Working Papers

The WNE Working Papers series has been published by the Faculty of Economic Sciences at the University of Warsaw since 2008.

The WNE Working Papers series provides a fast, open channel for disseminating research conducted at the Faculty of Economic Sciences, University of Warsaw. The papers hosted here are working versions (pre-prints) and may evolve as authors refine their analyses, incorporate feedback, or progress through formal peer-review. What you read is the current version released by the authors, timestamped and assigned DOI/ISSN identifiers to ensure precise citation and version tracking. Copyright remains with the authors, who may, at any time, upload a revised file or add a note directing readers to a later, peer-reviewed publication.

The Working Papers series accepts articles by research employees of the Faculty and publications from conferences organised at the Faculty of Economic Sciences at the University of Warsaw. Articles should be original research papers which have not been previously published, on the subject of economics.

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Number of Results 16

WP(27/2024)463. Informer in Algorithmic Investment Strategies on High Frequency Bitcoin Data

Authors: Filip Stefaniuk, Ślepaczuk Robert
The article investigates the usage of Informer architecture for building automated trading strategies for high frequency Bitcoin data. Three strategies using Informer model with different loss functions: Root Mean Squared Error (RMSE), Generalized Me…

WP463. Informer in Algorithmic Investment Strategies on High Frequency Bitcoin Data

The article investigates the usage of Informer architecture for building automated trading strategies for high frequency Bitcoin data. Three strategies using Informer model with different loss functions: Root Mean Squared Error (RMSE), Generalized Mean Absolute Directional Loss (GMADL) and Quantile loss, are proposed and evaluated against the Buy and Hold benchmark and two benchmark strategies based on technical indicators. The evaluation is conducted using data of various frequencies: 5 minute, 15 minute, and 30 minute intervals, over the 6 different periods. Although the Informer-based model with Quantile loss did not outperform the benchmark, two other models achieved better results. The performance of the model using RMSE loss worsens when used with higher frequency data while the model that uses novel GMADL loss function is benefiting from higher frequency data and when trained on 5 minute interval it beat all the other strategies on most of the testing periods. The primary contribution of this study is the application and assessment of the RMSE, GMADL and Quantile loss functions with the Informer model to forecast future returns, subsequently using these forecasts to develop automated trading strategies. The research provides evidence that employing an Informer model trained with the GMADL loss function can result in superior trading outcomes compared to the buy-and-hold approach.


DOI: https://doi.org/10.33138/2957-0506.2024.27.463
Filip Stefaniuk, Ślepaczuk Robert
Machine Learning Financial Series Forecasting Automated Trading Strategy Informer Transformer Bitcoin High Frequency Trading Statistics GMADL

WP(13/2024)449. The Hybrid Forecast of S&P 500 Volatility ensembled from VIX, GARCH and LSTM models

Authors: Natalia Roszyk, Ślepaczuk Robert
Predicting the S&P 500 index's volatility is crucial for investors and financial analysts as it helps in assessing market risk and making informed investment decisions. Volatility represents the level of uncertainty or risk related to the siz…

WP449. The Hybrid Forecast of S&P 500 Volatility ensembled from VIX, GARCH and LSTM models

Predicting the S&P 500 index's volatility is crucial for investors and financial analysts as it helps in assessing market risk and making informed investment decisions. Volatility represents the level of uncertainty or risk related to the size of changes in a security's value, making it an essential indicator for financial planning. This study explores four methods to improve the accuracy of volatility forecasts for the S&P 500: the established GARCH model, known for capturing historical volatility patterns; an LSTM network that utilizes past volatility and log returns; a hybrid LSTM-GARCH model that combines the strengths of both approaches; and an advanced version of the hybrid model that also factors in the VIX index to gauge market sentiment. This analysis is based on a daily dataset that includes data for S&P 500 and VIX index, covering the period from January 3, 2000, to December 21, 2023. Through rigorous testing and comparison, we found that machine learning approaches, particularly the hybrid LSTM models, significantly outperform the traditional GARCH model. The inclusion of the VIX index in the hybrid model further enhances its forecasting ability by incorporating real-time market sentiment. The results of this study offer valuable insights for achieving more accurate volatility predictions, enabling better risk management and strategic investment decisions in the volatile environment of the S&P 500.


