In Partial Fulfillment of the Course Applied Regression and Time Series Analysis for Financial Research
FINARTS K31

Submitted by Group 1:
ASIS, ALYSSA MARI E.
KAGAOAN, SOFIA MARIE Q.
SABADO, JAMES ETHAN R.

Submitted to:
Mr. Dioscoro P. Baylon Jr.

1 Company Background

1.1 SM Investments Corporation (SM)

SM Investments Corporation (SMIC) is one of the leading conglomerates in the Philippines, recognized for its diversified operations and significant contributions to the country’s economic growth. Founded by Henry Sy Sr., SMIC has evolved from a small shoe store in Manila to a major holding company with interests spanning various industries. Its core businesses include retail, banking, and property development, which are central to its sustained growth and influence in the Philippine economy (SM Investments Corporation, n.d.).

In the retail sector, SMIC operates one of the largest and most extensive networks of department stores, supermarkets, and specialty stores in the country through SM Retail, Inc. The company is known for its SM Supermalls, a chain of large shopping malls that serve as commercial, entertainment, and lifestyle hubs across the Philippines. As of recent years, SM Retail includes notable brands such as SM Store, SM Supermarket, and specialty brands like Watsons and Ace Hardware (SM Investments Corporation, n.d.).

In addition to retail, SMIC has a strong presence in the financial sector through its equity in BDO Unibank, the largest bank in the Philippines in terms of total assets, and China Banking Corporation. These institutions provide a broad range of financial services, including consumer and corporate banking, insurance, and investment services (BDO Unibank, 2022).

Furthermore, SMIC is a key player in real estate through SM Prime Holdings, one of Southeast Asia’s largest integrated property developers. SM Prime is involved in residential development, commercial properties, hotels, and convention centers. It has also been expanding its sustainability and green initiatives, integrating environmentally friendly practices into its developments (SM Prime Holdings, 2022).

Overall, SM Investments Corporation stands as a pillar of the Philippine business landscape, not only for its commercial success but also for its commitment to innovation, inclusive growth, and sustainable development.

Relevance of the time series analysis Forecasting stock prices and returns is a critical function that supports the strategic and financial decisions of various stakeholders including investors, corporate management, and financial regulators. These projections offer a forward-looking view of a company’s performance, market trends, and economic indicators, which help reduce uncertainty and support better planning. By anticipating potential fluctuations in the stock market, stakeholders can implement more proactive and informed strategies that align with both short-term goals and long-term sustainability.

For Investors Investors, both institutional and individual, depend heavily on financial forecasting to assess potential returns and risks associated with their investment portfolios. Accurate stock price predictions assist them in determining the optimal time to buy, hold, or sell shares. This reduces the likelihood of losses caused by market volatility and helps achieve better risk-adjusted returns. Technical analysis, which involves examining charts, price trends, and trading volumes, is commonly used to detect patterns and make short-term predictions. On the other hand, fundamental analysis considers a company’s financial health, competitive position, management effectiveness, and macroeconomic conditions to evaluate whether a stock is undervalued or overvalued (Boyles, 2022). These tools, when combined, enable investors to make evidence-based decisions that reflect both the historical context and future outlook of a stock.

For Management For company executives and financial planners, forecasting plays a vital role in corporate governance and long-term planning. It enables the leadership team to set realistic targets, assess internal capabilities, and adapt strategies based on anticipated changes in the financial landscape. For example, if forecasts suggest a decline in the company’s stock price due to industry trends or internal inefficiencies, management can respond by enhancing cost efficiency, diversifying revenue streams, or increasing investor engagement. Furthermore, forecasts help in capital budgeting, determining dividend policies, and evaluating mergers or expansion opportunities (Saini, 2021). These insights empower managers to act decisively and maintain competitiveness while also satisfying shareholder expectations.

For Regulators Financial regulators and policymakers rely on stock market forecasting as a tool to oversee market stability and economic health. Forecasts can signal upcoming downturns or speculative bubbles, prompting preventive interventions to avoid broader economic disruptions. For instance, a consistent misalignment between market expectations and real economic output may indicate irrational investor behavior or underlying structural issues. By analyzing forecast models and trends, regulators can introduce policy measures such as adjusting interest rates, tightening disclosure requirements, or imposing trade restrictions to mitigate systemic risks and ensure fair trading environments (Krylov, 2018). These practices help maintain investor confidence and safeguard the integrity of financial markets.

With that, forecasting stock prices and returns is not merely a technical exercise, it is a powerful tool that equips stakeholders with the knowledge needed to navigate the complexities of modern financial systems. Investors use it to enhance portfolio performance, management relies on it to align operations with market dynamics, and regulators depend on it to protect financial stability. Together, these practices promote transparency, reduce information asymmetry, and foster confidence in the financial ecosystem. Ultimately, accurate forecasting supports efficient capital allocation, sustainable business growth, and overall economic development.

1.2 Overview

This case study will apply several methods of time series analysis to predict the stock prices of SM Investments Corporation, namely: (1) Simple Moving Average, (2) Exponential Moving Average, (3) Simple Exponential Smoothing, (4) Holt Method, (5) Holt-Winters Method, (6) Kalman Filter, (7) Regression (OLS), and (8) ARIMA. Afterwards, diagnostics will be ran to assess each model’s performance. The diagnostics include the RMSE (Root Mean Squared Error), MAE (Mean Absolute Error), and R-squared.

2 Predicting Stock Prices

2.1 Simple Moving Average (SMA)

Figure 1.1. Simple Moving Average (SMA) Method

Figure 1.2. Simple Moving Average (SMA) Method (Diagnostics)

## SMA Evaluation Metrics:

## RMSE: 25.0938

## MAE: 17.8933

## R-squared: 0.8925

In analyzing the Simple Moving Average method, we can see that the Root Mean Squared Error (RMSE) of 25.0938 indicates the model’s predictions deviate from actual values by about 25 units on average, with larger errors having a more significant impact due to the squaring involved. Meanwhile, the Mean Absolute Error (MAE) of 17.8933 displays a more direct and straightforward measure, showing an average deviation of nearly 18 units. The high R-squared value of 0.8925 suggests that the model explains about 89.25% of the variability in the data, meaning that it is a strong fit.

The stock price chart from 2020 to 2024 plots the actual stock price in blue against a 10-day simple moving average (SMA) represented in red. The SMA effectively smooths the price data, providing a clearer view of overall trends by averaging out daily fluctuations. Visually, the red line follows the general direction of the blue line but with noticeably less volatility, which reflects the averaging effect. Noticeable deviations between the two lines indicate times wherein the actual price moves sharply away from its recent average. The SMA appears to generally capture major trends, such as the rise in early 2021 and the subsequent decline through 2022, while filtering out some of the shorter-term noise.

Overall, the model shows good performance but could benefit from addressing these outliers to improve its accuracy.

2.2 Exponential Moving Average (EMA)

Figure 2.1. Exponential Moving Average (EMA) Method

Figure 2.1 displays the movements of the stock prices in periods 2020 to 2024, with a 10-day exponential moving average overlaid. Throughout the observed period, the stock price experienced significant volatility, which is evident in the alternating upward and downward trend that can be seen. The green line represents the 10-day exponential moving average, which serves to smooth out the price fluctuations and provide a clearer indication of the overall trend. Overall, it aligns heavily with the findings in the evaluation metrics.

