{"id":182259,"date":"2022-11-16T15:47:00","date_gmt":"2022-11-16T20:47:00","guid":{"rendered":"https:\/\/ibkrcampus.com\/traders-insight\/theil-mixed-estimator\/"},"modified":"2023-03-13T13:09:50","modified_gmt":"2023-03-13T17:09:50","slug":"theil-mixed-estimator","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/theil-mixed-estimator\/","title":{"rendered":"Theil Mixed Estimator"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This post deals with the Theil mixed estimator which uses the prior information as well as the sample information.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Theil mixed estimator incorpoate stochastic non-sample information into a linear model. This estimator mixes sample and prior information in a generalized least squares sense. Information regarding GLS estimator can be found at the following previous post.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/kiandlee.blogspot.com\/2022\/09\/gls-estimator-for-black-litterman-model.html\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Generalized Least Squares (GLS) estimator<\/strong><\/a><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-linear-model-with-sample-information\">Linear model with sample information<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Sample information is represented by the following linear model.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-1.png\" alt=\"Linear model with sample information\" class=\"wp-image-166699 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Linear model with sample and prior information<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Prior information has the following form<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-3.png\" alt=\"Linear model with sample and prior information\" class=\"wp-image-166723 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">where&nbsp;<em><strong>R<\/strong><\/em>&nbsp;is&nbsp;<em><strong>J \u00d7 K<\/strong><\/em>&nbsp;and&nbsp;<em><strong>r<\/strong><\/em>&nbsp;is&nbsp;<em><strong>J \u00d7 1<\/strong><\/em>.&nbsp;<em><strong>\u03bd<\/strong><\/em>&nbsp;is a&nbsp;<em><strong>J \u00d7 1<\/strong><\/em>&nbsp;normally distributed random error vector.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Incorporating the prior information into the sample information leads to the following model specification.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-2.png\" alt=\"Incorporating the prior information into the sample information leads to the following model specification.\" class=\"wp-image-166700 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">GLS estimator<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The GLS estimator is as follows.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-4.png\" alt=\"GLS estimator\" class=\"wp-image-166701 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">By using matrix multiplications, this result can be simplified to<\/p>\n\n\n\n<figure class=\"wp-block-image img-twothird\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-5.png\" alt=\"matrix multiplications\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">where&nbsp;<em>\u03d5<\/em>&nbsp;stands for the precision of the regression model :&nbsp;<em>\u03d5 = 1\/\u03c3<sup>2<\/sup><\/em>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">By inverting <img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-11.png\" alt=\"=\"\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\"> and its variance can be obtained.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-7.png\" alt=\"where \u03d5 stands for the precision of the regression model : \u03d5 = 1\/\u03c32\" class=\"wp-image-166704 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">This is the Theil mixed estimator.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Theil mixed estimator<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">More generally, when <img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-8.png\" alt=\"theil mixed\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\">is used instead of <img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-9.png\" alt=\"thiel mixed estimator\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\"> the Theil mixed estimator can also be represented<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/02\/sh-fintech-theil-mixed-estimator-10.png\" alt=\"Theil mixed estimator\" class=\"wp-image-166708 lazyload\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Concluding Remarks<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">This post derived the Theil mixed estimator which uses the prior information as well as the sample information. This formulation will be used when deriving the Black-Litterman model.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Visit the SH Fintech Modeling Blog to read more about this topic: <a href=\"https:\/\/kiandlee.blogspot.com\/2022\/09\/theil-mixed-estimator.html\">https:\/\/kiandlee.blogspot.com\/2022\/09\/theil-mixed-estimator.html<\/a>.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post deals with the Theil mixed estimator which uses the prior information as well as the sample information.<\/p>\n","protected":false},"author":662,"featured_media":182270,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[339,343,338,350,341,344,342],"tags":[4922,14132,14133],"contributors-categories":[13728],"class_list":["post-182259","post","type-post","status-publish","format-standard","has-post-thumbnail","category-data-science","category-programing-languages","category-ibkr-quant-news","category-quant-asia-pacific","category-quant-development","category-quant-regions","category-r-development","tag-econometrics","tag-generalized-least-squares-gls-estimator","tag-theil-mixed-estimator","contributors-categories-sh-fintech-modeling"],"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 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