LECTURE 1
Introduction to Econometrics
Dali Laxton
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September 23, 2022

WHAT IS ECONOMETRICS?
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To beginning 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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q Econometrics is a set of statistical tools and techniques for  quantitative measurement of actual
economic and business  phenomena
q 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
q 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
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e 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)
e Theoretical functional relationship:
Q = f (P, Ps, Y)
e Econometrics allows us to    specify:
Q = 31.50 − 0.73P + 0.11Ps + 0.23Y

INTRODUCTORY ECONOMETRICS COURSE
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e Lecturer: Dali Laxton (CERGE-EI, Prague)  245603@mail.muni.cz
e Lectures / Seminars: Friday, 9:00-11:50 room VT 105
e Office hours: Saturday by appointment 17:00-18:00

INTRODUCTORY ECONOMETRICS COURSE
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e Course requirements:
Ø2 quizzes and 2 home assignments (account for 40 points)
ØMidterm exam (account for 30 points)
ØFinal exam/project (account for 30 points)
Øto pass the course, student has to get at least 50 points in total
e 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

COURSE CONTENT
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e Lectures:
§Lecture 1: Introduction, repetition of statistical background,  non-technical introduction to
regression
§Lectures 2 - 4: Linear regression models
§Lectures 5 - 11: Violations of standard assumptions
e In-class exercises:
§Will serve to clarify and apply concepts presented on  lectures
§We will use statistical software to solve the exercises

LECTURE 1.
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e Introduction, repetition of statistical background
§probability theory
§statistical inference
e 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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e 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.
e 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
e Continuous random variable: can take on any value in an  interval
Example: time until the first goal is scored in a football  match between Liverpool and Manchester
United

DISCRETE RANDOM VARIABLES
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e Described by listing the possible values and the associated  probability that it takes on each
value
e 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
.
.
e Cumulative distribution function (CDF) :

Example, probability that heads shows up less or equal than 3 times after flipping it 5 times =>
need to sum all probabilities: 0 times, once, twice and three times

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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When you increase the number of observations (100->1000) empirical distribution approaches the
theoretical one

CONTINUOUS RANDOM VARIABLES
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e Probability density function fX(x) (PDF) describes the  relative likelihood for the random
variable X to take on a  particular value x
e Cumulative distribution function (CDF) :
e Computational rule:
P(X > x) = 1 − P(X ≤ x)

EXPECTED VALUE AND MEDIAN
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e Expected value (mean) :
Mean is the (long-run) average value of random variable
Discrete variable Continuous variable
∫
Example: calculating mean wind speed given wind speed distribution and power curve
e Median : ”the value in the middle”

EXERCISE 1
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e 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.
e However, when entering these data into a statistical  software package, the researcher mistakenly
enters
$4,000,000 for the person with $400,000 wealth.
e How much does this error affect the mean and median?

VARIANCE AND STANDARD DEVIATION
e 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.
e Standard deviation :
§Note: Outliers influence on variance/sd.
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Example of variance, darts

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 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?

EXERCISE 2
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e 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?
e Explain your reasoning, without doing any calculations,  then verify, doing the calculations.

COVARIANCE, CORRELATION, INDEPENDENCE
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e 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]
e 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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e 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.
e If X and Y are independent, then Cov(X, Y) = 0 (not the  other way round in general)
e 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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e Example:
e Sometimes, we deal with vectors of random variables

e Expected value:
e Variance/covariance matrix:

STANDARDIZED RANDOM VARIABLES
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e Standardization is used for better comparison of different  variables
e Define Z to be the standardized variable of X:
Z = X − µX
σX
e The standardized variable Z measures how many standard  deviations X is below or above its mean
e No matter what are the expected value and variance of X,  it always holds that
E[Z] = 0 and Var[Z] = σZ2 = 1

NORMAL (GAUSSIAN) DISTRIBUTION
e Notation : X ∼ N(µ, σ2)
e  E[X] = µ e Var[X] = σ2
e Dancing statistics
https://www.youtube.com/watch?v=dr1DynUzjq0&index=2& list=PLEzw67WWDg82xKriFiOoixGpNLXK2GNs9
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EXERCISE 3
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e 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.
e 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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e A woman wrote to Dear Abby, saying that she had been  pregnant for 310 days before giving birth.
e Completed pregnancies are normally distributed with a  mean of 266 days and a standard deviation
of 16 days.
e 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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e Today, we revised some concepts from statistics that we  will use throughout our econometrics
classes
e It was a very brief overview, serving only for information  what students are expected to know
already
e The focus was on properties of statistical distributions and  on work with normal distribution
tables

NEXT LECTURE
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e We will go through terminology of sampling and  estimation
e We will start with regression analysis and introduce the  Ordinary Least Squares estimator