DOI: https://doi.org/10.33138/2957-0506.2024.13.449
Natalia Roszyk, Ślepaczuk Robert
volatility forecasting LSTM-GARCH S&P 500 index hybrid forecasting models VIX index machine learning financial time series analysis walk-forward process hyperparameters tuning deep learning recurrent neural networks

WP(27/2023)434. Predicting DJIA, NASDAQ and NYSE index prices using ARIMA and VAR models

Authors: Teymurzade Sahil, Ślepaczuk Robert
This paper implements automated trading strategies with buy/sell signals based on Autoregressive Integrated Moving Average (ARIMA) and Vector autoregression (VAR) models. ARIMA and VAR models are compared based on several forecast error measures and …

WP434. Predicting DJIA, NASDAQ and NYSE index prices using ARIMA and VAR models

This paper implements automated trading strategies with buy/sell signals based on Autoregressive Integrated Moving Average (ARIMA) and Vector autoregression (VAR) models. ARIMA and VAR models are compared based on several forecast error measures and investment performance statistics. The data used in this thesis are daily closing prices of Dow Jones Industrial Average, NASDAQ Composite and NYSE Composite indices. The trading period covers 20 years of data from 2000-11-30 to 2020-11-30. The sensitivity analysis is made by changing the initial parameters to test how robust the methods are to these changes. Results show that although ARIMA model performed remarkably well during the volatile periods, VAR based strategy had better investment performance and was less robust to the changes compared to the ARIMA based strategy. Additionally, we have found that error metrics might be insufficient to evaluate performance of forecasting models, as VAR with higher forecast errors outperformed ARIMA model in algorithmic trading strategies.


Teymurzade Sahil, Ślepaczuk Robert
ARIMA model VAR model time series analysis algorithmic trading strategies investment systems statistical models forecasting stock prices

WP(20/2023)427. Ensembling ARIMAX Model in Algorithmic Investment Strategies on Commodities Market

Authors: Jakubowski Paweł, Ślepaczuk Robert, Windorbski Franciszek
This paper presents the results of investment strategies based on predictions from an ARIMA with exogenous variables (ARIMAX/ARIMAX-Garch) model, using the prices of selected commodities and companies from the DJIA index as explanatory variables. The…

WP427. Ensembling ARIMAX Model in Algorithmic Investment Strategies on Commodities Market

This paper presents the results of investment strategies based on predictions from an ARIMA with exogenous variables (ARIMAX/ARIMAX-Garch) model, using the prices of selected commodities and companies from the DJIA index as explanatory variables. The explained variables are four Invesco ETF funds (DBE, DBA, DBP, DBB) corresponding to baskets of energy, agricultural, precious, and industrial metals. The models are optimized using the Walk-Forward technique, and the selection of exogenous variables is based on Granger causality tests. By analyzing the results, we conclude that ARIMAX/ARIMAX-Garch models are not useful tools for making buy or sell decisions for the selected commodity baskets. Out of the 80 estimated models, 44 outperform the Buy & Hold strategy, however, none achieved statistically significant results. Combining individual models into an investment portfolio reduced the risk without significantly reducing the profit, enabling us to consistently beat the benchmark. We also observe that using returns of commodities listed on stock exchanges is more effective than using stock returns. Sensitivity analysis shows instability in results with changes in the length of the training and testing windows. The highest annual return rate of 15.37% from 02.01.2008 to 01.12.2022 was characterized by an ARIMAX model with one commodity exogenous variable.


Jakubowski Paweł, Ślepaczuk Robert, Windorbski Franciszek
ARIMA(X) GARCH ARIMA(X)/GARCH Algorithmic Investment Strategies Granger Causality Investment Performance Evaluation Trading Systems Forecasting Models

WP(17/2023)424. Optimal Markowitz Portfolio Using Returns Forecasted with Time Series and Machine Learning Models

Authors: Ślusarczyk Damian, Ślepaczuk Robert
We aim to answer the question of whether using forecasted stock returns based on machine learning and time series models in a mean-variance portfolio framework yields better results than relying on historical returns. Nevertheless, the problem of the…

WP424. Optimal Markowitz Portfolio Using Returns Forecasted with Time Series and Machine Learning Models

We aim to answer the question of whether using forecasted stock returns based on machine learning and time series models in a mean-variance portfolio framework yields better results than relying on historical returns. Nevertheless, the problem of the efficient stock selection has been tested for more than 50 years, the issue of adequate construction of mean-variance portfolio framework and incorporating forecasts of returns in it has not been solved yet. Stock returns portfolios were created using ’raw’ historical returns and forecasted return based on ARIMA-GARCH and the XGBoost models. Two optimization problems were concerned: global maximum information ratio and global mini-mum variance. Then strategies were compared with two benchmarks – an equally weighted portfolio and buy and hold on the DJIA index. Strategies were tested on Dow Jones Industrial Average stocks in the period from 2007-01-01 to 2022-12-31 and daily data was used. The main portfolio performance metrics were information ratio* and information ratio**. The results showed that using forecasted returns we can enhance our portfolio selection based on Markowitz framework, but it is not a universal solution, and we have to control all the parameters and hyperparameters of selected models.


