The purpose of this in-class lab is to further practice your regression skills. To get credit, upload your .R script to the appropriate place on Canvas.

## For starters

Open up a new R script (named ICL4_XYZ.R, where XYZ are your initials) and add the usual “preamble” to the top:

# Add names of group members HERE
library(tidyverse)
library(broom)
library(wooldridge)
library(modelsummary)

For this lab, let’s use data on house prices. This is located in the hprice1 data set in the wooldridge package. Each observation is a house.

df <- as_tibble(hprice1)

Check out what’s in df by typing

glimpse(df)

Or for some summary statistics:

datasummary_skim(df,histogram=FALSE)

## Multiple Regression

Let’s estimate the following regression model: $price = \beta_0 + \beta_1 sqrft + \beta_2 bdrms + u$ where $$price$$ is the house price in thousands of dollars.

The code to do so is:

est1 <- lm(price ~ sqrft + bdrms, data=df)
modelsummary(est1)

You should get a coefficient of 0.128 on sqrft and 15.2 on bdrms. Interpret these coefficients. (You can type the interpretation as a comment in your .R script.) Do these numbers seem reasonable?

You should get $$R^2 = 0.632$$. Based on that number, do you think this is a good model of house prices?

Check that the average of the residuals is zero:

mean(est1$residuals) ## Adding in non-linearities The previous regression model had an estimated intercept of -19.3, meaning that a home with no bedrooms and no square footage would be expected to have a sale price of -$19,300.

To ensure that our model always predicts a positive sale price, let’s instead use $$log(price)$$ as the dependent variable. Let’s also add quadratic terms for sqrft and bdrms to allow those to exhibit diminishing marginal returns.

First, let’s use mutate() to add these new variables:

df <- df %>% mutate(logprice = log(price), sqrftSq = sqrft^2, bdrmSq = bdrms^2)

Now run the new model:

est2 <- lm(logprice ~ sqrft + sqrftSq + bdrms + bdrmSq, data=df)
modelsummary(est2)
# or, for more decimals:
modelsummary(est2, fmt = 10)

The new coefficients have much smaller magnitudes. Explain why that might be.

The new $$R^2 = 0.595$$ which is less than $$0.632$$ from before. Does that mean this model is worse?

## Using the Frisch-Waugh Theorem to obtain partial effects

Let’s experiment with the Frisch-Waugh Theorem, which says: $\hat{\beta}_{1} = \frac{\sum_{i=1}^{N} \hat{r}_{i1}y_{i}}{\sum_{i=1}^{N} \hat{r}_{i1}^2}$ where $$\hat{r}_{i1}$$ is the residual from a regression of $$x_{1}$$ on $$x_{2},\ldots,x_{k}$$

Let’s do this for the model we just ran. First, regress sqrft on the other $$X$$’s and store the residuals as a new column in df.

est <- lm(sqrft ~ sqrftSq + bdrms + bdrmSq, data=df)
df <- df %>% mutate(sqrft.resid = est$residuals) Now, if we run a simple regression of logprice on sqrft.resid we should get the same coefficient as that of sqrft in the original regression (=3.74e-4). est <- lm(logprice ~ sqrft.resid, data=df) modelsummary(est) ## Frisch-Waugh by hand We can also compute the Frisch-Waugh formula by hand: beta1 <- sum(df$sqrft.resid*df$logprice)/sum(df$sqrft.resid^2)
print(beta1)

Which indeed gives us what we expected.