{"id":45739,"date":"2023-08-19T09:52:46","date_gmt":"2023-08-19T09:52:46","guid":{"rendered":"https:\/\/analystprep.com\/cfa-level-1-exam\/?p=45739"},"modified":"2026-03-31T12:27:08","modified_gmt":"2026-03-31T12:27:08","slug":"predicted-value-and-prediction-interval-of-a-dependent-variable","status":"publish","type":"post","link":"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/predicted-value-and-prediction-interval-of-a-dependent-variable\/","title":{"rendered":"Predicted Value and Prediction Interval of a Dependent Variable"},"content":{"rendered":"\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"VideoObject\",\n  \"name\": \"Simple Linear Regression \u2013 Part II (2025 CFA\u00ae Level I Exam \u2013 Quantitative Methods \u2013 LM 10)\",\n  \"description\": \"CFA Level I Quantitative Methods lesson on Simple Linear Regression \u2013 Part II. Covers hypothesis testing of regression coefficients, ANOVA, dummy variables, prediction and confidence intervals, and functional forms including log-linear and semi-log models. Includes interpretation of regression fit measures, t-tests, F-tests, and standard error of estimate.\",\n  \"thumbnailUrl\": \"https:\/\/img.youtube.com\/vi\/FmVBqrMYreM\/maxresdefault.jpg\",\n  \"uploadDate\": \"2024-03-24\",\n  \"duration\": \"PT49M58S\",\n  \"contentUrl\": \"https:\/\/www.youtube.com\/watch?v=FmVBqrMYreM\",\n  \"embedUrl\": \"https:\/\/www.youtube.com\/embed\/FmVBqrMYreM\"\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"QAPage\",\n  \"mainEntity\": {\n    \"@type\": \"Question\",\n    \"name\": \"What is the predicted quantity supplied given the regression equation?\",\n    \"text\": \"The regression equation of the quantity of goods against the price is given by:\\n\\nY = \u2212159 + 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.\\nB. 155.\\nC. 471.\",\n    \"answerCount\": 3,\n    \"acceptedAnswer\": {\n      \"@type\": \"Answer\",\n      \"text\": \"153. Substituting X = 1,200 into the regression equation gives Y = \u2212159 + (0.26 \u00d7 1,200) = 153.\"\n    }\n  }\n}\n<\/script>\n<script type=\"application\/ld+json\">\n{\n  \"@context\": \"https:\/\/schema.org\",\n  \"@type\": \"ImageObject\",\n  \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12.jpg#image\",\n  \"url\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12.jpg\",\n  \"contentUrl\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12.jpg\",\n  \"name\": \"Predicted Value and Prediction Interval of a Dependent Variable\",\n  \"caption\": \"Illustration used in explaining predicted value and prediction interval concepts in regression analysis.\",\n  \"description\": \"Visual from the CFA Level I Quantitative Methods page showing the predicted value and prediction interval of a dependent variable based on a simple linear regression model. The page teaches how to calculate the predicted value using the regression equation and how to form a prediction interval around this estimate for a specified confidence level. :contentReference[oaicite:0]{index=0}\",\n  \"encodingFormat\": \"image\/jpeg\",\n  \"width\": 642,\n  \"height\": 612,\n  \"creator\": {\n    \"@type\": \"Organization\",\n    \"name\": \"AnalystPrep\"\n  },\n  \"creditText\": \"AnalystPrep\",\n  \"copyrightNotice\": \"\u00a9 AnalystPrep\",\n  \"mainEntityOfPage\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/predicted-value-and-prediction-interval-of-a-dependent-variable\/\"\n  },\n  \"isPartOf\": {\n    \"@type\": \"WebPage\",\n    \"@id\": \"https:\/\/analystprep.com\/cfa-level-1-exam\/quantitative-methods\/predicted-value-and-prediction-interval-of-a-dependent-variable\/\"\n  }\n}\n<\/script>\n\n\n\n<iframe loading=\"lazy\" width=\"560\" height=\"315\" src=\"https:\/\/www.youtube.com\/embed\/FmVBqrMYreM?si=CmQFIPY-EuAHnXCD\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n\n\n\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\n\n\n<p>$$\\hat{Y}=\\hat{b}_0+\\hat{b}_1X$$<\/p>\n\n\n\n<p>Where:<\/p>\n\n\n\n<p>\\(\\hat{Y}\\) = Predicted value of the dependent variable.