Notes on linear regression with Stata

1. LINEAR REGRESSION WITH ONE
REGRESSOR
Ryan Herzog, Ph.D.

2. HEDONIC PRICING IN REAL ESTATE
• What is the relationship between square footage of a house and
its price?
• If one square foot of living space is added to a house by how much will its
price increase?
• $50?
• $100?
• All of these questions can be answered with the help of a linear
regression. We will be using the ”real estate” dataset to help us
answer these questions.

3. THE RELATIONSHIP BETWEEN SQUARE FOOTAGE OF A
HOUSE AND ITS PRICE
• Stata: twoway scatter price sqft

4. SIDENOTE – LABEL VAR
• To clean up graphs (and other output) use the command ”label”
to clean up variable names.
• label variable sqft “Square Feet”
• label variable price “Price (thousands)”

5. THE RELATIONSHIP BETWEEN SQUARE FOOTAGE OF A
HOUSE AND ITS PRICE
• Add a line of best fit
• twoway (scatter price sqft) (lfit price sqft)

6. THE EQUATION OF THE LINE
Price = 153.18 + 0.195 * sq ft
For a 2,000 sq ft, expected price
would be
= 153.18 + 0.19537 * 2,000
= 543.92

7. ANOTHER GRAPHING COMMAND (AAPLOT)
• ssc install aaplot
• aaplot price sqft

8. LINEAR REGRESSION MODEL
Yi = β0 + β1Xi + ui, i = 1,…, n
• We have n observations, (Xi, Yi), i = 1,.., n.
• X is the independent variable or regressor or explanatory variable
• Y is the dependent variable
• β0 = intercept
• β1 = slope, coefficient
• ui = the regression error or disturbance term
• not a deterministic equation

9. EXAMPLE
• Name the components of the following linear
regression model
𝑝𝑟𝑖𝑐𝑒𝑖 = 𝛽0 + 𝛽1 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡𝑖 + 𝑒𝑖

10. FITTED EQUATION AND SLOPE INTERPRETATION
• 𝑝𝑟𝑖𝑐𝑒 = 153.18 + 0.195 ∗ 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡
• On average, a one unit (what is it?) increase in house size is associated
with about a 0.19 unit (what is it?) increase of its price
• If 50 square feet of living space are added to a house, on average its price
will increase by about _______
• If 100 square feet of living space are added to a house, on average its price
will increase by about ______

11. INTERPRETING BETA FOR THE CHANGE IN X AND AT A
POINT
• 𝑝𝑟𝑖𝑐𝑒 = 153.18 + 0.195 ∗ 𝑠𝑞𝑢𝑎𝑟𝑒 𝑓𝑒𝑒𝑡
• Change in X:
• If square feet (X) changes by 10, by how much will the price change on average?
• At a point:
• If the size of a house (X) is 1500 square feet what is its price?

12. MORE EXAMPLES
• 𝑇𝑒𝑠𝑡𝑠𝑐𝑜𝑟𝑒 = 520.4 − 5.82 ∗ 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒
• What is the regression’s prediction for a classroom of 20 students?
• What is the regression’s prediction for the change in the classroom average
test score if the classroom size increases by 4?
• 𝑊𝑒𝑖𝑔ℎ𝑡 = −99.41 + 3.94 ∗ 𝐻𝑒𝑖𝑔ℎ𝑡
• What is the regression’s weight prediction for someone who is 70 inches tall?
(weight is measured in pounds, height is measured in inches)
• If someone has a growth spurt of 1.5 inches over the course of a year what is
the regression’s prediction for the increase in this person’s weight?

