class: center, middle, inverse, title-slide # All About Regression ## EC 350: Labor Economics ### <a href="https://kyleraze.com">Kyle Raze</a> ### Winter 2022 --- # All About Regression ## **Econometrics** **The objective?** Identify the effect of a treatment variable `\(D\)` on an outcome variable `\(Y\)`..super[.hi-pink[<span>&#8224;</span>]] - **How?** Find a way to shut down .hi-pink[selection bias]. .footnote[.super[.hi-pink[<span>&#8224;</span>]] The other objective? Forecast future values of key outcome variables, such as unemployment, GDP, customer retention, *etc.* But that's a different subject for a different course.] -- ## **Regression analysis** > A set of statistical processes for quantifying the relationship between a dependent variable (*e.g.,* an outcome) and one or more independent variables (*e.g.,* a treatment or a control variable). A bundle of useful tools for doing econometrics! --- # All About Regression ## **Regression analysis** Economists often rely on regression analysis to make various statistical comparisons. - Can facilitate *other things equal* comparisons. - Can shut down .pink[selection bias] by explicitly **controlling for** .hi-pink[confounding variables]. - Failure to control for confounding variables? .mono[-->] .hi-pink[omitted-variable bias]. -- **Our objective?** Learn how to interpret the results of a regression analysis. 1. **Literal interpretation** - Interpret the size and statistical significance of regression coefficient estimates. - Know your way around a regression table. 2. **Big-picture interpretation** - What do the estimates imply about the effects of a treatment? - Should we trust the estimates? Do they reflect a causal relationship? --- class: inverse, middle # Simple linear regression --- # Simple linear regression <img src="04-All_About_Regression_files/figure-html/simple-1.svg" style="display: block; margin: auto;" /> --- count: false # Simple linear regression <img src="04-All_About_Regression_files/figure-html/simple_reg-1.svg" style="display: block; margin: auto;" /> --- # Simple linear regression ## **Model** We can express the relationship between the .hi-purple[outcome variable] and the .hi-green[treatment variable] as linear: $$ \color{#9370DB}{Y_i} = \alpha + \beta~\color{#007935}{D_i} + \varepsilon_i $$ - `\(i\)` indexes an individual. - `\(\alpha\)` .mono[=] the __intercept__ or constant. - `\(\beta\)` .mono[=] the __slope coefficient__. - Imagine for now that `\(D_i\)` can take on many different values (*e.g.,* more than just 0 or 1). - `\(\varepsilon_i\)` .mono[=] the __error term__. .footnote[ _Simple_ .mono[=] Only one independent variable. ] --- # Simple linear regression ## **Model** The .hi[intercept] tells us the expected value of `\(Y_i\)` when `\(D_i = 0\)`. $$ Y_i = \color{#e64173}{\alpha} + \beta ~ D_i + \varepsilon_i $$ Part of the regression line, but almost never the focus of an analysis. - In practice, omitting the intercept would bias estimates of the slope coefficient&mdash;the object we really care about. --- # Simple linear regression ## **Model** The .hi[slope coefficient] tells us the expected change in `\(Y_i\)` when `\(D_i\)` increases by one. $$ Y_i = \alpha + \color{#e64173}{\beta} ~ D_i + \varepsilon_i $$ "A one-unit increase in `\(D_i\)` *is associated with* a `\(\color{#e64173}{\beta}\)`-unit increase in `\(Y_i\)`." -- Under certain (strong) assumptions about the error term (*e.g.,* no selection bias), `\(\color{#e64173}{\beta}\)` represents the causal effect of `\(D_i\)` on `\(Y_i\)`. - "A one-unit increase in `\(D_i\)` *leads to* a `\(\color{#e64173}{\beta}\)`-unit increase in `\(Y_i\)`." - Otherwise, it's just the _association of_ `\(D_i\)` _with_ `\(Y_i\)`, representing a non-causal correlation. --- # Simple linear regression ## **Model** The .hi[error term] reminds us that `\(D_i\)` isn't the only variable that affects `\(Y_i\)`. $$ Y_i = \alpha + \beta ~ D_i + \color{#e64173}{\varepsilon_i} $$ -- The error term represents all other factors that explain `\(Y_i\)`. - **So what?** If some of those factors influence `\(D_i\)`, then omitted-variable bias will contaminate estimates of the slope coefficient. --- # Simple linear regression ## **Example** .pull-left[ **Q:** How does attendance affect performance? As a first attempt at an answer, we can estimate a regression of final exam scores on attendance: `$$\text{Final}_i = \alpha + \beta~\text{Attend}_i + \varepsilon_i$$` <table> <thead> <tr> <th style="text-align:left;"> Parameter </th> <th style="text-align:center;"> (1) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Intercept </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 56.82 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (2.19) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Attendance </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 0.3 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.08) </td> </tr> </tbody> </table> .center[*Standard errors in parentheses.*] ] .pull-right[ <img src="04-All_About_Regression_files/figure-html/attend_1_plot-1.svg" style="display: block; margin: auto;" /> ] --- # Simple linear regression ## **Example** .pull-left[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/campus_crime_1_plot-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ **Q:** Do police on college campuses reduce crime? - What does the slope coefficient tell us? ] --- count: false # Simple linear regression ## **Example** .pull-left[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/campus_crime_2_plot-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ **Q:** Do police on college campuses reduce crime? - What does the slope coefficient tell us? **Q:** Does this mean that police *cause* crime!? - Why or why not? ] --- count: false # Simple linear regression ## **Example** .pull-left[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/campus_crime_3_plot-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ **Q:** Do police on college campuses reduce crime? - What does the slope coefficient tell us? **Q:** Does this mean that police *cause* crime!? - Why or why not? .footnote[For an interesting discussion of the causal effects of police staffing on crime and arrests&mdash;and how those effects vary by