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<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<meta charset="utf-8" />
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<title>14 Ch 3 - MLR | R Programming Guidebook Project</title>
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<li class="chapter" data-level="" data-path="index.html"><a href="index.html"><i class="fa fa-check"></i>About</a></li>
<li class="part"><span><b>I DataCamp</b></span></li>
<li class="chapter" data-level="1" data-path="introduction-to-r.html"><a href="introduction-to-r.html"><i class="fa fa-check"></i><b>1</b> Introduction to R</a>
<ul>
<li class="chapter" data-level="1.1" data-path="introduction-to-r.html"><a href="introduction-to-r.html#intro-to-basics"><i class="fa fa-check"></i><b>1.1</b> Intro to basics</a></li>
<li class="chapter" data-level="1.2" data-path="introduction-to-r.html"><a href="introduction-to-r.html#vectors"><i class="fa fa-check"></i><b>1.2</b> Vectors</a></li>
<li class="chapter" data-level="1.3" data-path="introduction-to-r.html"><a href="introduction-to-r.html#matrices"><i class="fa fa-check"></i><b>1.3</b> Matrices</a></li>
<li class="chapter" data-level="1.4" data-path="introduction-to-r.html"><a href="introduction-to-r.html#factors"><i class="fa fa-check"></i><b>1.4</b> Factors</a></li>
<li class="chapter" data-level="1.5" data-path="introduction-to-r.html"><a href="introduction-to-r.html#data-frames"><i class="fa fa-check"></i><b>1.5</b> Data frames</a></li>
<li class="chapter" data-level="1.6" data-path="introduction-to-r.html"><a href="introduction-to-r.html#lists"><i class="fa fa-check"></i><b>1.6</b> Lists</a></li>
</ul></li>
<li class="chapter" data-level="2" data-path="intermediate-r.html"><a href="intermediate-r.html"><i class="fa fa-check"></i><b>2</b> Intermediate R</a>
<ul>
<li class="chapter" data-level="2.1" data-path="intermediate-r.html"><a href="intermediate-r.html#conditionals-and-control-flow"><i class="fa fa-check"></i><b>2.1</b> Conditionals And Control Flow</a></li>
<li class="chapter" data-level="2.2" data-path="intermediate-r.html"><a href="intermediate-r.html#loops"><i class="fa fa-check"></i><b>2.2</b> Loops</a></li>
<li class="chapter" data-level="2.3" data-path="intermediate-r.html"><a href="intermediate-r.html#functions"><i class="fa fa-check"></i><b>2.3</b> Functions</a></li>
<li class="chapter" data-level="2.4" data-path="intermediate-r.html"><a href="intermediate-r.html#the-apply-family"><i class="fa fa-check"></i><b>2.4</b> The apply family</a></li>
<li class="chapter" data-level="2.5" data-path="intermediate-r.html"><a href="intermediate-r.html#utilities"><i class="fa fa-check"></i><b>2.5</b> Utilities</a></li>
</ul></li>
<li class="chapter" data-level="3" data-path="intro-to-the-tidyverse.html"><a href="intro-to-the-tidyverse.html"><i class="fa fa-check"></i><b>3</b> Intro to the Tidyverse</a>
<ul>
<li class="chapter" data-level="3.1" data-path="intro-to-the-tidyverse.html"><a href="intro-to-the-tidyverse.html#data-wrangling"><i class="fa fa-check"></i><b>3.1</b> Data wrangling</a></li>
<li class="chapter" data-level="3.2" data-path="intro-to-the-tidyverse.html"><a href="intro-to-the-tidyverse.html#data-visualization"><i class="fa fa-check"></i><b>3.2</b> Data visualization</a></li>
<li class="chapter" data-level="3.3" data-path="intro-to-the-tidyverse.html"><a href="intro-to-the-tidyverse.html#grouping-and-summarizing"><i class="fa fa-check"></i><b>3.3</b> Grouping and summarizing</a></li>
<li class="chapter" data-level="3.4" data-path="intro-to-the-tidyverse.html"><a href="intro-to-the-tidyverse.html#types-of-visualizations"><i class="fa fa-check"></i><b>3.4</b> Types of visualizations</a></li>
</ul></li>
<li class="chapter" data-level="4" data-path="intro-to-data-visualization-with-ggplot2.html"><a href="intro-to-data-visualization-with-ggplot2.html"><i class="fa fa-check"></i><b>4</b> Intro to Data Visualization with ggplot2</a>
<ul>
<li class="chapter" data-level="4.1" data-path="intro-to-data-visualization-with-ggplot2.html"><a href="intro-to-data-visualization-with-ggplot2.html#introduction"><i class="fa fa-check"></i><b>4.1</b> Introduction</a></li>
<li class="chapter" data-level="4.2" data-path="intro-to-data-visualization-with-ggplot2.html"><a href="intro-to-data-visualization-with-ggplot2.html#aesthetics"><i class="fa fa-check"></i><b>4.2</b> Aesthetics</a></li>
<li class="chapter" data-level="4.3" data-path="intro-to-data-visualization-with-ggplot2.html"><a href="intro-to-data-visualization-with-ggplot2.html#geometries"><i class="fa fa-check"></i><b>4.3</b> Geometries</a></li>
<li class="chapter" data-level="4.4" data-path="intro-to-data-visualization-with-ggplot2.html"><a href="intro-to-data-visualization-with-ggplot2.html#themes"><i class="fa fa-check"></i><b>4.4</b> Themes</a></li>
</ul></li>
<li class="chapter" data-level="5" data-path="working-with-data-in-the-tidyverse.html"><a href="working-with-data-in-the-tidyverse.html"><i class="fa fa-check"></i><b>5</b> Working with Data in the Tidyverse</a>
<ul>