Figure 2.2. Exponential Moving Average (EMA) Method (Diagnostics)

## EMA Evaluation Metrics:

## RMSE: 21.1418

## MAE: 15.0074

## R-squared: 0.9237

The Exponential Moving Average (EMA) model shows a solid forecasting performance based on the given evaluation metrics. The Root Mean Squared Error (RMSE) is 21.14, which indicates that, on average, the forecasted values deviate from the actual stock prices by about 21 units. This metric focuses more weight to larger errors, so it suggests that while most predictions are close, a few of them may be shown to be farther off. The Mean Absolute Error (MAE) is 15.01, meaning that the typical error between the forecast and the actual values is around 15 units. This is a relatively more balanced measure that treats all errors equally. Lastly, the R-squared value is 0.9237, which implies that the EMA model explains approximately 92% of the variance in the actual stock prices.

Overall, these results suggest that the EMA model captures the trend well and provides reasonably accurate forecasts.

2.3 Simple Exponential Smoothing (SES)

Figure 3.1. Simple Exponential Smoothing (SES) Method

Figure 3.1 displays the stock price’s movement over time, with a forecast for the subsequent 30 days generated using simple exponential smoothing. The historical data (represented by the black line) depicts considerable volatility, evident in the rapid increases and decreases in price. Near the beginning of the sequence, the price experiences a sharp decline before recovering and fluctuating within a range between approximately 800 and 1100. As time progresses towards the present, the price generally trends downward. The blue line indicates the forecast for the next 30 days, showing an initial increase followed by a decrease and then stabilization. The grey area around the forecast represents the confidence interval, which widens as the forecast extends further into the future, which may reflect increased uncertainty.

Figure 3.2. Simple Exponential Smoothing (SES) Method (Diagnostics)

## SES Evaluation Metrics:

## RMSE: 47.0655

## MAE: 34.7557

## R-squared: -1.1963

The Simple Exponential Smoothing (SES) model appears to perform poorly based on its evaluation metrics. The Root Mean Squared Error (RMSE) is 47.07, which indicates that, on average, the forecasted values deviate from the actual stock prices by about 47 units. This high value suggests the model struggles to closely and accurately track the actual price movements. The Mean Absolute Error (MAE) is 34.76, which shows that the typical forecast error is around 35 units—more than double the error of the EMA model. Furthermore, the R-squared value is -1.1963, which signals that the SES model performs significantly worse than simply predicting the mean of the actual values. A negative R-squared value reflects an overall poor fit and suggests that the model fails to capture the underlying pattern of the data. To conclude, the evaluation of the metrics indicate that the SES model is not well-suited for forecasting this particular stock’s behavior.

2.4 Holt Method

Figure 4.1. Holt Method

The graph shows a Holt-Winters forecast for a stock price over approximately 200 time periods, with a focus on a 30-day forecast that considers weekly seasonality. The historical stock price data, represented by the black line, shows considerable volatility with upward and downward trends. The forecast itself, indicated by the blue line within the grey shaded area, predicts a slight increase in stock price over the next 30 days, but the shaded area suggests a high degree of uncertainty in this prediction.

Figure 4.2. Holt Method (Diagnostics)

## Holt Method Evaluation Metrics:

## RMSE: 50.6396

## MAE: 38.2514

## R-squared: -1.5426

The Holt’s Linear Trend model shows weak forecasting performance based on both the numerical metrics and the corresponding graph. The RMSE of 50.64 and MAE of 38.25 indicate a high average error in the model’s predictions, suggesting that it struggles to accurately follow the actual stock price movements. More notably, the R-squared value of -1.5426 implies that the model performs worse than a naive approach of simply predicting the mean, and it fails to account for the variability in the data. The graph supports these findings. The historical stock price, shown in black, exhibits high volatility and irregular patterns. In contrast, the 30-day forecast, shown in blue, appears unstable and inconsistent with recent trends. The forecast initially increases, then sharply declines, with a wide confidence interval (shaded in grey) indicating substantial uncertainty. Overall, while the model attempts to capture trend components, it does not effectively handle the volatility and complexity present in the stock price data, limiting its usefulness for forecasting in this case.

2.5 Holt-Winters Method

Figure 5.1. Holt-Winters Method

The graph shows how the Holt-Winters method (using an additive model with weekly seasonality) was applied to forecast the stock prices of SM Investments Corporation. In the plot, we can see the actual stock prices as the black line, while the blue section on the right side represents the forecasted values. The shaded area around the forecast indicates the confidence interval, which basically shows the level of uncertainty in the prediction—the wider the band, the less confident the model is as it predicts further into the future.

Figure 5.2. Holt-Winters Method (Diagnostics)

## Holt-Winters Evaluation Metrics:

## RMSE: 40.2161

## MAE: 27.1371

## R-squared: -0.6036

Looking at the evaluation metrics, we see that the RMSE (Root Mean Square Error) is 40.22 and the MAE (Mean Absolute Error) is 27.14. These values tell us how far off the model’s predictions are from the actual prices. RMSE gives more weight to bigger errors, while MAE gives the average size of all the errors. The numbers are not super high, but they are not very low either, especially considering how unpredictable stock prices like SM’s can be.

The biggest concern here is the R-squared value, which is -0.6036. R-squared normally tells us how well a model explains the variation in the data, and we usually want it close to 1. A negative R-squared, though, means the model is actually doing worse than if we had just guessed the average price for every point. So in this case, the Holt-Winters method didn’t really capture the pattern in SM Investments Corporation’s stock prices very well.

With that, while Holt-Winters is often helpful for forecasting data with trends and seasonality, it does not seem to be the best fit for this particular dataset. Stock prices tend to be noisy and volatile, so a more flexible or advanced model might give more accurate predictions.

2.6 Kalman Filter

Figure 6.1. Kalman Filter Method (Smoothed Close Prices)

The graph shows the results of applying the Kalman Filter to smooth the closing stock prices of SM Investments Corporation over time. The Kalman Filter is a popular algorithm used in time series analysis and control systems because of its ability to estimate values from noisy data. In this case, the purple line represents the smoothed version of the closing prices, which helps reduce short-term fluctuations and highlights the overall trend more clearly. This kind of smoothing is especially useful for financial data like stock prices, which tend to be volatile and noisy.

The evaluation metrics shown below the graph are based on the last 30 days of data. The Root Mean Square Error (RMSE) is 14.777, and the Mean Absolute Error (MAE) is 9.8251. These values are significantly lower compared to the results from the Holt-Winters model, which suggests that the Kalman Filter provided a more accurate estimate of the true price trend during this period. RMSE gives us a sense of how far off the model’s predictions are on average, especially emphasizing larger errors, while MAE gives a straightforward average of how much the predictions differ from actual prices.

Most importantly, the R-squared value is 0.7835, which is quite strong. This metric shows how well the model explains the variability of the actual data, and a value closer to 1 is ideal. In this case, an R-squared of around 0.78 means that the Kalman Filter was able to capture the underlying trend of the stock price quite effectively, especially compared to the Holt-Winters method, which had a negative R-squared.

