Machine Learning For Trading 09 Time Series Models 05 Cointegration

Leo Migdal
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machine learning for trading 09 time series models 05 cointegration

There was an error while loading. Please reload this page. We have seen how a time series can have a unit root that creates a stochastic trend and makes the time series highly persistent. When we use such an integrated time series in their original, rather than in differenced, form as a feature in a linear regression model, its relationship with the outcome will often appear statistically significant,... This phenomenon is called spurious regression (for details, see Chapter 18 in Wooldridge, 2008). Therefore, the recommended solution is to difference the time series so they become stationary before using them in a model.

However, there is an exception when there are cointegration relationships between the outcome and one or more input variables. To understand the concept of cointegration, let's first remember that the residuals of a regression model are a linear combination of the inputs and the output series. Usually, the residuals of the regression of one integrated time series on one or more such series yields non-stationary residuals that are also integrated, and thus behave like a random walk. However, for some time series, this is not the case: the regression produces coefficients that yield a linear combination of the time series in the form of the residuals that are stationary, even though... Such time series are cointegrated. A non-technical example is that of a drunken man on a random walk accompanied by his dog (on a leash).

Both trajectories are non-stationary but cointegrated because the dog will occasionally revert to his owner. In the trading context, arbitrage constraints imply cointegration between spot and futures prices. In other words, a linear combination of two or more cointegrated series has a stable mean to which this linear combination reverts. This also applies when the individual series are integrated of a higher order and the linear combination reduces the overall order of integration. �w��8��k�� n������C�C�~�>���9��;��[%α������X�c s��v[}���� q{�����yf����]ay�8���F�f��n�AGBn�Lj��O�x��f�k�ߎ]}J�4��?6؎��#k-��=��N���8�#w8�p%���A�b�DZ� p�]��������v�� ���L�oX3;e)UL��m�>B�CW�c���Jt�E^ ����{,��y]Ff���{��CT�lEq,� 8�.�q3v��Tˋ�Y�v�3��ҧ �hj����;Ns���3�1�A`R}\�����V8�w�0p � ��R\@W�c�q��l�X�3 ����8=�4+p ���1t��qv�.;�ǹ�'�Ⱔ�]��vv��c��S�����u���x�x��1b���zs�Tz�r9��k��^�t3pL3n�4r�z@We� ��� ��q�fv��[@���r��`�83�ܾ�ͱ������-�y��K�o5�2��q���� �R2�2K���9�k���x���D7�d~���U�CW����W�%r콦e�[!�9���!���<�e��&���y4��������dA�G� ��aj�x+���{p�n���)8V7��"|g �K���4p�z8�0p\�sjB�����B�g'��W�]O[^s��o& zH�����1�� �,���9�����*�/����G�����k��yg�r����xO����?v1�ߤ|��,~��+�l�q�D���M\��Ð3��?�J2s�����1�y��&:0��Vw9}\ocnI~Gp#��!�����>|�]��ō�F��Y�[~]�y��E]�9��ZP�Gp3ǖ����9��񮰸��i�c�鐆�� �a%�q�V�����+���9f=28���K�c��8�~p�v�9������9� ����9:C�+4�1?n�Ym�ʩ4� ?��oX���;��<'���J�XL�c s,��h��:C`9N�����Yeslf����񦾯����G�� ]�Qc�m����,��xʘ�]љJ��<�c��>=y"ϱ���9T�}_�?8�����k{rغ�� ��m�Y&�� �8����[r��w��&L��x����;� U�V � ���4qL��՗ҋ +Ь5�����ɻ ��q�V,�������αE`!p �U˱��4t8V?Ǩ�cp �%̱AP�1'��+4�qvr���n{�K!z9�... #N!8V5��� Lj ���1��ku�]�q�����͈ �e2~��|����p�p��A4�17� �)�v�<���=8�����?j��D�f������O�+:S5 ��E���s�n��zp S'�b��h��2��߀����c1����;� U�V�\�Kk�cN� �k�~�,P��I�|�?���Ʈ��_Sǘ?�Ӷ�wGQ��]��l��G4R9 ���18Γ����V���8��s)���� ��a]g?Q��o(�c����q�,1 ·�)�c|��eǚ��$o�p�j<�3�:�B������گ���n,?�#�I�⹤�ӡ!�}�dž�Yv�b8Ƽc���oS��Ə������z��z+5J6�[nr{���p S ��x� ��7�Yp�4 ��w�r�?3�ñ�3լ��6�ӣ9���m �D�yۼ����B�Aʩ��;S��cS�XK�F_���MFP��oz�d�On[���h�0�pl�����#�k­M��Ec]?�~��o��\��B7�qf����#����y�x��>֝ �F�mܹ���v�}X��Uq�T�No�[_G! ]z�X���m2S���N7�Kj�VΡ��3i��B�(8�� ��8?���T!�k�p��A�q>�੉N��(�a��f� ��ah;ݞeي����qa8���� ��yb���q� ��Z�X�YvAM����U��*�6q=w����l�t�{�x��3{+�c�@zp�]��q�Q�)�)fᒳ��;I� �|�/�h���B� �6��GT7�ֶ�M�R�cI���׹H�z�]�p��X�5f���K� ��ro��Ծ�+R�%�7��0=�(�&)�cB=O��X�������eT�ة�W��q��o�cX p,���ڠ+���-���$r����%�q�3�I�|=O�9dߓ��^p����~'D��|�8F�48�m:/ps\��~��G�sx}\=��7�1L �?�a���^�Uϱ0�c�s�E��X�c~p\:8F=�c8V�a΄~�s����9�q.Ewo@lj���6� r��!"@��;U��_�:Lj� �rة��|��D��z�Ωy�28漮�,�plXlZ ���q�w�q��1:p���vd�xl������6�+�k�yҝ� �µ\�?hS�WSL׌�����9F=����-�y��q�) 4а,r��� Wr...

