This paper provides a comprehensive review of time-series forecasting models for forecasting the performance of Indian mutual funds. Specifically, we evaluate the effectiveness of three popular approaches: ARIMA, deep learning, and Lasso regression. Using a dataset of historical mutual fund data from the Indian market, we compare the predictive accuracy of these models using various evaluation metrics. Our findings indicate that Lasso regression outperforms both ARIMA and Deep Learning (LSTM) models in capturing the complex patterns and dynamics of mutual fund data. These findings offer valuable insights for investors and financial practitioners, shedding light on the most effective modeling approaches for predicting Indian mutual fund performance. This study contributes to the field of time series forecasting by providing a comprehensive comparison of ARIMA, Deep Learning, and Lasso Regression models. The findings can guide researchers and practitioners in selecting the most suitable model for specific forecasting tasks based on the desired balance between accuracy, robustness, and computational efficiency. The proposed research focuses on providing sustainability in investment domain. Lasso Regression models exhibit superior accuracy and competitive performance with a lower computational cost. The popular methods MAE, RMSE, MAE, R2 Score, MAPE, and MPE are used to measure the accuracy of the models.
encouraging outcomes when applied to financial time-series data, showcasing their potential in forecasting stock prices, identifying market trends, and predicting various financial indicators. Sustainability in investment has gained significant traction in recent years as more investors recognize the importance of long-term sustainability for both financial returns and broader societal well-being. Various investment products, such as sustainable mutual funds, exchangetraded funds (ETFs), and green bonds, cater to investors looking to align their financial goals with their values. Ensuring sustainability in investment involves a combination of research, analysis, due diligence, and ongoing monitoring.
In recent years, various time-series forecasting models have gained prominence in the financial domain for their potential to capture the complex dynamics of financial data.
Zeroual, A et al. [1] studies five deep learning models to forecast the new and recovered cases of COVID-19. VAE (Variational Autoencoder) algorithm shows superior performance among all. Benevento, E et al. [2] evaluate the predictive performance of lasso regression, random forest, support vector regression, artificial neural networks, and ensemble methods using a range of error metrics and computation time measurements. The results reveal that the ensemble method surpasses other approaches in accurately predicting latency. Zhang, L et al. [3] introduced, where the stochastic trend data is eliminated from the SSE Composite Index to obtain de-noised training data for the SVM (Support Vector Machine). Subsequently, the SVM is trained using this de-noised data to make predictions on the test data. The SVM achieves a 25% hit rate when predicting with the noisy training data. Guo, K et al. [4] applying the ARIMA model, both the original data series and the logarithmic series of the S&P 500 Exponential Weekly Data Series and found that model predicts accurate stock price. Pandey, A et al. [5] developed a model for investors to accurately forecast prices, regardless of the employed strategy. The primary objective of this research is to analyze and predict changes in the stock market. By examining past historical trends, the model aims to identify and forecast emerging patterns that will manifest in the upcoming days. Xu, Y et al. [6] presents a predictive analysis conducted across various economic cycles, uncovering that the social media sentiment index demonstrates the strongest predictive ability during periods of economic expansion. Dai, Z et al. [7] predicts stock earnings volatility by utilizing the partially least squares technique, which identifies crucial predictors from a data-rich context. The research findings illustrate the efficacy of the partial least squares approach in improving the accuracy of stock return volatility predictions in data-rich environments. This approach surpasses alternative models and exhibits a significant advancement over benchmark models .Ma, F et al. [8] proposes the use of dimensionality reduction and contraction techniques to forecast stock market returns. This research provides fresh insights into stock market return projections by considering macroeconomic fundamentals as a basis for analysis. Li, X et al. [9] proposes a MS-MIDASLASSO model that shows superior predictive accuracy compared to both the conventional LASSO strategy and its regime-switching extension. Notably, the outstanding predictive performance of this model remains unchanged even in the face of the onset of the COVID-19 pandemic. Ren, X et al. [10] identify that the Fourier transform-based LSTM method enhances the prediction accuracy of stock price fluctuation dynamics. This improvement is observed from both statistical and economic standpoints, as we exploit the role of oil shocks in the analysis. Zhu, Z and He, K [11] Finding the best models to predict stock price trends has always been a topic of great interest and is closely related