<li class="chapter" data-level="5.1" data-path="working-with-data-in-the-tidyverse.html"><a href="working-with-data-in-the-tidyverse.html#explore-your-data"><i class="fa fa-check"></i><b>5.1</b> Explore your data</a></li>
<li class="chapter" data-level="5.2" data-path="working-with-data-in-the-tidyverse.html"><a href="working-with-data-in-the-tidyverse.html#tame-your-data"><i class="fa fa-check"></i><b>5.2</b> Tame your data</a></li>
<li class="chapter" data-level="5.3" data-path="working-with-data-in-the-tidyverse.html"><a href="working-with-data-in-the-tidyverse.html#tidy-your-data"><i class="fa fa-check"></i><b>5.3</b> Tidy your data</a></li>
<li class="chapter" data-level="5.4" data-path="working-with-data-in-the-tidyverse.html"><a href="working-with-data-in-the-tidyverse.html#transform-your-data"><i class="fa fa-check"></i><b>5.4</b> Transform your data</a></li>
</ul></li>
<li class="chapter" data-level="6" data-path="categorical-data-in-the-tidyverse.html"><a href="categorical-data-in-the-tidyverse.html"><i class="fa fa-check"></i><b>6</b> Categorical Data in the Tidyverse</a>
<ul>
<li class="chapter" data-level="6.1" data-path="categorical-data-in-the-tidyverse.html"><a href="categorical-data-in-the-tidyverse.html#introduction-to-factor-variables"><i class="fa fa-check"></i><b>6.1</b> Introduction to Factor Variables</a></li>
<li class="chapter" data-level="6.2" data-path="categorical-data-in-the-tidyverse.html"><a href="categorical-data-in-the-tidyverse.html#manipulating-factor-variables"><i class="fa fa-check"></i><b>6.2</b> Manipulating Factor Variables</a></li>
<li class="chapter" data-level="6.3" data-path="categorical-data-in-the-tidyverse.html"><a href="categorical-data-in-the-tidyverse.html#creating-factor-variables"><i class="fa fa-check"></i><b>6.3</b> Creating Factor Variables</a></li>
<li class="chapter" data-level="6.4" data-path="categorical-data-in-the-tidyverse.html"><a href="categorical-data-in-the-tidyverse.html#case-study-on-flight-etiquette"><i class="fa fa-check"></i><b>6.4</b> Case Study on Flight Etiquette</a></li>
</ul></li>
<li class="chapter" data-level="7" data-path="data-manipulation-with-dplyr.html"><a href="data-manipulation-with-dplyr.html"><i class="fa fa-check"></i><b>7</b> Data Manipulation with dplyr</a>
<ul>
<li class="chapter" data-level="7.1" data-path="data-manipulation-with-dplyr.html"><a href="data-manipulation-with-dplyr.html#transforming-data-with-dplyr"><i class="fa fa-check"></i><b>7.1</b> Transforming Data with dplyr</a></li>
<li class="chapter" data-level="7.2" data-path="data-manipulation-with-dplyr.html"><a href="data-manipulation-with-dplyr.html#aggregating-data"><i class="fa fa-check"></i><b>7.2</b> Aggregating Data</a></li>
<li class="chapter" data-level="7.3" data-path="data-manipulation-with-dplyr.html"><a href="data-manipulation-with-dplyr.html#selecting-and-transforming-data"><i class="fa fa-check"></i><b>7.3</b> Selecting and Transforming Data</a></li>
<li class="chapter" data-level="7.4" data-path="data-manipulation-with-dplyr.html"><a href="data-manipulation-with-dplyr.html#case-study-the-babynames-dataset"><i class="fa fa-check"></i><b>7.4</b> Case Study: The babynames Dataset</a></li>
</ul></li>
<li class="chapter" data-level="8" data-path="joining-data-with-dplyr.html"><a href="joining-data-with-dplyr.html"><i class="fa fa-check"></i><b>8</b> Joining Data with dplyr</a>
<ul>
<li class="chapter" data-level="8.1" data-path="joining-data-with-dplyr.html"><a href="joining-data-with-dplyr.html#joining-tables"><i class="fa fa-check"></i><b>8.1</b> Joining Tables</a></li>
<li class="chapter" data-level="8.2" data-path="joining-data-with-dplyr.html"><a href="joining-data-with-dplyr.html#left-and-right-joins"><i class="fa fa-check"></i><b>8.2</b> Left and Right Joins</a></li>
<li class="chapter" data-level="8.3" data-path="joining-data-with-dplyr.html"><a href="joining-data-with-dplyr.html#full-semi-and-anti-joins"><i class="fa fa-check"></i><b>8.3</b> Full, Semi, and Anti Joins</a></li>
<li class="chapter" data-level="8.4" data-path="joining-data-with-dplyr.html"><a href="joining-data-with-dplyr.html#case-study-joins-on-stack-overflow-data"><i class="fa fa-check"></i><b>8.4</b> Case Study: Joins on Stack Overflow Data</a></li>
</ul></li>
<li class="chapter" data-level="9" data-path="cleaning-data-in-r.html"><a href="cleaning-data-in-r.html"><i class="fa fa-check"></i><b>9</b> Cleaning Data in R</a>
<ul>
<li class="chapter" data-level="9.1" data-path="cleaning-data-in-r.html"><a href="cleaning-data-in-r.html#common-data-problems"><i class="fa fa-check"></i><b>9.1</b> Common Data Problems</a></li>
<li class="chapter" data-level="9.2" data-path="cleaning-data-in-r.html"><a href="cleaning-data-in-r.html#categorical-and-text-data"><i class="fa fa-check"></i><b>9.2</b> Categorical and Text Data</a></li>
<li class="chapter" data-level="9.3" data-path="cleaning-data-in-r.html"><a href="cleaning-data-in-r.html#advanced-data-problems"><i class="fa fa-check"></i><b>9.3</b> Advanced Data Problems</a></li>
<li class="chapter" data-level="9.4" data-path="cleaning-data-in-r.html"><a href="cleaning-data-in-r.html#record-linkage"><i class="fa fa-check"></i><b>9.4</b> Record Linkage</a></li>
