[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-118500-en":3,"doc-seo-118500-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},118500,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Measuring the Phillips Curve by Machine Learning","This paper measures the Phillips curve for Japan using Gaussian process regression within a Bayesian modeling framework. It contrasts Gaussian process regression with traditional parametric regression, emphasizing that Gaussian process regression is computationally heavier but more flexible and yields more accurate forecasts. The method treats regression weights as random variables and expresses uncertainty explicitly through predictive distributions. The paper motivates nonparametric approaches for big data and addresses limitations of frequentist maximum-likelihood solutions by leveraging tools such as MCMC.","􀀱  \nMeasuring the Phillips Curve by Machine Learning  \nKunio KAMA*  \nAbstract: In recent years, machine learning and Bayesian modeling have been applied in various fields. Machine learning includes supervised learning, unsupervised learning, and reinforcement learning, and regression analysis, which is often used in economics, is a typical example of supervised learning. Dynamic programming, a fundamental tool for dynamic macroeconomic models, is also an important topic in reinforcement learning. However, regression analysis in machine learning is fundamentally different from traditional regression models and is called Gaussian process regression. It is more computationally expensive than conventional methods, but it is more flexible and provides more accurate forecasts. This paper introduces Gaussian process regression and Bayesian modeling, which are unfamiliar to many economists, by estimating the Phillips curve for Japan. Gaussian process regression is a stochastic model of a functional relationship, in which the probability distribution of the function allows the uncertainty of the forecast to be made explicit. Traditionally, parametric models such as linear regression models have been used in empirical research in economics. However, various problems arise when applying conventional models to big data, which has recently come into use. A nonparametric approach should be applied to big data. In econometrics, approaches based on the frequentist approach of statistics have been predominant, with an emphasis on obtaining maximum likelihood estimators of parameters. Although the maximum likelihood estimator has consistency and invariance under general conditions, analytical solutions are often not available for nonlinear models. There is also the problem of multiple local maximum points depending on the shape of the likelihood function. Bayesian approaches based on alternatives to frequency-based approaches have been applied in many fields, but in economics, they are limited to a few, such as estimating the parameters of DSGE and VAR models. The main reason is that it requires a huge amount of computation, but powerful tools such as the MCMC method are now available. In Section 2, we estimate the Phillips curve for Japan using Bayesian methods.  \n* Professor emeritus, Soka University  \n􀀲 季刊 創 価 経 済 論 集 Vol. LIV, No. 1 ・ 2 ・ 3 ・ 4  \n1. Gaussian process regression  \nThe first method discussed in econometrics texts is the least squares method 1. The least squares method estimates coefficients and other parameters by fitting the following equation to observed data.  \ny = w0 + w1x1 + w2x2 + 􀂘 􀂘 􀂘 + wkxk + u  \nThe left-hand side represents the dependent variable, and the right-hand side indicates the independent variables, with u being the disturbance term. In some cases, it may be necessary to fita nonlinear function, such as  \n2 3  \ny = w0 + w1x + w2x + w3x + u  \nto the data. This is because the explanatory power of the model increases if the explanatory variables are transformed appropriately. In general, we fit an equation  \ny = w0 + w1􀁉1 (x) + w2 􀁉2 (x) + 􀂘 􀂘 􀂘 + wk 􀁉k (x) + u (1)  \nor  \n^y = Φw (2)  \nWhere ^y is the vector of estimated values for y, and  \n􀂧 􀁉0 (x1 ) 􀁉1 (x1 ) 􀁉2 (x1 ) 􀂘 􀂘 􀂘 􀁉k (x1 ) 􀂷 􀂧 w 􀂷􀂨 􀁉0 (x2 ) 􀁉1 (x2 ) 􀁉2 (x2 ) 􀂘 􀂘 􀂘 􀁉k (x2 ) 􀂸 􀂨 w01 􀂸  \n􀀩 = 􀂨 . . 􀂸 w = 􀂨 . 􀂸  \n􀂨 􀂸 , 􀂨 􀂸  \n􀂨 . . 􀂸 􀂨 . 􀂸  \n􀂨􀂩 􀁉0 (xN ) 􀁉1 (xN ) 􀁉2 (xN ) 􀂘 􀂘 􀂘 􀁉k (xN ) 􀂹􀂸 􀂩􀂨 wk 􀂹􀂸  \nwhere φ0 (x) ≡ 1. The value of variable x is given, and Φ is a constant matrix. As evident from the form of (2), ^y belongs to the function space created by φ0 (x), φ1 (x), ..., φk (x) . The functions that form the basis of the function space are called feature vectors of x. (1) is linear with respect to the coefficients, and the OLS estimates of the coefficients are given by  \nw = (􀀩T 􀀩)−1􀀩 Ty  \nIgnoring the disturbance term for simplification, (1) is expressed as:  \ny = Φw (3)  \nIn a typical regression model, w is considered a constant. On the other hand, in Gaussian  \n","cbCaidEbRf8vmAyY","https://ap.wps.com/l/cbCaidEbRf8vmAyY","pdf",1434503,1,13,"English","en",105,"# 1. Gaussian process regression\n## Regression and least squares foundation\n## Gaussian distributions and kernels\n## Gaussian processes and kernel functions\n## Noise handling and target modeling","[{\"question\":\"What is Gaussian process regression in this paper’s econometric context?\",\"answer\":\"It models a functional relationship stochastically, where the probability distribution of the function makes forecast uncertainty explicit. In regression, the weights are treated as Gaussian random variables, leading to a multivariate Gaussian distribution over outputs.\"},{\"question\":\"How does specifying the kernel function affect Gaussian process regression?\",\"answer\":\"The paper notes that once the kernel function k(xn, xm) is determined, there is no need to separately find corresponding feature vectors. The kernel determines the Gaussian process structure and the resulting covariance (kernel) matrix.\"},{\"question\":\"Why does the paper emphasize Bayesian methods and MCMC?\",\"answer\":\"The paper argues that Bayesian approaches can be limited by heavy computation, especially in economics. It highlights that powerful tools such as MCMC now make Bayesian estimation feasible for models like the Phillips curve.\"}]","Measuring the Phillips Curve by Machine Learning | PDF",1785683894,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"measuring-the-phillips-curve-by-machine-learning","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/measuring-the-phillips-curve-by-machine-learning/118500/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05","2026-08-02",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is Gaussian process regression in this paper’s econometric context?","Question",{"text":76,"@type":77},"It models a functional relationship stochastically, where the probability distribution of the function makes forecast uncertainty explicit. In regression, the weights are treated as Gaussian random variables, leading to a multivariate Gaussian distribution over outputs.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does specifying the kernel function affect Gaussian process regression?",{"text":81,"@type":77},"The paper notes that once the kernel function k(xn, xm) is determined, there is no need to separately find corresponding feature vectors. The kernel determines the Gaussian process structure and the resulting covariance (kernel) matrix.",{"name":83,"@type":74,"acceptedAnswer":84},"Why does the paper emphasize Bayesian methods and MCMC?",{"text":85,"@type":77},"The paper argues that Bayesian approaches can be limited by heavy computation, especially in economics. It highlights that powerful tools such as MCMC now make Bayesian estimation feasible for models like the Phillips curve.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":46,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":46,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]