<\/p>\n\n\n\n<p>\\(X\\) = Estimated value of the independent variable.<\/p>\n\n\n\n<p><strong>Example: Calculating the Predicted Value of a Dependent Variable<\/strong><\/p>\n\n\n\n<p>Refer to the example of regressed inflation rates against unemployment rates from 2011 to 2020.<\/p>\n\n\n\n<p>$$ \\begin{array}{c|c|c|c|c|c|c|c}<br>\\text{Year} &amp; \\text{Unemployment} &amp; \\text{Inflation} &amp; \\text{Predicted} &amp; \\text{Variation} &amp; \\text{Variation} &amp; \\text{Variation} &amp; (X_i \\\\<br>&amp; {\\text{Rate } \\% (X_i)} &amp; {\\text{Rate }\\%} &amp; \\text{Unemployment} &amp; \\text{to be} &amp; \\text{Unexplained} &amp; \\text{Explained} &amp; -\\bar{X})^2 \\\\<br>&amp; &amp; ({{Y}}_i) &amp; {\\text{rate } (\\hat Y_i)} &amp; \\text{Explained.} &amp; &amp; &amp; \\\\<br>&amp; &amp; &amp; &amp; \\left(Y_i-\\bar{Y}\\right)^2 &amp; \\left(Y_i- \\hat{Y}_i\\right)^2 &amp; \\left({\\hat{Y}}_i-\\bar{Y}\\right)^2 &amp; \\\\ \\hline<br>2011 &amp; 6.1 &amp; 1.7 &amp; 1.610 &amp; 0.410 &amp; 0.008 &amp; 0.533 &amp; 0.656 \\\\ \\hline<br>2012 &amp; 7.4 &amp; 1.2 &amp; 0.437 &amp; 1.300 &amp; 0.582 &amp; 3.621 &amp; 4.452 \\\\ \\hline<br>\\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots \\\\ \\hline<br>2019 &amp; 4.0 &amp; 4.7 &amp; 3.504 &amp; 5.570 &amp; 1.430 &amp; 1.355 &amp; 1.664 \\\\ \\hline<br>2020 &amp; 3.9 &amp; 3.6 &amp; 3.594 &amp; 1.588 &amp; 0.000 &amp; 1.573 &amp; 1.932 \\\\ \\hline<br>\\textbf{Sum} &amp; \\bf{52.90} &amp; \\bf{23.4} &amp; &amp; \\bf{13.704} &amp; \\bf{3.136} &amp; \\bf{10.568} &amp; \\bf{12.989} \\\\ \\hline<br>\\textbf{Arithmetic} &amp; \\bf{5.29} &amp; \\bf{2.34} &amp; &amp; &amp; &amp; &amp; \\\\<br>\\textbf{Mean} &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\<br>\\end{array} $$<\/p>\n\n\n\n<p>The estimated regression model is illustrated below.<\/p>\n\n\n\n<p>$$ \\hat{Y}=7.112-0.9020X_i+\\varepsilon_i $$<\/p>\n\n\n\n<p>Calculate the predicted inflation rate value if the forecasted value of the unemployment rate is 4.5%.<\/p>\n\n\n\n<p><strong>Solution<\/strong><\/p>\n\n\n\n<p>The predicted value of the inflation rate is determined as follows:<\/p>\n\n\n\n<p>$$ \\hat{Y}=7.112-0.9020\\times4.5=3.053\\% $$<\/p>\n\n\n\n<div style=\"text-align: center; margin: 22px 0;\">\n  <div style=\"max-width: 680px; margin: 0 auto;\">\n    <a href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\"\n       style=\"display: flex; align-items: center; justify-content: center;\n       width: 100%; padding: 10px 18px;\n       border: 2px solid #1e5bd8; color: #1e5bd8;\n       border-radius: 9999px; text-decoration: none; font-weight: 600;\">\n      Practice predicted values and prediction intervals with our free trial\n    <\/a>\n  <\/div>\n<\/div>\n\n\n<p>\u00a0<\/p>\n<h2>Confidence Interval for Predicted Values<\/h2>\n<p>The confidence interval calculation 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>$$ s_f^2=s_e^2\\left[1+\\frac{1}{n}+\\frac{\\left(X_f-\\bar{X}\\right)^2}{\\left(n-1\\right)s_x^2}\\right]=s_e^2\\left[1+\\frac{1}{n}+\\frac{\\left(X_f-\\bar{X}\\right)^2}{\\sum_{i\\ =\\ 1}^{n}\\left(X_i-\\bar{X}\\right)^2}\\right] $$<\/p>\n<p>Where:<\/p>\n<p>\\(s_e^2\\) = The squared standard error of the estimate.<\/p>\n<p>\\(n\\) = Number of observations.<\/p>\n<p>\\(s_X^2\\)= Variance of the independent variable.<\/p>\n<p>\\(X_f\\) = Value of the independent variable.