3.6 — Regression with Categorical Data

# 3.6 — Regression with Categorical Data
## ECON 480 • Econometrics • Fall 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/metricsF21"><i class="fa fa-github fa-fw"></i>ryansafner/metricsF21</a><br> <a href="https://metricsF21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>metricsF21.classes.ryansafner.com</a><br>

---

# Outline

### [Working with `Factor` Variables in R](#3)

### [Regression with Dummy Variables](#12)

### [Recoding Dummies](#36)

### [Categorical Variables (More than 2 Categories)](#47)

---
# Categorical Data

- .hi[Categorical data] place an individual into one of several possible *categories*
    - e.g. sex, season, political party
    - may be responses to survey questions
    - can be quantitative (e.g. age, zip code)

- `R` calls these `factors`
]

![](../images/categoricaldata.png)

]

---

# Working with `factor` Variables in `R`

---

# Factors in R

.quitesmall[
- `factor` is a special type of `character` object class that indicates membership in a category (called a `level`)

- Suppose I have data on students:

```r
students %>% head(n = 5)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["ID"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["Rank"],"name":[2],"type":["chr"],"align":["left"]},{"label":["Grade"],"name":[3],"type":["dbl"],"align":["right"]}],"data":[{"1":"1","2":"Sophomore","3":"77"},{"1":"2","2":"Senior","3":"72"},{"1":"3","2":"Freshman","3":"73"},{"1":"4","2":"Senior","3":"73"},{"1":"5","2":"Junior","3":"84"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]

---

# Factors in R

.quitesmall[
- We can make `Rank` a `factor` variable, to indicate a student is a member of one of the possible categories: (freshman, sophomore, junior, senior)

```r
students <- students %>%
  mutate(Rank = as.factor(Rank)) # overwrite and change class of Rank to factor

students %>% head(n = 5)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["ID"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["Rank"],"name":[2],"type":["fct"],"align":["left"]},{"label":["Grade"],"name":[3],"type":["dbl"],"align":["right"]}],"data":[{"1":"1","2":"Sophomore","3":"77"},{"1":"2","2":"Senior","3":"72"},{"1":"3","2":"Freshman","3":"73"},{"1":"4","2":"Senior","3":"73"},{"1":"5","2":"Junior","3":"84"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]

---

# Factors in R

```r
# what are the categories?
students %>%
  group_by(Rank) %>%
  count()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["Rank"],"name":[1],"type":["fct"],"align":["left"]},{"label":["n"],"name":[2],"type":["int"],"align":["right"]}],"data":[{"1":"Freshman","2":"1"},{"1":"Junior","2":"4"},{"1":"Senior","2":"2"},{"1":"Sophomore","2":"3"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

```r
# note the order is arbitrary! This is an "unordered" factor
```

]

---

# Ordered Factors in R

.quitesmall[
- If there is a rank order you wish to preserve, you can make an `ordered` (`factor`) variable
  - list the `levels` from 1st to last

```r
students <- students %>%
  mutate(Rank = ordered(Rank, # overwrite and change class of Rank to ordered
                        # next, specify the levels, in order
                        levels = c("Freshman", "Sophomore", "Junior", "Senior")
                        )
         )

students %>% head(n = 5)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["ID"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["Rank"],"name":[2],"type":["ord"],"align":["right"]},{"label":["Grade"],"name":[3],"type":["dbl"],"align":["right"]}],"data":[{"1":"1","2":"Sophomore","3":"77"},{"1":"2","2":"Senior","3":"72"},{"1":"3","2":"Freshman","3":"73"},{"1":"4","2":"Senior","3":"73"},{"1":"5","2":"Junior","3":"84"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]