</ul></li>
<li class="chapter" data-level="10" data-path="introduction-to-sql.html"><a href="introduction-to-sql.html"><i class="fa fa-check"></i><b>10</b> Introduction to SQL</a>
<ul>
<li class="chapter" data-level="10.1" data-path="introduction-to-sql.html"><a href="introduction-to-sql.html#selecting-columns"><i class="fa fa-check"></i><b>10.1</b> Selecting columns</a></li>
<li class="chapter" data-level="10.2" data-path="introduction-to-sql.html"><a href="introduction-to-sql.html#filtering-rows"><i class="fa fa-check"></i><b>10.2</b> Filtering rows</a></li>
<li class="chapter" data-level="10.3" data-path="introduction-to-sql.html"><a href="introduction-to-sql.html#aggregate-functions"><i class="fa fa-check"></i><b>10.3</b> Aggregate Functions</a></li>
<li class="chapter" data-level="10.4" data-path="introduction-to-sql.html"><a href="introduction-to-sql.html#sorting-and-grouping"><i class="fa fa-check"></i><b>10.4</b> Sorting and grouping</a></li>
</ul></li>
<li class="chapter" data-level="11" data-path="joining-data-in-sql.html"><a href="joining-data-in-sql.html"><i class="fa fa-check"></i><b>11</b> Joining Data in SQL</a>
<ul>
<li class="chapter" data-level="11.1" data-path="joining-data-in-sql.html"><a href="joining-data-in-sql.html#introduction-to-joins"><i class="fa fa-check"></i><b>11.1</b> Introduction to joins</a></li>
<li class="chapter" data-level="11.2" data-path="joining-data-in-sql.html"><a href="joining-data-in-sql.html#outer-joins-and-cross-joins"><i class="fa fa-check"></i><b>11.2</b> Outer joins and cross joins</a></li>
<li class="chapter" data-level="11.3" data-path="joining-data-in-sql.html"><a href="joining-data-in-sql.html#set-theory-clauses"><i class="fa fa-check"></i><b>11.3</b> Set theory clauses</a></li>
<li class="chapter" data-level="11.4" data-path="joining-data-in-sql.html"><a href="joining-data-in-sql.html#subqueries"><i class="fa fa-check"></i><b>11.4</b> Subqueries</a></li>
</ul></li>
<li class="chapter" data-level="12" data-path="web-scraping-in-r.html"><a href="web-scraping-in-r.html"><i class="fa fa-check"></i><b>12</b> Web Scraping in R</a>
<ul>
<li class="chapter" data-level="12.1" data-path="web-scraping-in-r.html"><a href="web-scraping-in-r.html#introduction-to-html-and-web-scraping"><i class="fa fa-check"></i><b>12.1</b> Introduction to HTML and Web Scraping</a></li>
<li class="chapter" data-level="12.2" data-path="web-scraping-in-r.html"><a href="web-scraping-in-r.html#navigation-and-selection-with-css"><i class="fa fa-check"></i><b>12.2</b> Navigation and Selection with CSS</a></li>
<li class="chapter" data-level="12.3" data-path="web-scraping-in-r.html"><a href="web-scraping-in-r.html#advanced-selection-with-xpath"><i class="fa fa-check"></i><b>12.3</b> Advanced Selection with XPATH</a></li>
<li class="chapter" data-level="12.4" data-path="web-scraping-in-r.html"><a href="web-scraping-in-r.html#scraping-best-practices"><i class="fa fa-check"></i><b>12.4</b> Scraping Best Practices</a></li>
</ul></li>
<li class="part"><span><b>II Econometrics</b></span></li>
<li class="chapter" data-level="13" data-path="ch-2---slr.html"><a href="ch-2---slr.html"><i class="fa fa-check"></i><b>13</b> Ch 2 - SLR</a>
<ul>
<li class="chapter" data-level="13.1" data-path="ch-2---slr.html"><a href="ch-2---slr.html#notes"><i class="fa fa-check"></i><b>13.1</b> Notes</a></li>
<li class="chapter" data-level="13.2" data-path="ch-2---slr.html"><a href="ch-2---slr.html#example-2.3-ceo-salary-and-return-on-equity"><i class="fa fa-check"></i><b>13.2</b> Example 2.3: CEO Salary and Return on Equity</a></li>
<li class="chapter" data-level="13.3" data-path="ch-2---slr.html"><a href="ch-2---slr.html#example-2.4-wage-and-education"><i class="fa fa-check"></i><b>13.3</b> Example 2.4: Wage and Education</a></li>
<li class="chapter" data-level="13.4" data-path="ch-2---slr.html"><a href="ch-2---slr.html#example-2.5-voting-outcomes-and-campaign-expenditures"><i class="fa fa-check"></i><b>13.4</b> Example 2.5: Voting Outcomes and Campaign Expenditures</a></li>
<li class="chapter" data-level="13.5" data-path="ch-2---slr.html"><a href="ch-2---slr.html#example-of-fitted-values-haty"><i class="fa fa-check"></i><b>13.5</b> Example of Fitted Values (<span class="math inline">\(\hat{y}\)</span>)</a></li>
</ul></li>
<li class="chapter" data-level="14" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html"><i class="fa fa-check"></i><b>14</b> Ch 3 - MLR</a>
<ul>
<li class="chapter" data-level="14.1" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#data"><i class="fa fa-check"></i><b>14.1</b> Data</a>
<ul>
<li class="chapter" data-level="14.1.1" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#summary-statistics"><i class="fa fa-check"></i><b>14.1.1</b> Summary Statistics</a></li>
</ul></li>
<li class="chapter" data-level="14.2" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#regression-model-comparisons"><i class="fa fa-check"></i><b>14.2</b> Regression model comparisons</a></li>