to investor investment behavior. However, LSTM models still need to be improved in terms of performance to reduce distortion. We expect to discover more models for predicting stock prices in the future. Lee, H. Y et al. [12] purpose of this study was to extract valuable outlier information from the residuals of ARIMA modeling using the Continuous Wavelet Transform (CWT). The obtained CWT information was then incorporated into the ARIMA forecasts, resulting in the creation of long-term heterogeneous forecasts. Liu, T et al. [13] suggests a new stock price forecast model named VML with the aim of enhancing forecast accuracy and achieving improved forecast results. The proposed approach involves splitting the decomposed subseries into multiple tasks using the MAML algorithm. This facilitates the training of the LSTM model with initial parameters that possess strong generalization capabilities. Experimental outcomes obtained from Chinese and American stock market datasets demonstrate that the proposed method significantly enhances prediction accuracy. Nair, A. V and Narayanan, J [14] suggest a stock market forecasting model was suggested to anticipate the future performance of a company's stock. The incorporation of machine learning techniques represents the latest advancement in market analysis technology, enabling the determination of current stock index values by leveraging past values. Zeng, L et al. [15] proposes an optimal combinatorial framework for agricultural commodity price forecasting was introduced. This framework integrates a decomposition-reconstruction ensemble technique and an enhanced global optimization algorithm, inspired by natural processes. Wu, D et al. [16] introduces a hybrid stock market forecasting model that merges a multilayer artificial neural perceptron network (MLP-ANN) with the conventional Altman Zscore model. Empirical analysis demonstrates that the hybrid neural network model achieves a notable average correct classification rate. Isabona, J et al. [17] study indicate that the prediction errors of the suggested MLP model, when compared to the measured data, are highly favorable and surpass those obtained through the conventional logarithmic distance-based path loss model. Li, G et al. [18] proposes a technique called the PCC-BLS framework was suggested to choose multi-indicator functions for predicting stock prices. This approach utilizes the Pearson's correlation coefficient (PCC) and the broad learning system (BLS). Initially, PCC was employed to select input features from a pool of 35 options, which encompassed original stock prices, technical indicators, and financial indicators. Banerjee, S and Mukherjee, D [19] emphasis his study on the utilization of nonparametric approaches like stacked multilayer perceptions (MLP), long short-term memory (LSTM), and gated recurrent units (GRU). Specifically, long-term short-term bidirectional memory (BLSTM) and gated bidirectional recurrent units (BGRU) were employed to forecast short-term stock prices for three NSE-listed banks. The performance of these models was then compared against a flat neural network benchmark. Ji, X et al. [20] proposes a novel forecasting approach was introduced, which combines conventional financial indicators with social media text features as inputs for predictive models. Additionally, a unique stock price prediction model incorporating both traditional financial variables and social media text features extracted through deep learning methods was suggested in this study. Kumar, D [21] proposes that stock market prediction is a cohesive process, implying the need for a closer examination of specific parameters relevant to stock market forecasting. Tanwar, R et al. [22] proposed a hybrid deep learning approach, specifically a model combining Convolutional Neural Network and Long Short-Term Memory (CNN-LSTM), designed for the identification of stress. Tanwar, R et al.[23] introduced a hybrid deep learning model that incorporates an attention mechanism. This allows for thorough feature extraction and dynamic prioritization of information. Makwana, Y et al.[24] Conducts a comparative analysis of different methods and technologies, with a particular focus on the effectiveness of Convolutional Neural Network (CNN) in food recognition. The research reveals insights into various CNN models, showcasing their accuracy and outcomes in the context of food recognition.
The problem at hand is the lack of a comprehensive assessment of time-series forecasting models for predicting the performance of Indian mutual funds. Although various approaches, such as ARIMA, deep learning (LSTM), and Lasso regression, have shown promise in other domains, their effectiveness and comparative performance in the context of Indian mutual funds remain unclear. The evaluation seeks to address this research gap by conducting a comprehensive assessment of the ARIMA, deep learning, and Lasso regression approaches.
i. This analysis will provide insights into the models' ability to accurately predict mutual fund performance.
ii. This evaluation will help determine the models' ability to adapt and provide reliable forecasts under different circumstances.
iii. This analysis will provide insights into how well the models can generalize their predictions beyond the training data and make accurate forecasts for unseen mutual fund performance.
