Binary Variables and Functional Form

class: center, middle, inverse, title-slide

.title[
# Binary Variables and Functional Form
]
.subtitle[
## i.e. what you actually do most of the time
]
.date[
### Updated 2025-10-21
]

---

pre[class] {
  max-height: 100px;
}
</style>

# The Right Hand Side

- Today we'll be focusing on the *right hand side* of a regression
- Economists generally refer to a regression as having a "left-hand-side" of the dependent variable `$Y$`, and a "right-hand-side" of all the independent stuff, like `$\beta_0 + \beta_1X + \beta_2Z + \varepsilon$`.
- So far, we've just tossed stuff on the right-hand side and called it our treatment variable or a control variable without thinking too much harder about it
- Today we will think harder about it!

---

# The Right Hand Side

We will look at three features of the right-hand side

- What if the variable is *categorical* or *binary*? (binary variables)
- What if the variable has a *nonlinear effect* on `$Y$` (polynomials and logarithms)
- What if the effect of one variable *depends on the value of another variable?* (interaction terms)

---

# The Overriding Key Concept

**The derivative** is magical here

- `$\partial Y/\partial X$` will give you an expression that tells you the effect of a change in `$X$` on `$Y$`
- Whatever form `$X$` enters the equation as!
- (then if it's binary, imagine that as a change from 0 to 1, or off to on)
- If you're confused **try taking the derivative, I'm serious, it helps**
- Your options with this stuff: EITHER commit to memory a bunch of different interpretation rules for each different case, OR just take the darn derivative

---

# Binary Data

- A variable is binary if it only has two values - 0 or 1 (or "No" or "Yes", etc.)
- Binary variables are super common in econometrics!
- Did you get the treatment? Yes / No
- Do you live in the US? Yes / No
- Is a floating exchange rate in effect? Yes / No

---

# Comparison of Means

- When a binary variable is an independent variable, what we are often interested in doing is *comparing means*
- Is mean income higher inside the US or outside?
- Is mean height higher for kids who got a nutrition supplement or those who didn't?
- Is mean GDP growth higher with or without a floating exchange rate?

---

# Comparison of Means

- Let's compare log earnings in 1993 between married people 30 or older vs. never-married people 30 or older

``` r
data(PSID, package = 'Ecdat')
PSID <- PSID %>%
  filter(age >= 30, married %in% c('married','never married'), earnings > 0) %>%
  mutate(married  = married == 'married')
PSID %>%
  group_by(married) %>%
  summarize(log_earnings = mean(log(earnings)))
```

```
## # A tibble: 2 × 2
##   married log_earnings
##   <lgl>          <dbl>
## 1 FALSE           9.26
## 2 TRUE            9.47
```

---

# Comparison of Means

- Seems to be a slight favor to the married men

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-3-1.png)

---

# Comparison of Means

- The *difference between the means* follows a t-distribution under the null that they're identical
- So of course we can do a hypothesis test of whether they're different

``` r
t.test(log(earnings) ~ married, data = PSID, var.equal = TRUE)
```

```
## 
## 	Two Sample t-test
## 
## data:  log(earnings) by married
## t = -3.683, df = 2828, p-value = 0.0002348
## alternative hypothesis: true difference in means between group FALSE and group TRUE is not equal to 0
## 95 percent confidence interval:
##  -0.32598643 -0.09947606
## sample estimates:
## mean in group FALSE  mean in group TRUE 
##            9.255160            9.467891
```

---

# Comparison of Means

- But why bother trotting out a specific test when we can just do a regression?
- (In fact, a lot of specific tests can be replaced with basic regression, see [this explainer](https://lindeloev.github.io/tests-as-linear/))

``` r
feols(log(earnings) ~ married, data = PSID)
```

```
## OLS estimation, Dep. Var.: log(earnings)
## Observations: 2,830
## Standard-errors: IID 
##             Estimate Std. Error   t value   Pr(>|t|)    
## (Intercept) 9.255160   0.052891 174.98636  < 2.2e-16 ***
## marriedTRUE 0.212731   0.057760   3.68305 0.00023478 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 1.13028   Adj. R2: 0.004422
```

---

# Comparison of Means

Notice:

- The intercept gives the mean for the *non-married* group
- The coefficient on *marriedTRUE* gives the married minus non-married difference
- The t-stat and p-value on that coefficient are exactly the same as that `t.test` (except the t is reversed; same deal)
- i.e. *the coefficient on a binary variable in a regression gives the difference in means*
- If we'd defined it the other way, with "not married" as the independent variable, the intercept would be the mean for the *married* group (i.e. "not married = 0"), and the coefficient would be the exact same but times `$-1$` (same difference, just opposite direction!)

