ECON 0150 | Economic Data Analysis

The economist’s data analysis pipeline.


Part 5.3 | Numerical Control Variables

Pittsburgh Housing

What is the value of an additional square foot of living area?


House prices depend on:

  • Square foot
  • Age of the house
  • Number of bedrooms
  • Neighborhood
  • Many other factors!

Pittsburgh Housing

What is the value of an additional square foot of living area?


Pittsburgh housing data with 10,000 homes.


Variables:

  • SALEPRICE: Sale price of the home
  • LOGPRICE: Log of sale price
  • FINISHEDLIVINGAREA: Square footage of finished living space
  • YEARBLT: Year the home was built

Pittsburgh Housing: Separate Models

Lets predict price using Living Area and Year Built.

Both living area and year built have positive relationships with price.

Pittsburgh Housing: Multiple Regression

We can model multiple predictor variables simultaneously.

The Multiple Regression Equation

Extending the best-fitting line to multiple dimensions


Single Variable:

\[\text{LogPrice} = \beta_0 + \beta_1 \times \text{LivingArea} + \epsilon\]

Multiple Variables:

\[\text{LogPrice} = \beta_0 + \beta_1 \times \text{LivingArea} + \beta_2 \times \text{YearBuilt} + \epsilon\]

Interpretation:

  • \(\beta_0\) = Base log price (intercept)
  • \(\beta_1\) = Effect of one more square foot, holding year built constant
  • \(\beta_2\) = Effect of being built one year later, holding living area constant

Pittsburgh Housing

What is the value of an additional square foot?

Without controlling for year built:

  • We compare small (old) houses to large (new) houses
  • The size effect includes the age effect
  • Overestimates the true value of square footage

With controlling for year built:

  • We compare houses of similar age but different sizes
  • Isolates the pure size effect
  • Gives us the true value of square footage

This is what we call a control variable.

Multiple Regression with Control

We can adjust for multiple variables simultaneously. 

Model 1: Living Area Only
======================================================================================
                         coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------------
Intercept             10.2029      0.031    328.476      0.000      10.142      10.264
FINISHEDLIVINGAREA     0.0005   1.65e-05     33.260      0.000       0.001       0.001
======================================================================================


Model 3: Living Area + Year Built Control
======================================================================================
                         coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------------
Intercept            -17.9337      0.741    -24.213      0.000     -19.386     -16.482
FINISHEDLIVINGAREA     0.0005   1.56e-05     29.095      0.000       0.000       0.000
YEARBLT                0.0145      0.000     38.018      0.000       0.014       0.015
======================================================================================

Comparing Models

Compare coefficients with and without the control.

Without control: size effect was inflated by age effect

With control: we get the size effect between houses of equal age

Residual Diagnostics

Checking if our model improved

Residuals look somewhat more random with the control.

Log Interpretation

For log outcomes, coefficients represent proportional changes.


For small coefficients (< 0.1):

\[\beta \times 100 \approx \text{percentage change}\]


Interpretation (holding year built constant):

  - Each additional sq ft increases price by 0.045%
  - 100 additional sq ft increases price by 4.55%

Interpretation (holding living area constant):

  - Each year newer increases price by 1.454%
  - A house 10 years newer increases price by 14.54%

The “Holding Constant” Intuition

A control variable compares two similar observations.

Without control (simple regression):

\[\text{LogPrice} = \beta_0 + \beta_1 \times \text{LivingArea} + \epsilon\]

  • Compares all small houses to all large houses
  • Large houses are newer, so \(\beta_1\) captures both size AND age effects

With control (multiple regression):

\[\text{LogPrice} = \beta_0 + \beta_1 \times \text{LivingArea} + \beta_2 \times \text{YearBuilt} + \epsilon\]

  • Compares small and large houses built in the same year
  • \(\beta_1\) now captures only the pure size effect
  • \(\beta_2\) captures only the pure age effect

Summary

Main ideas about numerical control variables


  1. Control variables help isolate the effect of your main predictor
  2. “Holding constant” means comparing observations with similar control values
  3. Coefficients change when you add controls because you remove confounding
  4. Choose controls carefully: confounders yes, mediators no
  5. Check residuals to see if controls improved the model

Exercise 5.3 | Pittsburgh Housing Prices


Lets find the value of an extra square feet in the Pittsburgh housing market.

Step 1: Load and Explore the Data


# Load the data
data = pd.read_csv(file_path + 'pgh_housing.csv', index_col=0)
data['LOGPRICE'] = np.log(data['SALEPRICE'])
data.head()

Step 2: Modeling Relationships Separately


# Model 1: Living area only
model_area = smf.ols('LOGPRICE ~ FINISHEDLIVINGAREA', data=data).fit()
print(model_area.summary().tables[1])


# Model 2: Year built only
model_year = smf.ols('LOGPRICE ~ YEARBLT', data=data).fit()
print(model_year.summary().tables[1])

Step 3: Compare Living Area and Year Built


# Plot the relationship between living area and year built
plt.figure(figsize=(8, 5))
sns.scatterplot(data, y='FINISHEDLIVINGAREA', x='YEARBLT', alpha=0.5)
plt.title('Relationship Between Living Area and Year Built')
plt.ylabel('Finished Living Area (sq ft)')
plt.xlabel('Year Built')

Step 4: Multiple Regression with Control Variable


# Model 3: Both variables (with control)
model_both = smf.ols('LOGPRICE ~ FINISHEDLIVINGAREA + YEARBLT', data=data).fit()
print(model_both.summary().tables[1])

Step 5: Checking Residuals


# Create residual plots for both models
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Model 1: Without control
residuals_area = model_area.resid
predictions_area = model_area.fittedvalues
ax1.scatter(predictions_area, residuals_area, alpha=0.5, color='#4C72B0')
ax1.axhline(y=0, color='red', linestyle='--', linewidth=2)
ax1.set_title('Model 1: Without Year Built Control')

# Model 3: With control
residuals_both = model_both.resid
predictions_both = model_both.fittedvalues
ax2.scatter(predictions_both, residuals_both, alpha=0.5, color='#4C72B0')
ax2.axhline(y=0, color='red', linestyle='--', linewidth=2)
ax2.set_title('Model 3: With Year Built Control')