---

# Ordered Factors in R

```r
students %>%
  group_by(Rank) %>%
  count()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["Rank"],"name":[1],"type":["ord"],"align":["right"]},{"label":["n"],"name":[2],"type":["int"],"align":["right"]}],"data":[{"1":"Freshman","2":"1"},{"1":"Sophomore","2":"3"},{"1":"Junior","2":"4"},{"1":"Senior","2":"2"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]

---

# Example Research Question

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How much higher wages, on average, do men earn compared to women?
]

]

---

# The Pure Statistics of Comparing Group Means

.smallest[
- Basic statistics: can test for statistically significant difference in group means with a **t-test**<sup>.magenta[†]</sup>, let:

- .blue[`\$Y_M\$`]: average earnings of a sample of .blue[`\$n_M\$`] men

- .magenta[`\$Y_W\$`]: average earnings of a sample of .magenta[`\$n_W\$`] women

- **Difference** in group averages: `$d=$` .blue[`\$\bar{Y}_M\$`] `$-$` .magenta[`\$\bar{Y}_W\$`]

- The hypothesis test is:
  - `$H_0: d=0$`
  - `$H_1: d \neq 0$`

]
]
.pull-right[
.center[
![:scale 100%](../images/genderpaygap.jpg)
]
]

---

# Plotting Factors in R

- If I plot a `factor` variable, e.g. `Gender` (which is either `Male` or `Female`), the scatterplot with `wage` looks like this
  - effectively `R` treats values of a factor variable as integers
  - in this case, `"Female"` = 0, `"Male"` = 1

- Let’s make this more explicit by making a .hi[dummy variable] to stand in for Gender
]

---

# Regression with Dummy Variables

---

# Comparing Groups with Regression

.smallest[
- In a regression, we can easily compare across groups via a .hi[dummy variable]<sup>.magenta[†]</sup>

- Dummy variable *only* `$=0$` or `$=1$`, if a condition is `TRUE` vs. `FALSE`

- Signifies whether an observation belongs to a category or not
]

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i \quad \quad \text{ where } Female_i =
 		 \begin{cases}
    		1 & \text{if individual } i \text{ is } Female \\
   			0 & \text{if individual } i \text{ is } Male\\
  		\end{cases}$$`

]
]
--

]

---

# Comparing Groups in Regression: Scatterplot

- `Female` is our dummy `$x$`-variable

- Hard to see relationships because of **overplotting**

]

---

# Comparing Groups in Regression: Scatterplot

- `Female` is our dummy `$x$`-variable

- Hard to see relationships because of **overplotting**

- Tip: use `geom_jitter()` instead of `geom_point()` to *randomly* nudge points to see them better!
  - Only used for *plotting*, does not affect actual data, regression, etc.

]

---

# Comparing Groups in Regression: Scatterplot

- `Female` is our dummy `$x$`-variable

- Hard to see relationships because of **overplotting**

- Use `geom_jitter()` instead of `geom_point()` to *randomly* nudge points
  - *Only* for plotting purposes, does not affect actual data, regression, etc.

]

---

# Dummy Variables as Group Means

.smallest[
`$$\hat{Y_i}=\hat{\beta_0}+\hat{\beta_1} D_i \quad \text{ where }D_i=\{\color{#6A5ACD}{0},\color{#e64173}{1}\}$$`
]
--

.smallest[
- .purple[When `\$D_i=0\$` (“Control group”):]
  - `$\hat{Y_i}=\hat{\beta_0}$`
  - `$\color{#6A5ACD}{E[Y_i|D_i=0]}=\hat{\beta_0}$` `$\iff$` the mean of `$Y$` when `$D_i=0$`
]

.smallest[
- .hi[When `\$D_i=1\$` (“Treatment group”):]
  - `$\hat{Y_i}=\hat{\beta_0}+\hat{\beta_1} D_i$`
  - `$\color{#e64173}{E[Y_i|D_i=1]}=\hat{\beta_0}+\hat{\beta_1}$` `$\iff$` the mean of `$Y$` when `$D_i=1$`
]

- So the **difference** in group means:

`$$\begin{align*}
		&=\color{#e64173}{E[Y_i|D_i=1]}-\color{#6A5ACD}{E[Y_i|D_i=0]}\\
		&=(\hat{\beta_0}+\hat{\beta_1})-(\hat{\beta_0})\\
		&=\hat{\beta_1}\\
\end{align*}$$`

]

---

# Dummy Variables as Group Means: Our Example

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`
]
]

- Mean wage for men: 
]

---

# Dummy Variables as Group Means: Our Example

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`
]
]

- Mean wage for men:
`$$E[Wage|Female=0]=\hat{\beta_0}$$`
]

---

# Dummy Variables as Group Means: Our Example

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`
]
]

- Mean wage for men: 
`$$E[Wage|Female=0]=\hat{\beta_0}$$`

- Mean wage for women: 
]

---

# Dummy Variables as Group Means: Our Example

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`
]
]

- Mean wage for men:
`$$E[Wage|Female=0]=\hat{\beta_0}$$`

- Mean wage for women:
`$$E[Wage|Female=1]=\hat{\beta_0}+\hat{\beta_1}$$`
]

---

# Dummy Variables as Group Means: Our Example

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`
]
]

- Mean wage for men:
`$$E[Wage|Female=0]=\hat{\beta_0}$$`

- Mean wage for women:
`$$E[Wage|Female=1]=\hat{\beta_0}+\hat{\beta_1}$$`

- Difference in wage between men & women:
]

---

# Dummy Variables as Group Means: Our Example

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Female_i$$`

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`
]
]

- Mean wage for men: 
`$$E[Wage|Female=0]=\hat{\beta_0}$$`

- Mean wage for women: 
`$$E[Wage|Female=1]=\hat{\beta_0}+\hat{\beta_1}$$`

- Difference in wage between men & women: 
`$$\hat{\beta_1}$$`
]

---

# Comparing Groups in Regression: Scatterplot

`$$\text{where } Female_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is }Female \\
   			0 & \text{if } i \text{ is } Male\\
  		\end{cases}$$`

]

---

# The Data