Therefore, the Kalman Filter appears to be a much better fit for modeling the recent price movements of SM Investments Corporation. Its ability to handle noise and continuously update predictions based on new data makes it a valuable tool for smoothing and tracking financial time series like stock prices.

Figure 6.2. Kalman Filter Method (Diagnostics)

## Kalman Filter Evaluation Metrics (Last 30 Days):

## RMSE: 14.777

## MAE: 9.8251

## R-squared: 0.7835

This graph specifically presents the Kalman Filter’s performance over the most recent 30-day window. The gray line shows the actual observed stock prices of SM Investments Corporation during that period, while the purple line represents the smoothed predictions or estimates generated by the Kalman Filter.

From the visual pattern, we can observe that the Kalman Filter effectively captures the underlying trend of the stock movement during this time. In the first half of the graph (around mid-November to early December), both lines show moderate fluctuations, and the filter manages to follow the price movements quite closely, albeit with a slightly lagged response in rapid changes. This is a normal behavior for smoothing models, which trade off a bit of responsiveness to reduce noise.

In the second half (early to mid-December), the stock undergoes a sharp upward surge followed by a slight dip toward the end of the month. The Kalman Filter tracks this sharp movement with noticeable accuracy, although it slightly underestimates the peaks and smooths over smaller short-term oscillations.

The performance metrics provided at the top—RMSE: 14.78, MAE: 9.83, and R²: 0.7835—quantitatively confirm the filter’s effectiveness: (1) The RMSE (Root Mean Squared Error) and MAE (Mean Absolute Error) values are relatively low, suggesting modest deviations between the actual prices and the filtered estimates. (2)The R-squared value of 0.7835 reflects a strong goodness of fit, meaning the filter successfully explains around 78% of the variance in the stock price during this short-term period.

Therefore, the graph gives a zoomed-in, short-term view of how well the Kalman Filter models the most recent stock price activity. It complements the earlier long-term chart, which shows how the filter smooths out closing prices over years. Together, both visuals confirm that the Kalman Filter is a reliable method for tracking stock trends in both long-term trend detection and short-term forecasting accuracy.

2.7 Regression

Figure 7.1. Regression Method (Linear Regression on Close Prices)

The graph above illustrates the application of linear regression to model the closing stock prices of SM Investments Corporation over time. In this visualization, the blue line represents the actual historical closing prices, while the orange line corresponds to the linear regression trendline fitted to the data. The x-axis is indexed over time (not labeled by date but by data point index), and the y-axis represents the stock’s closing price.

From the graph, it is evident that the linear regression line has a slightly downward slope, indicating a general decline in the closing price of the stock over the full dataset. However, the actual price data fluctuates significantly above and below the regression line, which suggests a high level of volatility and non-linearity in the stock’s movement, something that linear regression does not capture well due to its simplicity and assumption of a constant linear relationship.

Figure 7.2. Regression Method (Diagnostics)

## Linear Regression Evaluation Metrics:

## RMSE: 70.03

## MAE: 57.9599

## R-squared: 0.1827

The accompanying evaluation metrics further support this observation. The Root Mean Square Error (RMSE) is 70.03, and the Mean Absolute Error (MAE) is 57.96, both of which are relatively high. These values indicate that the predictions made by the linear regression model deviate considerably from the actual values. RMSE, in particular, penalizes larger errors more heavily, and such a high value implies that the model performs poorly in estimating more extreme price movements. MAE offers a more direct interpretation of the average prediction error, which again is quite substantial here.

Most telling is the R-squared (R²) value of 0.1827, which signifies that the linear regression model explains only about 18% of the variance in the actual closing prices. This is a rather weak performance and highlights that a simple linear model is insufficient for modeling such a complex, dynamic financial time series. Financial data like stock prices typically exhibit nonlinear patterns, trends, and abrupt shifts due to market sentiment, macroeconomic events, and investor behavior—all of which cannot be captured by a straight-line model.

2.8 ARIMA

Figure 8.1. ARIMA Method

The graph presents the ARIMA (AutoRegressive Integrated Moving Average) model’s forecast of the closing stock prices for SM Investments Corporation. This time series model attempts to capture patterns in the data such as trends, cycles, and seasonality in order to make predictions. The chart shows historical stock price data (black line), followed by a forecast (blue line) extending into the future. The shaded blue area around the forecast represents the confidence interval, indicating the uncertainty of the prediction—the wider it gets, the less certain the model is.

Figure 8.2. ARIMA Method (Diagnostics)

## RMSE:  39.07064

## MAE:  36.26935

## R-squared:  -0.5135253

The ARIMA model’s evaluation metrics indicate moderate predictive accuracy, with an RMSE of 39.07 and an MAE of 36.27. These values are significantly lower than those of the linear regression model, suggesting that the ARIMA model fits the data more accurately and handles time-based fluctuations better. Specifically, the MAE indicates that, on average, the ARIMA model’s predicted closing price deviates by about 36 points from the actual value, which is a marked improvement over the linear regression model’s MAE of nearly 58.

However, the R-squared value of -0.5135 is unusually low and negative. In typical regression models, R-squared values range from 0 to 1, where higher values indicate a better fit. A negative R-squared means the model performs worse than simply predicting the mean of the data, which suggests that while the ARIMA model captures short-term trends reasonably well (as reflected in the lower RMSE and MAE), it may not effectively model the overall structure or variability of the dataset. This could indicate overfitting or that the model doesn’t align well with the broader time span of the stock price data.

Overall, the ARIMA model performs better than linear regression in minimizing average and squared errors, suggesting it adapts more effectively to stock price fluctuations. However, the negative R-squared value reveals significant limitations, such as an inability to explain the full variability of the data or a lack of generalizability. This mixed result highlights that while ARIMA can be a powerful tool for forecasting time series data, it requires careful parameter tuning and may still fall short when applied to highly volatile or non-stationary financial series without proper preprocessing.

2.9 Comparison of Model Results

Table 1. Comparison of Model Results

##                 Model    RMSE     MAE R_squared
## 1                 SMA 25.0938 17.8933    0.8925
## 2                 EMA 21.1418 15.0074    0.9237
## 3                 SES 47.0655 34.7557   -1.1963
## 4         Holt Method 50.6396 38.2514   -1.5426
## 5 Holt-Winters Method 40.2161 27.1371   -0.6036
## 6       Kalman Filter 14.7770  9.8251    0.7835
## 7          Regression 70.0300 57.9599    0.1827
## 8               ARIMA 39.0706 36.2694   -0.5135

3 Bonus - Log Returns

Extracting SM Daily Log Returns

# Extract Closing Prices
SM_closing <- Cl(data)

# Compute Log Returns
SM_log_returns <- diff(log(SM_closing))

# Remove NA Values (First row will be NA)
SM_log_returns <- na.omit(SM_log_returns)

SM_log_returns[is.na(SM_log_returns)] <- 0

View(SM_log_returns)

3.1 Simple Moving Average (SMA)

Figure 9.1. Simple Moving Average Method with LOG RETURNS

This graph illustrates the performance of a 10-day Simple Moving Average (SMA) plotted against the actual closing prices of a stock over a span of four years, from 2020 to 2024. The blue line represents the daily closing prices, which are highly volatile, while the red line shows the smoother 10-day moving average. This moving average helps filter out short-term fluctuations and provides a clearer view of the stock’s general direction. Through visual observation, it’s evident that the SMA follows the overall trend of the stock quite well, lagging slightly behind the actual price due to its nature as an average of past data. This method is particularly helpful for identifying short-term trends and assessing the general movement of the stock without being distracted by daily noise.