<li class="chapter" data-level="14.3" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#adjusted-r-squared"><i class="fa fa-check"></i><b>14.3</b> Adjusted R-Squared</a></li>
<li class="chapter" data-level="14.4" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#multicollinearity"><i class="fa fa-check"></i><b>14.4</b> Multicollinearity</a>
<ul>
<li class="chapter" data-level="14.4.1" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#variance-inflation-factor-vif"><i class="fa fa-check"></i><b>14.4.1</b> Variance Inflation Factor (VIF)</a></li>
<li class="chapter" data-level="14.4.2" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#joint-hypotheses-test"><i class="fa fa-check"></i><b>14.4.2</b> Joint hypotheses test</a></li>
</ul></li>
<li class="chapter" data-level="14.5" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#testing-linear-combinations-of-parameters"><i class="fa fa-check"></i><b>14.5</b> Testing linear combinations of parameters</a></li>
<li class="chapter" data-level="14.6" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#a-note-on-presentation"><i class="fa fa-check"></i><b>14.6</b> A note on presentation</a></li>
<li class="chapter" data-level="14.7" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#log-transformations"><i class="fa fa-check"></i><b>14.7</b> Log transformations</a>
<ul>
<li class="chapter" data-level="14.7.1" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#histograms"><i class="fa fa-check"></i><b>14.7.1</b> Histograms</a></li>
<li class="chapter" data-level="14.7.2" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#scatter-plots"><i class="fa fa-check"></i><b>14.7.2</b> Scatter plots</a></li>
<li class="chapter" data-level="14.7.3" data-path="ch-3---mlr.html"><a href="ch-3---mlr.html#regression-models-with-levels-and-logs"><i class="fa fa-check"></i><b>14.7.3</b> Regression models with levels and logs</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="15" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html"><i class="fa fa-check"></i><b>15</b> Dummy Variables Part 1</a>
<ul>
<li class="chapter" data-level="15.1" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#obtain-and-prepare-data"><i class="fa fa-check"></i><b>15.1</b> Obtain and prepare data</a></li>
<li class="chapter" data-level="15.2" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#define-dummy-variables"><i class="fa fa-check"></i><b>15.2</b> Define dummy variables</a>
<ul>
<li class="chapter" data-level="15.2.1" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#alpha-model"><i class="fa fa-check"></i><b>15.2.1</b> Alpha Model</a></li>
<li class="chapter" data-level="15.2.2" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#beta-model"><i class="fa fa-check"></i><b>15.2.2</b> Beta Model</a></li>
</ul></li>
<li class="chapter" data-level="15.3" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#compare-the-regressions-side-by-side"><i class="fa fa-check"></i><b>15.3</b> Compare the regressions side-by-side</a></li>
<li class="chapter" data-level="15.4" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#compare-the-predictions-of-each-model"><i class="fa fa-check"></i><b>15.4</b> Compare the predictions of each model</a>
<ul>
<li class="chapter" data-level="15.4.1" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#group-averages"><i class="fa fa-check"></i><b>15.4.1</b> Group averages</a></li>
<li class="chapter" data-level="15.4.2" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#causal-estimates"><i class="fa fa-check"></i><b>15.4.2</b> Causal estimates?</a></li>
<li class="chapter" data-level="15.4.3" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#what-about-age"><i class="fa fa-check"></i><b>15.4.3</b> What about age?</a></li>
<li class="chapter" data-level="15.4.4" data-path="dummy-variables-part-1.html"><a href="dummy-variables-part-1.html#what-about-sex"><i class="fa fa-check"></i><b>15.4.4</b> What about sex?</a></li>
</ul></li>
</ul></li>
<li class="chapter" data-level="16" data-path="dummy-variables-part-2.html"><a href="dummy-variables-part-2.html"><i class="fa fa-check"></i><b>16</b> Dummy Variables Part 2</a>
<ul>
<li class="chapter" data-level="16.1" data-path="dummy-variables-part-2.html"><a href="dummy-variables-part-2.html#size-only"><i class="fa fa-check"></i><b>16.1</b> Size only</a></li>
<li class="chapter" data-level="16.2" data-path="dummy-variables-part-2.html"><a href="dummy-variables-part-2.html#number-of-bathrooms-and-size"><i class="fa fa-check"></i><b>16.2</b> Number of bathrooms and size</a></li>
<li class="chapter" data-level="16.3" data-path="dummy-variables-part-2.html"><a href="dummy-variables-part-2.html#slope-dummy"><i class="fa fa-check"></i><b>16.3</b> Slope dummy</a></li>
<li class="chapter" data-level="16.4" data-path="dummy-variables-part-2.html"><a href="dummy-variables-part-2.html#intercept-and-slope-dummies"><i class="fa fa-check"></i><b>16.4</b> Intercept and slope dummies</a></li>
<li class="chapter" data-level="16.5" data-path="dummy-variables-part-2.html"><a href="dummy-variables-part-2.html#models-with-the-number-of-bedrooms"><i class="fa fa-check"></i><b>16.5</b> Models with the number of bedrooms</a></li>
