{"id":214987,"date":"2024-11-12T12:53:37","date_gmt":"2024-11-12T17:53:37","guid":{"rendered":"https:\/\/ibkrcampus.com\/campus\/?p=214987"},"modified":"2024-11-12T12:53:19","modified_gmt":"2024-11-12T17:53:19","slug":"a-time-varying-parameter-vector-autoregression-model-with-stochastic-volatility-part-i","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/a-time-varying-parameter-vector-autoregression-model-with-stochastic-volatility-part-i\/","title":{"rendered":"A Time-Varying-Parameter Vector Autoregression Model with Stochastic Volatility &#8211; Part I"},"content":{"rendered":"\n<p>The basic Vector Autoregression (VAR) model is heavily used in macro-econometrics for explanatory purposes and forecasting purposes in trading. In recent years, a VAR model with time-varying parameters has been used to understand the interrelationships between macroeconomic variables. Since Primiceri (2005), econometricians have been applying these models using macroeconomic variables such as:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Japan time series (Nakahima, 2011)<\/li>\n\n\n\n<li>US Bond yields (Fischer et al., 2022)<\/li>\n\n\n\n<li>Monthly Stock Indices from industrialized countries (Gupta et al., 2020)<\/li>\n\n\n\n<li>Peruvian exchange rate (Rodriguez et al., 2024)<\/li>\n\n\n\n<li>Indian exchange rate (Kumar, M., 2010)<\/li>\n<\/ul>\n\n\n\n<p>This article extends the model usage to something our audience greatly cares about: trading! You\u2019ll learn the basics of the estimation procedure and how to create a trading strategy based on the model.<\/p>\n\n\n\n<p>Are you excited? I was when I started writing this article. Let me share what I\u2019ve learned with you!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-what-is-the-difference-between-a-basic-var-and-a-tvp-var-sv-model\">What is the difference between a basic VAR and a TVP-VAR-SV model?<\/h2>\n\n\n\n<p>All the explanations of the basic VAR can be found in our previous&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/vector-autoregression\/\">article<\/a>. Here, we\u2019ll provide the system of equations and compare them with our new model.<\/p>\n\n\n\n<p>Let\u2019s remember the basic model. For example, a basic bivariate VAR(1) can be described as a system of equations:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"843\" height=\"102\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-1.jpg\" alt=\"\" class=\"wp-image-214997 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-1.jpg 843w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-1-700x85.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-1-300x36.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-1-768x93.jpg 768w\" data-sizes=\"(max-width: 843px) 100vw, 843px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 843px; aspect-ratio: 843\/102;\" \/><\/figure>\n\n\n\n<p>Or,<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"822\" height=\"49\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-2.jpg\" alt=\"\" class=\"wp-image-214999 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-2.jpg 822w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-2-700x42.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-2-300x18.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-2-768x46.jpg 768w\" data-sizes=\"(max-width: 822px) 100vw, 822px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 822px; aspect-ratio: 822\/49;\" \/><\/figure>\n\n\n\n<p>Where<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"820\" height=\"288\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-3.jpg\" alt=\"\" class=\"wp-image-215001 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-3.jpg 820w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-3-700x246.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-3-300x105.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-3-768x270.jpg 768w\" data-sizes=\"(max-width: 820px) 100vw, 820px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 820px; aspect-ratio: 820\/288;\" \/><\/figure>\n\n\n\n<p>A time-varying parameter VAR would be something like the following:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"826\" height=\"100\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-4.jpg\" alt=\"\" class=\"wp-image-215002 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-4.jpg 826w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-4-700x85.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-4-300x36.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-4-768x93.jpg 768w\" data-sizes=\"(max-width: 826px) 100vw, 826px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 826px; aspect-ratio: 826\/100;\" \/><\/figure>\n\n\n\n<p>Do you get to see the difference between the two models? Not yet?<\/p>\n\n\n\n<p>Let\u2019s use matrices to see it clearly.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"822\" height=\"57\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-5.jpg\" alt=\"\" class=\"wp-image-215003 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-5.jpg 822w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-5-700x49.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-5-300x21.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-5-768x53.jpg 768w\" data-sizes=\"(max-width: 822px) 100vw, 822px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 822px; aspect-ratio: 822\/57;\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"838\" height=\"292\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-6.jpg\" alt=\"\" class=\"wp-image-215004 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-6.jpg 838w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-6-700x244.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-6-300x105.