{"id":36745,"date":"2021-10-29T17:57:18","date_gmt":"2021-10-29T17:57:18","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=36745"},"modified":"2026-03-19T19:59:32","modified_gmt":"2026-03-19T19:59:32","slug":"predicted-value-of-a-dependent-variable","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/predicted-value-of-a-dependent-variable\/","title":{"rendered":"Predicted Value of a Dependent Variable"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"Predict quantity supplied using regression equation\",\n    \"text\": \"The regression equation of quantity of goods against price is given by:\\n\\nY = -159 + 0.26X\\n\\nWhere:\\nY = Quantity supplied\\nX = Price per unit of the product\\n\\nThe predicted value of the quantity supplied when the price equals 1,200 is closest to:\\n\\nA. 153\\n\\nB. 155\\n\\nC. 471\",\n    \"answerCount\": 1,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"The correct answer is A.\\n\\nSubstitute X = 1,200 into the regression equation:\\n\\nY = -159 + 0.26 \u00d7 1,200 = -159 + 312 = 153.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"url\": \"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/02\/cfa-level-2-t-table-2.3.png\",\n  \"caption\": \"t-distribution critical values table\",\n  \"width\": 735,\n  \"height\": 720,\n  \"copyrightNotice\": \"\u00a9 2024 AnalystPrep\",\n  \"acquireLicensePage\": \"https:\/\/analystprep.com\/license-info\",\n  \"creditText\": \"AnalystPrep Design Team\",\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  }\n}\n<\/script>\n\n\n<p><iframe loading=\"lazy\" src=\"\/\/www.youtube.com\/embed\/6ouI_27iDVM\" width=\"611\" height=\"343\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We calculate the predicted value of the dependent variable, \\(Y\\), by inserting the estimated value of the independent variable, \\(X\\), into the regression equation. The predicted value of the dependent variable, \\(Y\\), is determined using the following formula:<\/p>\n<p>$$\\hat{Y}=\\widehat{b}_0+\\widehat{b}_1X$$<\/p>\n<p>Where:<\/p>\n<p>\\(\\widehat{Y}\\) = Predicted value of the dependent variable.<\/p>\n<p>\\(X\\) = Estimated value of the independent variable.<\/p>\n<h4><strong>Example: Calculating the Predicted Value of a Dependent Variable<\/strong><\/h4>\n<p>The estimated linear regression equation for inflation and unemployment rates for a certain country is:<\/p>\n<p>$$\\hat{Y}=0.070953-0.90405X_i$$<\/p>\n<p>Calculate the predicted value of the inflation rate if the forecasted value of the unemployment rate is 7.5%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>The predicted value of the inflation rate is determined as follows:<\/p>\n<p>$$\\begin{align}\\hat{Y}&amp;=\\widehat{b}_0+\\widehat{b}_1X\\\\ &amp;=0.070953-0.90405\\times 7.5\\%=0.3\\%\\end{align}$$<\/p>\n<h1><strong>Confidence Interval for Predicted Values<\/strong><\/h1>\n<p>The calculation of the confidence interval for the predicted value of a dependent variable is the same as that of the confidence interval for regression coefficients. The confidence interval for a predicted value of the dependent variable is given by:<\/p>\n<p>$$\\text{Prediction Interval}=\\ \\hat{Y}\\pm t_cs_f$$<\/p>\n<p>Where:<\/p>\n<p>\\(t_c\\)= Two-tailed critical t-value at the given significance level with n &#8211; 2 df.<\/p>\n<p>\\(\\hat{Y}\\) = Predicted value of a dependent variable.<\/p>\n<p>\\(s_f^2\\)= The estimated variance of the prediction error.<\/p>\n<p>$$s_f^2=s_e^2\\left[1+\\frac{1}{n}+\\frac{\\left(X-\\bar{X}\\right)^2}{\\left(n-1\\right)s_x^2}\\right]=s_e^2\\left[1+\\frac{1}{n}+\\frac{\\left(X-\\bar{X}\\right)^2}{\\sum_{i\\ =\\ n}^{n}\\left(X_i-\\bar{X}\\right)^2}\\right]$$<\/p>\n<p>Where:<\/p>\n<p>\\(s_e^2\\) = The squared standard error of estimate.<\/p>\n<p>\\(n\\) = Nnumber of observations.<\/p>\n<p>\\(s_x^2\\)= Variance of the independent variable.<\/p>\n<p>\\(X\\) = Value of the independent variable.