{"id":187547,"date":"2023-03-29T09:43:20","date_gmt":"2023-03-29T13:43:20","guid":{"rendered":"https:\/\/ibkrcampus.com\/?p=187547"},"modified":"2023-03-29T09:43:36","modified_gmt":"2023-03-29T13:43:36","slug":"generalized-least-squares-gls-estimator","status":"publish","type":"post","link":"https:\/\/www.interactivebrokers.com\/campus\/ibkr-quant-news\/generalized-least-squares-gls-estimator\/","title":{"rendered":"Generalized Least Squares (GLS) Estimator"},"content":{"rendered":"\n<p>This post deals with the generalized least squares (GLS) estimator. When deriving the Black-Litterman (BL) model, the Theil mixed estimator is used, which is a kind of GLS.<\/p>\n\n\n\n<p><strong>Generalized Least Squares (GLS)<\/strong><\/p>\n\n\n\n<p>Black-Litterman (BL) model is based on Theil mixed estimator which is a kind of the GLS estimator.<\/p>\n\n\n\n<p>Since BL model combines two regressions (market + views) which have different variance terms respectively, the conbined BL regression model has two different variance terms naturally. Unlike OLS with a constant (homogenous) variance term, when variance terms are not constant (heterogeneous), it is natural to apply the GLS estimator to get estimated parameters.<\/p>\n\n\n\n<p>The starting point is a linear model.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"h-linear-model\">Linear model<\/h3>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"813\" height=\"195\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-1.png\" alt=\"\" class=\"wp-image-187550 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-1.png 813w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-1-700x168.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-1-300x72.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-1-768x184.png 768w\" data-sizes=\"(max-width: 813px) 100vw, 813px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 813px; aspect-ratio: 813\/195;\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">OLS estimator<\/h3>\n\n\n\n<p>As we are familar, OLS estimator has the following form with a constant variance term across all observations.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"824\" height=\"57\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-2.png\" alt=\"\" class=\"wp-image-187551 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-2.png 824w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-2-700x48.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-2-300x21.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-2-768x53.png 768w\" data-sizes=\"(max-width: 824px) 100vw, 824px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 824px; aspect-ratio: 824\/57;\" \/><\/figure>\n\n\n\n<p>Hance, OLS estimator is of the following formula.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"833\" height=\"115\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-3.png\" alt=\"\" class=\"wp-image-187552 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-3.png 833w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-3-700x97.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-3-300x41.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-3-768x106.png 768w\" data-sizes=\"(max-width: 833px) 100vw, 833px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 833px; aspect-ratio: 833\/115;\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">GLS estimator<\/h3>\n\n\n\n<p>Our focus in on the GLS estimator for the linear regression model with non-constant variance terms (\u03a3 = \u03c3<sup>2<\/sup>\u03a9) across all observations, which means that residuals are heteroscedastic and\/or serially dependent.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"838\" height=\"82\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-4.png\" alt=\"\" class=\"wp-image-187554 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-4.png 838w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-4-700x68.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-4-300x29.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-4-768x75.png 768w\" data-sizes=\"(max-width: 838px) 100vw, 838px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 838px; aspect-ratio: 838\/82;\" \/><\/figure>\n\n\n\n<p>GLS estimator is of the following formula.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"819\" height=\"106\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-5.png\" alt=\"\" class=\"wp-image-187556 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-5.png 819w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-5-700x91.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-5-300x39.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-5-768x99.png 768w\" data-sizes=\"(max-width: 819px) 100vw, 819px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 819px; aspect-ratio: 819\/106;\" \/><\/figure>\n\n\n\n<p>or<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"801\" height=\"118\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-6.png\" alt=\"\" class=\"wp-image-187557 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-6.png 801w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-6-700x103.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-6-300x44.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-6-768x113.png 768w\" data-sizes=\"(max-width: 801px) 100vw, 801px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 801px; aspect-ratio: 801\/118;\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Derivation of GLS estimator<\/h3>\n\n\n\n<p>When&nbsp;\u03a9&nbsp;is symmetric, using eigen decomposition,&nbsp;\u03a9&nbsp;can be expressed as follows,<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"823\" height=\"178\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-7.png\" alt=\"\" class=\"wp-image-187558 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-7.png 823w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-7-700x151.