Efficiency

1. STATISTICS
ECON 355 – Regression Analysis

2. DISTRIBUTION REVIEW
• We use distribution to better understand what data looks like
• A police unit records speed of passing vehicles on a road with speed limit of 45
miles/hour
• Use speed.csv dataset on blackboard to find out the mean, standard deviation, and
histogram of the results
• For the histogram (in Stata) set the number of bins to 5, Y-axis should display frequency,
add height label to bars, change Range/value (major tick-label properties) of X-axis to
min 25, max 90 and delta of 5

3. POPULATION VS. SAMPLE
• Most of the time our questions are about the universe (population).
• For example:
• “do people who sleep more drink less coffee?”
• Or “are smaller class sizes associated with higher test scores?”
• Or “what is the relationship between price and age of residential real estate?”
• Unfortunately, we are not able to collect data from every single person, class size, or
house in the world
• We use samples of data to answer the question about the population

4. SOME OTHER QUESTIONS WE CAN ANSWER BY USING
SAMPLES
• What are the mean earnings of people in their 20s?
• What is the standard deviation of the earnings of people in their 20s?
• What is the variance of the earnings of people in their 20s?
• What is the average unemployment rate in New York city?
• What is the average height of a GU student?

5. PARAMETER, STATISTIC, AND ESTIMATOR
• When we want to find out the mean earnings (𝜇) of people in their 20s we are looking
for a population parameter
• When we collect answers from a sample of people in their 20s and calculate the average
earnings ( 𝑥) we have found the sample statistic
• We use sample statistics to estimate population parameters
• Sample statistics are estimators of population parameters
• In this specific example 𝑥 is an estimator of 𝜇
• Estimate – numerical value of the estimator when it is actually computed using data
from a specific sample

6. PARAMETER, STATISTIC, AND ESTIMATOR EXAMPLE
• In the regression of test scores on class size
• What is the general regression?
• What is the fitted regression?
• What is the population here?
• What is the sample?
• What is the parameter we are trying to estimate?
• What is the estimator of that parameter?
• What is the estimate?
• Difference between estimator and estimate?

7. IN CLASS
You run a regression of coffee consumption on the number of hours one sleeps at night.
And find that the coefficient on the number of hours slept at night is -0.75. Please choose
the answer that lists the following in the correct order: population in question; sample;
parameter; estimator; estimate
A. Everyone on the planet, the group of people you interview for your homework; 𝛽1; 1; -0.75
B. Everyone on the planet, the group of people you interview for your homework; 1; 𝛽1; -0.75
C. The group of people you interview for your homework, everyone on the planet; 𝛽1; 1; -0.75
D. Everyone on the planet, the group of people you interview for your homework; 1; -0.75; 𝛽1

8. REPEATED SAMPLES
• Suppose you are trying to evaluate an estimator by repeatedly drawing random samples
from a population
• Example 1.
• To estimate the relationship between hours slept and caffeine consumed you divided into 8
groups, collected data and ran 8 regressions. Here we have 8 estimates of the relationship. What
is the estimator here?
• Example 2.
• To estimate the average height of a GU student each of you asks 30 students how tall they are.
This way we will get 24 estimates of the average height of a GU student. What is the estimator
here?
• Repeated samples mean that the estimator is not one number but rather a series of
numbers. We are going to be working with its distribution.

9. RANDOM SAMPLE
• If we estimate the height of GU students by using the sample of people we meet outside
of basketball locker room we are not going to have a random sample
• If we estimate the unemployment rate by going to a park at 10 am on a weekday and
ask people we run into if they are employed we are not going to have a random sample
• I am trying to figure out the relationship between the number of DUIs in a state and
alcohol consumption. What sample should I use?
• Random sample means that everyone in the population has an equal chance of ending
up in the sample

10. ESTIMATOR PROPERTY 1. UNBIASEDNESS
• If you evaluate an estimator many times by using repeatedly drawn samples, it is
reasonable to hope that on average you would get the right answer.
• Example: we are trying to estimate the average height of a GU student. We collect data
from random samples 5 times and get the following results
Sample number Average height in the
sample
1 167.64
2 175.26
3 160.02
4 152.4
5 177.8

11. UNBIASEDNESS CONT’D
• What is the average number across the samples?
• If our estimator 𝑥 is unbiased then the average height of a GU student is about 5 feet 6
inches (166 cm)
• If you decided to estimate the average height in your sample by using the following
formula:
• 𝑖=1
𝑁
ℎ𝑒𝑖𝑔ℎ𝑡 𝑖
𝑁
+ 2
• you would have a biased estimator

12. ESTIMATOR PROPERTY 2. CONSISTENCY.
• The larger the sample the closer the estimator is to the population parameter
• Example: The larger the sample of GU students you survey about their height the closer
the estimate of the height will be to the actual average height of a GU student

13. ESTIMATOR PROPERTY 3. VARIANCE AND EFFICIENCY
• We want the estimator with the smaller variance. That estimator will be the most
efficient.
• Example: we are trying to estimate the average height of a GU student. We collect data
from random samples 5 times and get the following results
Sample number Average height in the
sample
Median height in the
sample
1 167.64 167.64
2 175.26 172.72
3 160.02 162.56
4 152.4 165.1
5 177.8 162.56
Variance 111.6 18.06

14. EFFICIENCY CONT’D
• Median appears to have a smaller variance, which means it is a more efficient estimator
of the average height of a GU student

15. REVIEW.
• Please explain unbiasedness, consistency, and variance by applying the ideas to the
example of the relationship between coffee consumed and hours slept.
• What is the difference between parameter, estimator and estimate?