Figure 9.2. Simple Moving Average Method with LOG RETURNS (Diagnostics)

## SMA Evaluation Metrics:

## RMSE: 0.0205

## MAE: 0.0149

## R-squared: 0.0884

This graph shows the same 10-day SMA overlayed on the actual stock prices, but this time it includes performance metrics: RMSE (Root Mean Square Error), MAE (Mean Absolute Error), and R² (coefficient of determination). These values give a numerical assessment of how well the SMA models or approximates the actual data. With an RMSE of 0.02 and MAE of 0.01, the errors between the predicted and actual values are relatively small, indicating that the SMA closely follows the stock’s daily movements. However, the R² value of 0.0884 is quite low, suggesting that the moving average does not explain much of the variance in the data. In other words, while the SMA stays close to the actual values in terms of magnitude, it does not capture all the dynamic patterns in the price movement. This highlights the limitation of simple moving averages: they are useful for smoothing and trend-following, but not for prediction or explaining complex market behavior.

Overall, while the SMA does not account for complex patterns such as seasonality or structural breaks, it provides a valuable visual tool for observing short-term trends. The evaluation metrics support that the SMA offers a reasonably accurate and stable estimate of the underlying movement in the log returns, making it useful for trend-following strategies and initial exploratory analysis in financial forecasting.

3.2 Exponential Moving Average (EMA)

Figure 10.1. Exponential Moving Average with LOG RETURNS

The graph illustrates the 10-day Exponential Moving Average (EMA) applied to the log returns of SM Investments Corporation’s stock from 2020 to 2024. The blue line represents the actual log return values, while the green line indicates the EMA. Compared to the Simple Moving Average (SMA), the EMA gives more weight to recent data points, making it more sensitive to recent price movements and more effective in capturing short-term trends.

Figure 10.2. Exponential Moving Average with LOG RETURNS (Diagnostics)

## EMA Evaluation Metrics:

## RMSE: 0.0219

## MAE: 0.0159

## R-squared: -0.0409

The EMA closely follows the actual log returns, particularly during periods of sharp fluctuation, which highlights its responsiveness. However, this also makes it more prone to reacting to short-term noise, as seen in the minor jagged movements of the green line throughout the graph. In terms of performance, both the Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) are 0.02, indicating a relatively low average error between the EMA and actual values. Despite this, the R-squared value is -0.0409, suggesting that the EMA performs worse than simply predicting the mean, which reflects its limited explanatory power when applied to log returns.

Overall, the EMA remains a useful tool for tracking short-term trends in financial data. While it does not perform well as a predictive model, its ability to highlight recent trend changes in a volatile market like that of SM Investments Corporation makes it a valuable method for smoothing time series data.

3.3 Simple Exponential Smoothing (SES)

Figure 11.1. Simple Exponential Smoothing with LOG RETURNS

The plot displays the Simple Exponential Smoothing (SES) forecast applied to the log returns of SM Investments Corporation’s stock data. The black line represents the actual log returns, while the blue line toward the right side shows the fitted forecast values, with a surrounding grey band indicating the forecast’s confidence interval.

The chart reveals noticeable volatility in the earlier part of the time series, which gradually stabilizes over time—a common trait in financial data influenced by market shocks or firm-specific events. Since SES is most appropriate for data without clear trends or seasonality, its main role here is to smooth the series and generate short-term forecasts rather than capture significant directional changes.

Toward the end of the series, the forecast line flattens, reflecting the SES model’s emphasis on recent observations and its assumption that future values will resemble the recent past. The widening grey band around the forecast illustrates growing uncertainty as the forecast extends further, which aligns with the model’s limitations in long-term predictions.

With that, SES proves useful for smoothing the noisy log return data and providing a quick estimate of short-term movements. However, its lack of trend or seasonal components limits its suitability for more complex or longer-term forecasting tasks in financial contexts.

Figure 11.2. Simple Exponential Smoothing with LOG RETURNS

## [1] 0.01629634

## SES Evaluation Metrics:

## RMSE: 0.0163

## MAE: 0.0106

## R-squared: 0

The chart presents a 30-day Simple Exponential Smoothing (SES) forecast applied to the log returns of SM Investments Corporation’s stock price. The black line represents the historical log return data, while the blue line on the right side shows the forecasted values generated by the SES model. Surrounding the forecast is a shaded grey area that reflects the confidence interval, indicating the range within which future values are expected to fall and highlighting the model’s uncertainty.

The SES forecast produces a flat, nearly constant line—an expected outcome, given that SES assumes no trend or seasonality. It relies heavily on recent data, assigning more weight to the latest values to predict future movements. This approach results in a steady forecast that assumes future prices will resemble recent past behavior. The widening of the confidence interval as the forecast extends further suggests increasing uncertainty, which is typical in time series forecasting models.

The model’s evaluation metrics provide more insight into its performance: RMSE is 0.02, MAE is 0.01, and R² is 0. These low RMSE and MAE values indicate minimal average error, suggesting that the forecast is relatively close to the actual data. However, the R² value of 0 reveals that the model does not explain any variance in the actual log returns. This supports the visual observation that the SES forecast does not capture any directional movement or fluctuation, only projecting a stable continuation of recent behavior.

With that, while SES is effective for smoothing out noise in volatile financial data and offering a basic short-term forecast, it lacks the ability to model more complex patterns like trends or cycles. This limits its predictive power in dynamic market conditions, but its simplicity makes it a practical starting point for time series analysis before applying more advanced forecasting techniques.

3.4 Holt Method

Figure 12.1. Holt Method with LOG RETURNS

The graph presents the forecasted stock prices of SM Investments Corporation using Holt’s Linear Trend Method, applied to log-transformed returns. Unlike Simple Exponential Smoothing, Holt’s method incorporates both level and trend, making it more appropriate for time series data with noticeable upward or downward movements. In this context, the model attempts to identify and project a linear trend based on recent patterns.

The black line shows the actual log-return-transformed stock prices, while the blue line at the right end represents the forecasted values. The surrounding grey, fan-shaped area indicates the confidence interval, illustrating the range of possible future outcomes with increasing uncertainty over time. The forecast shows a slight downward slope, consistent with the recent trend of gradual decline in the log returns. This suggests that the model effectively identified the ongoing negative trend and extended it into the forecast horizon.

Despite the trend adjustment, the confidence interval widens as the forecast progresses, reflecting the volatility typical of financial data—even after log transformation. The graph still displays sharp fluctuations, which likely contribute to the broad confidence bounds. However, compared to SES, Holt’s method offers a more realistic forecast by accounting for trend direction.