</ul></li>
<li class="chapter" data-level="17" data-path="fixed-effects.html"><a href="fixed-effects.html"><i class="fa fa-check"></i><b>17</b> Fixed Effects</a>
<ul>
<li class="chapter" data-level="17.1" data-path="fixed-effects.html"><a href="fixed-effects.html#variables"><i class="fa fa-check"></i><b>17.1</b> Variables</a></li>
<li class="chapter" data-level="17.2" data-path="fixed-effects.html"><a href="fixed-effects.html#ols"><i class="fa fa-check"></i><b>17.2</b> OLS</a></li>
<li class="chapter" data-level="17.3" data-path="fixed-effects.html"><a href="fixed-effects.html#country-fixed-effects"><i class="fa fa-check"></i><b>17.3</b> Country Fixed Effects</a></li>
<li class="chapter" data-level="17.4" data-path="fixed-effects.html"><a href="fixed-effects.html#year-fixed-effects"><i class="fa fa-check"></i><b>17.4</b> Year Fixed Effects</a></li>
<li class="chapter" data-level="17.5" data-path="fixed-effects.html"><a href="fixed-effects.html#country-and-year-fixed-effects"><i class="fa fa-check"></i><b>17.5</b> Country and Year Fixed Effects</a></li>
<li class="chapter" data-level="17.6" data-path="fixed-effects.html"><a href="fixed-effects.html#comparison-of-all-models"><i class="fa fa-check"></i><b>17.6</b> Comparison of all models</a>
<ul>
<li class="chapter" data-level="17.6.1" data-path="fixed-effects.html"><a href="fixed-effects.html#within-transformation"><i class="fa fa-check"></i><b>17.6.1</b> Within Transformation</a></li>
<li class="chapter" data-level="17.6.2" data-path="fixed-effects.html"><a href="fixed-effects.html#plm-package"><i class="fa fa-check"></i><b>17.6.2</b> PLM Package</a></li>
<li class="chapter" data-level="17.6.3" data-path="fixed-effects.html"><a href="fixed-effects.html#dummy-variables"><i class="fa fa-check"></i><b>17.6.3</b> Dummy Variables</a></li>
</ul></li>
<li class="chapter" data-level="17.7" data-path="fixed-effects.html"><a href="fixed-effects.html#data-summary-by-country"><i class="fa fa-check"></i><b>17.7</b> Data Summary by Country</a>
<ul>
<li class="chapter" data-level="17.7.1" data-path="fixed-effects.html"><a href="fixed-effects.html#average-values-for-each-country"><i class="fa fa-check"></i><b>17.7.1</b> Average Values for Each Country</a></li>
<li class="chapter" data-level="17.7.2" data-path="fixed-effects.html"><a href="fixed-effects.html#variable-specific-values-and-within-transformation-for-each-country"><i class="fa fa-check"></i><b>17.7.2</b> Variable-Specific Values and Within Transformation for Each Country</a></li>
<li class="chapter" data-level="17.7.3" data-path="fixed-effects.html"><a href="fixed-effects.html#life-expectancy"><i class="fa fa-check"></i><b>17.7.3</b> Life expectancy</a></li>
<li class="chapter" data-level="17.7.4" data-path="fixed-effects.html"><a href="fixed-effects.html#gdp-per-capita"><i class="fa fa-check"></i><b>17.7.4</b> GDP per capita</a></li>
<li class="chapter" data-level="17.7.5" data-path="fixed-effects.html"><a href="fixed-effects.html#population"><i class="fa fa-check"></i><b>17.7.5</b> Population</a></li>
<li class="chapter" data-level="17.7.6" data-path="fixed-effects.html"><a href="fixed-effects.html#percent-female"><i class="fa fa-check"></i><b>17.7.6</b> Percent female</a></li>
<li class="chapter" data-level="17.7.7" data-path="fixed-effects.html"><a href="fixed-effects.html#percent-rural"><i class="fa fa-check"></i><b>17.7.7</b> Percent rural</a></li>
</ul></li>
<li class="chapter" data-level="17.8" data-path="fixed-effects.html"><a href="fixed-effects.html#bookdown-style-note"><i class="fa fa-check"></i><b>17.8</b> Bookdown Style Note</a></li>
</ul></li>
<li class="chapter" data-level="18" data-path="difference-in-differences.html"><a href="difference-in-differences.html"><i class="fa fa-check"></i><b>18</b> Difference-in-Differences</a>
<ul>
<li class="chapter" data-level="18.1" data-path="difference-in-differences.html"><a href="difference-in-differences.html#data-1"><i class="fa fa-check"></i><b>18.1</b> Data</a></li>
<li class="chapter" data-level="18.2" data-path="difference-in-differences.html"><a href="difference-in-differences.html#model-1"><i class="fa fa-check"></i><b>18.2</b> Model 1</a>
<ul>
<li class="chapter" data-level="18.2.1" data-path="difference-in-differences.html"><a href="difference-in-differences.html#equivalent-model-1"><i class="fa fa-check"></i><b>18.2.1</b> Equivalent model 1</a></li>
</ul></li>
<li class="chapter" data-level="18.3" data-path="difference-in-differences.html"><a href="difference-in-differences.html#model-2"><i class="fa fa-check"></i><b>18.3</b> Model 2</a></li>
<li class="chapter" data-level="18.4" data-path="difference-in-differences.html"><a href="difference-in-differences.html#comparison-of-models"><i class="fa fa-check"></i><b>18.4</b> Comparison of models</a></li>
<li class="chapter" data-level="18.5" data-path="difference-in-differences.html"><a href="difference-in-differences.html#additional-questions"><i class="fa fa-check"></i><b>18.5</b> Additional questions</a>
<ul>
