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The framework applies broadly to persistent stochastic processes beyond the standard models.",{"@graph":14,"@context":73},[15,34,56],{"@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/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/low-frequency-econometrics-inference-on-long-run-trend-variation/203236/",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/low-frequency-econometrics-inference-on-long-run-trend-variation/203236.png","ImageObject",300,407,{"name":42,"@type":43},"Finn","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-17","2026-09-04",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",5,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How does the paper propose extracting long-run information from time series?","Question",{"text":63,"@type":64},"It isolates long-run sample information using a small number of low-frequency trigonometric weighted averages based on projections onto trigonometric functions.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Why can inference procedures be simplified in this approach?",{"text":68,"@type":64},"The low-frequency weighted averages have large-sample normal distributions, allowing valid large-sample inference via familiar normal small-sample style methods.",{"name":70,"@type":61,"acceptedAnswer":71},"Does the method apply only to specific time-series models like I(0) and I(1)?",{"text":72,"@type":64},"No. The general approach is applicable to a wide range of persistent stochastic processes beyond the standard I(0) and I(1) models.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},203236,1788557477,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":55,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144,"preview_page_count":4},34359740700684,"https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487","Low-Frequency Econometrics∗  \nUlrich K. Müller and Mark W. Watson Princeton University August 2015 (Revised August 2016)  \nAbstract  \nMany questions in economics involve long-run or “trend” variation and covariation in time series. Yet, time series of typical lengths contain only limited information about this long-run variation. This paper suggests that long-run sample information can be isolated using a small number of low-frequency trigonometric weighted averages, which in turn can be used to conduct inference about long-run variability and covariability. Because the low-frequency weighted averages have large sample normal distributions, large sample valid inference can often be conducted using familiar small sample normal inference procedures. Moreover, the general approach is applicable for a wide range of persistent stochastic processes that go beyond the familiar 􀁌(0) and 􀁌(1) models.  \nJEL classification: C22, C32, C12  \nKeywords: persistent time series, HAC, long-run covariance matrix  \n∗We thank Frank Schorfheide for a thoughtful discussion at the 2015 World Congress in Montreal. Support was provided by the National Science Foundation through grant SES-1226464 . Replication files are available at [http://www.princeton.edu/](http://www.princeton.edu/) ~mwatson.  \n1 Introduction  \nThis paper discusses inference about trends in economic time series. By “trend” we mean the low-frequency variability evident in a time series after forming moving averages such as low-pass (cf. Baxter and King (1999)) or Hodrick and Prescott (1997) filters. To measure this low-frequency variability we rely on projections of the series onto a small number of trigonometric functions (e.g., discrete Fourier, sine, or cosine transforms) . The fact that a small number of projection coefficients capture low-frequency variability reflects the scarcity of low-frequency information in the data, leading to what is effectively a “small-sample”econometric problem. As we show, it is still relatively straightforward to conduct statistical inference using the small sample of low-frequency data summaries. Moreover, these lowfrequency methods are appropriate for both weakly and highly persistent processes. Before getting into the details, it is useful to fix ideas by looking at some data.  \nFigure 1 plots the value of per-capita GDP growth rates (panel A) and price inflation (panel B) for the United States using quarterly data from 1947 through 2014, and where both are expressed in percentage points at an annual rate.1 The plots show the raw series and two“trends”. The first trend was constructed using a band-pass moving average filter designed to pass cyclical components with periods longer than 􀁗􀁀6 ≈ 11 years, and the second is the full-sample projection of the series onto a constant and twelve cosine functions with periods 2􀁗􀁀􀁭 for 􀁭 = 1 􀀾􀀽􀀽􀀽􀀾 12, also designed to capture variability for periods longer than 11 years.2 A glance at the figure shows that the band-pass and projection trends essentially coincide both for GDP, for which there is only moderate trend variability, and inflation, for which there is substantial trend variability. This paper focuses on trends computed by projection methods because, as we will see, they give rise to simple methods for statistical inference.  \nFigure 1 raises several empirical questions. For example, the trend in GDP growth has  \n1 Three data series are used in this paper. All are quarterly series from 1947:Q1 to 2014:Q4 for the United States. Real per capita gross domestic product is available from FRED as series A939RX0Q048SBEA. Inflation is measured using the price deflator for personal consumption expenditures (FRED series DPCERD3Q086SBEA) . Total factor productivity is from Fernald (2014) and is available at [http://www.frbsf.org/economic-research/economists/john-fernald/](http://www.frbsf.org/economic-research/economists/john-fernald/) .  \n2 The low-pass values were computed using an ideal low-pass filter truncated after 􀁗􀁀2 ter","cbCainTKNV3pUogs","https://ap.wps.com/l/cbCainTKNV3pUogs","pdf",1315847,47,"English","# Introduction\n## Trends and Low-Frequency Variation in Economic Time Series\n## Empirical Motivation Using U.S. Data\n# Discrete Cosine Transforms and Trigonometric Projections\n## Notation and Properties\n# Multivariate Inference on Trend Covariation","[{\"question\":\"How does the paper propose extracting long-run information from time series?\",\"answer\":\"It isolates long-run sample information using a small number of low-frequency trigonometric weighted averages based on projections onto trigonometric functions.\"},{\"question\":\"Why can inference procedures be simplified in this approach?\",\"answer\":\"The low-frequency weighted averages have large-sample normal distributions, allowing valid large-sample inference via familiar normal small-sample style methods.\"},{\"question\":\"Does the method apply only to specific time-series models like I(0) and I(1)?\",\"answer\":\"No. The general approach is applicable to a wide range of persistent stochastic processes beyond the standard I(0) and I(1) models.\"}]","Low-Frequency Econometrics - Inference on Long-Run Trend Variation | PDF",118]