{"id":21011,"date":"2024-02-24T20:35:52","date_gmt":"2024-02-24T19:35:52","guid":{"rendered":"https:\/\/complex-systems-ai.com\/?page_id=21011"},"modified":"2024-02-24T20:43:26","modified_gmt":"2024-02-24T19:43:26","slug":"diebold-mariano","status":"publish","type":"page","link":"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/diebold-mariano\/","title":{"rendered":"Diebold-Mariano test"},"content":{"rendered":"<div data-elementor-type=\"wp-page\" data-elementor-id=\"21011\" class=\"elementor elementor-21011\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4337180 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4337180\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-839db02\" data-id=\"839db02\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f4a59ea elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"f4a59ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Forecasting<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-3cf1979\" data-id=\"3cf1979\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-c0f427f elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"c0f427f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/complex-systems-ai.com\/en\/\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Home page<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-top-column elementor-element elementor-element-51ebf7c\" data-id=\"51ebf7c\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2066d70 elementor-align-justify elementor-widget elementor-widget-button\" data-id=\"2066d70\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/plat.ai\/blog\/difference-between-prediction-and-forecast\/\" target=\"_blank\" rel=\"noopener\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Wiki<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-29c0d72 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"29c0d72\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-2fc4853\" data-id=\"2fc4853\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b2c79be elementor-widget elementor-widget-heading\" data-id=\"b2c79be\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewbox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewbox=\"0 0 24 24\" version=\"1.2\" baseprofile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/diebold-mariano\/#Test-de-Diebold-Mariano-et-HLN\" >Diebold-Mariano test and HLN<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/diebold-mariano\/#Bases\" >Bases<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/diebold-mariano\/#Calcul-avec-Excel\" >Calculation with Excel<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/complex-systems-ai.com\/en\/prediction-forecast\/diebold-mariano\/#Test-de-HLN\" >HLN test<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Test-de-Diebold-Mariano-et-HLN\"><\/span>Diebold-Mariano test and HLN<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-afe397a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"afe397a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-bf729bb\" data-id=\"bf729bb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-d466042 elementor-widget elementor-widget-text-editor\" data-id=\"d466042\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Here in detail the Diebold-Mariano test and the measurement of <span style=\"font-size: 1.125rem;\">Harvey, Leybourne, and Newbold.<\/span><\/p><p><img decoding=\"async\" class=\"aligncenter wp-image-11096 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2020\/09\/cropped-Capture.png\" alt=\"Diebold-Mariano\" width=\"97\" height=\"97\" title=\"\"><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-2a9cefc elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2a9cefc\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-41be1bb\" data-id=\"41be1bb\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ae71704 elementor-widget elementor-widget-heading\" data-id=\"ae71704\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Bases\"><\/span>Bases<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0a1e34b elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0a1e34b\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1c3e07d\" data-id=\"1c3e07d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f8b8421 elementor-widget elementor-widget-text-editor\" data-id=\"f8b8421\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Suppose we have two forecasts f1, \u2026, fn and g1, \u2026, gn for a time series if y1, \u2026, yn and we want to see which forecast is better, in the sense that it has the best predictive accuracy. The obvious approach is to select the forecast that has the smallest error measure based on one of the error measures described in Forecast Errors. But we need to go further and determine whether this difference is significant (for predictive purposes) or simply due to the specific choice of data values in the sample.<\/p><p>We use the Diebold-Mariano test to determine whether the two predictions are significantly different. Let ei and ri be the residuals of the two forecasts, i.e.<\/p><p><img decoding=\"async\" class=\"aligncenter size-full wp-image-26004\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image033d.png\" alt=\"Forecast residuals\" width=\"182\" height=\"20\" title=\"\"><\/p><p>and is defined as one of the following measures (or other similar measures):<\/p><p><img decoding=\"async\" class=\"aligncenter size-full wp-image-26005\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image034d.png\" alt=\"Loss differential\" width=\"222\" height=\"21\" title=\"\"><\/p><p>The time series di is called differential loss. Obviously, the first of these formulas is related to the MSE error statistic and the second is related to the MAE error statistic. We now define<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26006\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image035d.png\" alt=\"Loss-differential mean\" width=\"176\" height=\"51\" title=\"\"><\/p><p>For n &gt; k \u2265 1, we define<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26007\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image036d.png\" alt=\"Autocovariance at lag k\" width=\"200\" height=\"51\" title=\"\"><\/p><p>As described in the autocorrelation function, \u03b3k is the autocovariance at lag k.<\/p><p>For h \u2265 1, define the Diebold-Mariano statistic as follows:<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26008\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image037d.png\" alt=\"Diebold Mariano statistic\" width=\"170\" height=\"60\" title=\"\"><\/p><p>It is generally sufficient to use the value h = n1\/3 + 1.<\/p><p>Under the assumption that \u03bc = 0 (the null hypothesis), DM follows a standard normal distribution:<\/p><p>DM \u223c N(0, 1)<\/p><p>There is therefore a significant difference between the forecasts if |DM| &gt; zcrit where zcrit is the two-sided critical value for the standard normal distribution; that&#039;s to say.