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Our goal is to identify systematic patterns in time series that help us predict how the time series will behave in the future. More specifically, we focus on models that extract signals from a historical sequence of the output and, optionally, other contemporaneous or lagged input variables to predict future values of the output. For example, we might try to predict future returns for a stock using past returns, combined with historical returns of a benchmark or macroeconomic variables. We focus on linear time-series models before turning to nonlinear models like recurrent or convolutional neural networks in Part 4. Time-series models are very popular given the time dimension inherent to trading. Key applications include the prediction of asset returns and volatility, as well as the identification of co-movements of asset price series.

Time-series data are likely to become more prevalent as an ever-broader array of connected devices collects regular measurements with potential signal content. We first introduce tools to diagnose time-series characteristics and to extract features that capture potential patterns. Then we introduce univariate and multivariate time-series models and apply them to forecast macro data and volatility patterns. We conclude with the concept of cointegration and how to apply it to develop a pairs trading strategy. Most of the examples in this section use data provided by the Federal Reserve that you can access using the pandas datareader that we introduced in Chapter 2, Market and Fundamental Data. Sarah Lee AI generated o4-mini 6 min read · April 19, 2025

Time series data often exhibit common trends and shared long‑run relationships. Traditional regression of non‑stationary series can give spurious results, but cointegration unlocks the ability to model these relationships properly. In this article, we dive into the theory, step‑by‑step methods, and practical R/Python implementations for: Definition: Two or more non‑stationary series XtX_tXt​ and YtY_tYt​ are cointegrated if a linear combination, Zt=Yt−βXt,Z_t = Y_t - \beta X_t,Zt​=Yt​−βXt​, is stationary (I(0)I(0)I(0)) even though XtX_tXt​ and YtY_tYt​ are I(1)I(1)I(1) (unit root processes).

We briefly mentioned cointegration in the previous section on multivariate time-series models. Let's now explain this concept and how to diagnose its presence in more detail before leveraging it for a statistical arbitrage trading strategy. We have seen how a time series can have a unit root that creates a stochastic trend and makes the time series highly persistent. When we use such an integrated time series in their original, rather than in differenced, form as a feature in a linear regression model, its relationship with the outcome will often appear statistically significant,... This phenomenon is called spurious regression (for details, see Chapter 18, CNNs for Financial Time Series and Satellite Images, in Wooldridge, 2008). Therefore, the recommended solution is to difference the time series so they become stationary before using them in a model.

However, there is an exception when there... Cointegrated Time Series Analysis for Mean Reversion Trading with R Cointegrated Time Series Analysis for Mean Reversion Trading with R A while back we considered a trading model based on the application of the ARIMA and GARCH time series models to daily S&P500 data. We mentioned in that article as well as other previous time series analysis articles that we would eventually be considering mean reverting trading strategies and how to construct them. In this article I want to discuss a topic called cointegration, which is a time series concept that allows us to determine if we are able to form a mean reverting pair of assets.

We will cover the time series theory related to cointegration here and in the next article we will show how to apply that to real trading strategies using the new open source backtesting framework:... We will proceed by discussing mean reversion in the traditional "pairs trading" framework. This will lead us to the concept of stationarity of a linear combination of assets, ultimately leading us to cointegration and unit root tests. Once we have outlined these tests we will simulate various time series in the R statistical environment and apply the tests in order to assess cointegration. There was an error while loading. Please reload this page.

The concept of an integrated multivariate series is complicated by the fact that all the component series of the process may be individually integrated but the process is not jointly integrated in the sense... In other words, a combination of two co-integrated series has a stable mean to which this linear combination reverts. A multivariate series with this characteristic is said to be co-integrated. This also applies when the individual series are integrated of a higher order and the linear combination reduces the overall order of integration. Cointegration is different from correlation: two series ...

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There was an error while loading. Please reload this page. We have seen how a time series can have a unit root that creates a stochastic trend and makes the time series highly persistent. When we use such an integrated time series in their original, rather than in differenced, form as a feature in a linear regression model, its relationship with the outcome will often appear statistically signific...

However, There Is An Exception When There Are Cointegration Relationships

However, there is an exception when there are cointegration relationships between the outcome and one or more input variables. To understand the concept of cointegration, let's first remember that the residuals of a regression model are a linear combination of the inputs and the output series. Usually, the residuals of the regression of one integrated time series on one or more such series yields ...

Both Trajectories Are Non-stationary But Cointegrated Because The Dog Will

Both trajectories are non-stationary but cointegrated because the dog will occasionally revert to his owner. In the trading context, arbitrage constraints imply cointegration between spot and futures prices. In other words, a linear combination of two or more cointegrated series has a stable mean to which this linear combination reverts. This also applies when the individual series are integrated ...

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