Overall, Holt’s Linear Trend Method provides a reasonable and trend-aware forecast of the stock’s log returns. While it does not model seasonality or nonlinear behaviors, it serves as a stronger alternative to SES for trending data and offers a clearer view of expected short-term price direction in volatile market conditions.

Figure 12.2. Holt Method with LOG RETURNS (Diagnostics)

## Holt Method Evaluation Metrics:

## RMSE: 0.0165

## MAE: 0.0109

## R-squared: -0.022

Figure 13.1. Holt-Winters Method with LOG RETURNS ## Holt-Winters Method

The graph displays a Holt-Winters additive model with weekly seasonality applied to the log returns of SM Investments Corporation’s stock. The y-axis reflects logarithmic returns rather than raw prices, which is standard in financial analysis to stabilize variance and improve statistical reliability. The x-axis represents time in trading days, and the blue segment at the right side of the plot shows the model’s forecast, enclosed by a grey confidence interval.

This model is intended for time series data with both trend and seasonality. In this case, weekly seasonality was assumed, aligning with the typical five-day trading week. The historical data, represented by the black line, shows significant volatility, with frequent sharp jumps up and down. These fluctuations illustrate the erratic nature of daily stock returns.

Despite incorporating seasonal and trend components, the forecast line remains relatively flat and hovers around zero, indicating a neutral expectation of future returns. There are slight oscillations in the forecast, suggesting the model detected weak seasonal patterns, though these are not visually dominant. The confidence interval, while accommodating some variability, is relatively narrow, which may signal the model’s low certainty yet stable outlook under the given assumptions.

The limited movement in the forecast and the narrow range of the confidence band may indicate either a lack of clear seasonality in the data or that the noise in daily returns overshadows any recurring patterns. This is common in financial time series, where market behavior is often driven by unpredictable external factors rather than consistent cycles. With that, the Holt-Winters additive model offers a structured approach to analyzing short-term movements, but in this case, it produces a modest and largely neutral forecast. While the method is statistically sound, its practical utility is constrained by the unpredictable and noisy nature of log return data in stock markets.

Figure 13.2. Holt Method with LOG RETURNS (Diagnostics)

## Holt-Winters Evaluation Metrics:

## RMSE: 0.0163

## MAE: 0.011

## R-squared: 0.0047

The 30-day Holt-Winters additive model forecast for SM Investments Corporation’s stock log returns shows the actual historical log returns (black line) and the 30-day forecast (blue line), with a prediction interval (grey shaded). The model assumes weekly seasonality, which is common in financial data due to trading behavior. However, the historical returns appear highly volatile and irregular, with sharp spikes and drops, suggesting weak or inconsistent seasonality. This volatility limits the model’s ability to capture meaningful patterns, reflected in the low R² value of 0.0047, indicating almost no explanatory power in terms of variance.

Despite this, the model maintains modest error rates, with an RMSE of 0.02 and MAE of 0.01, which suggests that the short-term point forecasts are reasonably close to the observed values. However, these small errors are more a reflection of the small magnitude of the returns rather than the model’s strength. The forecast projects a mean-reverting behavior, where returns hover around zero. Although the model attempts to integrate weekly seasonality, the erratic nature of the data limits its accuracy. These results reinforce that predicting log returns is inherently challenging, especially when market movements are influenced by unpredictable external factors.

3.5 Kalman Filter

Figure 14.1. Kalman Filter with LOG RETURNS

The thick line shows the Kalman Filter’s adaptive interpretation of price movements, clearly dampening extreme volatility (like 2020’s COVID swings) while tracking major directional shifts. We see it adjusting its responsiveness - tighter smoothing during calm periods (2023) versus wider bands during turbulent phases. The model particularly struggles during black swan events where the smoothed line lags behind actual spikes, reminding us that even sophisticated filters can’t perfectly predict market chaos. Traders value this view to distinguish meaningful trends from daily noise, though the Y-axis scale (where 0.03 ≈ 3% daily move) confirms we’re still dealing with significant volatility. The widening gaps during crises highlight where human judgment must supplement algorithmic outputs.

Figure 14.2. Kalman Filter with LOG RETURNS (Diagnostics)

## Kalman Filter Evaluation Metrics (Last 30 Days):

## RMSE: 0.0164

## MAE: 0.0109

## R-squared: -0.017

The model achieves low error metrics (RMSE: 0.02, MAE: 0.01) but has a negative R² (-0.017), indicating it fails to explain the variance better than a simple mean model. The forecast (likely the smooth line) appears to track the general trend of the volatile price data (jagged line), though the widening confidence interval suggests increasing uncertainty in predictions over time. The Kalman Filter adapts to new data efficiently but struggles with the inherent noise in financial time series.

LMdata <- data[-1 , ]
LMdata$Logret <- SM_log_returns

3.6 Regression

Figure 15.1. Regression with LOG RETURNS

Figure 15.2. Regression with LOG RETURNS (Diagnostics)

## Linear Regression Evaluation Metrics:

## RMSE: 918.8477

## MAE: 915.5817

## R-squared: 0

3.7 ARIMA

Figure 15.1. ARIMA with LOG RETURNS

This graph shows the results of an ARIMA forecast applied to a time series of log returns. The black line represents historical log returns, which are characterized by high volatility, rapid fluctuations, and a mean-reverting tendency around zero. The blue line represents the out-of-sample forecast, while the shaded grey region indicates the confidence interval.

The forecast remains near zero, reflecting the stationary nature of the data and the ARIMA model’s assumption that future returns will behave similarly to past returns. The tight confidence intervals suggest limited deviation from the mean, though this may underestimate risk due to the large spikes and noise earlier in the series.

The model captures the broad structure of the data, but its predictive value should be interpreted cautiously. While ARIMA models are effective for modeling stationary time series, their ability to forecast financial returns is constrained by the random, non-linear nature of such data.

Figure 15.2. ARIMA with LOG RETURNS (Diagnostics)

## RMSE:  0.01640586

## MAE:  0.01088867

## R-squared:  -0.01038861

The ARIMA model shows low error values (RMSE: 0.0164, MAE: 0.0109), which is reasonable given the small scale of log returns. However, the R-squared value of -0.0104 suggests the model performs worse than a simple mean predictor, indicating that it does not explain meaningful variation in the data. This highlights the challenge of capturing patterns in volatile and noisy financial return series.

4 Stock Price Forecasting Conclusions

Table 2. Comparison of Model Results for Log-Returns

##                 Model     RMSE      MAE R_squared
## 1                 SMA   0.0205   0.0149    0.0884
## 2                 EMA   0.0219   0.0159   -0.0409
## 3                 SES   0.0163   0.0106    0.0000
## 4         Holt Method   0.0165   0.0109   -0.0220
## 5 Holt-Winters Method   0.0163   0.0110    0.0047
## 6       Kalman Filter   0.0164   0.0109   -0.0170
## 7          Regression 918.8477 915.5817    0.0000
## 8               ARIMA   0.0164   0.1099   -0.0104

The Simple Exponential Smoothing (SES) and Holt-Winters methods deliver the best overall results, with the lowest RMSE and MAE (SES: RMSE = 0.0163, MAE = 0.0106; Holt-Winters: RMSE = 0.0163, MAE = 0.0110). SES, despite a near-zero R-squared, is the most accurate due to its minimal error.

