Econometrics
Anna Donina
Lecture 1
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INTRODUCTORY ECONOMETRICS COURSE
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• Lecturer: Anna Donina (Junior Researcher, CERGE-EI, Prague)
anna.donina@gmail.com
• Lectures / Seminars: Friday, 14:00 – 15:50 / 16:00 – 17:50 (VT203)
Grading
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• Quizzes/Home assignment: 40 %
• Midterm exam: 30 %
• Final Exam/Project: 30 %
to pass the course, student has to get at least 20 points in the
final exam and 50 points in total
INTRODUCTORY ECONOMETRICS COURSE
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Recommended literature:
▪ Studenmund, A. H., Using Econometrics: A Practical Guide
▪ Wooldridge, J. M., Introductory Econometrics: A Modern
Approach
▪ Adkins, L., Using gretl for Principles of Econometrics
WHAT IS ECONOMETRICS?
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Tobeginning students, it may seem as if econometrics is an overly
complex obstacle to an otherwise useful education. (. . .) To professionals
in the field, econometrics is a fascinating set of techniques that allows
the measurement and analysis of economic phenomena and the
prediction of future economic trends.
Studenmund (Using Econometrics: A Practical Guide)
WHAT IS ECONOMETRICS?
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• Econometrics is a set of statistical tools and techniques for
quantitative measurement of actual economic and business
phenomena
• It attempts to
1. quantify economic reality
2. bridge the gap between the abstract world of economic
theory and the real world of human activity
• It has three major uses:
1. describing economic reality
2. testing hypotheses about economic theory
3. forecasting future economic activity
EXAMPLE
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• Consumer demand for a particular commodity can be
thought of as a relationship between
▪ quantity demanded (Q)
▪ commodity’s price (P)
▪ price of substitute good (Ps)
▪ disposable income (Y)
• Theoretical functionalrelationship:
Q = f(P, Ps, Y)
• Econometrics allows us to specify:
Q = 31.50 − 0.73P + 0.11Ps +0.23Y
LECTURE 1
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Introduction, repetition of statisticalbackground
▪ probability theory
▪ statistical inference
Readings:
▪ Studenmund, A. H., Using Econometrics: A Practical Guide,
Chapter 16
▪ Wooldridge, J. M., Introductory Econometrics: A Modern
Approach, Appendix B and C
RANDOM VARIABLES
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• Random variable X is a variable whose numerical value is
determined by chance. It is a quantification of the outcome of a
random phenomenon.
• Discrete random variable has a countable number of possible
values
Example: the number of times that a coin will be flipped before a
heads is obtained
• Continuous random variable can take on any value in an interval
Example: time until the first goal is scored in a football match
DISCRETE RANDOM VARIABLES
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• Described by listing the possible values and the
associated probability that it takes on each value
• Probability distribution of a variable X that can
take values x1, x2, x3, ... :
P(X = x1) = p1
P(X = x2) = p2
P(X = x3) = p3
..
• Cumulative distribution function (CDF):
SIX-SIDED DIE: PROBABILITY DISTRIBUTION FUNCTION
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SIX-SIDED DIE: HISTOGRAM OF DATA (100 ROLLS)
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SIX-SIDED DIE: HISTOGRAM OF DATA (1000 ROLLS)
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CONTINUOUS RANDOM VARIABLES
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• Probability density function fX(x) (PDF) describes the
relative likelihood for the random variable X to take on
a particular value x
• Cumulative distribution function (CDF):
• Computational rule:
P(X ≥ x) = 1 − P(X ≤x)
EXPECTED VALUE AND MEDIAN
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• Expected value (mean):
Mean is the (long-run) average value of random variable
Discrete variable Continuous variable
∫
Example: calculating mean of six-sided die
• Median : ”the value in themiddle”
EXERCISE 1
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• A researcher is analyzing data on financial wealth of 100
professors at a small liberal arts college. The values of
their wealth range from $400 to $400,000, with a mean of
$40,000, and a median of $25,000.
• However, when entering these data into a statistical
software package, the researcher mistakenly enters
$4,000,000 for the person with $400,000 wealth.
• How much does this error affect the mean andmedian?
VARIANCE AND STANDARD DEVIATION
• Variance:
Measures the extent to which the values of a random
variable are dispersed from the mean.