race&mdash;check out [episode 55](https://www.probablecausation.com/podcasts/episode-55-morgan-williams-jr) of the [*Probable Causation*](https://www.probablecausation.com/) podcast.] ] --- # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-2-1.svg" style="display: block; margin: auto;" /> ] --- count: false # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-3-1.svg" style="display: block; margin: auto;" /> ] --- count: false # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-4-1.svg" style="display: block; margin: auto;" /> ] --- count: false # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - Every "fitted line" produces .hi-pink[residuals]. - Residual .mono[=] actual .mono[-] .hi-purple[predicted] ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-5-1.svg" style="display: block; margin: auto;" /> ] --- # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - Some fitted lines generate bigger residuals than others. ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 58.2 .mono[+] -2.2 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-7-1.svg" style="display: block; margin: auto;" /> ] --- count: false # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - Some fitted lines generate bigger residuals than others. ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 20.5 .mono[+] 3.15 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-9-1.svg" style="display: block; margin: auto;" /> ] --- count: false # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - Some fitted lines generate bigger residuals than others. ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 1.3 .mono[+] 0.75 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-11-1.svg" style="display: block; margin: auto;" /> ] --- # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - The "line of best fit" is the line that **minimizes** the **sum of squared residuals**. - **Q:** Why squared? ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-12-1.svg" style="display: block; margin: auto;" /> ] --- count: false # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - The "line of best fit" is the line that **minimizes** the **sum of squared residuals**. - **Q:** Why squared? - Using math you'll see in EC 320 or matrix algebra, OLS does this without the guesswork. ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-13-1.svg" style="display: block; margin: auto;" /> ] --- # Simple linear regression ## **Estimation** .pull-left[ **Q:** Where does the regression line come from? <br> **A:** A routine called **ordinary least squares (OLS)**. **How does OLS work?** - **"Squares?"** Sum of squared residuals. - **"Least?"** Minimize that sum. - **"Ordinary?"** Oldest, most common way of estimating a regression. ] .pull-right[ .center[.purple[Crime.sub[*i*] .mono[=] 18.41 .mono[+] 1.76 Police.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-14-1.svg" style="display: block; margin: auto;" /> ] --- # Simple linear regression ## **Example: Returns to education** The optimal investment in education by students, parents, and legislators depends in part on the monetary *return to education*. -- .hi-purple[Thought experiment:] - Randomly select an individual. - Give her an additional year of education. - How much do her earnings increase? The change in her earnings describes the .hi-slate[causal effect] of education on earnings. --- # Simple linear regression ## **Example: Returns to education** .pull-left[ .center[.purple[Earnings.sub[*i*] .mono[=] 146.95 .mono[+] 60.21 Schooling.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-16-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ **Q:** How much extra money can a worker in this sample expect from an additional year of education? - How do you know? ] --- count: false # Simple linear regression ## **Example: Returns to education** .pull-left[ .center[.purple[Earnings.sub[*i*] .mono[=] 146.95 .mono[+] 60.21 Schooling.sub[*i*]]] <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-17-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ **Q:** How much extra money can a worker in this sample expect from an additional year of education? - How do you know? **Q:** Does this number represent the causal return to an additional year of education? - What other variables could be driving the relationship? ] --- class: inverse, middle # Making adjustments --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-18-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ We can produce a fitted line by estimating a regression of an outcome on a treatment: `$$Y_i = \alpha + \beta~D_i + \varepsilon_i$$` `\(\beta\)` describes how the outcome changes, *on average*, when treatment changes. <table> <thead> <tr> <th style="text-align:left;"> Parameter </th> <th style="text-align:center;"> (1) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Intercept </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 1.22 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.18) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Treatment </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 0.56 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.08) </td> </tr> </tbody> </table> .center[*Standard errors in parentheses.*] ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-20-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ However, we might worry that a third variable `\(W_i\)` confounds our estimate of the effect of the treatment on the outcome. ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-21-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` ] **Q:** How does OLS "adjust" for the confounder? --- count: false # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-22-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` **Q:** How does OLS "adjust" for the confounder? - **Step 1:** Figure out what differences in D are explained by W. ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-23-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` **Q:** How does OLS "adjust" for the confounder? - **Step 2:** Remove differences in D explained by W. ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-24-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` **Q:** How does OLS "adjust" for the confounder? - **Step 3:** Figure out what differences in Y are explained by W. ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-25-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` **Q:** How does OLS "adjust" for the confounder? - **Step 4:** Remove differences in Y explained by W. ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-26-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` **Q:** How does OLS "adjust" for the confounder? - **Step 5:** Fit a regression through the adjusted data. ] --- # Making adjustments .pull-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-27-1.svg" style="display: block; margin: auto;" /> ] .pull-right[ If data on the confounder exists, it can be added to the regression model: `$$Y_i = \alpha + \beta~D_i + \gamma~W_i + \varepsilon_i$$` <table> <thead> <tr> <th style="text-align:left;"> Parameter </th> <th style="text-align:center;"> (1) </th> <th style="text-align:center;"> (2) </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Intercept </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1.22 </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 0.9 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (0.18) </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.1) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Treatment </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0.56 </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> -0.42 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (0.08) </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.07) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Confounder </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 3.91 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.2) </td> </tr> </tbody> </table> .center[*Standard errors in parentheses.*] ] --- # Omitted-variable bias ## **Example: Returns to education** .pull-left[ <br> <table> <caption>Outcome: Weekly Earnings</caption> <thead> <tr> <th style="text-align:left;"> Parameter </th> <th style="text-align:center;"> 1 </th> <th style="text-align:center;"> 2 </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Intercept </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 146.95 </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -128.89 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (77.72) </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (92.18) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Schooling (Years) </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 60.21 </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 42.06 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (5.70) </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (6.55) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> IQ Score (Points) </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5.14 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (0.96) </td> </tr> </tbody> </table> .center[*Standard errors in parentheses.*] ] .pull-right[ ] --- count: false # Omitted-variable bias ## **Example: Returns to education** .pull-left[ <br> <table> <caption>Outcome: Weekly Earnings</caption> <thead> <tr> <th style="text-align:left;"> Parameter </th> <th style="text-align:center;"> 1 </th> <th style="text-align:center;"> 2 </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Intercept </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 146.95 </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> -128.89 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (77.72) </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (92.18) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> Schooling (Years) </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 60.21 </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 42.06 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (5.70) </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (6.55) </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;line-height: 110%;font-style: italic;color: black !important;"> IQ Score (Points) </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;"> </td> <td style="text-align:center;color: #272822 !important;line-height: 110%;font-weight: bold;"> 5.14 </td> </tr> <tr> <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> </td> <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.96) </td> </tr> </tbody> </table> .center[*Standard errors in parentheses.*] ] -- .pull-right[ <br> <br> .orange[Bias] from omitting IQ score <br> `\(\quad\)` .mono[=] .pink["short"] .mono[-] .purple["long"] <br> `\(\quad\)` .mono[=] .pink[60.21] .mono[-] .purple[42.06] <br> `\(\quad\)` .mono[=] .orange[18.15] The first regression mistakenly attributes some of the influence of intelligence to education. ] --- # Omitted-variable bias .more-left[ <img src="04-All_About_Regression_files/figure-html/venn2-1.svg" style="display: block; margin: auto;" /> ] .less-right[ .hi-purple[Y] .mono[=] Outcome .hi-green[D] .mono[=] Treatment .hi-orange[W] .mono[=] Omitted variable If .hi-orange[W] is correlated with both .hi-green[D] and .hi-purple[Y] .mono[-->] omitted variable bias .mono[-->] regression fails to isolate the causal effect of .hi-green[D] on .hi-purple[Y]. ] --- # Omitted-variable bias .more-left[ <img src="04-All_About_Regression_files/figure-html/unnamed-chunk-31-1.svg" style="display: block; margin: auto;" /> ] .less-right[ .hi-purple[Y] .mono[=] Outcome .hi-green[D] .mono[=] Treatment .hi-orange[W] .mono[=] Omitted variable If .hi-orange[W] is correlated with both .hi-green[D] and .hi-purple[Y] .mono[-->] omitted variable bias .mono[-->] regression fails to isolate the causal effect of .hi-green[D] on .hi-purple[Y]. ] --- # Housekeeping **MLK Jr. Day:** No class or office hours on Monday the 17th. **Pre-recorded lecture** for Wednesday the 19th. - I will try to post it sometime next week. - In the meantime, enjoy your weekend! **Assigned reading for next week:** [Snapping back: Food stamp bans and criminal recidivism](https://www.aeaweb.org/articles?id=10.1257/pol.20170490) by Cody Tuttle (2019). - Best to read it *after* you watch next week's lecture. - Reading Quiz 3 due the following week (Monday the 24th). **Problem Set 1** due on Friday the 21st by 11:59pm. - Covers everything though next Wednesday.