---

# Comparison of Means

Why does OLS give us a comparison of means when you give it a binary variable?

- The only `$X$` values are 0 (FALSE) and 1 (TRUE)
- Because of this, OLS no longer really fits a *line*, it's more of two separate means
- And when you're estimating to minimize the sum of squared errors separately for each group, can't do any better than to predict the mean!
- So you get the mean of each group as each group's prediction

---

# Binary with Controls

- Obviously this is handy for including binary controls, but why do this for binary treatments? Because we can add controls!

```
##                 feols(log(earning..
## Dependent Var.:       log(earnings)
##                                    
## Constant          8.740*** (0.1478)
## marriedTRUE      0.3404*** (0.0579)
## kids            -0.2259*** (0.0159)
## age              0.0223*** (0.0038)
## _______________ ___________________
## S.E. type                       IID
## Observations                  2,803
## R2                          0.07609
## Adj. R2                     0.07510
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

# Multicollinearity

- Why is just one side of it on the regression? Why aren't "married" and "not married" BOTH included?
- Because regression couldn't give an answer!
- Mean of married is `$9.47$` and of non-married is `$9.26$`.

$$ \log(Earnings) = 0 + 9.47Married + 9.26NonMarried $$
$$ \log(Earnings) = 9.26 + .21Married + 0NonMarried $$
$$ \log(Earnings) = 3 + 6.47Married + 6.26NonMarried $$

- These (and infinitely many other options) all give the exact same predictions! OLS can't pick between them. There's no single best way to minimize squared residuals
- So we pick one with convenient properties, setting one of the categories to have a coefficient of 0 (dropping it) and making the coefficient on the other *the difference relative to* the one we left out

---

# More than Two Categories

- That interpretation - dropping one and making the other relative to *that*, conveniently extends to *multi-category variables*
- Why stop at binary categorical variables? There are plenty of categorical variables with more than two values
- What is your education level? What is your religious denomination? What continent are you on?
- We can put these in a regression by turning *each value* into its own binary variable
- (and then dropping one so the coefficients on the others give you the difference with the omitted one)

---

# More than Two Categories

- Make the mean of group A be 1, of group B be 2, etc.

``` r
tib <- tibble(group = sample(LETTERS[1:4], 10000, replace = TRUE)) %>%
  mutate(Y = rnorm(10000) + (group == "A") + 2*(group == "B") + 3*(group == "C") + 4*(group == "D"))
feols(Y ~ group, data = tib)
```

```
## OLS estimation, Dep. Var.: Y
## Observations: 10,000
## Standard-errors: IID 
##             Estimate Std. Error  t value  Pr(>|t|)    
## (Intercept) 1.024508   0.020043  51.1159 < 2.2e-16 ***
## groupB      0.962644   0.028038  34.3335 < 2.2e-16 ***
## groupC      2.006833   0.028003  71.6649 < 2.2e-16 ***
## groupD      2.976867   0.028115 105.8832 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.986189   Adj. R2: 0.55817
```

---

# More than Two Categories

- By changing the reference group, the coefficients change because they're "different from" a different group!
- And notice that, as before, the intercept is the mean of the omitted group (although this changes once you add controls; the intercept is the predicted mean when all right-hand-side variables are 0)

``` r
tib <- tib %>% mutate(group = factor(group, levels = c('B','A','C','D')))
feols(Y ~ group, data = tib)
```

```
## OLS estimation, Dep. Var.: Y
## Observations: 10,000
## Standard-errors: IID 
##              Estimate Std. Error  t value  Pr(>|t|)    
## (Intercept)  1.987152   0.019607 101.3515 < 2.2e-16 ***
## groupA      -0.962644   0.028038 -34.3335 < 2.2e-16 ***
## groupC       1.044189   0.027692  37.7067 < 2.2e-16 ***
## groupD       2.014223   0.027805  72.4404 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 0.986189   Adj. R2: 0.55817
```

---

# More than Two Categories

- Some Interpretations: Controlling for number of kids and age, people with a high school degree have log earnings .324 higher than those without a high school degree (earnings 32.4% higher). BA-holders have earnings 84.8% higher than those without a HS degree
- Controlling for kids and age, a graduate degree earns (.976 - .848 =) 12.8% more than someone with a BA (`glht()` could help!)

