close to normal. As a result, this model can be considered the best thus far.

As seen in the table (Table 1), the worst model was the Average model, and the best model was the ARIMA model developed using auto.arima

Forecasting

The last best model was used to forecast future values for 6 consecutive periods as directed. The results presented in Table 2 and displayed as a plot in Fig 2.4

Period	Values	80% CI	95% CI
Jan 2023	3982.025	+- 135.988	+- 207.976
Feb 2023	4003.167	+- 192.316	+- 294.122
Mar 2023	4024.309	+- 235.539	+- 360.225
Apr 2023	4045.450	+- 271.976	+- 415.952
May 2023	4066.592	+- 304.079	+- 465.048
Jun 2023	4087.734	+- 333.101	+- 509.434

Table 2 - Point Forecasts along with 80% and 95% CI from final ARIMA model

Fig. 12 - Forecast Plot for final ARIMA model