Ślusarczyk Damian, Ślepaczuk Robert
Algorithmic Investment Strategies Markowitz framework portfolio optimization forecasting ARIMA GARCH XGBoost minimum variance

WP(30/2022)406. Quantile regression analysis to predict GDP distribution using data from the US and UK

Authors: Tran Thi Huyen, Ślepaczuk Robert
This paper aims to find the best models to forecast one-quarter-ahead and one-year-ahead US and UK real GDP growth distributions by employing quantile regression with skewed-t distribution on different sets of relevant near-term predictors. The resea…

WP406. Quantile regression analysis to predict GDP distribution using data from the US and UK

This paper aims to find the best models to forecast one-quarter-ahead and one-year-ahead US and UK real GDP growth distributions by employing quantile regression with skewed-t distribution on different sets of relevant near-term predictors. The research data period starts in 1947Q1/1955Q1 for US/UK data and ends in 2021Q3/2020Q4 for one-quarter-ahead/one-year-ahead prediction. The out-of-sample period ranges from 1996Q3 to 2021Q3 for one-quarter-ahead prediction and to 2020Q4 for one-year-ahead forecasting. The author applies a two-step testing procedure, in which models with the lowest average error in out-of-sample period are selected to the next step where the cumulative distribution functions of probability integral transforms are computed for the out-of-sample period, to select the best models. The improvement in the final forecasts of the tested models results, among others, from the use of new macroeconomic data with a higher frequency and focusing on the specific properties of the tested models separately for the US and UK. The chosen best models indicate that there exist better models than the model proposed by Adrian et al. (2016) to predict US growth distributions and that near-term predictors can produce good UK growth forecasts. Additionally, some simplified models associated with significantly lower portion of model risk are detected to produce meaningful forecasts for both US and UK case. For the US data, there exist several models that can produce timely predicted results.


Tran Thi Huyen, Ślepaczuk Robert
GDP growth density forecast quantile regression US GDP UK GDP cumulative distribution function probability integral transform out-of-sample forecasting

WP(29/2022)405. The efficiency of various types of input layers of LSTM model in investment strategies on S&P500 index

Authors: Nguyen Thi Thu Giang, Ślepaczuk Robert
The study compares the use of various Long Short-Term Memory (LSTM) variants to conventional technical indicators for trading the S&P 500 index between 2011 and 2022. Two methods were used to test each strategy: a fixed training data set from 200…

WP405. The efficiency of various types of input layers of LSTM model in investment strategies on S&P500 index

The study compares the use of various Long Short-Term Memory (LSTM) variants to conventional technical indicators for trading the S&P 500 index between 2011 and 2022. Two methods were used to test each strategy: a fixed training data set from 2001–2010 and a rolling train–test window. Due to the input sensitivity of LSTM models, we concentrated on data processing and hyperparameter tuning to find the best model. Instead of using the traditional MSE function, we used the Mean Absolute Directional Loss (MADL) function based on recent research to enhance model performance. The models were assessed using the Information Ratio and the Modified Information Ratio, which considers the maximum drawdown and the sign of the annualized return compounded (ARC). LSTM models' performance was compared to benchmark strategies using the SMA, MACD, RSI, and Buy&Hold strategies. We rejected the hypothesis that algorithmic investment strategy using signals from LSTM model consisting only from daily returns in its input layer is more efficient. However, we could not reject the hypothesis that signals generated by LSTM model combining daily returns and technical indicators in its input layer are more efficient. The LSTM Extended model that combined daily returns with MACD and RSI in the input layer generated a better result than Buy&Hold and other strategies using a single technical indicator. The results of the sensitivity analysis show how sensitive this model is to inputs like sequence length, batch size, technical indicators, and the length of the rolling train - test window.