<li class="chapter" data-level="18.5.1" data-path="difference-in-differences.html"><a href="difference-in-differences.html#question-1"><i class="fa fa-check"></i><b>18.5.1</b> Question 1</a></li>
<li class="chapter" data-level="18.5.2" data-path="difference-in-differences.html"><a href="difference-in-differences.html#question-2"><i class="fa fa-check"></i><b>18.5.2</b> Question 2</a></li>
<li class="chapter" data-level="18.5.3" data-path="difference-in-differences.html"><a href="difference-in-differences.html#question-3"><i class="fa fa-check"></i><b>18.5.3</b> Question 3</a></li>
<li class="chapter" data-level="18.5.4" data-path="difference-in-differences.html"><a href="difference-in-differences.html#question-4"><i class="fa fa-check"></i><b>18.5.4</b> Question 4</a></li>
</ul></li>
<li class="chapter" data-level="18.6" data-path="difference-in-differences.html"><a href="difference-in-differences.html#polynomials"><i class="fa fa-check"></i><b>18.6</b> Polynomials</a></li>
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<h1><span class="header-section-number">14</span> Ch 3 - MLR</h1>
<p>This chapter walks through a few basics of working with MLR models. The last section at the bottom, <a href="ch-3---mlr.html#log-transformations">Log transformations</a>, asks you to create a few graphs, estimate a few models, and formally interpret a few coefficients.</p>
<p>You’ll need to make sure you have the following packages installed:</p>
<div class="sourceCode" id="cb678"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb678-1"><a href="ch-3---mlr.html#cb678-1" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
<span id="cb678-2"><a href="ch-3---mlr.html#cb678-2" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(stargazer)</span>
<span id="cb678-3"><a href="ch-3---mlr.html#cb678-3" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(multcomp)</span>
<span id="cb678-4"><a href="ch-3---mlr.html#cb678-4" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(pander)</span>
<span id="cb678-5"><a href="ch-3---mlr.html#cb678-5" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(car)</span>
<span id="cb678-6"><a href="ch-3---mlr.html#cb678-6" aria-hidden="true" tabindex="-1"></a><span class="fu">library</span>(ggplot2)</span></code></pre></div>
<div id="data" class="section level2" number="14.1">
<h2><span class="header-section-number">14.1</span> Data</h2>
<style>
.variableTable th {
text-align: left;
color: orange;
}
</style>
<p>The dataset <code>ceo.csv</code> has the following variables:</p>
<div class="variableTable">
<table>
<thead>
<tr class="header">
<th>Variable</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code>salary</code></td>
<td>1990 compensation in dollars ($)</td>
</tr>
<tr class="even">
<td><code>age</code></td>
<td>age of CEO in years</td>
</tr>
<tr class="odd">
<td><code>comten</code></td>
<td>years CEO with company</td>
</tr>
<tr class="even">
<td><code>ceoten</code></td>
<td>years as ceo with company</td>
</tr>
<tr class="odd">
<td><code>sales</code></td>
<td>1990 firm sales in $millions</td>
</tr>
<tr class="even">
<td><code>profits</code></td>
<td>1990 profits in $millions</td>
</tr>
<tr class="odd">
<td><code>mktval</code></td>
<td>1990 market value in $millions</td>
</tr>
</tbody>
</table>
</div>
<p>Note that the table above has custom formatting applied by adding an HTML class name “variableTable” (via a “div” element that surrounds the table), formatting the table header (th) to be left-aligned and have an orange font so you can easily see what was affected.</p>
<style>
.sumstats td {
text-align: right;
color: green;
}
</style>
<div id="summary-statistics" class="section level3 sumstats" number="14.1.1">
<h3><span class="header-section-number">14.1.1</span> Summary Statistics</h3>
<p>The <code>stargazer</code> package provides an easy way to display summary statistics. You need the <code>result='asis'</code> code chunk option so that it displays the html table in the knit output file (and you need the <code>type="html"</code> stargazer option to produce the html table). The <code>warning=FALSE</code> and <code>message=FALSE</code> keep it from displaying some of the messages it displays by default.</p>
<div class="sourceCode" id="cb679"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb679-1"><a href="ch-3---mlr.html#cb679-1" aria-hidden="true" tabindex="-1"></a>mydata <span class="ot"><-</span> <span class="fu">read.csv</span>(<span class="st">"https://raw.githubusercontent.com/econ380w21/380data/main/ceosal.csv"</span>)</span>
<span id="cb679-2"><a href="ch-3---mlr.html#cb679-2" aria-hidden="true" tabindex="-1"></a><span class="fu">stargazer</span>(mydata, <span class="at">type =</span> <span class="st">"html"</span>,<span class="at">summary.stat =</span> <span class="fu">c</span>(<span class="st">"mean"</span>,<span class="st">"sd"</span>, <span class="st">"min"</span>, <span class="st">"p25"</span>, <span class="st">"median"</span>, <span class="st">"p75"</span>, <span class="st">"max"</span>))</span></code></pre></div>
<table style="text-align:center">
<tr>
<td colspan="8" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
Statistic
</td>
<td>
Mean
</td>
<td>
St. Dev.