<\/p>\n<p>We can, therefore, calculate the standard error of forecast as shown below:<\/p>\n<p>$$s_f=s_e\\sqrt{1+\\frac{1}{n}+\\frac{\\left(X_f-\\bar{X}\\right)^2}{\\sum_{i=1}^{n}\\left(X_i-\\bar{X}\\right)^2}}$$<\/p>\n<p>From the formula above, we can observe that:<\/p>\n<ul>\n<li>A better fit of the regression analysis leads to a smaller standard error of the estimate \\((s_e)\\), subsequently resulting in a lower standard error of the forecast.<\/li>\n<li>When the sample size \\((n)\\) in the regression calculation increases, it directly corresponds to a reduction in the standard error of the forecast.<\/li>\n<li>If the forecasted independent variable \\((X_f)\\) approaches the mean of the independent variable \\((\\bar{X})\\) utilized in the regression analysis, it decreases the standard error of the forecast.<\/li>\n<\/ul>\n<p><strong>Example: Calculating the Confidence Interval of the Predicted Value<\/strong><\/p>\n<p>Refer to the example of regressed inflation rates against unemployment rates from 2011 to 2020.<\/p>\n<p>$$ \\begin{array}{c|c|c|c|c|c|c|c}<br \/>\\text{Year} &amp; \\text{Unemployment} &amp; \\text{Inflation} &amp; \\text{Predicted} &amp; \\text{Variation} &amp; \\text{Variation} &amp; \\text{Variation} &amp; (X_i \\\\<br \/>&amp; {\\text{Rate } \\% (X_i)} &amp; {\\text{Rate }\\%} &amp; \\text{Unemployment} &amp; \\text{to be} &amp; \\text{Unexplained} &amp; \\text{Explained} &amp; -\\bar{X})^2 \\\\<br \/>&amp; &amp; ({{Y}}_i) &amp; {\\text{rate } (\\hat Y_i)} &amp; \\text{Explained.} &amp; &amp; &amp; \\\\<br \/>&amp; &amp; &amp; &amp; \\left(Y_i-\\bar{Y}\\right)^2 &amp; \\left(Y_i- \\hat{Y}_i\\right)^2 &amp; \\left({\\hat{Y}}_i-\\bar{Y}\\right)^2 &amp; \\\\ \\hline<br \/>2011 &amp; 6.1 &amp; 1.7 &amp; 1.610 &amp; 0.410 &amp; 0.008 &amp; 0.533 &amp; 0.656 \\\\ \\hline<br \/>2012 &amp; 7.4 &amp; 1.2 &amp; 0.437 &amp; 1.300 &amp; 0.582 &amp; 3.621 &amp; 4.452 \\\\ \\hline<br \/>\\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots &amp; \\vdots \\\\ \\hline<br \/>2019 &amp; 4.0 &amp; 4.7 &amp; 3.504 &amp; 5.570 &amp; 1.430 &amp; 1.355 &amp; 1.664 \\\\ \\hline<br \/>2020 &amp; 3.9 &amp; 3.6 &amp; 3.594 &amp; 1.588 &amp; 0.000 &amp; 1.573 &amp; 1.932 \\\\ \\hline<br \/>\\textbf{Sum} &amp; \\bf{52.90} &amp; \\bf{23.4} &amp; &amp; \\bf{13.704} &amp; \\bf{3.136} &amp; \\bf{10.568} &amp; \\bf{12.989} \\\\ \\hline<br \/>\\textbf{Arithmetic} &amp; \\bf{5.29} &amp; \\bf{2.34} &amp; &amp; &amp; &amp; &amp; \\\\<br \/>\\textbf{Mean} &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\\\<br \/>\\end{array} $$<\/p>\n<p>Consider the results of the regression analysis of inflation rates on unemployment rates:<\/p>\n<p>$$ \\begin{array}{lcccc}<br \/>\\bf{\\textit{Regression Statistics} } &amp; &amp; &amp; &amp; \\\\ \\hline<br \/>\\text{R Square} &amp; 0.7711 &amp; &amp; &amp; \\\\<br \/>\\text{Standard Error} &amp; 0.6261 &amp; &amp; &amp; \\\\<br \/>\\text{Observations} &amp; 10 &amp; &amp; &amp; \\\\ \\hline \\\\ \\hline<br \/>\\text{ANOVA} &amp; &amp; &amp; &amp; \\\\ \\hline<br \/>&amp; \\textbf{df} &amp; \\textbf{Sum of} &amp; \\textbf{Mean} &amp; \\textbf{F} \\\\<br \/>&amp; &amp; \\textbf{Squares} &amp; \\textbf{Square} &amp; \\\\ \\hline<br \/>\\text{Regression} &amp; 1 &amp; 10.568 &amp; 10.568 &amp; 26.9565 \\\\<br \/>\\text{Residual} &amp; 8 &amp; 3.136 &amp; 0.392 &amp; \\\\<br \/>\\text{Total} &amp; 9 &amp; 13.704 &amp; &amp; \\\\ \\hline<br \/>\\\\ \\hline<br \/>&amp; \\textbf{Coefficients} &amp; \\textbf{Standard} &amp; \\textbf{t Stat} &amp; \\textbf{p-value} \\\\<br \/>&amp; &amp; \\textbf{Error} &amp; &amp; \\\\ \\hline<br \/>\\text{Intercept} &amp; 7.112 &amp; 0.940 &amp; 7.565 &amp; 0.000 \\\\<br \/>\\text{Unemployment} &amp; -0.902 &amp; 0.174 &amp; -5.192 &amp; 0.001 \\\\<br \/>\\text{rate (%)} &amp; &amp; &amp; &amp; \\\\ \\hline<br \/>\\end{array} $$<\/p>\n<p>Given that the forecasted unemployment rate is 4.5%, calculate the 95% confidence interval for the predicted inflation rate value.