In contrast, the Simple Moving Average (SMA) and Exponential Moving Average (EMA) models show higher RMSE and MAE values, with EMA having a negative R-squared (-0.0409), indicating poor performance. The Regression model performs poorly, with extremely high error metrics (RMSE = 918.85, MAE = 915.58) and an R-squared of 0, highlighting its unsuitability for forecasting log returns.

The ARIMA, Holt, and Kalman Filter models are similar in performance to SES and Holt-Winters, but their low or negative R-squared values suggest limited explanatory power. In conclusion, SES stands out as the most reliable model, offering consistent and accurate forecasts, while Holt-Winters is a close second.

4.1 Analysis and Discussion

The comparative analysis reveals distinct patterns in how each model handles trends and noise in SM Investments Corporation’s stock data. SMA and EMA effectively smooth short-term volatility, with EMA (R²: 0.92) outperforming SMA (R²: 0.89) due to its responsiveness to recent data, though both lag during abrupt market shifts. SES excels in minimizing errors (lowest RMSE/MAE) but fails to explain variance (R²: 0), producing flat forecasts ideal for stable trends but inadequate for volatile markets. Holt’s method captures linear trends better than SES, yet its negative R² (-1.54) highlights oversimplification of complex price dynamics. The Holt-Winters model, while incorporating seasonality, struggles with financial data’s erratic nature (R²: -0.60), reflecting the challenge of modeling non-recurring shocks.

The Kalman Filter stands out for adaptive smoothing (R²: 0.78), dynamically adjusting to new data while reducing noise, though it lags during extreme events like the 2020 crash. Conversely, Regression and ARIMA models appear to underperform, with Regression’s linear assumptions (R²: 0.18) and ARIMA’s negative R² (-0.51) which shows their mismatch with nonlinear, volatile returns.

5 Appendices

Load Necessary Libraries

library(readxl)
library(tidyverse)
library(forecast)
library(TTR)
library(zoo)
library(dlm)
library(Metrics)
library(KFAS)
library(forecast)  
library(quantmod)
library(tseries)
library(timeSeries)
library(xts)
library(ggplot2)
library(urca)
library(plotly)
library(ggfortify)

Load Data and convert date

# Load Data
data <- read.csv("SM-historical-prices.csv")

# Convert date
data$Date <- as.Date(data$Date, format = "%m/%d/%y")

# Sort by date just in case
data <- data[order(data$Date), ]

Generate Simple Moving Average + Evaluation Metrics

data$SMA_10 <- SMA(data$Close, n = 10)

# Remove rows with NA values from SMA
eval_data <- data %>% filter(!is.na(SMA_10))

# Calculate evaluation metrics
rmse_val <- rmse(eval_data$Close, eval_data$SMA_10)
mae_val <- mae(eval_data$Close, eval_data$SMA_10)
r_squared <- 1 - sum((eval_data$Close - eval_data$SMA_10)^2) / 
                 sum((eval_data$Close - mean(eval_data$Close))^2)

# Print the results
cat("SMA Evaluation Metrics:\n")

## SMA Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 25.0938

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 17.8933

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0.8925

# Plot the SMA vs actual
ggplot(data, aes(x = Date)) +
  geom_line(aes(y = Close), color = "blue") +
  geom_line(aes(y = SMA_10), color = "red") +
  labs(title = "Simple Moving Average (10-day)",
       subtitle = paste("RMSE:", round(rmse_val, 2), 
                        "MAE:", round(mae_val, 2), 
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

## Warning: Removed 9 rows containing missing values or values outside the scale range
## (`geom_line()`).

Generate Exponential Moving Average (EMA) + Evaluation Metrics

# Calculate 10-day Exponential Moving Average
data$EMA_10 <- EMA(data$Close, n = 10)

# Remove rows with NA values
eval_data <- data %>% filter(!is.na(EMA_10))

# Compute evaluation metrics
rmse_val <- rmse(eval_data$Close, eval_data$EMA_10)
mae_val <- mae(eval_data$Close, eval_data$EMA_10)
r_squared <- 1 - sum((eval_data$Close - eval_data$EMA_10)^2) /
                 sum((eval_data$Close - mean(eval_data$Close))^2)

# Print results
cat("EMA Evaluation Metrics:\n")

## EMA Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 21.1418

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 15.0074

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0.9237

# Plot the EMA vs actual
ggplot(data, aes(x = Date)) +
  geom_line(aes(y = Close), color = "blue") +
  geom_line(aes(y = EMA_10), color = "green") +
  labs(title = "Exponential Moving Average (10-day)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

## Warning: Removed 9 rows containing missing values or values outside the scale range
## (`geom_line()`).

Generate Simple Exponential Smoothing (SES) + Evaluation Metrics

# Set forecast horizon (e.g., last 30 days as test set)
h <- 30
n <- nrow(data)

# Training and test sets
train <- ts(data$Close[1:(n - h)])
test <- data$Close[(n - h + 1):n]

# Fit SES model and forecast
ses_model <- ses(train, h = h)
forecast_vals <- as.numeric(ses_model$mean)

# Evaluation metrics
rmse_val <- rmse(test, forecast_vals)
mae_val <- mae(test, forecast_vals)
r_squared <- 1 - sum((test - forecast_vals)^2) / sum((test - mean(test))^2)

# Print results
cat("SES Evaluation Metrics:\n")

## SES Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 47.0655

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 34.7557

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: -1.1963

# Convert test set into a time series (match start time and frequency)
test_ts <- ts(test, start = end(train)[1] + 1, frequency = frequency(train))

# Plot forecast and actual values
autoplot(ses_model) +
  autolayer(test_ts, series = "Actual", color = "blue") +
  labs(title = "Simple Exponential Smoothing Forecast (30 days)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

Generate Holt Method + Evaluation Metrics

# Forecast horizon (30 days)
h <- 30
n <- nrow(data)

# Split data
train <- ts(data$Close[1:(n - h)])
test <- data$Close[(n - h + 1):n]

# Fit Holt model
holt_model <- holt(train, h = h)

# Extract forecasted values
forecast_vals <- as.numeric(holt_model$mean)

# Calculate evaluation metrics
rmse_val <- rmse(test, forecast_vals)
mae_val <- mae(test, forecast_vals)
r_squared <- 1 - sum((test - forecast_vals)^2) / sum((test - mean(test))^2)

# Print metrics
cat("Holt Method Evaluation Metrics:\n")

## Holt Method Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 50.6396

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 38.2514

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: -1.5426

# Convert test set to ts object for plotting
test_ts <- ts(test, start = end(train)[1] + 1, frequency = frequency(train))

# Plot forecast vs actual
autoplot(holt_model) +
  autolayer(test_ts, series = "Actual", color = "blue") +
  labs(title = "Holt's Linear Trend Forecast (30 Days)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

