# Many X Variables Alex Cardazzi All materials can be found at alexcardazzi.github.io. ## Housing Price Models While we love pretending to be guidance counselors, we *really* love pretending to be real estate agents. For (maybe?) the last time, let’s read in our `ames` housing data ([data](https://vincentarelbundock.github.io/Rdatasets/csv/openintro/ames.csv); [documentation](https://vincentarelbundock.github.io/Rdatasets/doc/modeldata/ames.html)). We are going to keep all of the variables we did last time plus one more: `BsmtFin.Type.1`.

Definition/Details of `BsmtFin.Type.1`

BsmtFin Type 1 (Ordinal): Rating of basement finished area GLQ Good Living Quarters ALQ Average Living Quarters BLQ Below Average Living Quarters Rec Average Rec Room LwQ Low Quality Unf Unfinshed NA No Basement Source:

Let’s load in our data and re-name the columns. ``` r ames <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/openintro/ames.csv") keep_columns <- c("price", "area", "Bedroom.AbvGr", "Year.Built", "Overall.Cond", "BsmtFin.Type.1") ames <- ames[,keep_columns] colnames(ames) <- c("price", "sqft", "bedrooms", "yr_built", "condition", "bsmnt") ames$age <- 2011 - ames$yr_built ``` ``` r ames <- read.csv("https://vincentarelbundock.github.io/Rdatasets/csv/openintro/ames.csv") keep_columns <- c("price", "area", "Bedroom.AbvGr", "Year.Built", "Overall.Cond", "BsmtFin.Type.1") ames <- ames[,keep_columns] colnames(ames) <- c("price", "sqft", "bedrooms", "yr_built", "condition", "bsmnt") ames$age <- 2011 - ames$yr_built ames$bsmnt <- ifelse(is.na(ames$bsmnt) | ames$bsmnt == "", "No Basement", ames$bsmnt) ames$has_bsmnt <- ifelse(ames$bsmnt == "No Basement", 0, 1) ``` ## Basements Let’s take a peek at the `bsmnt` variable to see what all it contains. To do this, we are going to use `table()`, but add in `useNA = "always"` just in case there are any missing data elements in the column. ``` r table() ``` According to this tabulation, there are 80 observations with a blank entry, and 0 observations that R reads as missing (`NA`). According to [this resource](https://jse.amstat.org/v19n3/decock/DataDocumentation.txt#:~:text=BsmtFin%20Type%201%09(Ordinal)%3A%20Rating%20of%20basement%20finished%20area%0A%0A%20%20%20%20%20%20%20GLQ%09Good%20Living%20Quarters%0A%20%20%20%20%20%20%20ALQ%09Average%20Living%20Quarters%0A%20%20%20%20%20%20%20BLQ%09Below%20Average%20Living%20Quarters%09%0A%20%20%20%20%20%20%20Rec%09Average%20Rec%20Room%0A%20%20%20%20%20%20%20LwQ%09Low%20Quality%0A%20%20%20%20%20%20%20Unf%09Unfinshed%0A%20%20%20%20%20%20%20NA%09No%20Basement), `NA` values in the original data (here, they are blank) indicate no basement. Therefore, we can convert these blank values to say `"No Basement"`. ``` r ames$bsmnt <- ifelse(ames$bsmnt == "", "No Basement", ames$bsmnt) ``` If we were real estate agents, it would be really nice to have some way to quantify this basement variable so we could use it in our model. To start, let’s just create a dummy variable equal to 1 if the property has a basement and a zero otherwise. ``` r ames$has_bsmnt <- ifelse(ames$bsmnt == "No Basement", 0, 1) ``` Similar to how we controlled for gender differences, which is a categorical variable, we can include this variable into our housing price model. For simplicity, we are going to just use square footage as our other explanatory variable. Students are encouraged to explore the inclusion of other variables by themselves. Estimating our model: ``` r # Use log(price) as the outcome variable # Use log(sqft) and has_bsmnt as explanatory variables reg1 <- lm(log(price) ~ , ames) reg1 ``` This model suggests that a 1% increase in square footage increases expected sale price by 0.89%, controlling for whether the property has a basement. Second, this model says that having a basement (controlling for square footage) increases price by 38.8% relative to no basement.

**A note on percentages.** The rule of thumb from Module 4.5 is to multiply a coefficient from a log model by 100 to get a percent change, which is what we just did. The exact formula is $(e^{\beta} - 1) \times 100$, which gives 47.4% here. The rule of thumb works well for small coefficients, but the two drift apart for larger ones like this.

Let’s think about this for a second. A home that is 2,000 square feet *including a basement* is worth more than a 2,000 square foot home without a basement? Put differently, a 1,600 square foot home with a 400 square foot basement is worth more than a 2,000 square foot home? If this sounds fishy to you, check out [the definition of the `area` column](https://vincentarelbundock.github.io/Rdatasets/doc/openintro/ames.html#:~:text=area,area%20square%20feet.). It seems that `area` only measures the non-basement square footage. Therefore, this coefficient represents a composite effect of both adding a basement as well as adding the square footage that comes with the basement. In other words, this regression is *really* saying that a 2,000 square foot home with a 400 square foot basement (for a total of 2,400 ft²) is worth more than a 2,000 square foot home without a basement. This should sound awfully similar to the same omitted variable bias we had in the bedroom situation! However, let’s turn a blind eye to this for now. ## Basement Quality Let’s return to the original model that said adding a basement increases prices by 38.8%. This is helpful, but of course there are many different “ratings” of basement quality reported in the `bsmnt` variable that we are ignoring. The lowest quality is “Unf”, or unfinished. Let’s make a dummy variable for this type of basement. ``` r ames$unfinished <- ifelse(ames$bsmnt == "Unf", 1, 0) ``` Now, let’s re-estimate the model with this variable included:

Code

``` r reg2 <- lm(log(price) ~ log(sqft) + has_bsmnt + unfinished, ames) reg2 ```

Output

Call: lm(formula = log(price) ~ log(sqft) + has_bsmnt + unfinished, data = ames) Coefficients: (Intercept) log(sqft) has_bsmnt unfinished 5.0663 0.9062 0.4363 -0.1679

We find that having a basement, relative to no basement, increases the expected price by 43.6%. Now, what if that basement was unfinished? The coefficient suggests -16.8%, but this is not the end of the story. Notice that `has_bsmnt` is *always* equal to one whenever `unfinished` is equal to one. Therefore, the overall effect is 43.6% - 16.8% = 26.8%. In other words, an unfinished basement is worth more than no basement, but a finished basement is worth more than an unfinished basement ($0 < 26.8 < 43.6$). This interpretation is a bit clunky because it requires us to sum two coefficients to get our total estimate. To make the interpretation easier, we can redefine `has_bsmnt` to exclude unfinished basements:

Code

``` r ames$has_bsmnt <- ifelse(ames$bsmnt %in% c("No Basement", "Unf"), 0, 1) reg2 <- lm(log(price) ~ log(sqft) + has_bsmnt + unfinished, ames) reg2 ```

Output

Call: lm(formula = log(price) ~ log(sqft) + has_bsmnt + unfinished, data = ames) Coefficients: (Intercept) log(sqft) has_bsmnt unfinished 5.0663 0.9062 0.4363 0.2683

This interpretation is much easier. An unfinished basement, relative to no basement, is worth a 26.8% increase. A non-unfinished basement, relative to no basement, is worth a 43.6% increase. If we wanted to estimate the effect of some other basement quality level, we would have to again redefine `has_bsmnt` to exclude that basement type, and create a new dummy variable for it. If we take this logic to the finish line, we’d end up $K-1$ dummy variables where $K$ is the number of possible categories. We subtract one because one group must be the reference group (no basement in this case). I’m sure you’ve heard this one before: *work smarter, not harder*. In programming, they say: *A good programmer is a lazy programmer*. While one of these is earnest and the other sarcastic, they mean similar things. In our case, making a bunch of dummy variables can be a lot of work! Luckily, R has a ~~lazy~~ **efficient** solution for us. ## Factors To create many dummies at once, we can use the `factor()` command. Moreover, we can even select which factor becomes our reference group! To do this in R, we would do the following: ``` r # I am putting a "_" to differentiate between the original variable. ames$bsmnt_ <- as.factor(ames$bsmnt) head(ames$bsmnt_) cat("\n") ames$bsmnt_ <- relevel(ames$bsmnt_, ref = "No Basement") head(ames$bsmnt_) ```

Output

[1] BLQ Rec ALQ ALQ GLQ GLQ Levels: ALQ BLQ GLQ LwQ No Basement Rec Unf [1] BLQ Rec ALQ ALQ GLQ GLQ Levels: No Basement ALQ BLQ GLQ LwQ Rec Unf

Then, if we throw this into a regression, R will create all of the dummies for us. I will also include bedrooms and age into this model for completeness. ``` r reg3 <- lm(log(price) ~ log(sqft) + bsmnt_ + bedrooms + age, ames) summary(reg3) ```

Output

Call: lm(formula = log(price) ~ log(sqft) + bsmnt_ + bedrooms + age, data = ames) Residuals: Min 1Q Median 3Q Max -1.86737 -0.11004 0.00482 0.12182 0.77719 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.8752614 0.1011122 58.11 <2e-16 *** log(sqft) 0.8568861 0.0146948 58.31 <2e-16 *** bsmnt_ALQ 0.3529548 0.0242718 14.54 <2e-16 *** bsmnt_BLQ 0.3285093 0.0253126 12.98 <2e-16 *** bsmnt_GLQ 0.3932176 0.0240384 16.36 <2e-16 *** bsmnt_LwQ 0.2831736 0.0274313 10.32 <2e-16 *** bsmnt_Rec 0.2922540 0.0251362 11.63 <2e-16 *** bsmnt_Unf 0.2459743 0.0233557 10.53 <2e-16 *** bedrooms -0.0694749 0.0055325 -12.56 <2e-16 *** age -0.0047742 0.0001461 -32.69 <2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.1987 on 2920 degrees of freedom Multiple R-squared: 0.763, Adjusted R-squared: 0.7623 F-statistic: 1045 on 9 and 2920 DF, p-value: < 2.2e-16

The model spits out a bunch of numbers for `bsmnt_`, but it’s important to remember that they are all interpreted as having that “type” of basement relative to not having a basement, controlling for the property’s age, size and number of bedrooms. Moreover, we can plot the coefficients and show that they’re generally increasing in the order that we’d expect.

Plot

Basement Rating Reminder

It is important to point out the assumptions of this model. We are assuming that changing the square footage (or age or bedrooms) of a home impacts sale price independently of the type of basement. Of course, we could estimate a fully interacted model, but the output would be huge and uninterpretable.