LECTURE 1
Introduction to Econometrics
Gega Todua
September 22, 2017
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WHAT IS ECONOMETRICS?
To beginning students, it may seem as if econometrics is an overly
complex obstacle to an otherwise useful education. (. . .) To
professionals in the field, econometric 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)
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WHAT IS ECONOMETRICS?
Econometrics is a set of statistical tools and techniques for
quantitative measurement of actual economic and business
phenomena
It attempts to
quantify economic reality
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
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EXAMPLE
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 functional relationship:
Q = f(P, Ps, Y)
Econometrics allows us to specify:
Q = 31.50 − 0.73P + 0.11Ps + 0.23Y
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INTRODUCTORY ECONOMETRICS COURSE
Lecturer: Gega Todua (CERGE-EI, Prague)
gega.todua@cerge-ei.cz
Lectures: Friday, 9,20-10,05, room VT 203
Friday, 10,15-11,50, room VT 203
Office hours: Friday, after Seminar by appointment
Web: https://is.muni.cz/auth/course/econ/
podzim2017/BPE_AIEC?lang=en
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INTRODUCTORY ECONOMETRICS COURSE
Course requirements:
NO EXAMS! :)
3 home assignments (account for 3 × 20 = 60 points)
written Empirical Project (accounts for 40 points).
Details will be announced during following weeks
to pass the course, student has to achieve at least 20 points
in the project and 50 points in total
Recommended literature:
Studenmund, A. H., Using Econometrics: A Practical Guide
Adkins, L., Using gretl for Principles of Econometrics
Wooldridge, J. M., Introductory Econometrics: A Modern
Approach
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IMPORTANT DATES
24.11.2017: Last Lecture
15.12.2017 00:00 The deadline for the Empirical Project
17.11.2017: Public Holiday
29.09.2017: No Lectures (away for the conference)
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COURSE CONTENT
Lectures:
Lecture 1: Introduction, repetition of statistical background,
non-technical introduction to regression
Lectures 2 - 4: Linear regression models
Lectures 5 - 12: Violations of standard assumptions
In-class exercises:
Will serve to clarify and apply concepts presented on
lectures
We will use statistical software (Gretl) to solve the exercises
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LECTURE 1.
Introduction, repetition of statistical background
probability theory
statistical inference
Readings:
Studenmund, A. H., Using Econometrics: A Practical
Guide, Chapter 17
Wooldridge, J. M., Introductory Econometrics: A Modern
Approach, Appendix B and C
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RANDOM VARIABLES
A 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 shot in a football match
between FC Barcelona and Real Madrid
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DISCRETE RANDOM VARIABLES
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) :
FX(x) = P(X ≤ x) =
i=1,xi≤x
P(X = xi)
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SIX-SIDED DIE: PROBABILITY DENSITY 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
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) :
FX(x) = P(X ≤ x) =
x
−∞
fX(t)dt
Computational rule:
P(X ≥ x) = 1 − P(X ≤ x)
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EXPECTED VALUE AND MEDIAN
Expected value (mean) :
Mean is the (long-run) average value of random variable
Discrete variable
E [X] =
i=1
xiP(X = xi)
Continuous variable
E [X] =
+∞
−∞
x fX(x)dx
Example: calculating mean of six-sided die
Median : ”the value in the middle”
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EXERCISE 1
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 and median?
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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.
Var[X] = E (X − E [X])2
= E[X2] − (E[X])2
Standard deviation : σX = Var[X]
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DANCING STATISTICS
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 these questions:
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?
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EXERCISE 2
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.
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COVARIANCE, CORRELATION, INDEPENDENCE
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
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INDEPENDENCE OF VARIABLES
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
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COMPUTATIONAL RULES
E (aX + b) = aE(X) + b
Var(aX + b) = a2
Var(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]
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RANDOM VECTORS
Sometimes, we deal with vectors of random variables
Example: X =


X1
X2
X3


Expected value: E [X] =


E[X1]
E[X2]
E[X3]


Variance/covariance matrix:
Var [X] =


Var[X1] Cov(X1, X2) Cov(X1, X3)
Cov(X2, X1) Var[X2] Cov(X2, X3)
Cov(X3, X1) Cov(X3, X2) Var[X3]


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STANDARDIZED RANDOM VARIABLES
Standardization is used for better comparison of different
variables
Define Z to be the standardized variable of X:
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 = 1
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NORMAL (GAUSSIAN) DISTRIBUTION
Notation : X ∼ N(µ, σ2) E[X] = µ Var[X] = σ2
Dancing statistics
https://www.youtube.com/watch?v=dr1DynUzjq0&index=2&
list=PLEzw67WWDg82xKriFiOoixGpNLXK2GNs9
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EXERCISE 3
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?
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EXERCISE 4
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
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SUMMARY
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
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NEXT LECTURE
We will go through terminology of sampling and
estimation
We will start with regression analysis and introduce the
Ordinary Least Squares estimator
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