13. CORRELATION DOES NOT IMPLY CAUSATION
• Proper language: is associated/correlated with, suggests
*Check out the book “Spurious Correlation by Tyler Vigen
https://www.tylervigen.com/spurious-correlations

14. THE RELATIONSHIP BETWEEN PRICE AND SQUARE FEET OF
A HOUSE

15. MORE EXAMPLES
• Regress the price on the following variables, write out
the population regression equation, fitted equation, and
interpret the results:
• Number of bedrooms
• Number of bathrooms
• Age
• Year built

16. EVEN MORE EXAMPLES. DIY TIME.
• Use California test score dataset
• Regress test score on the following variables, write out
the population regression equation, the fitted equation,
and interpret the results.
• Class size
• Percent of ESL students
• Computers per student
• Percent of students qualified for free lunches

17. MEANINGFUL INTERCEPT
• Pay attention to the following
Intuitively can one of the X variable observations be zero?
- The zero value for the X variable is in the sample
Example: In the following two regressions is the intercept
meaningful?
𝑡𝑒𝑠𝑡 𝑠𝑐𝑜𝑟𝑒𝑖 = 𝛽0 + 𝛽1 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝐸𝑆𝐿 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠𝑖 + 𝑢𝑖
𝑡𝑒𝑠𝑡 𝑠𝑐𝑜𝑟𝑒𝑖 = 𝛽0 + 𝛽1 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑖𝑛𝑐𝑜𝑚𝑒𝑖 + 𝑢𝑖

18. MEANINGFUL INTERCEPT
• Is the intercept meaningful in the following two
regressions?
𝑡𝑒𝑠𝑡 𝑠𝑐𝑜𝑟𝑒𝑖 = 𝛽0 + 𝛽1 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒𝑖 + 𝑢𝑖
𝑡𝑒𝑠𝑡 𝑠𝑐𝑜𝑟𝑒𝑖 = 𝛽0 + 𝛽1 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑟𝑠 𝑝𝑒𝑟 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑖 + 𝑢𝑖

19. MEANINGFUL SLOPE COEFFICIENT. INTERPRETING THE
MAGNITUDE
Is the increase in X associated with the large increase in Y? if so => the
coefficient is economically meaningful (large)
• We can’t simply compare coefficients from two different regressions, since
each variable has a different standard deviation. We want to find out by how
much Y changes when X changes by its standard deviation
• Steps to find out the magnitude:
1. Find the standard deviation of X
2. Multiply the standard deviation of X by the coefficient
3. Compare the product from (2) to the standard deviation of Y (often by
dividing it by the standard deviation of Y)

20. INTERPRETING THE MAGNITUDE. EXAMPLES
• Is the effect of class size on test scores large?
• Is the effect of the percent of ESL students on test scores large?
• If we compare the coefficients from the two regressions what
conclusion might we arrive to?
• If class size increases by 1 standard deviation by how much will the test
scores increase?
• If the percent of ESL students increases by 1 standard deviation by how
much will the test scores increase?
• How do these changes compare to the standard deviation of test
scores?

21. INTERPRETING THE MAGNITUDE.
• Is the effect of the independent variable in the following two
regressions on the dependent variable economically meaningful?
• With one standard deviation change in each of the independent
variables by how much will the dependent variable change in
terms of its standard deviation?
𝑡𝑒𝑠𝑡 𝑠𝑐𝑜𝑟𝑒𝑖
= 𝛽0 + 𝛽1 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑜𝑓 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑠 𝑞𝑢𝑎𝑙𝑖𝑓𝑖𝑒𝑑 𝑓𝑜𝑟 𝑓𝑟𝑒𝑒 𝑙𝑢𝑛𝑐ℎ𝑒𝑠𝑖 + 𝑢𝑖
𝑡𝑒𝑠𝑡 𝑠𝑐𝑜𝑟𝑒𝑖 = 𝛽0 + 𝛽1 𝑐𝑜𝑚𝑝𝑢𝑡𝑒𝑟𝑠 𝑝𝑒𝑟 𝑠𝑡𝑢𝑑𝑒𝑛𝑡𝑖 + 𝑢𝑖

22. REVIEW
• Please name the components of a linear regression (based on an example of
your own choosing)
• Why do we need to have an error term in the regression equation?
• What is a fitted equation? (What is the difference between the fitted equation
and the population equation)? Please give examples.
• How do you interpret the results of a regression at a point/ based on the
change in X?
• Command in Stata to run a regression
• What does a meaningful intercept mean? Please give an example
• How do we interpret the magnitude of the coefficient and decide if it is
economically meaningful?