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-7-300x65.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-7-768x166.png 768w\" data-sizes=\"(max-width: 823px) 100vw, 823px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 823px; aspect-ratio: 823\/178;\" \/><\/figure>\n\n\n\n<p>Here,&nbsp;A&nbsp;and&nbsp;\u039b&nbsp;are eigenvector and eigenvalue matrix respectively.<\/p>\n\n\n\n<p>Now&nbsp;\u03a9&nbsp;can be transformed into the identity matrix (I<sub>n<\/sub>) by multiplying&nbsp;P<sup>\u22121<\/sup>&nbsp;in both sides<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"793\" height=\"64\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-8.png\" alt=\"\" class=\"wp-image-187559 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-8.png 793w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-8-700x56.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-8-300x24.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-8-768x62.png 768w\" data-sizes=\"(max-width: 793px) 100vw, 793px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 793px; aspect-ratio: 793\/64;\" \/><\/figure>\n\n\n\n<p>Multiplying&nbsp;P<sup>\u22121<\/sup>&nbsp;on both sides of&nbsp;y = \u03b2X + u &nbsp;results in<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"837\" height=\"384\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-9.png\" alt=\"\" class=\"wp-image-187560 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-9.png 837w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-9-700x321.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-9-300x138.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-9-768x352.png 768w\" data-sizes=\"(max-width: 837px) 100vw, 837px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 837px; aspect-ratio: 837\/384;\" \/><\/figure>\n\n\n\n<p>Therefore, the linear regression model above is rewritten as<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"729\" height=\"57\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-10.png\" alt=\"\" class=\"wp-image-187561 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-10.png 729w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-10-700x55.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-10-300x23.png 300w\" data-sizes=\"(max-width: 729px) 100vw, 729px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 729px; aspect-ratio: 729\/57;\" \/><\/figure>\n\n\n\n<p>Least squares as the minimization problem is implemented as follows.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"756\" height=\"178\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-11.png\" alt=\"\" class=\"wp-image-187563 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-11.png 756w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-11-700x165.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-11-300x71.png 300w\" data-sizes=\"(max-width: 756px) 100vw, 756px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 756px; aspect-ratio: 756\/178;\" \/><\/figure>\n\n\n\n<p>GLS estimator is<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"768\" height=\"94\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-12.png\" alt=\"\" class=\"wp-image-187564 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-12.png 768w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-12-700x86.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-12-300x37.png 300w\" data-sizes=\"(max-width: 768px) 100vw, 768px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 768px; aspect-ratio: 768\/94;\" \/><\/figure>\n\n\n\n<p>GLS estimator can be also expressed with the OLS estimator<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"825\" height=\"133\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-13.png\" alt=\"\" class=\"wp-image-187565 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-13.png 825w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-13-700x113.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-13-300x48.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-13-768x124.png 768w\" data-sizes=\"(max-width: 825px) 100vw, 825px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 825px; aspect-ratio: 825\/133;\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"785\" height=\"183\" data-src=\"\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-14-1.png\" alt=\"\" class=\"wp-image-187567 lazyload\" data-srcset=\"https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-14-1.png 785w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-14-1-700x163.png 700w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-14-1-300x70.png 300w, https:\/\/ibkrcampus.com\/campus\/wp-content\/uploads\/sites\/2\/2023\/03\/gls-estimator-sh-fintech-14-1-768x179.png 768w\" data-sizes=\"(max-width: 785px) 100vw, 785px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 785px; aspect-ratio: 785\/183;\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\">Concluding Remarks<\/h3>\n\n\n\n<p>This post derived the GLS estimator which will be used when deriving the Black-Litterman model.<\/p>\n\n\n\n<p><em>Originally posted on <a href=\"https:\/\/kiandlee.blogspot.com\/2022\/09\/gls-estimator-for-black-litterman-model.html\">SH Fintech Modeling<\/a> blog.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This post deals with the generalized least squares (GLS) 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