</td>
<td>
Min
</td>
<td>
Pctl(25)
</td>
<td>
Median
</td>
<td>
Pctl(75)
</td>
<td>
Max
</td>
</tr>
<tr>
<td colspan="8" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
salary
</td>
<td>
865,864.400
</td>
<td>
587,589.300
</td>
<td>
100,000
</td>
<td>
471,000
</td>
<td>
707,000
</td>
<td>
1,119,000
</td>
<td>
5,299,000
</td>
</tr>
<tr>
<td style="text-align:left">
age
</td>
<td>
56.429
</td>
<td>
8.422
</td>
<td>
33
</td>
<td>
52
</td>
<td>
57
</td>
<td>
62
</td>
<td>
86
</td>
</tr>
<tr>
<td style="text-align:left">
comten
</td>
<td>
22.503
</td>
<td>
12.295
</td>
<td>
2
</td>
<td>
12
</td>
<td>
23
</td>
<td>
33
</td>
<td>
58
</td>
</tr>
<tr>
<td style="text-align:left">
ceoten
</td>
<td>
7.955
</td>
<td>
7.151
</td>
<td>
0
</td>
<td>
3
</td>
<td>
6
</td>
<td>
11
</td>
<td>
37
</td>
</tr>
<tr>
<td style="text-align:left">
sales
</td>
<td>
3,529.463
</td>
<td>
6,088.654
</td>
<td>
29
</td>
<td>
561
</td>
<td>
1,400
</td>
<td>
3,500
</td>
<td>
51,300
</td>
</tr>
<tr>
<td style="text-align:left">
profits
</td>
<td>
207.831
</td>
<td>
404.454
</td>
<td>
-463
</td>
<td>
34
</td>
<td>
63
</td>
<td>
208
</td>
<td>
2,700
</td>
</tr>
<tr>
<td style="text-align:left">
mktval
</td>
<td>
3,600.316
</td>
<td>
6,442.276
</td>
<td>
387
</td>
<td>
644
</td>
<td>
1,200
</td>
<td>
3,500
</td>
<td>
45,400
</td>
</tr>
<tr>
<td colspan="8" style="border-bottom: 1px solid black">
</td>
</tr>
</table>
<p>In addition to providing you with an example of how to display a summary statistics table, this section also provides an example of how you can control aspects of how the table displays using HTML/CSS. In bookdown, the horizontal space is already modified for you, but if you use this in a regular RMD file that you knit, there will be no spacing at all between cells (so the numbers bump right up next to each other). To fix this, you can add padding (e.g., padding-left, padding-right). Often you want to right-align summary statistics. To do this, I’ve added an HTML class name “sumstats” to this section by including .sumstats in curly brackets in the section header (the part started with ##). I included CSS styling to align the text right. I also make the font green so that you could easily see what was modified.</p>
<p>Beyond simply formatting a table, this example, and the one above that uses a “div” and class name, are the two main ways you can control the formatting of specific parts of your book (without affecting the entire book).</p>
<p>Now back to econometrics…</p>
</div>
</div>
<div id="regression-model-comparisons" class="section level2" number="14.2">
<h2><span class="header-section-number">14.2</span> Regression model comparisons</h2>
<p>Consider four models:</p>
<div class="sourceCode" id="cb680"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb680-1"><a href="ch-3---mlr.html#cb680-1" aria-hidden="true" tabindex="-1"></a>model1 <span class="ot"><-</span> <span class="fu">lm</span>(salary<span class="sc">~</span>sales<span class="sc">+</span>mktval<span class="sc">+</span>profits,<span class="at">data=</span>mydata)</span>
<span id="cb680-2"><a href="ch-3---mlr.html#cb680-2" aria-hidden="true" tabindex="-1"></a>model2 <span class="ot"><-</span> <span class="fu">lm</span>(salary<span class="sc">~</span>sales<span class="sc">+</span>mktval<span class="sc">+</span>profits<span class="sc">+</span>age,<span class="at">data=</span>mydata)</span>
<span id="cb680-3"><a href="ch-3---mlr.html#cb680-3" aria-hidden="true" tabindex="-1"></a>model3 <span class="ot"><-</span> <span class="fu">lm</span>(salary<span class="sc">~</span>sales<span class="sc">+</span>mktval<span class="sc">+</span>profits<span class="sc">+</span>ceoten,<span class="at">data=</span>mydata)</span>
<span id="cb680-4"><a href="ch-3---mlr.html#cb680-4" aria-hidden="true" tabindex="-1"></a>model4 <span class="ot"><-</span> <span class="fu">lm</span>(salary<span class="sc">~</span>sales<span class="sc">+</span>mktval<span class="sc">+</span>profits<span class="sc">+</span>ceoten<span class="sc">+</span>age,<span class="at">data=</span>mydata)</span></code></pre></div>