```r
# comes from wooldridge package

# install.packages("wooldridge")
library(wooldridge)

# data is called "wage1", save as a dataframe I'll call "wages"
wages <- wage1

wages %>% head()
```
]

---

# Get Group Averages & Std. Devs.

```r
# Summarize for Men

wages %>%
  filter(female==0) %>%
  summarize(mean = mean(wage),
            sd = sd(wage))
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["mean"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["sd"],"name":[2],"type":["dbl"],"align":["right"]}],"data":[{"1":"7.099489","2":"4.160858"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]

```r
# Summarize for Women

wages %>%
  filter(female==1) %>%
  summarize(mean = mean(wage),
            sd = sd(wage))
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["mean"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["sd"],"name":[2],"type":["dbl"],"align":["right"]}],"data":[{"1":"4.587659","2":"2.529363"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>

]
]

---

# Visualize Differences

---

# The Regression I

```r
femalereg <- lm(wage ~ female, data = wages)
summary(femalereg)
```

```
## 
## Call:
## lm(formula = wage ~ female, data = wages)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.5995 -1.8495 -0.9877  1.4260 17.8805 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   7.0995     0.2100  33.806  < 2e-16 ***
## female       -2.5118     0.3034  -8.279 1.04e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.476 on 524 degrees of freedom
## Multiple R-squared:  0.1157,	Adjusted R-squared:  0.114 
## F-statistic: 68.54 on 1 and 524 DF,  p-value: 1.042e-15
```
]
]
]

```r
library(broom)

tidy(femalereg)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["term"],"name":[1],"type":["chr"],"align":["left"]},{"label":["estimate"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["std.error"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["statistic"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["p.value"],"name":[5],"type":["dbl"],"align":["right"]}],"data":[{"1":"(Intercept)","2":"7.099489","3":"0.2100082","4":"33.805777","5":"8.971839e-134"},{"1":"female","2":"-2.511830","3":"0.3034092","4":"-8.278688","5":"1.041764e-15"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

---

# Dummy Regression vs. Group Means

| Gender | Avg. Wage | Std. Dev.   | `$n$`   |
|--------|-------------|-----------|-------|
| Female | `$4.59$`    | `$2.33$`      | `$252$` |
| Male   | `$7.10$`    | `$4.16$`      | `$274$` |
| Difference | `$2.51$` | `$0.30$` | `$-$` |

From `$t$`-test of difference in group means
]

]