This paper focuses on analyzing historical mutual fund data of TATAPOWER. The data,
which can be obtained from the yahoo finance site, encompasses the period from January 1,
2011, to April 28, 2023. To facilitate analysis, the data is divided into training and testing
segments, with 80% allocated for training and 20% for testing. Prediction tasks are then carried
out on this dataset using ARIMA (
Sample Dataset (TATAPOWER) Date Open High Low Close Adj Close Volume 2011-01-03 133.558380 133.558380 132.014343 132.665741 102.871704 1747585 2011-01-04 132.506500 133.558380 131.584915 133.235092 103.313179 2267182 2011-01-05 132.979370 135.777908 132.120499 135.189255 104.828468 3228574 2011-01-06 134.619888 136.163925 133.321945 135.034851 104.708755 2761494 2011-01-07 133.881653 135.763443 132.796005 134.065002 103.956696 3027490 ... ... ... ... ... ... ... 2023-04-24 196.500000 196.699997 194.800003 195.850006 194.042862 5017631 2023-04-25 195.850006 198.800003 195.350006 197.649994 195.826233 5957551 2023-04-26 197.649994 198.949997 196.149994 198.199997 196.371170 4910837 2023-04-27 198.449997 199.949997 197.649994 198.500000 196.668396 5215692 2023-04-28 199.500000 201.550003 199.000000 201.100006 199.244415 7951645 Dataset contains 3038 rows × 6 columns from TATAPOWER mutual fund from dated 201-01-03 to 2023-04-28.
The Autoregressive Integrated Moving Average (ARIMA) model is a commonly employed technique for time-series forecasting. It incorporates three essential components: auto regression (AR), differencing (I), and moving average (MA). The ARIMA model is defined by the order assigned to each component, denoted as ARIMA (p, d, q). In this notation, 'p' represents the autoregressive order, ’d’ represents the differencing order, and 'q' represents the moving average order.
5.1.1.
The autoregressive component of the model captures the linear association between the present observation and its previous values. The AR component of order p is represented by the equation:
AR(p): Xt = c + Σ(ϕi ∗ Xt − i) + εt
Here, Xt represents the current observation, c is a constant term, ϕi represents the
autoregressive coefficients for lagged values X t-i, and εt is the error term at time t.
(
The moving average component addresses the interdependence between the current observation and the error terms within the model. It acknowledges the relationship between them. The MA component of order q is represented by the equation:
MA (q): Xt = c + εt + Σ(θi ∗ εt − i)
Here, θi represents the moving average coefficients for the lagged error terms εt-i. Combining the three components, the ARIMA (p, d, and q) model is given by:
ARIMA (p, d, q): Xt = c + Σ(ϕi ∗ Xt − i) + εt + Σ(θi ∗ εt − i)
(
The ARIMA model aims to estimate the optimal values of the parameters (p, d, q) that minimizes the disparity between the observed values and the predicted values. This estimation is commonly accomplished through techniques like maximum likelihood estimation.
Deep learning, a branch of machine learning, concentrates on training artificial neural networks with multiple layers to acquire knowledge and make predictions based on intricate data. At the heart of deep learning lies artificial neural networks, consisting of interconnected layers of artificial neurons (also referred to as nodes or units). Each neuron conducts a weighted summation of its inputs, applies an activation function, and generates an output.
The mathematical representation of the output of a neuron can be expressed as:
z = w₁x₁ + w₂x₂ + . . . + wₙxₙ + b
(
In this context, x₁, x₂, ..., xₙ denote the input values or activations from the preceding layer, w₁, w₂, ..., wₙ refer to the respective weights, b represents the bias term, and z denotes the weighted sum of inputs.
To train a deep learning model, a loss or cost function is necessary, which measures the
disparity between the predicted output and the true output. The objective is to minimize this
difference using an optimization algorithm called backpropagation. Backpropagation calculates
the gradient of the loss function concerning the weights and biases in the network, enabling
their adjustment in a manner that reduces the error. The gradient descent algorithm is
commonly employed for this purpose. The process of updating the weights and biases is
governed by the following equations:
wᵢ(new) = wᵢ(old) − learning rate ∗ ∂loss/ ∂wᵢ
b(new) = b(old) − learning rate ∗ ∂loss/ ∂b
(
Here, wᵢ(new) and b(new) represent the updated weights and biases, wᵢ(old) and b(old) are the current weights and biases, learning rate is a hyper parameter that determines the step size of the update, and ∂loss/∂wᵢ and ∂loss/∂b represent the derivatives of the loss function with respect to the weights and biases.
Lasso Regression, which stands for Least Absolute Shrinkage and Selection Operator, is a linear regression technique that integrates regularization to enhance model performance and select relevant features. Given a dataset with n observations and p features, let X be an n x p matrix representing the predictor variables, y is an n-dimensional vector representing the response variable, and β be a p-dimensional vector representing the coefficients to be estimated.