<p>The package you should use for 380 to report the results of multiple models side-by-side is the <code>stargazer</code> package. Each column is a separate model. If a variable does not have a coefficient displayed in a column, that means it was not included as an explanatory variable in that model.</p>
<div class="sourceCode" id="cb681"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb681-1"><a href="ch-3---mlr.html#cb681-1" aria-hidden="true" tabindex="-1"></a><span class="fu">stargazer</span>(model1, model2, model3, model4, </span>
<span id="cb681-2"><a href="ch-3---mlr.html#cb681-2" aria-hidden="true" tabindex="-1"></a> <span class="at">type =</span> <span class="st">"html"</span>, </span>
<span id="cb681-3"><a href="ch-3---mlr.html#cb681-3" aria-hidden="true" tabindex="-1"></a> <span class="at">report=</span>(<span class="st">'vc*p'</span>),</span>
<span id="cb681-4"><a href="ch-3---mlr.html#cb681-4" aria-hidden="true" tabindex="-1"></a> <span class="at">keep.stat =</span> <span class="fu">c</span>(<span class="st">"n"</span>,<span class="st">"rsq"</span>,<span class="st">"adj.rsq"</span>), </span>
<span id="cb681-5"><a href="ch-3---mlr.html#cb681-5" aria-hidden="true" tabindex="-1"></a> <span class="at">notes =</span> <span class="st">"<em>&#42;p&lt;0.1;&#42;&#42;p&lt;0.05;&#42;&#42;&#42;p&lt;0.01</em>"</span>, </span>
<span id="cb681-6"><a href="ch-3---mlr.html#cb681-6" aria-hidden="true" tabindex="-1"></a> <span class="at">notes.append =</span> <span class="cn">FALSE</span>)</span></code></pre></div>
<table style="text-align:center">
<tr>
<td colspan="5" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td colspan="4">
<em>Dependent variable:</em>
</td>
</tr>
<tr>
<td>
</td>
<td colspan="4" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td colspan="4">
salary
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
(1)
</td>
<td>
(2)
</td>
<td>
(3)
</td>
<td>
(4)
</td>
</tr>
<tr>
<td colspan="5" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
sales
</td>
<td>
15.984
</td>
<td>
15.369
</td>
<td>
18.233<sup>*</sup>
</td>
<td>
18.061
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
p = 0.152
</td>
<td>
p = 0.169
</td>
<td>
p = 0.100
</td>
<td>
p = 0.106
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
mktval
</td>
<td>
23.831
</td>
<td>
23.842
</td>
<td>
21.157
</td>
<td>
21.231
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
p = 0.137
</td>
<td>
p = 0.137
</td>
<td>
p = 0.183
</td>
<td>
p = 0.183
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
profits
</td>
<td>
31.703
</td>
<td>
28.100
</td>
<td>
48.636
</td>
<td>
47.528
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
p = 0.909
</td>
<td>
p = 0.920
</td>
<td>
p = 0.859
</td>
<td>
p = 0.863
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
age
</td>
<td>
</td>
<td>
4,528.026
</td>
<td>
</td>
<td>
823.966
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
p = 0.353
</td>
<td>
</td>
<td>
p = 0.873
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
ceoten
</td>
<td>
</td>
<td>
</td>
<td>
12,730.860<sup>**</sup>
</td>
<td>
12,390.430<sup>**</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
p = 0.026
</td>
<td>
p = 0.042
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td style="text-align:left">
Constant
</td>
<td>
717,062.400<sup>***</sup>
</td>
<td>
464,424.700<sup>*</sup>
</td>
<td>
613,958.800<sup>***</sup>
</td>
<td>
570,743.300<sup>**</sup>
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
p = 0.000
</td>
<td>
p = 0.093
</td>
<td>
p = 0.000
</td>
<td>
p = 0.042
</td>
</tr>
<tr>
<td style="text-align:left">
</td>
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td>
</tr>
<tr>
<td colspan="5" style="border-bottom: 1px solid black">
</td>
</tr>
<tr>
<td style="text-align:left">
Observations
</td>
<td>
177
</td>
<td>
177
</td>
<td>
177
</td>
<td>
177
</td>
</tr>
<tr>
<td style="text-align:left">
R<sup>2</sup>
</td>
<td>
0.178
</td>
<td>
0.182
</td>
<td>
0.201
</td>
<td>
0.202
</td>
</tr>
<tr>
<td style="text-align:left">
Adjusted R<sup>2</sup>