Nguyen Thi Thu Giang, Ślepaczuk Robert
algorithmic investment strategies machine learning testing architecture deep learning recurrent neural networks LSTM technical indicators forecasting financial-time series technical indicators hyperparameter tuning S&P 500 Index

WP(12/2022)388. Investment Portfolio Optimization Based on Modern Portfolio Theory and Deep Learning Models

Authors: Wysocki Maciej, Sakowski Paweł
This paper investigates an important problem of an appropriate variance-covariance matrix estimation in the Modern Portfolio Theory. In this study we propose a novel framework for variance-covariance matrix estimation for purposes of the portfolio op…

WP388. Investment Portfolio Optimization Based on Modern Portfolio Theory and Deep Learning Models

This paper investigates an important problem of an appropriate variance-covariance matrix estimation in the Modern Portfolio Theory. In this study we propose a novel framework for variance-covariance matrix estimation for purposes of the portfolio optimization, which is based on deep learning models. We employ the long short-term memory (LSTM) recurrent neural networks (RNN) along with two probabilistic deep learning models: DeepVAR and GPVAR to the task of one-day ahead multivariate forecasting. We then use these forecasts to optimize portfolios that consisted of stocks and cryptocurrencies. Our analysis presents results across different combinations of observation windows and rebalancing periods to compare performances of classical and deep learning variance-covariance estimation methods. The conclusions of the study are that although the strategies (portfolios) performance differed significantly between different combinations of parameters, generally the best results in terms of the information ratio and annualized returns are obtained using the LSTM-RNN models. Moreover, longer observation windows translate into better performance of the deep learning models indicating that these methods require longer windows to be able to efficiently capture the long-term dependencies of the variance-covariance matrix structure. Strategies with less frequent rebalancing typically perform better than these with the shortest rebalancing windows across all considered methods.


Wysocki Maciej, Sakowski Paweł
Portfolio Optimization Deep Learning Variance-Covariance Matrix Forecasting Investment Strategies Recurrent Neural Networks Long Short-Term Memory Neural Networks

WP(10/2020)316. Investing in VIX futures based on rolling GARCH models forecasts

Authors: Bilyk Oleh, Sakowski Paweł, Ślepaczuk Robert
The aim of this work is to compare the performance of VIX futures trading strategies built across different GARCH model volatility forecasting techniques. Long and short signals for VIX futures are produced by comparing one-day ahead volatility forec…

WP316. Investing in VIX futures based on rolling GARCH models forecasts

The aim of this work is to compare the performance of VIX futures trading strategies built across different GARCH model volatility forecasting techniques. Long and short signals for VIX futures are produced by comparing one-day ahead volatility forecasts with current historical volatility. We found out that using the daily data over the seven-year period (2013-2019), strategy based on the fGARCH-TGARCH and GJR-GARCH specifications outperformed those of the GARCH and EGARCH models, and performed slightly below the "buy-and-hold" S&P 500 strategy. For the base GARCH(1,1) model, the training window size and the type gave stable results, whereas the performance across refit frequency, conditional distribution of returns, and historical volatility estimators varies significantly. Despite non-robustness of some investment strategies and some space for improvements, the presented strategies show their potential in competing with the equity and volatility benchmarks.


Bilyk Oleh, Sakowski Paweł, Ślepaczuk Robert
GARCH VIX index volatility futures rolling forecasting volatility investment strategies volatility exposure

WP(6/2016)197. EWS-GARCH: New Regime Switching Approach to Forecast Value-at-Risk

Authors: Chlebus Marcin
In the study a proposal of two-step EWS-GARCH models to forecast Value-at-Risk is presented. The EWS-GARCH allows different distributions of returns to be used in Value-at-Risk forecasting depending on a forecasted state of the financial time series.…

WP197. EWS-GARCH: New Regime Switching Approach to Forecast Value-at-Risk

In the study a proposal of two-step EWS-GARCH models to forecast Value-at-Risk is presented. The EWS-GARCH allows different distributions of returns to be used in Value-at-Risk forecasting depending on a forecasted state of the financial time series. In the study EWS-GARCH with GARCH(1,1) and GARCH(1,1) with the amendment to the empirical distribution of returns as a Value-at-Risk model in a state of tranquillity and empirical tail, exponential or Pareto distributions used to forecast Value-at-Risk in a state of turbulence were considered. The evaluation of the quality of the Value-at-Risk forecasts was based on the Value-at-Risk forecasts adequacy (the excess ratio, the Kupiec test, the Christoffersen test, the asymptotic test of unconditional coverage and the back-testing criteria defined by the Basel committee) and the analysis of loss functions (the Lopez quadratic loss function, the Abad & Benito absolute loss function, the 3rd version of Caporin loss function and proposed in the study the function of excessive costs). Obtained results indicate that the EWS-GARCH models may improve the quality of the Value-at-Risk forecasts generated using benchmark models. However, the choice of best assumptions for an EWS-GARCH model should depend on the goals of the Value-at-Risk forecasting model. The final selection may depend on an expected level of adequacy, conservatism and costs of a model.


Chlebus Marcin
Value-at-Risk GARCH forecasting state of turbulence regime switching risk management risk measure market risk.
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