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-6-768x268.jpg 768w\" data-sizes=\"(max-width: 838px) 100vw, 838px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 838px; aspect-ratio: 838\/292;\" \/><\/figure>\n\n\n\n<p>Now you see it?<\/p>\n\n\n\n<p>The only difference is that the model&#8217;s parameters vary as time passes. Hence, it\u2019s referred to as a &nbsp;\u201ctime-varying-parameter\u201d model.<\/p>\n\n\n\n<p>Even though the difference appears simple, &nbsp;the estimation procedure is much more complex than the basic VAR estimation.<\/p>\n\n\n\n<p>You now say: I know we can have time-varying parameters, but where is the stochastic volatility in the previous equations?<\/p>\n\n\n\n<p>Wait for it, my friend! We\u2019ll see it later!<\/p>\n\n\n\n<p>Don\u2019t worry, we\u2019ll keep it simple!<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"the-tvp-var-sv-model-variables\">The TVP-VAR-SV model variables<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"the-system-of-equations-of-the-model\">The system of equations of the model<\/h3>\n\n\n\n<p>Using a new notation provided by Primiceri (2005):<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"823\" height=\"64\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-7.jpg\" alt=\"\" class=\"wp-image-215005 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-7.jpg 823w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-7-700x54.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-7-300x23.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-7-768x60.jpg 768w\" data-sizes=\"(max-width: 823px) 100vw, 823px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 823px; aspect-ratio: 823\/64;\" \/><\/figure>\n\n\n\n<p>Where:<\/p>\n\n\n\n<p>Y: The vector of time series<\/p>\n\n\n\n<p>B: The parameters of the lagged time series of this reduced model<\/p>\n\n\n\n<p>A: The contemporary parameters of the time series vector<\/p>\n\n\n\n<p>Sigma: The time-varying standard deviation (volatility) of each equation in the VAR.<\/p>\n\n\n\n<p>Epsilon: A vector of shocks of each equation in the VAR.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"what-is-the-reduced-model-and-what-are-contemporary-parameters\">What is the reduced model and what are contemporary parameters?<\/h3>\n\n\n\n<p>Well, in macroeconometrics, the reduced model can be understood as a simple VAR as modeled in our previous&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/vector-autoregression\/\">article<\/a>. In this model, today\u2019s time series values of the VAR vector are impacted only by their lag versions.<\/p>\n\n\n\n<p>However, economists also talk about the impact that the same today\u2019s time series values have on each other today\u2019s time series values. This can be modeled as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"825\" height=\"56\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-8.jpg\" alt=\"\" class=\"wp-image-215006 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-8.jpg 825w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-8-700x48.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-8-300x20.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-8-768x52.jpg 768w\" data-sizes=\"(max-width: 825px) 100vw, 825px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 825px; aspect-ratio: 825\/56;\" \/><\/figure>\n\n\n\n<p>This can shown as a matrix below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"826\" height=\"91\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-9.jpg\" alt=\"\" class=\"wp-image-215008 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-9.jpg 826w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-9-700x77.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-9-300x33.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-9-768x85.jpg 768w\" data-sizes=\"(max-width: 826px) 100vw, 826px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 826px; aspect-ratio: 826\/91;\" \/><\/figure>\n\n\n\n<p>Which can also be presented as a system of equations:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"827\" height=\"77\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-10.jpg\" alt=\"\" class=\"wp-image-215010 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-10.jpg 827w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-10-700x65.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-10-300x28.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-10-768x72.jpg 768w\" data-sizes=\"(max-width: 827px) 100vw, 827px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 827px; aspect-ratio: 827\/77;\" \/><\/figure>\n\n\n\n<p>The above model is understood in econometrics as a structural model to comprehend the time series interrelationships, contemporary or not, between the time series analyzed.<\/p>\n\n\n\n<p>So, assuming we have daily data, the first question, which belongs to y1, has a12*y2 as today\u2019s y2 impact on today\u2019s y1. The same is true for the second question, which belongs to y2, where we see a21*y1, which is today\u2019s time series y1 impact on y2. In a VAR, we have lag periods impacting today\u2019s variables, in a structural VAR we have today\u2019s variables impacting today\u2019s other variables.<\/p>\n\n\n\n<p>Due to these contemporary relationships, there is a problem called endogeneity, where the error terms epsilons are correlated with Y_t-1. To estimate a structural VAR, we need to clearly identify the matrix A variables. As Eric (2021) explained, there are 3 ways in the economic literature. But it\u2019s not only that, as per this model, A is also time-varying. We\u2019ll see later how this variables are estimated.<\/p>\n\n\n\n<p>When you pre-multiply the system of equations by A^-1, you get something like:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"795\" height=\"60\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-11.jpg\" alt=\"\" class=\"wp-image-215011 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-11.jpg 795w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-11-700x53.