.pull-right[
.quitesmall[
<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["term"],"name":[1],"type":["chr"],"align":["left"]},{"label":["estimate"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["std.error"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["statistic"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["p.value"],"name":[5],"type":["dbl"],"align":["right"]}],"data":[{"1":"(Intercept)","2":"7.099489","3":"0.2100082","4":"33.805777","5":"8.971839e-134"},{"1":"female","2":"-2.511830","3":"0.3034092","4":"-8.278688","5":"1.041764e-15"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]

`$$\widehat{\text{Wages}_i}=7.10-2.51 \, \text{Female}_i$$`
]

---

# Recoding Dummies

---

# Recoding Dummies

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]:

- Suppose instead of `$female$` we had used:

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i \quad \quad \text{ where } Male_i =
 		 \begin{cases}
    		1 & \text{if person } i \text{ is } Male \\
   			0 & \text{if person } i \text{ is } Female\\
  		\end{cases}$$`

]

---

# Recoding Dummies with Data

```r
wages<-wages %>%
  mutate(male = ifelse(female == 0, # condition: is female equal to 0?
                       yes = 1, # if true: code as "1"
                       no = 0)) # if false: code as "0"

# verify it worked
wages %>% 
  select(wage, female, male) %>%
  head()
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":[""],"name":["_rn_"],"type":[""],"align":["left"]},{"label":["wage"],"name":[1],"type":["dbl"],"align":["right"]},{"label":["female"],"name":[2],"type":["int"],"align":["right"]},{"label":["male"],"name":[3],"type":["dbl"],"align":["right"]}],"data":[{"1":"3.10","2":"1","3":"0","_rn_":"1"},{"1":"3.24","2":"1","3":"0","_rn_":"2"},{"1":"3.00","2":"0","3":"1","_rn_":"3"},{"1":"6.00","2":"0","3":"1","_rn_":"4"},{"1":"5.30","2":"0","3":"1","_rn_":"5"},{"1":"8.75","2":"0","3":"1","_rn_":"6"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]

---

# Scatterplot with Male

---

# Dummy Variables as Group Means: With Male

`$$\widehat{Wage_i}=\hat{\beta_0}+\hat{\beta_1}Male_i$$`

`$$\text{where } Male_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is } Male \\
   			0 & \text{if } i \text{ is } Female\\
  		\end{cases}$$`
]
]

- Mean wage for men: 
`$$E[Wage|Male=1]=\hat{\beta_0}+\hat{\beta_1}$$`

- Mean wage for women: 
`$$E[Wage|Male=0]=\hat{\beta_0}$$`

- Difference in wage between men & women: 
`$$\hat{\beta_1}$$`
]

---

# Scatterplot with Male

--
.pull-right[
<img src="3.6-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" />
]

---

# The Regression with Male I

```r
malereg <- lm(wage ~ male, data = wages)

summary(malereg)
```

```
## 
## Call:
## lm(formula = wage ~ male, data = wages)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.5995 -1.8495 -0.9877  1.4260 17.8805 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   4.5877     0.2190  20.950  < 2e-16 ***
## male          2.5118     0.3034   8.279 1.04e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.476 on 524 degrees of freedom
## Multiple R-squared:  0.1157,	Adjusted R-squared:  0.114 
## F-statistic: 68.54 on 1 and 524 DF,  p-value: 1.042e-15
```
]
]
]

```r
library(broom)
tidy(malereg)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["term"],"name":[1],"type":["chr"],"align":["left"]},{"label":["estimate"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["std.error"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["statistic"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["p.value"],"name":[5],"type":["dbl"],"align":["right"]}],"data":[{"1":"(Intercept)","2":"4.587659","3":"0.2189834","4":"20.949802","5":"3.012371e-71"},{"1":"male","2":"2.511830","3":"0.3034092","4":"8.278688","5":"1.041764e-15"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]
]

---

# The Dummy Regression: Male or Female

.pull-left[
.quitesmall[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-26">
<col><col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1)</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(2)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Constant</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">4.59 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">7.10 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.22)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.21)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Female</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-2.51 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.30)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Male</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">2.51 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.30)   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526       </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.12    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.12    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.48    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.48    </td></tr>
<tr>
<th colspan="3" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]
]

- Note it doesn't matter if we use `male` or `female`, males always earn $2.51 more than females

- Compare the constant (average for the `$D=0$` group)

- Should you use `male` AND `female`? We'll come to that...