```
## 
## No High School Degree    High School Degree          Some College 
##                   333                  1105                   708 
##     Bachelor's Degree       Graduate Degree 
##                   356                   301
```

```
## OLS estimation, Dep. Var.: log(earnings)
## Observations: 2,803
## Standard-errors: IID 
##                              Estimate Std. Error  t value   Pr(>|t|)    
## (Intercept)                  8.360245   0.156293 53.49092  < 2.2e-16 ***
## educationHigh School Degree  0.323807   0.067175  4.82036 1.5096e-06 ***
## educationSome College        0.576478   0.071976  8.00934 1.6758e-15 ***
## educationBachelor's Degree   0.848200   0.082697 10.25666  < 2.2e-16 ***
## educationGraduate Degree     0.976291   0.086842 11.24212  < 2.2e-16 ***
## kids                        -0.149679   0.015689 -9.54045  < 2.2e-16 ***
## age                          0.023137   0.003741  6.18453 7.1388e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## RMSE: 1.05675   Adj. R2: 0.123473
```

---

# Concept Checks

- If `$X$` is binary, in sentences interpret the coefficients from the estimated OLS equation `$Y = 4 + 3X + 2Z$`
- How might a comparison of means come in handy if you wanted to analyze the results of a randomized policy experiment?
- If you had a data set of people from every continent and added "continent" as a control, how many coefficients would this add to your model?
- If in that regression you really wanted to compare Europe to Asia specifically, what might you do so that the regression made this easy?

---

# Interpreting OLS

- To think more about the right-hand-side, let's go back to our original interpretation of an OLS coefficient

$$ Y = \beta_0 + \beta_1X + \varepsilon $$

- A one-unit change in `$X$` is associated with a `$\beta_1$`-unit change in `$Y$`
- This logic still works with binary variables since "a one-unit change in `$X$`" means "changing `$X$` from No to Yes"
- We can also think of this as `$\partial Y/\partial X = \beta_1$` in calculus terms
- Notice that this assumes that a one-unit change in `$X$` *always has the same effect* on `$\beta_1$` no matter what else is going on
- What if that's not true?

---

# Functional Form

- We talked before about times when a *linear model* like standard OLS might not be sufficient
- However, as long as those *non*-linearities are on the right hand side, we can fix the problem easily but just having `$X$` enter non-linearly! Run it through a *transformation*!
- The most common transformations by far are *polynomials* and *logarithms*

---

# Functional Form

- Why do this? Because sometimes a straight line is clearly not going to do the trick!

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-10-1.png)

---

# Polynomials

- `$\beta_1X$` is a "first order polynomial" - there's one term
- `$\beta_1X + \beta_2X^2$` is a "second order polynomial" or a "quadratic" - two terms (note both included, it's not just `$X^2$`)
- `$\beta_1X + \beta_2X^2 + \beta_3X^3$` is a third-order or cubic, etc.

What do they do?

- The more polynomial terms, the more flexible the line can be. With enough terms you can mimic *any* shape of relationship
- Of course, if you just add a whole buncha terms, it gets very noisy, and prediction out-of-sample gets very bad
- Keep it minimal - quadratics are almost always enough, unless you have reason to believe there's a true more-complex relationship. You can try adding higher-order terms and see if they make a difference

---

# Polynomials

- The true relationship is quadratic

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-11-1.png)

---

# Polynomials

- Higher-order terms don't do anything for us here (because a quadratic is sufficient!)

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-12-1.png)

---

# Polynomials

- Interpret polynomials using the derivative
- `$\partial Y/\partial X$` will be different depending on the value of `$X$` (as it should! Notice in the graph that the slope changes for different values of `$X$`)

$$ Y = \beta_1X + \beta_2X^2 $$
$$ \partial Y/\partial X = \beta_1 + 2\beta_2X $$

So at `$X = 0$`, the effect of a one-unit change in `$X$` is `$\beta_1$`. At `$X = 1$`, it's `$\beta_1 + \beta_2$`. At `$X = 5$` it's `$\beta_1 + 5\beta_2$`.

- **IMPORTANT**: when you have a polynomial, *the coefficients on each individual term mean very little on their own*. You have to consider them alongisde the other coefficients from the polynomial! **Never** interpret `$\beta_1$` here without thinking about `$\beta_2$` alongside. Also, the significance of the individual terms doesn't really matter - consider doing an F-test of all of them at once.

---

# Polynomials in R

- We can add an `I()` function to our regression to do a calculation on a variable before including it. So `I(X^2)` adds a squared term
- There's also a `poly()` function but you **must** set `raw=TRUE` if you do