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-11-300x23.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-11-768x58.jpg 768w\" data-sizes=\"(max-width: 795px) 100vw, 795px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 795px; aspect-ratio: 795\/60;\" \/><\/figure>\n\n\n\n<p>Which can be further simplified as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"817\" height=\"61\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-12.jpg\" alt=\"\" class=\"wp-image-215013 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-12.jpg 817w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-12-700x52.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-12-300x22.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-12-768x57.jpg 768w\" data-sizes=\"(max-width: 817px) 100vw, 817px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 817px; aspect-ratio: 817\/61;\" \/><\/figure>\n\n\n\n<p>So,<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"818\" height=\"81\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-13.jpg\" alt=\"\" class=\"wp-image-215015 lazyload\" style=\"--smush-placeholder-width: 818px; aspect-ratio: 818\/81;width:950px;height:auto\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-13.jpg 818w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-13-700x69.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-13-300x30.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-13-768x76.jpg 768w\" data-sizes=\"(max-width: 818px) 100vw, 818px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"time-varying-volatilities\">Time-varying volatilities?<\/h3>\n\n\n\n<p>Yes! In a basic VAR, the error terms are homoskedastic, meaning, they present constant variance. In this case, we have variances that change over time; they\u2019re time-variant.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"the-time-varying-parameter-stochastic-behaviors\">The time-varying parameter stochastic behaviors<\/h3>\n\n\n\n<p>The basic VAR had its parameters constant. In this TVP-VAR-SV, we have almost all of our parameters time-variant. Due to this, we need to assign them stochastic processes. As in Primiceri (2005), we define them as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img decoding=\"async\" width=\"825\" height=\"114\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-14.jpg\" alt=\"\" class=\"wp-image-215017 lazyload\" style=\"--smush-placeholder-width: 825px; aspect-ratio: 825\/114;width:825px;height:auto\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-14.jpg 825w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-14-700x97.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-14-300x41.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-14-768x106.jpg 768w\" data-sizes=\"(max-width: 825px) 100vw, 825px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p>We can then specify the matrix of variances of all the model\u2019s shocks as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"833\" height=\"142\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-15.jpg\" alt=\"\" class=\"wp-image-215018 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-15.jpg 833w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-15-700x119.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-15-300x51.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-15-768x131.jpg 768w\" data-sizes=\"(max-width: 833px) 100vw, 833px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 833px; aspect-ratio: 833\/142;\" \/><\/figure>\n\n\n\n<p>Where I_n is the identity matrix and n is the number of time series in the VAR (in our case it\u2019s 2). Q, S, and W are square positive-definite covariance matrices with a number of rows (or columns) equal to the number of parameters in B, A, and Sigma, respectively.<\/p>\n\n\n\n<p>Something else to note: sigma is stochastic-based, which can be interpreted as stochastic volatility as, e.g., the Heston&nbsp;<a href=\"https:\/\/blog.quantinsti.com\/heston-model\/\">model<\/a>&nbsp;is.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"the-priors\">The priors<\/h2>\n\n\n\n<p>For a Bayesian inference, you always need priors. In the Primiceri (2005) algorithm, the priors are computed using your data sample&#8217;s first \u201cT1\u201d observations.<\/p>\n\n\n\n<p>Using our previously defined variables, you can specify the priors (following Primiceri, 2005, and Del Negro and Primiceri, 2015):<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"858\" height=\"216\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-16.jpg\" alt=\"\" class=\"wp-image-215020 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-16.jpg 858w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-16-700x176.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-16-300x76.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-16-768x193.jpg 768w\" data-sizes=\"(max-width: 858px) 100vw, 858px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 858px; aspect-ratio: 858\/216;\" \/><\/figure>\n\n\n\n<p>Where<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>N(): Normal distribution<\/li>\n\n\n\n<li>B_ols: This is the point estimate of the B parameters obtained by estimating a basic time-invariant VAR using the first T1 observations of the data sample.<\/li>\n\n\n\n<li>V(B_ols): This is the point estimate of the B parameters\u2019 variances obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample. In B_0, the variance is multiplied by 4. This value can be named k_B.<\/li>\n\n\n\n<li>A_ols: This is the point estimate of the A parameters obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample.<\/li>\n\n\n\n<li>V(A_ols): This is the point estimate of the A parameters\u2019 variances obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample. In A_0, this variance is multiplied by 4. This value can be named k_A.<\/li>\n\n\n\n<li>log(sigma_0): This is the point estimate of the standard errors obtained by estimating a basic time-invariant structural VAR using the first T1 observations of the data sample.}<\/li>\n\n\n\n<li>I_n: This is the identity matrix with \u201cnxn\u201d dimensions, where \u201cn\u201d is the number of time series used to estimate the VAR on them. Contrary to to A_0 and B_0, this variance is just multiplied by 1, where this value can be named k_sig.<\/li>\n\n\n\n<li>IW: The inverse Wishart distribution<\/li>\n\n\n\n<li>Q_0 follows an IW distribution with a scale matrix of k_Q^2 times 40 times V(B_ols) and 40 degrees of freedom<\/li>\n\n\n\n<li>W_0 follows an IW distribution with a scale matrix of k_W^2 times 2 times V(B_ols) and 2 degrees of freedom<\/li>\n\n\n\n<li>Q_0 follows an IW distribution with a scale matrix of k_S^2 times 2 times V(B_ols) and 2 degrees of freedom<\/li>\n\n\n\n<li>k_Q^2, k_W^2 and k_S^2 are 1, 0.01 and 0.1, respectively.<\/li>\n<\/ul>\n\n\n\n<p>Once you estimate the priors with the first T1 observations, then you get the posterior distribution using the rest of the data sample.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"the-mixture-of-indicators\">The mixture of indicators<\/h2>\n\n\n\n<p>Before we dive into the algorithm, let\u2019s learn something else. Do you remember the reduced-form model:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"830\" height=\"50\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-17.jpg\" alt=\"\" class=\"wp-image-215021 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-17.jpg 830w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-17-700x42.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-17-300x18.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-17-768x46.jpg 768w\" data-sizes=\"(max-width: 830px) 100vw, 830px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 830px; aspect-ratio: 830\/50;\" \/><\/figure>\n\n\n\n<p>To clear the error term, we get<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"827\" height=\"54\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-18.jpg\" alt=\"\" class=\"wp-image-215022 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-18.jpg 827w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-18-700x46.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-18-300x20.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-18-768x50.jpg 768w\" data-sizes=\"(max-width: 827px) 100vw, 827px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 827px; aspect-ratio: 827\/54;\" \/><\/figure>\n\n\n\n<p>Primiceri (2005), appendix A.2 explains that the above model has a Gaussian non-linear state space representation. The difficulty with drawing Sigma_t is that they enter the model multiplicatively.<\/p>\n\n\n\n<p>This presents the issue of not making it easy for the Kalman filter estimation done inside the whole estimation algorithm (The Kalman filter is linear-based). To overcome this issue, Primiceri (2005) applies squaring and takes the logarithms of every element of the previous equation. As a consequence of this transformation, the resulting state-space form becomes non-Gaussian, because the log(epsilon_t^2) has a&nbsp;<a href=\"https:\/\/www.oreilly.com\/library\/view\/bayesian-statistics-an\/9781118359778\/OEBPS\/c1-sec1-0007.htm\">log chi-squared distribution<\/a>. To finally get a normal distribution for the error terms, Kim et al. (1998) use a mixture of normals to approximate each element of log(epsilon_t^2). Thus, the estimation algorithm uses the mixture indicators for each error term and each date.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"808\" height=\"55\" data-src=\"https:\/\/www.interactivebrokers.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-19.jpg\" alt=\"\" class=\"wp-image-215024 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-19.jpg 808w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-19-700x48.jpg 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-19-300x20.jpg 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2024\/11\/quantinsti-time-varying-parameter-autoregression-19-768x52.jpg 768w\" data-sizes=\"(max-width: 808px) 100vw, 808px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 808px; aspect-ratio: 808\/55;\" \/><\/figure>\n\n\n\n<p><\/p>\n\n\n\n<p><em>Stay tuned for the next installment to read about the TVP-VAR-SV model estimation algorithm.<\/em><\/p>\n\n\n\n<p><em>Originally posted on <a href=\"https:\/\/blog.quantinsti.com\/time-varying-parameter-var-model-stochastic-volatility\/\">QuantInsti<\/a> blog.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The basic Vector Autoregression (VAR) model is heavily used in macro-econometrics for explanatory purposes and forecasting purposes in trading.<\/p>\n","protected":false},"author":825,"featured_media":169675,"comment_status":"open","ping_status":"closed","sticky":true,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[339,338,341],"tags":[806,4922,3918,18006,18005],"contributors-categories":[13654],"class_list":{"0":"post-214987","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-data-science","8":"category-ibkr-quant-news","9":"category-quant-development","10":"tag-data-science","11":"tag-econometrics","12":"tag-financial-modeling","13":"tag-stochastic-volatility","14":"tag-time-varying-parameter-vector-autoregression-model","15":"contributors-categories-quantinsti"},"pp_statuses_selecting_workflow":false,"pp_workflow_action":"current","pp_status_selection":"publish","acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v26.9 (Yoast SEO v27.5) - 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