]

---

# Categorical Variables (More than 2 Categories)

---

# Categorical Variables with More than 2 Categories

- A .hi[categorical variable] expresses membership in a category, where there is no ranking or hierarchy of the categories
    - We've looked at categorical variables with 2 categories only
    - e.g. Male/Female, Spring/Summer/Fall/Winter, Democratic/Republican/Independent

- Might be an .hi[ordinal variable] expresses rank or an ordering of data, but not necessarily their relative magnitude
    - e.g. Order of finalists in a competition (1st, 2nd, 3rd)
    - e.g. Highest education attained (1=elementary school, 2=high school, 3=bachelor's degree, 4=graduate degree)

---

# Using Categorical Variables in Regression I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How do wages vary by region of the country? Let `$Region_i=\{Northeast, \, Midwest, \, South, \, West\}$`
]

- Can we run the following regression?

`$$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Region_i$$`

---

# Using Categorical Variables in Regression II

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How do wages vary by region of the country?

]

Code region numerically: 
`$$Region_i=
\begin{cases}1 & \text{if } i \text{ is in }Northeast\\
   			2 & \text{if } i \text{ is in } Midwest\\
    		3 & \text{if } i \text{ is in } South \\
   			4 & \text{if } i \text{ is in } West\\ 
\end{cases}$$`

- Can we run the following regression?

`$$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Region_i$$`

---

# Using Categorical Variables in Regression III

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: How do wages vary by region of the country?

Create a dummy variable for *each* region:

- `$Northeast_i = 1$` if `$i$` is in Northeast, otherwise `$=0$`
- `$Midwest_i = 1$` if `$i$` is in Midwest, otherwise `$=0$`
- `$South_i = 1$` if `$i$` is in South, otherwise `$=0$`
- `$West_i = 1$` if `$i$` is in West, otherwise `$=0$`

]
]

`$$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i+\hat{\beta_4}West_i$$`
]

---

# The Dummy Variable Trap

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i+\hat{\beta_4}West_i$`

]

- If we include *all* possible categories, they are .hi-purple[perfectly multicollinear], an exact linear function of one another:

`$$Northeast_i+Midwest_i+South_i+West_i=1 \quad \forall i$$`

- This is known as the .hi[dummy variable trap], a common source of perfect multicollinearity

---

# The Reference Category

- To avoid the dummy variable trap, always omit one category from the regression, known as the .hi[“reference category”]

- It does not matter which category we omit!

- .hi-purple[Coefficients on each dummy variable measure the *difference* between the *reference* category and each category dummy]

---

# The Reference Category: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i$`
]

- `$West_i$` is omitted (arbitrarily chosen)

- `$\hat{\beta_0}$`:

---

# The Reference Category: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i$`
]

- `$West_i$` is omitted (arbitrarily chosen)

- `$\hat{\beta_0}$`: average wage for `$i$` in the West

- `$\hat{\beta_1}$`:

---

# The Reference Category: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i$`
]

- `$West_i$` is omitted (arbitrarily chosen)

- `$\hat{\beta_0}$`: average wage for `$i$` in the West

- `$\hat{\beta_1}$`: difference between West and Northeast

- `$\hat{\beta_2}$`:

---

# The Reference Category: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i$`
]

- `$West_i$` is omitted (arbitrarily chosen)

- `$\hat{\beta_0}$`: average wage for `$i$` in the West

- `$\hat{\beta_1}$`: difference between West and Northeast

- `$\hat{\beta_2}$`: difference between West and Midwest

- `$\hat{\beta_3}$`:

---

# The Reference Category: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$\widehat{Wages_i}=\hat{\beta_0}+\hat{\beta_1}Northeast_i+\hat{\beta_2}Midwest_i+\hat{\beta_3}South_i$`
]