``` r
# Linear
feols(Y ~ X, data = df)
# Quadratic
feols(Y ~ X + I(X^2), data = df)
# Cubic
feols(Y ~ X + I(X^2) + I(X^3), data = df)
feols(Y ~ poly(X, 3, raw = TRUE), data = df)
```

---

# Concept Check

- What's the effect of a one-unit change in `$X$` at `$X = 0$`, `$X = 1$`, and `$X = 2$` for each of these?

```
##                 feols(Y ~ X, dat.. feols(Y ~ X + I(.. feols(Y ~ X + I(...1
## Dependent Var.:                  Y                  Y                    Y
##                                                                           
## Constant         7.285*** (0.5660)   -0.1295 (0.3839)      0.0759 (0.5091)
## X               -8.934*** (0.1953)   0.9779* (0.3831)      0.4542 (0.9331)
## X square                           -2.003*** (0.0752)   -1.738*** (0.4368)
## X cube                                                    -0.0354 (0.0574)
## _______________ __________________ __________________   __________________
## S.E. type                      IID                IID                  IID
## Observations                   200                200                  200
## R2                         0.91357            0.98122              0.98126
## Adj. R2                    0.91313            0.98103              0.98097
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

# Logarithms

- Another common transformation, both for dependent and independent variables, is to take the logarithm
- This has the effect of pulling in extreme values from strongly right-skewed data and making linear relationships pop out
- Income, for example, is almost always used with a logarithm
- It also gives the coefficients a nice percentage-based interpretation

---

# Logarithms

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-15-1.png)

---

# Or if you prefer...

- Notice the change in axes

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-16-1.png)

---

# Logarithms

- How can we interpret them?
- The key is to remember that `$\log(X) + a \approx \log((1+a)X)$`, meaning that a `$a$`-unit change in `$log(X)$` is similar to a `$a\times100%$` change in `$X$`
- So, walk through our "one-unit change in the variable" logic from before, but whenever we hit a log, change that into a percentage!
- Note this *only works for a small `$a$`*! With natural logs, the approximation breaks down above `$a = .1$` or so
- Can interpret exactly instead: a `$a$`-unit change in `$\log(X)$` is really a `$(\log(1+a)-1)%$` change in `$X$`
- Or you can change the base - a 1-unit change in `$\log_{1+a}(X)$` is exactly a `$a$`% change in `$X$`.

---

# Logarithms

- `$Y = \beta_0 + \beta_1\log(X)$` A one-unit change in `$\log(X)$`, or a 100% change in `$X$`, is associated with a `$\beta_1$`-unit change in `$Y$`
- `$\log(Y) = \beta_0 + \beta_1X$` a one-unit change in `$X$` is associated with a `$\beta_1\times 100$`% change in `$Y$`
- `$\log(Y) = \beta_0 + \beta_1\log(X)$` A one-unit change in `$\log(X)$`, or a or a 100% change in `$X$`, is associated with a `$\beta_1$`-unit change in `$\log(Y)$`, or a `$\beta_1\times100$`% change in `$Y$`.
- (Try also with changes smaller than one unit - that's usually more reasonable)

---

# Logarithms

Downsides:

- Logarithms require that all data be positive. No negatives or zeroes!
- Fairly rare that a variable with negative values wants a log anyway
- But zeroes are common! A common practice is to just do `$log(X+1)$` or `$asinh(X)=\ln(X+\sqrt{X^2+1})$` but these are pretty arbitrary
- On the left-hand side at least, better to deal with 0s with Poisson regression

---

# Functional Form

- In general, you want the shape of your function to match the shape of the relationship in the data (or, even better, the true relationship)
- Polynomials and logs can usually get you there!
- Which to use? Use logs for highly skewed data or variables with exponential relationships
- Use polynomials if it doesn't look straight! **Check that scatterplot and see how not-straight it is!**

---

# Concept Checks

- Which of the following variables would you likely want to log before using them? Income, height, wealth, company size, home square footage
- In each of the following estimated OLS lines, interpret the coefficient by filling in "A [blank] change in X is associated with a [blank] change in Y":

$$ Y = 1 + 2\log(X) $$
$$ \log(Y) = 3 + 2\log(X) $$

$$ \log(Y) = 4 + 3X $$

---

# Interactions

- For both polynomials and logarithms, the effect of a one-unit change in `$X$` differs depending on its current value (for logarithms, a 1-unit change in `$X$` is different percentage changes in `$X$` depending on current value)
- But why stop there? Maybe the effect of `$X$` differs depending on the current value of *other variables!*
- Enter interaction terms!

$$ Y = \beta_0 + \beta_1X + \beta_2Z + \beta_3X\times Z + \varepsilon $$
- Interaction terms are *a little tough* but also **extremely important**. Expect to come back to these slides, as you're almost certainly going to use interaction terms in both of your major projects this term

---

# Interactions

- Change in the value of a *control* can shift a regression line up and down
- Using the model `$Y = \beta_0 + \beta_1X + \beta_2Z$`, estimated as `$Y = .01 + 1.2X + .95Z$`:

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-17-1.png)

---

# Interactions

- But an interaction can both shift the line up and down AND change its slope
- Using the model `$Y = \beta_0 + \beta_1X + \beta_2Z + \beta_3X\times Z$`, estimated as `$Y = .035 + 1.14X + .94Z + 1.02X\times Z$`:

![](data:image/png;base64,#Week_05_1_Binary_Variables_and_Interactions_files/figure-html/unnamed-chunk-18-1.png)

---

# Interactions

- How can we interpret an interaction?
- The idea is that the interaction shows how *the effect of one variable changes as the value of the other changes*
- The derivative helps!

$$ Y = \beta_0 + \beta_1X + \beta_2Z + \beta_3X\times Z $$
$$ \partial Y/\partial X = \beta_1 + \beta_3 Z $$

- The effect of `$X$` is `$\beta_1$` when `$Z = 0$`, or `$\beta_1 + \beta_3$` when `$Z = 1$`, or `$\beta_1 + 3\beta_3$` if `$Z = 3$`!

---

# Interactions

- Often we are doing interactions with binary variables to see how an effect differs across groups
- In these cases, we combine what we know about binary variables with what we know about interactions!
- Now, instead of the intercept giving the baseline and the binary coefficient giving the difference, the coefficient on `$X$` is the baseline effect of `$X$` and the interaction is the difference in the effect of `$X$`
- The interaction coefficient becomes "the difference in the effect of `$X$` between the `$Z$` = "No" and `$Z$` = "Yes" groups"
- (What if it's continuous? Mathematically the same but the thinking changes - the interaction term is the difference in the effect of `$X$` you get when increasing `$Z$` by one unit)