Generate Holt Winters + Evaluation Metrics

# Set weekly frequency (assuming daily data, 5 trading days/week)
data_ts <- ts(data$Close, frequency = 5)

# Forecast horizon
h <- 30
n <- length(data_ts)

# Split into training and test sets
train <- window(data_ts, end = c(floor((n - h) / 5), (n - h) %% 5 + 1))
test <- window(data_ts, start = c(floor((n - h) / 5) + 1, ((n - h) %% 5 + 1)))

# Fit Holt-Winters model (additive seasonality)
hw_model <- hw(train, seasonal = "additive", h = h)

# Extract forecasts
forecast_vals <- as.numeric(hw_model$mean)

# Evaluation metrics
rmse_val <- rmse(test, forecast_vals)
mae_val <- mae(test, forecast_vals)
r_squared <- 1 - sum((test - forecast_vals)^2) / sum((test - mean(test))^2)

# Print metrics
cat("Holt-Winters Evaluation Metrics:\n")

## Holt-Winters Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 40.2161

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 27.1371

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: -0.6036

# Plot
autoplot(hw_model) +
  autolayer(test, series = "Actual", color = "blue") +
  labs(title = "Holt-Winters Forecast (30 Days, Weekly Seasonality)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

Generate Kalman Filter Smoothed Close Prices

# Define a state space model with local level (random walk) trend
model_kf <- SSModel(data$Close ~ SSMtrend(degree = 1, Q = NA), H = NA)

# Estimate the model parameters
fit_kf <- fitSSM(model_kf, inits = c(0.1, 0.1))

# Get the smoothed values
kf_smoothed <- KFS(fit_kf$model)

# Add the Kalman smoothed estimates to the data
data$Kalman <- kf_smoothed$a[1:nrow(data), 1]

# Plot
ggplot(data, aes(x = Date)) +
  geom_line(aes(y = Close), color = "gray") +
  geom_line(aes(y = Kalman), color = "purple") +
  labs(title = "Kalman Filter Smoothed Close Prices")

Generate Kalman Filter + Evaluation Metrics

# Forecast horizon
h <- 30
n <- nrow(data)

# Define test set (last 30 days of actual vs smoothed values)
actual <- data$Close[(n - h + 1):n]
predicted <- data$Kalman[(n - h + 1):n]

# Calculate metrics
rmse_val <- rmse(actual, predicted)
mae_val <- mae(actual, predicted)
r_squared <- 1 - sum((actual - predicted)^2) / sum((actual - mean(actual))^2)

# Print metrics
cat("Kalman Filter Evaluation Metrics (Last 30 Days):\n")

## Kalman Filter Evaluation Metrics (Last 30 Days):

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 14.777

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 9.8251

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0.7835

# Plot actual vs smoothed for just the last 30 days
library(ggplot2)
kalman_eval_df <- data.frame(
  Date = data$Date[(n - h + 1):n],
  Actual = actual,
  Smoothed = predicted
)

ggplot(kalman_eval_df, aes(x = Date)) +
  geom_line(aes(y = Actual), color = "gray", linewidth = 1.1) +
  geom_line(aes(y = Smoothed), color = "purple", linewidth = 1.1) +
  labs(title = "Kalman Filter Evaluation (Last 30 Days)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

Generate Linear Regression of Close Prices

data$Index <- 1:nrow(data)
lm_model <- lm(Close ~ Index, data = data)
data$Pred_LM <- predict(lm_model)

ggplot(data, aes(x = Index, y = Close)) +
  geom_line(color = "blue") +
  geom_line(aes(y = Pred_LM), color = "orange") +
  labs(title = "Linear Regression on Close Prices")

Generate Linear Regression Evaluation Metrics

# Actual and predicted values
actual <- data$Close
predicted <- data$Pred_LM

# Calculate metrics
rmse_val <- rmse(actual, predicted)
mae_val <- mae(actual, predicted)
r_squared <- summary(lm_model)$r.squared

# Print results
cat("Linear Regression Evaluation Metrics:\n")

## Linear Regression Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 70.03

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 57.9599

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0.1827

Generate ARIMA Forecast

auto_fit <- auto.arima(data$Close)
forecast_arima <- forecast(auto_fit, h = 30)
autoplot(forecast_arima) + labs(title = "ARIMA Forecast")

Generate ARIMA diagnostics

# Fit the ARIMA model
auto_fit <- auto.arima(data$Close)

# Generate forecasts for the next 30 periods
forecast_arima <- forecast(auto_fit, h = 30)

# Assuming you have a test set (actual values for comparison)
# Replace 'actual_test_values' with the actual values for the test period
actual_test_values <- data$Close[(length(data$Close)-29):length(data$Close)]

# Get the predicted values from the forecast
predicted_values <- forecast_arima$mean

# RMSE (Root Mean Squared Error)
rmse <- sqrt(mean((predicted_values - actual_test_values)^2))

# MAE (Mean Absolute Error)
mae <- mean(abs(predicted_values - actual_test_values))

# R-squared (Coefficient of Determination)
sst <- sum((actual_test_values - mean(actual_test_values))^2)  # Total Sum of Squares
sse <- sum((predicted_values - actual_test_values)^2)  # Sum of Squares due to Error
rsq <- 1 - (sse / sst)

# Print the evaluation metrics
cat("RMSE: ", rmse, "\n")

## RMSE:  39.07064

cat("MAE: ", mae, "\n")

## MAE:  36.26935

cat("R-squared: ", rsq, "\n")

## R-squared:  -0.5135253

Generate Table for the Results of each model

library(knitr)

results <- data.frame(
  Model = c("SMA", "EMA", "SES", "Holt Method", "Holt-Winters Method", "Kalman Filter", "Regression", "ARIMA"),
  RMSE = c(25.0938, 21.1418, 47.0655, 50.6396, 40.2161, 14.7770, 70.0300, 39.0706),
  MAE = c(17.8933, 15.0074, 34.7557, 38.2514, 27.1371, 9.8251, 57.9599, 36.2694),
  R_squared = c(0.8925, 0.9237, -1.1963, -1.5426, -0.6036, 0.7835, 0.1827, -0.5135)
)

[BONUS APPENDICES]

Extract SM Daily Log Returns

# Extract Closing Prices
SM_closing <- Cl(data)

# Compute Log Returns
SM_log_returns <- diff(log(SM_closing))

# Remove NA Values (First row will be NA)
SM_log_returns <- na.omit(SM_log_returns)

SM_log_returns[is.na(SM_log_returns)] <- 0

View(SM_log_returns)

Generate Simple Moving Average

SM_SMA <- SMA(SM_log_returns, n = 10)
SM_SMA <- na.omit(SM_SMA)
ggplot(data, aes(x = Date)) +
  geom_line(aes(y = Close), color = "blue") +
  geom_line(aes(y = SMA_10), color = "red") +
  labs(title = "Simple Moving Average (10-day)")

## Warning: Removed 9 rows containing missing values or values outside the scale range
## (`geom_line()`).

Generate Simple Moving Average + Evaluation Metrics

# Remove rows with NA values from SMA
eval_data <- data %>% filter(!is.na(SMA_10))

# Calculate evaluation metrics
rmse_val <- rmse(SM_log_returns, SM_SMA)

## Warning in actual - predicted: longer object length is not a multiple of
## shorter object length

mae_val <- mae(SM_log_returns, SM_SMA)