<\/p>\n<p>We can, therefore, calculate the standard error of forecast as shown below:<\/p>\n<p>$$s_f=s_e\\sqrt{1+\\frac{1}{n}+\\frac{\\left(X-\\bar{X}\\right)^2}{\\sum_{i=n}^{n}\\left(X_i-\\bar{X}\\right)^2}}$$<\/p>\n<h4>Example: Calculating the Confidence Interval of the Predicted Value<\/h4>\n<p>Consider the results of the regression analysis of inflation on unemployment performed by an analyst:<\/p>\n<p>$$\\small{\\begin{array}{llll}\\hline{}&amp; \\textbf{Regression Statistics} &amp; &amp; \\\\ \\hline {}&amp; \\text{Multiple R} &amp; 0.8766 &amp; {}\\\\ {}&amp; \\text{R Square} &amp; 0.7684 &amp; {}\\\\ {}&amp; \\text{Adjusted R Square} &amp; 0.7394 &amp; {}\\\\ {}&amp; \\text{Standard Error} &amp; 0.0063 &amp; {}\\\\ {}&amp; \\text{Observations} &amp; 10 &amp; {}\\\\ \\hline{}&amp; \\textbf{Coefficients} &amp; \\textbf{Standard Error} &amp; \\text{t-Stat} \\\\ \\hline\\text{Intercept} &amp; 0.0710 &amp; 0.0094 &amp; 7.5160\\\\ \\text{Forecast (Slope)} &amp; -0.9041 &amp; 0.1755 &amp; -5.1516\\\\ \\hline\\end{array}}$$<\/p>\n<p>Calculate the 95% confidence interval of the predicted value of the inflation rate given that the forecasted unemployment rate is 7.5%.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$\\text{Prediction Interval} = \\widehat{Y}\u00b1t_{c}S_{f}$$<\/p>\n<p>The estimated variance of the prediction error is:<\/p>\n<p>$$S_{f}^{2}=S^{2}\\bigg[1+\\frac{1}{2}+\\frac{(X-\\bar{X})^2}{(n-1)S_{x}^{2}}\\bigg]$$<\/p>\n<p>$$\\begin{align*}S_f^{2}&amp;=0.0063^2\\bigg[1+\\frac{1}{10} +\\frac{(0.075-0.0526)^2}{(10-1)0.0013}\\bigg]\\\\&amp;=0.006733^2\\end{align*}$$<\/p>\n<p>From the regression results, the estimated regression equation is:<\/p>\n<p>$$\\widehat{Y}=0.0710-0.9041X$$<\/p>\n<p>The predicted value of the inflation rate given an unemployment rate of 7.5% is:<\/p>\n<p>$$\\begin{align*}\\widehat{Y}&amp;= 0.0710-0.9041\\times7.5\\%\\\\&amp;=0.3\\%\\end{align*}$$<\/p>\n<p>The two-tailed critical t-value with 8 (n-2) degrees of freedom at the 5% significance level is 2.306.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-12774 aligncenter\" src=\"https:\/\/analystprep.com\/study-notes\/wp-content\/uploads\/2021\/02\/cfa-level-2-t-table-2.3.png\" alt=\"\" width=\"735\" height=\"720\" \/><\/p>\n<p>The prediction interval at the 95% confidence level is:<\/p>\n<p>$$\\text{Prediction Interval (PI)} = \\widehat{Y}\u00b1t_{c}S_{f}$$<\/p>\n<p>$$\\text{PI} = 0.3\\%\u00b12.306\\times 0.6735\\% = -1.25\\% \\text{ to } 1.85\\%$$<\/p>\n<p><strong>Interpretation<\/strong><\/p>\n<p>Given an unemployment rate of 7.5%, we are <strong>95% confident<\/strong> that <strong>the inflation rate will lie between <\/strong><strong>-1.25% <\/strong>and<strong> 1.85%<\/strong>.<\/p>\n<blockquote>\n<h2><strong>Question 1<\/strong><\/h2>\n<p>The regression equation of quantity of goods against the price is given by:<\/p>\n<p>$$Y =-159+0.26X$$<\/p>\n<p>Where:<\/p>\n<p>\\(Y\\) = Quantity supplied.<\/p>\n<p>\\(X\\) = Price per unit of the product.<\/p>\n<p>The predicted value of the quantity supplied when the price equals 1,200 is <em>closest<\/em> to:<\/p>\n<ol style=\"list-style-type: upper-alpha;\">\n<li>153.<\/li>\n<li>155.<\/li>\n<li>471.<\/li>\n<\/ol>\n<p>The correct answer is<strong> A.<\/strong><\/p>\n<p>$$Y\\ =\\ -159\\ +\\ 0.26\\times1,200=153$$<\/p><\/blockquote>","protected":false},"excerpt":{"rendered":"<p>We calculate the predicted value of the dependent variable, \\(Y\\), by inserting the estimated value of the independent variable, \\(X\\), into the regression equation. The predicted value of the dependent variable, \\(Y\\), is determined using the following formula: $$\\hat{Y}=\\widehat{b}_0+\\widehat{b}_1X$$ Where:&#8230;<\/p>\n","protected":false},"author":15,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[],"class_list":["post-36745","post","type-post","status-publish","format-standard","hentry","category-quantitative-methods","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Predicted Value of a Dependent Variable | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn to calculate the predicted value of a dependent variable using regression equations. 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