If values (outcomes) are far away from the mean, variance is
high. If they are close to the mean, variance is low.
• Standard deviation :
Note: Outliers influence onvariance/sd.
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DANCING STATISTICS
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Watch the video ”Dancing statistics: Explaining the statistical
concept of variance through dance”:
https://www.youtube.com/watch?v=pGfwj4GrUlA&list=
PLEzw67WWDg82xKriFiOoixGpNLXK2GNs9&index=4
Use the ’dancing’ terminology to answer thesequestions:
1. How do we define variance?
2. How can we tell if variance is large or small?
3. What does it mean to evaluate variance within a set?
4. What does it mean to evaluate variance between sets?
5. What is the homogeneity of variance?
6. What is the heterogeneity of variance?
EXERCISE 2
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• Which has a higher expected value and which has a higher
standard deviation:
• a standard six-sided die or
• a four-sided die with the numbers 1 through 4 printed on
the sides?
• Explain your reasoning, without doing any calculations,
then verify, doing the calculations.
COVARIANCE, CORRELATION, INDEPENDENCE
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• Covariance:
▪ How, on average, two random variables vary with
one another.
▪ Do the two variables move in the same or
opposite direction?
▪ Measures the amount of linear dependence between two
variables.
Cov(X, Y) = E [(X − E[X]) (Y − E[Y])] = E [XY] − E[X]E[Y]
• Correlation:
Similar concept to covariance, but easier to interpret.
It has values between -1 and 1.
Corr(X, Y)=
Cov(X, Y)
σXσY
INDEPENDENCE OF VARIABLES
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• Independence: X and Y are independent if the conditional
probability distribution of X given the observed value of Y
is the same as if the value of Y had not been observed.
• If X and Y are independent, then Cov(X, Y) = 0 (not
the other way round in general)
• Dancing statistics: explaining the statistical concept
of correlation through dance
https://www.youtube.com/watch?v=VFjaBh12C6s&index=3&
list=PLEzw67WWDg82xKriFiOoixGpNLXK2GNs9
COMPUTATIONAL RULES
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E(aX + b) = aE(X) +b
Var(aX +b) = a2Var(X)
Var(X +Y) = Var(X) + Var(Y) + 2Cov(X, Y)
Cov(aX,bY) = Cov(bY, aX) = abCov(X, Y)
Cov(X + Z,Y) = Cov(X, Y) + Cov(Z, Y)
Cov(X,X) = Var[X]
RANDOM VECTORS
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Example:
Sometimes, we deal with vectors of randomvariables
Expected value:
Variance/covariancematrix:
STANDARDIZED RANDOM VARIABLES
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• Standardization is used for better comparison of
different variables
• Define Z to be the standardized variable ofX:
Z =
X − µX
σX
• The standardized variable Z measures how many standard
deviations X is below or above its mean
• No matter what are the expected value and variance of
X, it always holds that
E[Z] = 0 and Var[Z] = σZ
2 = 1
NORMAL (GAUSSIAN) DISTRIBUTION
Notation : X ∼ N(µ,σ2)
• E[X]= µ
• Var[X] = σ2
Dancingstatistics
https://www.youtube.com/watch?v=dr1DynUzjq0&index=2&
list=PLEzw67WWDg82xKriFiOoixGpNLXK2GNs9
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OTHER DISTRIBUTIONS
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EXERCISE 3
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• The heights of U.S. females between age 25 and 34 are
approximately normally distributed with a mean of 66
inches and a standard deviation of 2.5 inches.
• What fraction of U.S. female population in this age bracket is
taller than 70 inches, the height of average adult U.S. male of
this age?
EXERCISE 4
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• A woman wrote to Dear Abby, saying that she had been
pregnant for 310 days before giving birth.
• Completed pregnancies are normally distributed with a
mean of 266 days and a standard deviation of 16 days.
• Use statistical tables to determine the probability that a
completed pregnancy lasts
- at least 270 days
- at least 310 days
SUMMARY
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• Today, we revised some concepts from statistics that we
will use throughout our econometrics classes
• It was a very brief overview, serving only for
information what students are expected to know already
• The focus was on properties of statistical distributions and
on work with normal distribution tables
NEXT LECTURE
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• Wewill go through terminology of sampling and
estimation
• Wewill start with regression analysis and introduce the
Ordinary Least Squares estimator