This code has been run in the background for you to facilitate your use
of the following WebR chunks.
In addition, we are going to create a new variable for age.
``` r
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")
ames <- ames[,keep_columns]
colnames(ames) <- c("price", "sqft", "bedrooms", "yr_built", "condition")
ames$age <- 2011 - ames$yr_built
```
## Simple Correlations
We’ll start by estimating a simple model where price is determined by
age. Next, we’ll estimate a model where price is determined by age *and*
square footage. Before we estimate anything, though, let’s first
establish some of the correlations between the three variables in our
data.
``` r
cor(ames[,c("price", "age", "sqft")])
```
1. The correlation between price and age is -0.558. This suggests that
newer homes are typically more expensive.
2. The correlation between price and square footage is 0.707. We saw
this last module – as square footage increases, so does price. This
makes sense; bigger houses typically command higher prices.
3. The correlation between age and square footage is *-0.242*. This
suggests that newer houses are bigger than older homes. **This is an
important data artifact!**
Why is this so important? Let’s first estimate a simple model:
$$\text{Price}_i = \alpha_0 + \alpha_1 \times \text{Age}_i + \epsilon_i$$
``` r
reg_alpha <- lm(price ~ age, ames)
cat("Alpha0 (intercept):", coef(reg_alpha)[1],
"\n", "Alpha1 (slope):", coef(reg_alpha)[2])
```
$\alpha_1$ = -1474.96 suggests that for every one year increase in
property age, housing price decreases by \$1474.96. However, remember,
when we are talking about an older house, we are also talking about a
generally smaller home (as evidenced via the correlation from before).
This means that the regression coefficient of -1474.96 is actually a
“composite” effect of both increased age *and* decreased square footage!
In other words, when we increase age, two things happen: not only does
age increase, but *square footage also decreases*. So which of these two
effects are we really picking up with this coefficient?
Obviously, in reality, houses do not shrink as they age, but the model
doesn’t know that. It only understands the correlations in the data. In
these data, older houses are also smaller, so it’s important we
*control* the size of a home to get a true measure of the effect of age
on prices.
Another way to think about this is that we are trying to compare houses
that are the same in every way *except* for their ages. *Controlling*
for square footage gets us one step closer to keeping everything about
the home constant while we vary only its age.
How, then, do we control for square footage?
## Breaking Down the Composite
To get an estimate of the isolated effect of age on price, we need some
extra information. First, we need the initial regression where price is
a function of age. Next, we need to know how square footage will respond
to changes in age. To get this information, we’ll estimate a regression
where square footage is the outcome and age is the explanatory variable.
Lastly, we need to know how price responds to changes in square footage.
``` r
coef(lm( ~ , ames)) # price vs age
coef(lm( ~ , ames)) # sqft vs age
coef(lm( ~ , ames)) # price vs sqft
```
Without going through all the
math
,
OLS will simultaneously find *two* lines of best fit. You can even think
of this as a surface, or a “plane”, since there are two dimensions
instead of just one. $\beta_0, \beta_1, \text{and} \ \beta_2$ will still
minimize the sum of squared residuals and provide an average residual of
zero.
Before estimating this specific model, however, let’s visualize the data
in three dimensions. On the $Z$-axis (the one going up and down), we’ll
plot the home’s sale price. Then, on the $X$- and $Y$-axes, we are going
to plot age and square footage. Feel free to click on and spin the
following visualization:
*(Interactive 3D plot omitted from the markdown version.)*
The following figures are 2D representations of what you would see if
you were to spin the 3D figure in certain directions. The first panel
shows us the relationship between price and square footage. The second
panel depicts the relationship between age and price. The final panel
shows the relationship between square footage and age.