## Warning in actual - predicted: longer object length is not a multiple of
## shorter object length

r_squared <- 1 - sum((SM_log_returns - SM_SMA)^2) / 
                 sum((SM_log_returns - mean(SM_SMA))^2)

## Warning in SM_log_returns - SM_SMA: longer object length is not a multiple of
## shorter object length

# Print the results
cat("SMA Evaluation Metrics:\n")

## SMA Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 0.0205

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 0.0149

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0.0884

# Plot the SMA vs actual
ggplot(data, aes(x = Date)) +
  geom_line(aes(y = Close), color = "blue") +
  geom_line(aes(y = SMA_10), color = "red") +
  labs(title = "Simple Moving Average (10-day)",
       subtitle = paste("RMSE:", round(rmse_val, 2), 
                        "MAE:", round(mae_val, 2), 
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

## Warning: Removed 9 rows containing missing values or values outside the scale range
## (`geom_line()`).

Generate Exponential Moving Average Diagnostics + Evaluation Metrics

# Calculate 10-day Exponential Moving Average
SM_EMA <- EMA(SM_log_returns, n = 10)
SM_EMA <- na.omit(SM_EMA)

# Compute evaluation metrics
rmse_val <- rmse(SM_log_returns, SM_EMA)

## Warning in actual - predicted: longer object length is not a multiple of
## shorter object length

mae_val <- mae(SM_log_returns, SM_EMA)

## Warning in actual - predicted: longer object length is not a multiple of
## shorter object length

r_squared <- 1 - sum((SM_log_returns - SM_EMA)^2) /
                 sum((SM_log_returns - mean(SM_EMA))^2)

## Warning in SM_log_returns - SM_EMA: longer object length is not a multiple of
## shorter object length

# Print results
cat("EMA Evaluation Metrics:\n")

## EMA Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 0.0219

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 0.0159

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: -0.0409

# Plot the EMA vs actual
ggplot(data, aes(x = Date)) +
  geom_line(aes(y = Close), color = "blue") +
  geom_line(aes(y = EMA_10), color = "green") +
  labs(title = "Exponential Moving Average (10-day)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

## Warning: Removed 9 rows containing missing values or values outside the scale range
## (`geom_line()`).

Generate Simple Exponential Smoothing Forecast

SM_SES <- ses(ts(SM_log_returns), h = 30)
SM_SES <- na.omit(SM_SES)
autoplot(SM_SES) + labs(title = "Simple Exponential Smoothing Forecast")

**Generate Simple Exponential Smoothing (SES) + Evaluation Metrics)

# Set forecast horizon (e.g., last 30 days as test set)
h <- 30
n <- nrow(data)

# Training and test sets
train <- ts(SM_log_returns[1:(n - h)])
train <- na.omit(train)
train[is.na(train)] <- 0

test <- SM_log_returns[(n - h + 1):n]
train <- na.omit(test)
test[is.na(test)] <- 0

# Fit SES model and forecast
SM_SES <- ses(train, h = h)
forecast_vals <- as.numeric(SM_SES$mean)
forecast_vals <- na.omit(forecast_vals)

# Evaluation metrics
rmse_val <- rmse(test, forecast_vals)
rmse_val

## [1] 0.01629634

mae_val <- mae(test, forecast_vals)
r_squared <- 1 - sum((test - forecast_vals)^2) / sum((test - mean(test))^2)

# Print results
cat("SES Evaluation Metrics:\n")

## SES Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 0.0163

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 0.0106

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0

# Convert test set into a time series (match start time and frequency)
test_ts <- ts(test, start = end(train)[1] + 1, frequency = frequency(train))

# Plot forecast and actual values
autoplot(ses_model) +
  autolayer(test_ts, series = "Actual", color = "blue") +
  labs(title = "Simple Exponential Smoothing Forecast (30 days)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

Generate Holt’s Linear Trend Forecast + Evaluation Metrics

h <- 30
n <- nrow(data)

train <- ts(SM_log_returns[1:(n - h)])
train <- na.omit(train)
train[is.na(train)] <- 0

test <- SM_log_returns[(n - h + 1):n]
train <- na.omit(test)
test[is.na(test)] <- 0

holt_model <- holt(train, h = h)

forecast_vals <- as.numeric(holt_model$mean)

rmse_val <- rmse(test, forecast_vals)
mae_val <- mae(test, forecast_vals)
r_squared <- 1 - sum((test - forecast_vals)^2) / sum((test - mean(test))^2)

cat("Holt Method Evaluation Metrics:\n")

## Holt Method Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 0.0165

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 0.0109

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: -0.022

test_ts <- ts(test, start = end(train)[1] + 1, frequency = frequency(train))

autoplot(holt_model) +
  autolayer(test_ts, series = "Actual", color = "blue") +
  labs(title = "Holt's Linear Trend Forecast (30 Days)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")

Generate Holt Linear Trend Forecast

SM_holt <- holt(ts(data$Close), h = 30)
autoplot(holt_model) + labs(title = "Holt's Linear Trend Forecast")

Generate Holt-Winters Forecast

# Convert the closing prices into a time series with weekly seasonality (5 trading days per week)
ts_close <- ts(SM_log_returns, frequency = 5)

# Apply Holt-Winters method with additive seasonality
hw_model <- hw(ts_close, seasonal = "additive", h = 30)

# Plot the forecast
autoplot(hw_model) +
  labs(title = "Holt-Winters Forecast (Additive, Weekly Seasonality)",
       y = "Stock Price", x = "Time")

Generate Holt-Winters Forecast + Evaluation Metrics

# Set weekly frequency (assuming daily data, 5 trading days/week)
data_ts <- ts(SM_log_returns, frequency = 5)

# Forecast horizon
h <- 30
n <- length(data_ts)

# Split into training and test sets
train <- window(data_ts, end = c(floor((n - h) / 5), (n - h) %% 5 + 1))
test <- window(data_ts, start = c(floor((n - h) / 5) + 1, ((n - h) %% 5 + 1)))

# Fit Holt-Winters model (additive seasonality)
hw_model <- hw(train, seasonal = "additive", h = h)

# Extract forecasts
forecast_vals <- as.numeric(hw_model$mean)

# Evaluation metrics
rmse_val <- rmse(test, forecast_vals)
mae_val <- mae(test, forecast_vals)
r_squared <- 1 - sum((test - forecast_vals)^2) / sum((test - mean(test))^2)

# Print metrics
cat("Holt-Winters Evaluation Metrics:\n")

## Holt-Winters Evaluation Metrics:

cat("RMSE:", round(rmse_val, 4), "\n")

## RMSE: 0.0163

cat("MAE:", round(mae_val, 4), "\n")

## MAE: 0.011

cat("R-squared:", round(r_squared, 4), "\n")

## R-squared: 0.0047

# Plot
autoplot(hw_model) +
  autolayer(test, series = "Actual", color = "blue") +
  labs(title = "Holt-Winters Forecast (30 Days, Weekly Seasonality)",
       subtitle = paste("RMSE:", round(rmse_val, 2),
                        "MAE:", round(mae_val, 2),
                        "R²:", round(r_squared, 4)),
       y = "Stock Price")