The formulation of the Lasso Regression model can be expressed as follows:
y = β₀ + β₁x₁ + β₂x₂ + . . . + βₚxₚ + ɛ
(
The primary goal of Lasso Regression is to minimize the total of squared residuals while adhering to a constraint on the absolute sum of the coefficients:
minimize: (1/2n) ∗ Σ(yᵢ − (β₀ + β₁x₁ᵢ + β₂x₂ᵢ + . . . + βₚxₚᵢ))² (
The constraint Σ|βⱼ| ≤ t encourages sparsity in the model, meaning it promotes the selection of a subset of relevant features by driving some coefficients to zero. The characteristic of Lasso Regression makes it valuable for the purpose of feature selection since it automatically conducts variable selection by reducing the coefficients of irrelevant features towards zero.
In Fig. actual closing price of TATAPOWER mutual fund and predicted closing price of TATAPOWER mutual fund taken into consideration. The fig. shows forecasted and actual closing price of mutual fund results i.e. very closed to each other. So we can say that performance of model is very adequate. The MAE value is 3.020% and RMSE is 4.764% also shows the accuracy of model.
Fig- shows the real closing price of TATAPOWER mutual fund and predicted closing price of TATAPOWER mutual fund. The graph shows that proposed model the actual value and predicted value of this mutual fund is very close to each other. Forecasting analysis also proves the accuracy of model with MAE value is 03.3140% and RMSE is 04.7740% these values slightly differ from ARIMA model.
Fig- shows the real closing price of TATAPOWER mutual fund and predicted closing price of TATAPOWER mutual fund. The graph shows that proposed Lasso Regression model’s actual closing price and predicted value of this mutual fund is very close to each other. Forecasting analysis also proves the accuracy of model with MAE value is 0. 0.0274% and RMSE is 0.0333% these values slightly differ from ARIMA model. This model performs more actuate than both above models.
MSE is another way to calculate the accuracy and error of the forecast model used:
MSE = Yi – Ŷi )2
Ŷi is the predicted ith value and Yi is the actual/ observed value.
(
RMSE is another way to calculate the accuracy of proposed model but it considers the error calculation based on standard deviation. The final output is one standard deviation of the magnitude of the error, and the individual calculations are reported as residuals: RMSE =
Yi – Ŷi)2
(
MAPE may be a formula for calculating the precision of estimates. The calculation is done by taking the contrast between the real value and the anticipated esteem and separating the distinction by the actual value.
Mean Absolute Error (MAE) R2 Score Explained Variance Score Mean Percentage (MAPE) Mean Percentage Error (MPE)
Error 68.207455468409
(
In conclusion, this study aimed to perform a Comparative Analysis of ARIMA, Deep Learning, and Lasso Regression Models for Time Series Forecasting on an Indian mutual fund dataset. Through a comprehensive evaluation and comparison of these models, several significant findings have emerged. Firstly, the ARIMA model exhibited robust performance in capturing the temporal patterns and trends in the mutual fund data. Secondly, the deep learning models, particularly the long short-term memory (LSTM) networks, demonstrated comparable predictive capabilities to ARIMA. Lastly, the Lasso regression approach, which leverages regularization techniques, offered a unique perspective by incorporating variable selection and regularization into the forecasting process. It proved to be effective in handling multicollinearity and identifying significant predictors for mutual fund performance.Table-5 shows the accuracy results of different models Lasso Regression Model outperforms over Deep Learning and ARIMA model. Sustainability in investment refers to the practice of considering environmental, social, and governance (ESG) factors when making investment decisions. It goes beyond traditional financial analysis by evaluating how a company's operations and practices impact the planet, society, and its long-term performance. The goal of sustainable investing is to generate positive financial returns while also promoting positive outcomes for the environment and society. It is crucial to acknowledge that the choice of an appropriate forecasting model should consider multiple factors, such as the specific objectives, characteristics of the data, and the desired balance between accuracy and interpretability. Researchers and practitioners can leverage the insights gained from this study to make informed decisions when selecting a time-series forecasting model for Indian mutual fund performance analysis. Additionally, further research could explore ensemble techniques that combine the strengths of different models to enhance forecasting accuracy and robustness. [16] Wu, D., Ma, X., & Olson, D. L. (2022). Financial distress prediction using integrated Z-score and multilayer perceptron neural networks. Decision Support Systems, 159, 113814. [17] Isabona, J., Imoize, A. L., Ojo, S., Karunwi, O., Kim, Y., Lee, C. C., & Li, C. T. (2022).
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