- `$West_i$` is omitted (arbitrarily chosen)

- `$\hat{\beta_0}$`: average wage for `$i$` in the West

- `$\hat{\beta_1}$`: difference between West and Northeast

- `$\hat{\beta_2}$`: difference between West and Midwest

- `$\hat{\beta_3}$`: difference between West and South

---

# Dummy Variable Trap in R

```r
lm(wage ~ noreast + northcen + south + west, data = wages) %>% summary()
```

```
## 
## Call:
## lm(formula = wage ~ noreast + northcen + south + west, data = wages)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -6.083 -2.387 -1.097  1.157 18.610 
## 
## Coefficients: (1 not defined because of singularities)
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   6.6134     0.3891  16.995  < 2e-16 ***
## noreast      -0.2436     0.5154  -0.473  0.63664    
## northcen     -0.9029     0.5035  -1.793  0.07352 .  
## south        -1.2265     0.4728  -2.594  0.00974 ** 
## west              NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.671 on 522 degrees of freedom
## Multiple R-squared:  0.0175,	Adjusted R-squared:  0.01185 
## F-statistic: 3.099 on 3 and 522 DF,  p-value: 0.02646
```
]
]

---

# Using Different Reference Categories in R

```r
# let's run 4 regressions, each one we omit a different region
no_noreast_reg <- lm(wage ~ northcen + south + west, data = wages)
no_northcen_reg <- lm(wage ~ noreast + south + west, data = wages)
no_south_reg <- lm(wage ~ noreast + northcen + west, data = wages)
no_west_reg <- lm(wage ~ noreast + northcen + south, data = wages)

# now make an output table
library(huxtable)
huxreg(no_noreast_reg,
       no_northcen_reg,
       no_south_reg,
       no_west_reg,
       coefs = c("Constant" = "(Intercept)",
                 "Northeast" = "noreast",
                 "Midwest" = "northcen",
                 "South" = "south",
                 "West" = "west"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 3)
```
]

---

# Using Different Reference Categories in R II

.pull-left[
.tiny[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-30">
<col><col><col><col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1)</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(2)</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(3)</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(4)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Constant</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">6.370 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">5.710 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">5.387 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">6.613 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.338)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.320)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.268)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.389)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Northeast</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.659    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.983 *  </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.244    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.465)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.432)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.515)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Midwest</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.659    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.324    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.903    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.465)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.417)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.504)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">South</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.983 *  </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.324    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1.226 ** </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.432)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.417)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.473)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">West</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.244    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.903    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">1.226 ** </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.515)   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.504)   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.473)   </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526        </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526        </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526        </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526        </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.017    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.017    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.017    </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.017    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.671    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.671    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.671    </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.671    </td></tr>
<tr>
<th colspan="5" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]
]

- Compare coefficients between Midwest in (1) and Northeast in (2)...

- Compare coefficients between West in (3) and South in (4)...

- Does not matter which region we omit!
  - Same `$R^2$`, SER, coefficients give same results
]

]

---

# Dummy *Dependent* (Y) Variables

.smallest[
- In many contexts, we will want to have our *dependent* `$(Y)$` variable be a dummy variable
]
--

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: 
`$$\widehat{Admitted_i}=\hat{\beta_0}+\hat{\beta_1}GPA_i \quad \text{ where } Admitted_i =
 		 \begin{cases}
    		1 & \text{if } i \text{ is Admitted} \\
   			0 & \text{if } i \text{ is  Not Admitted}\\
  		\end{cases}$$`
]
]
--

.smallest[
- A model where `$Y$` is a dummy is called a .hi[linear probability model], as it measures the .hi-purple[probability of `\$Y\$` occuring `\$(=1)\$` given the X's, i.e. `\$P(Y_i=1|X_1, \cdots, X_k)\$`]
    - e.g. the probability person `$i$` is Admitted to a program with a given GPA
]
--

.smallest[
- Requires special tools to properly interpret and extend this (**logit**, **probit**, etc)

- Feel free to write papers that have dummy `$Y$` variables (but you may have to ask me some more questions)!
]