[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572662-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-572662-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","handout-11-overview-of-the-kalman-filter-cheat-sheet","Handout 11 - Overview of the Kalman Filter - cheat sheet","","Handout 11 provides a structured overview of the Kalman filter, starting from its state space representation with a state equation and an observation equation. It derives the core linear least squares forecasting setup, including the Kalman filter recursion with Kalman gain and the associated error covariance update. The handout also explains steady-state conditions, builds the likelihood function via forecast errors, and covers several initialization schemes, including pre-sample training and augmented Kalman filter setup. Finally, it summarizes filtering vs smoothing and connects notation between Hamilton and Durbin-Koopman, including backward smoothing recursions for state variables and shocks.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/general/","General",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/handout-11-overview-of-the-kalman-filter-cheat-sheet/572662/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/handout-11-overview-of-the-kalman-filter-cheat-sheet/572662.png","ImageObject",442,249,{"name":42,"@type":43},"Himbo","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-20","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":22},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What are the state and observation equations used in the Kalman filter setup?","Question",{"text":62,"@type":63},"The setup includes a state equation describing evolution of state variables and an observation equation relating state variables to observed variables. The handout gives the state equation ξt+1 = Fξt + vt+1 with covariance E(vt+1vt+1)=Q and an observation equation yt = H′ξt.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the Kalman filter recursion compute the one-step-ahead forecast?",{"text":67,"@type":63},"It recursively updates the projected state using ˆξt+1|t = Fˆξt|t−1 + Kt(yt − H′ˆξt|t−1), where Kt is the Kalman gain matrix. The handout also provides the corresponding update for the mean square error matrix Pt+1|t.",{"name":69,"@type":60,"acceptedAnswer":70},"How are filtering and smoothing estimates different in the Kalman filter?",{"text":71,"@type":63},"Filtered estimates use the information set up to time t−1 and update as new observations arrive. Smoothed estimates condition on all observations available, and the handout explains how to perform smoothing using backward recursions while addressing notation differences between Hamilton and Durbin-Koopman.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},572662,1792524679,{"code":4,"msg":81,"data":82},"success",[83,88,93,98,103,108,113,118,123],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},11,"Presentations",90,"presentations",{"id":89,"doc_module":22,"doc_module_name":25,"category_name":90,"show_sort_weight":91,"slug":92},12,"Resumes",80,"resumes",{"id":94,"doc_module":22,"doc_module_name":25,"category_name":95,"show_sort_weight":96,"slug":97},14,"Invoices",70,"invoices",{"id":99,"doc_module":22,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},15,"Posters",60,"posters",{"id":104,"doc_module":22,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},16,"Social Media",50,"social-media",{"id":109,"doc_module":22,"doc_module_name":25,"category_name":110,"show_sort_weight":111,"slug":112},17,"Forms",40,"forms",{"id":114,"doc_module":22,"doc_module_name":25,"category_name":115,"show_sort_weight":116,"slug":117},18,"Letters",30,"letters",{"id":119,"doc_module":22,"doc_module_name":25,"category_name":120,"show_sort_weight":121,"slug":122},21,"Paper Templates",5,"papers-templates",{"id":124,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":4,"slug":125},158,"general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":124,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":22,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":135,"language":136,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":12,"update_tm":140,"read_time":26},687197100911,"https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1785132997149421697","Handout 11  \n1 Overview of the Kalman Filter  \nA good reference to pick up the basics of the Kalman filter is the chapter on the Kalman filterin Hamilton’s time series volume. A more in depth treatment of the Kalman filter can be found in Durbin and Koopman.  \nThe state space representation used in the setup of the Kalman filter includes two components: a state equation, which describes the evolution of the state variables, and an observation equation, which relates the state variables to the observed variables.  \nState equation  \n′  \nξt+1 = Fξt + vt+1 , where E(vt+1vt+1) = Q.  \nAssume that vt is multivariate normal.  \nObservation equation  \nyt = H′ξt.  \n1.1 Linear Least Squares Forecasts  \nDefine ˆξt+1 as the projection of ξt+1 on the history of yt.  \nDefine Pt+1|t to be the mean squared error matrix associated with the projection Pt+1|t = E[(ξt+1 − ˆξt+1|t)(ξt+1 − ˆξt+1|t)′]  \nThe Kalman filter gives us a way to recursively calculate ˆξt+1|t .  \nThe Kalman Filter Recursion  \nˆξt+1|t = Fˆξt|t−1 + Kt (yt − H′ˆξt|t−1) where Kt is the Kalman gain matrix  \nKt = FPt|t−1H (H′ Pt|t−1H)−1  \nIn turn, the mean square error matrix is updated following Pt+1|t = (F − KtH′)Pt|t−1(F′ − HK′t) + Q  \nSteady-State Kalman Filter  \nIf the eigenvalues of F are all within the unit circle and Q is strictly positive definite then Pt+1/t will converge to P.  \nKt+1 will converge to K.  \nThus the Kalman filter recursion becomes  \nˆξt+1|t = Fˆξt|t−1 + K(yt − H′ˆξt|t−1)  \nThis fact can be used to speed the recursion when dealing with large matrices and large datasets.  \n1.2 Likelihood Function  \nIf all the state variables were observed, we could form the likelihood using vt+1 = ξt+1 − Fξt ,  \nbut strategy fails when ξt is not directly observed.  \nInstead use the fact that  \nat = K1 (ˆξt+1|t − Fˆξt|t−1)  \nwhere at = (yt − H′ˆξt|t−1) is the forecast error for the observed variables.  \nNotice that at is uncorrelated over time by construction. If vt is multivariate normal, then at is also multivariate normal.  \nThe var/cov matrix of at is given by H′ Pt|t−1H.  \ng (at ) = (2π) − n2 􀀌 H′ Pt|t−1H 􀀌 − 12 exp 􀀚 − ~~ ~~12 [a′t (H′ Pt|t−1H)−1at]􀀛 (1)  \n1.3 Initialization schemes  \nTo form the unconditional likelihood, set ˆξ1|0 = Eξt and set P1|0 to unconditional variance covariance of ξt , where the variance is  \nE [ξtξ′t] = E [(Fξt−1 + vt )(Fξt−1 + vt)′], which can be written as:  \nΣ = FΣF′ + Q.  \nSee Hamilton on how to use the properties of the vec operator to solve the equation above for Sigma.  \nThis strategy has the drawback that ξ = 0 might imply structural innovations so outlandish that they have a disproportionate weight in the likelihood when the sample is small. Another drawback is that this scheme is viable only when ξt is stationary.  \n1.4 Fixed and known initial point  \nSet ξ0 = 0 .  \nSet P0 = 0 .  \nTo avoid polluting the likelihood, train the filter by running the updates on a pre-sample, but not incorporating the information into the likelihood.  \n1.5 The augmented Kalman filter  \nlet ξ0 = z0  \nConsider two possibilities:  \n1) let var (z0 ) = kI where k → ∞ (Rosenberg)  \n2) let z0 be non-stochastic but unknown (de Jong), estimate jointly with other parameters These schemes also work when some of the state variables are trending.  \n1.6 Filtering and Smoothing  \nThe history of ˆξt|t−1 is referred to as the filtered estimates of the state variables. Notice that the information set used in forming these estimates varies through time, and incorporates new information as we go along.  \nThe Kalman filter can also help us produce estimates of history of the state variables conditional on the information set implied by all of the observations available. The history of the state variables conditional on all of the observations is referred to as the smoothed estimates of the state variables.  \nThe setup for the Kalman filter in Hamilton’s “Time Series Analysis” assumes that we the state equation has as many shocks as state variables. The smoothing recursion derived in Hamilt","cbCaicMWDlVMipp7","https://ap.wps.com/l/cbCaicMWDlVMipp7","pdf",49918,6,"English","# Overview of the Kalman Filter\n## Linear Least Squares Forecasts\n## Likelihood Function\n## Initialization schemes\n## Fixed and known initial point\n## The augmented Kalman filter\n## Filtering and Smoothing","[{\"question\":\"What are the state and observation equations used in the Kalman filter setup?\",\"answer\":\"The setup includes a state equation describing evolution of state variables and an observation equation relating state variables to observed variables. The handout gives the state equation ξt+1 = Fξt + vt+1 with covariance E(vt+1vt+1)=Q and an observation equation yt = H′ξt.\"},{\"question\":\"How does the Kalman filter recursion compute the one-step-ahead forecast?\",\"answer\":\"It recursively updates the projected state using ˆξt+1|t = Fˆξt|t−1 + Kt(yt − H′ˆξt|t−1), where Kt is the Kalman gain matrix. The handout also provides the corresponding update for the mean square error matrix Pt+1|t.\"},{\"question\":\"How are filtering and smoothing estimates different in the Kalman filter?\",\"answer\":\"Filtered estimates use the information set up to time t−1 and update as new observations arrive. Smoothed estimates condition on all observations available, and the handout explains how to perform smoothing using backward recursions while addressing notation differences between Hamilton and Durbin-Koopman.\"}]","Handout 11 - Overview of the Kalman Filter - cheat sheet | PDF",1791928761]