---

# Interactions

- Marriage for those without a college degree raises earnings by 24%. A college degree reduces the marriage premium by 25%. Marriage for those with a college degree reduces earnings by .24 - .25 = -1%

```
##                           feols(log(earnin..
## Dependent Var.:                log(earnings)
##                                             
## Constant                   9.087*** (0.0583)
## marriedTRUE               0.2381*** (0.0638)
## collegeTRUE               0.8543*** (0.1255)
## marriedTRUE x collegeTRUE  -0.2541. (0.1363)
## _________________________ __________________
## S.E. type                                IID
## Observations                           2,803
## R2                                   0.06253
## Adj. R2                              0.06153
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

# Notes on Interactions

- Like with polynomials, the coefficients on their own now have little meaning and must be evaluated alongside each other. `$\beta_1$` by itself is just "the effect of `$X$` when `$Z = 0$`", not "the effect of `$X$`"
- Yes, you *do* almost always want to include both variables in un-interacted form and interacted form. Otherwise the interpretation gets very thorny
- Interaction effects are *poorly powered*. You need a *lot* of data to be able to tell whether an effect is different in two groups. If `$N$` observations is adequate power to see if the effect itself is different from zero, you need a sample of roughly `$16\times N$` to see if the difference in effects is nonzero. Sixteen times!!
- It's tempting to try interacting your effect with everything to see if it's bigger/smaller/nonzero in some groups, but because it's poorly powered, this is a bad idea! You'll get a lot of false positives

---

# In R!

- Binary variables in R (on the right-hand-side) you can just treat as normal variables
- Categorical variables too (although if it's numeric you may need to run it through `factor()` first, or `i()` in `feols()`)
- In `feols()` you can specify which group gets dropped using `i()` and setting `ref` in it

``` r
# drops married = FALSE
feols(log(earnings) ~ married, data = PSID)
# drops married = TRUE
feols(log(earnings) ~ i(married, ref = 'TRUE'), data = PSID)
```

---

# Binary variables

- You can also use `I()` to specify binary variables in-model, or `case_when()` to create categorical variables
- `case_when` works in steps - the first one that applies to you, you get, so that `TRUE` at the end catches "everyone else"

``` r
PSID <- PSID %>%
  mutate(education  = case_when(
    educatn < 12 ~ 'No High School Degree',
    educatn == 12 ~ 'High School Degree',
    educatn < 16 ~ 'Some College',
    educatn == 16 ~ 'Bachelor\'s Degree',
    TRUE ~ 'Graduate Degree'
  ))
feols(log(earnings) ~ education + I(kids > 0), data = PSID) |> etable()
```

---

# Binary Variables

```
##                             feols(log(earning..
## Dependent Var.:                   log(earnings)
##                                                
## Constant                      10.06*** (0.0701)
## educationGraduateDegree        0.1673* (0.0842)
## educationHighSchoolDegree   -0.5433*** (0.0658)
## educationNoHighSchoolDegree -0.9404*** (0.0826)
## educationSomeCollege        -0.2893*** (0.0699)
## I(kids>0)TRUE               -0.2922*** (0.0548)
## ___________________________ ___________________
## S.E. type                                   IID
## Observations                              2,803
## R2                                      0.09907
## Adj. R2                                 0.09746
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

# Polynomials and Logarithms

- As previously discussed, `$I()$` will let us do functions like `$X^2$`
- We can also do `log()` straight in the regression.

``` r
lm(Y ~ X + I(X^2) + I(X^3), data = df)
lm(log(Y) ~ log(X), data = df)
```

---

# Interactions

- `X*Z` will include `X`, `Z`, and also their interaction
- If necessary, `X:Z` is the interaction only, but you rarely need this. However, it's handy for referring to the interaction term in `linearHypothesis`!
- In `feols()` the `i()` function is a very powerful way of doing interactions

``` r
feols(Y ~ X*Z, data = df)
feols(Y ~ X + X:Z, data = df)
feols(Y ~ i(Z, X), data = df)
feols(Y ~ i(Z, X), data = df) |> iplot()
```

---

# Tests

- `wald()` can be handy for testing groups of binary variables for a categorical
- Also good for testing all the polynomial terms, or testing if the effect of `$X$` is significant at a certain value of `$Z$`

``` r
# Is the education effect zero overall?
m1 <- feols(log(earnings)~educatn, data = PSID)
wald(m1, 'educatn')

# Does X have any effect?
m2 <- feols(Y ~ X + I(X^2) + I(X^3), data = df)
wald(m2, 'X') # Gets all coefficients with an 'X' anywhere in the name - check this is right!

# Is the effect of X significant when Z = 5?
library(multcomp)
m3 <- lm(Y ~ X*Z, data = df)
glht(m3, 'X + 5*X:Z= 0') %>% summary()
```