<\/p><p>zcrit = NORM.S.DIST(1\u2013\u03b1\/2, TRUE)<\/p><p>The key assumption for using the Diebold-Mariano test is that the differential loss time series di is stationary (see Stationary time series).<\/p><p>Use the Diebold-Mariano test to determine whether there is a significant difference in the predictions in columns B and C of Figure 1 for the data in column A.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21014 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB1.png\" alt=\"Diebold-Mariano\" width=\"768\" height=\"507\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB1.png 768w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB1-300x198.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB1-18x12.png 18w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-0b7d1c6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"0b7d1c6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-44abd3d\" data-id=\"44abd3d\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-86f773d elementor-widget elementor-widget-heading\" data-id=\"86f773d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Calcul-avec-Excel\"><\/span>Calculation with Excel<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7866ef6 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7866ef6\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1879785\" data-id=\"1879785\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-bac0e24 elementor-widget elementor-widget-text-editor\" data-id=\"bac0e24\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>We start by calculating the residuals for the 20 data items based on the two predictions (columns F and G). For example. cell F4 contains the formula =A4-B4 and cell G4 contains =A4-C4. From these values we can calculate the loss differentials in column H. E.g. cell H4 contains =F4^2-G4^2. We can now calculate the mean and variance of the time series di via the formulas =AVERAGE(H4:H23) and =VAR.P(H4:H23), as shown in cells H25 and K25.<\/p><p>The \u03b3i values can now be calculated as shown in column I. E.g. cell I4 contains the array formula<\/p><p>=SUMPRODUCT(H5:H$23-H$25,OFFSET(H$4:H$23,0,0,E$23-E4)-H$25)\/E$23<\/p><p>Alternatively, cell I4 can be calculated using the formula =ACVF(H$4:H$23,E4), as described in Autocorrelation function. Standard errors (column J) and Diebold-Mariano statistics (column K) can then be calculated. For example. cell J4 contains the formula =SQRT(J25\/E23), cell J5 contains =SQRT((J$25+2*SUMPRODUCT(I$4:I4))\/E$23) (and the same for the other cells in column J) and cell K4 contains the formula =G$25\/J4.<\/p><p>We can read the DM statistical values from column K. For example. the DM statistic for order h = 4 is 1.005199, as shown in cell K7. Note that since h = n1\/3 + 1 = 201\/3 + 1 = 3.7, h = 4 seems like a good order value to use. Since the p-value = 2*(1-NORM.S.DIST(K7,TRUE)) = 0.3148 &gt; 0.05 = \u03b1, we conclude that there is no significant difference between the two predictions.<\/p><p>In the figure, we plot both predictions for the data in column A, and so you can judge for yourself whether the graph is consistent with the results of the Diebold-Mariano test.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21015 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB2.png\" alt=\"Diebold-Mariano\" width=\"488\" height=\"293\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB2.png 488w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB2-300x180.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB2-18x12.png 18w\" sizes=\"(max-width: 488px) 100vw, 488px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-6566497 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6566497\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-54208cc\" data-id=\"54208cc\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-ad770d1 elementor-widget elementor-widget-heading\" data-id=\"ad770d1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\"><span class=\"ez-toc-section\" id=\"Test-de-HLN\"><\/span>HLN test<span class=\"ez-toc-section-end\"><\/span><\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-acc0ce9 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"acc0ce9\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-8f947ad\" data-id=\"8f947ad\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-5b66854 elementor-widget elementor-widget-text-editor\" data-id=\"5b66854\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>The Diebold-Mariano test tends to reject the null hypothesis too often for small samples. A better test is the Harvey, Leybourne, and Newbold (HLN) test, which is based on the following:<\/p><h2><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-26009\" src=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image038d.png\" sizes=\"(max-width: 321px) 100vw, 321px\" srcset=\"https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image038d.png 321w, https:\/\/real-statistics.com\/wp-content\/uploads\/2018\/01\/image038d-300x22.png 300w\" alt=\"HLN Test\" width=\"321\" height=\"24\" title=\"\"><\/h2><p>Especially since for example we have a small sample, we use the HLN test as shown in the figure. Once again, we see that there is no significant difference between the forecasts.<\/p><p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-21016 size-full\" src=\"http:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB3.png\" alt=\"Harvey, Leybourne, and Newbold\" width=\"436\" height=\"268\" title=\"\" srcset=\"https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB3.png 436w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB3-300x184.png 300w, https:\/\/complex-systems-ai.com\/wp-content\/uploads\/2024\/02\/DB3-18x12.png 18w\" sizes=\"(max-width: 436px) 100vw, 436px\" \/><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>","protected":false},"excerpt":{"rendered":"<p>Forecasting Home page Wiki Diebold-Mariano test and HLN Here in detail the Diebold-Mariano test and the Harvey, Leybourne, and Newbold measurement. Basics\u2026 <\/p>","protected":false},"author":1,"featured_media":0,"parent":20753,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-21011","page","type-page","status-publish","hentry"],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/21011","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/comments?post=21011"}],"version-history":[{"count":4,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/21011\/revisions"}],"predecessor-version":[{"id":21019,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/21011\/revisions\/21019"}],"up":[{"embeddable":true,"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/pages\/20753"}],"wp:attachment":[{"href":"https:\/\/complex-systems-ai.com\/en\/wp-json\/wp\/v2\/media?parent=21011"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}