The document discusses model specification for multiple regression analysis, focusing on measures of fit including R-squared and standard error of regression, and how to properly interpret these statistics. It emphasizes the importance of random sampling to establish causal relationships and warns of potential biases from non-random samples, such as when evaluating mutual fund performance or estimating political support based on telephone and automobile owners.

1. MODEL SPECIFICATION FOR
MULTIPLE REGRESSION
Ryan Herzog, Ph.D.
Gonzaga University
ECON 355 – Regression Analysis

2. MEASURES OF FIT
• R-squared
• Standard Error of the Regression

3. R-SQUARED
• The regression R2 measures the fraction of the variance of Y that is
explained by X; it is unitless and ranges between zero (no fit) and one
(perfect fit)
• By simply looking at R-squared of a regression we will not be able to say
much, we need to have at least two regressions to compare them
• All else equal we want to be able to explain a higher share of variance in Y
• Stata:
• Use California school dataset
• Regress test scores on class size. The R-squared is 0.0512. This means that class size
explains about 5% of the variance in test scores.
• Regress test scores on expenditure per student. What is the R-squared? How would
you interpret it?

4. INTERPRETING R-SQUARED
• An increase in R-squared does not necessarily mean that an added variable
is statistically significant
• A high R-squared does not mean that the regressors are a true cause of the
dependent variable
• A high R-squared does not mean that the coefficients on the regressors are
true
• A low R-squared does not mean that the coefficients on the regressors are
wrong
• A high R-squared does not necessarily mean that you have the most
appropriate set of regressors, nor does a low R-squared necessarily mean
that you have an inappropriate set of regressors.

5. R-SQUARED RULES
• Same dependent variable
• Same number of independent variables
• Stata:
• generate ltestscr=ln(testscr)
Generate a table with the regressions below (use outreg)
• Regress test score on class size and expenditure per student
• Regress natural log of test score on class size
• Regress test score on class size and average district income
• Regress expenditure per student on average district income
• Regress natural log of test score on average district income
• Regress natural log of test score on class size and expenditure per student

6. STANDARD ERROR OF THE REGRESSION
• The SER is a measure of the spread of the observations around the regression line
measured in the units of the dependent variable
• SER is an estimator of the standard deviation of the regression error 𝑢𝑖
• All else equal we want to have a smaller spread of the observations around the
regression line
• In Stata SER is called root MSE (mean squared error)
• In the regression of class size on test score the SER is about 18.6. This means that the
standard deviation of the regression residuals around the regression line is 18.6 points.
• We can use SER to compare models
• What are the SERs for the rest of the regressions you have run? What does that mean?

7. CAUSAL EFFECTS AND IDEALIZED
EXPERIMENTS
• Most of our questions concern causal relationships among variables, i.e.
does lower class size lead to higher test scores?
• Causality means that a specific action leads to a specific measurable
consequence
• The best way to measure a causal effect is by conducting an experiment
• In a randomized controlled experiment there is both a control group and a
treatment group. Assignment to a group happens randomly
• We would like to be able to show that the only systematic reason for
differences in outcomes between the treatment and control groups is the
treatment itself
• In practice, it is not possible to perform ideal experiment. This however, gives
us a benchmark.

8. NONRANDOM SAMPLE EXAMPLE - 1
• In 1936 the Literary Gazette polled a “random” sample of households chosen
from telephone records and automobile registration
• In 1936 many households did not have cars or telephones, and those that
did tended to be richer – and were also more likely to be Republican
• The results of the poll indicated the Landon (a republican presidential
candidate) would defeat an incumbent (Roosevelt) by a landslide – 57% to
43% in the 1936 election
• Roosevelt ended up winning by 59% to 41%
• Do you think surveys conducted using social media might have a similar
problem with bias?

9. NONRANDOM SAMPLE EXAMPLE - 2
• Some mutual funds simply track the market, some are actively managed by full-time
professionals.
• Do the latter mutual funds outperform the former?
• One way to answer the question is to use historical data on funds currently available
for purchase, however this means that the most poorly underperforming funds would
not be represented.
• The sample is selected based on the value of the dependent variable, returns, because
funds with the lowest returns are eliminated
• The mean return of all funds would then be lower than the mean return of those still in
existence. This is also called a survivorship bias.
• When corrected for survivorship bias it turns out actively managed funds do not
outperform the market

10. NON-RANDOM SAMPLE EXAMPLE 3
• Does the class size affect the test scores with only districts where
average class size is above 20 students included?
• What is the average height of a GU student measured outside of a
basketball locker room?