<\/p>\n<p><strong>Solution<\/strong><\/p>\n<p>$$\\text{Prediction Interval} = \\hat{Y} \\pm t_{c}s_{f}$$<\/p>\n<p>The estimated variance of the prediction error is:<\/p>\n<p>$$ \\begin{align*}<br \/>s_f^2 &amp; =s_e^2\\left[1+\\frac{1}{n}+\\frac{\\left(X_f-\\bar{X}\\right)^2}{\\left(n-1\\right)s_X^2}\\right] \\\\ &amp; =s_e^2\\left[1+\\frac{1}{n}+\\frac{\\left(X_f-\\bar{X}\\right)^2}{\\sum_{i=1}^{n}\\left(X_i-\\bar{X}\\right)^2}\\right] \\\\ &amp; ={0.6261}^2\\left[1+\\frac{1}{10}+\\frac{\\left(4.5-5.29\\right)^2}{12.989}\\right]=0.450 \\end{align*} $$<\/p>\n<p>As such, the standard error of forecast is:<\/p>\n<p>$$ s_f=\\sqrt{0.450}=0.6708 $$<\/p>\n<p>The predicted value of the inflation rate given an unemployment rate of 4.5% is:<\/p>\n<p>$$ \\hat{Y}=7.112-0.9020\\times4.5=3.05\\% $$<\/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\" width=\"624\" height=\"612\" class=\"aligncenter size-full wp-image-45742\" style=\"max-width: 100%;\" src=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12.jpg\" alt=\"\" srcset=\"https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12.jpg 624w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12-300x294.jpg 300w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12-400x392.jpg 400w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12-24x24.jpg 24w, https:\/\/analystprep.com\/cfa-level-1-exam\/wp-content\/uploads\/2023\/08\/LM10_IMG12-48x48.jpg 48w\" sizes=\"auto, (max-width: 624px) 100vw, 624px\" \/><\/p>\n<p>The prediction interval at the 95% confidence level is:<\/p>\n<p>$$\\text{Prediction Interval (PI)} = \\hat{Y} \\pm t_{c}s_{f}$$<\/p>\n<p>$$\\text{PI} = 3.05 \\pm 2.306\\times 0.6708= 1.50\\% \\text{ to } 4.60\\%$$<\/p>\n<p><strong>Interpretation<\/strong><\/p>\n<p>Given an unemployment rate of 4.5%, we are <strong>95% confident<\/strong> that <strong>the inflation rate will lie between 1.50%<\/strong> and <strong>4.60%<\/strong>.<\/p>\n<blockquote>\n<h2>Question 1<\/h2>\n<p>The regression equation of the 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 to<\/em>:<\/p>\n<ol type=\"A\">\n<li>153.<\/li>\n<li>155.<\/li>\n<li>471.<\/li>\n<\/ol>\n<p><strong>The correct answer is A<\/strong>.<\/p>\n<p>$$Y = -159 + 0.26\\times1,200=153$$<\/p>\n<\/blockquote>\n\n\n<div style=\"text-align: center; margin: 30px 0;\">\n  <a style=\"display: inline-flex; align-items: center; justify-content: center; padding: 12px 26px; border-radius: 9999px; background: #1e5bd8; color: #ffffff; font-weight: bold; text-decoration: none;\" href=\"https:\/\/analystprep.com\/free-trial\/\" target=\"_blank\" rel=\"noopener noreferrer\">\n    Start Free Trial \u2192\n  <\/a>\n  <p style=\"margin-top: 12px; font-size: 16px; line-height: 1.5;\">\n    Practice regression predictions, confidence intervals, and model interpretation with CFA Level I exam-style questions.\n  <\/p>\n<\/div>\n","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: $$\\hat{Y}=\\hat{b}_0+\\hat{b}_1X$$ Where:&#8230;<\/p>\n","protected":false},"author":7,"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-45739","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Predicted Value &amp; Prediction Interval | CFA Level 1<\/title>\n<meta name=\"description\" content=\"Learn how to calculate and interpret predicted values in regression models, including prediction interval formulas for dependent variables.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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