Introduction to econometrics
I. Introduction to econometrics and working with data
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 1 / 69
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 2 / 69
Course organization
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 3 / 69
Course organization
Contact informations
E-mail: nemecd@econ.muni.cz
Office hours: Wed. 13:00–15:00 and by appointment.
Department of Economics, room 612.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 4 / 69
Course organization
Course books and software
Main book: Gary Koop (2008), „Introduction to Econometrics“.
Other econometric books available (cover similar topics) – Hill et al.
(nice exercises), Wooldridge, Gujarati, Dougherty etc.
Main software: GRETL (other possibilitie available – e.g. Matlab).
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 5 / 69
Course organization
Software review
Platform – Name Developer
Open source
gretl „By econometricians, for econometricians“
JMulTi Benkwitz, Krätzig
Octave University of Wisconsin
R/Rmetrics Free Software Foundation, Inc.
Commercial
EViews QMS Software, Inc.
GAUSS Aptech Systems, Inc.
LIMDEP Econometric Software, Inc.
Matlab MathWorks, Inc.
RATS Estima
SAS SAS Institute
SHAZAM Northwest Econometrics, Ltd.
SPSS SPSS, Inc.
Stata StataCorp
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 6 / 69
Course organization
Literature review
Level – Authors Title
For beginners
Koop (2009) Analysis of Economic Data
Introductory
Koop (2008) Introduction to Econometrics
Hill, Griffiths, Lim (2008) Principless of Econometrics
Stock, Watson (2007) Introduction to Econometrics
Stock, Watson (2008) Introduction to Econometrics
Wooldridge (2009) Introductory Econometrics (A Modern Approach)
Intermediate
Brooks (2008) Introductory Econometrics for Finance
Dougherty (2007) Introduction to Econometrics
Enders (2005) Applied Econometric Time Series
Gujarati, Porter (2009) Basic Econometrics
Kennedy (2008) A Guide to Econometrics
Verbeek (2008) A Guide to Modern Econometrics
Advanced
Baltagi (2008) Econometric Analysis of Panel Data
Hayashi (2001) Econometrics
Greene (2008) Econometric Analysis
Davidson, Russel, MacKinnon (2004) Econometric Theory and Methods
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 7 / 69
Course organization
Course grading
50 % – homeworks (mostly computer exercises).
50 % – final (written) exam (including econometric theory and
interpreting estimation results).
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 8 / 69
Course organization
Final grading
Grade Points
A 86–100
B 80–85
C 73–79
D 67–72
E 60–66
F 0–59
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 9 / 69
Introduction to econometrics
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 10 / 69
Introduction to econometrics
What is econometrics?
„Econometrics is what the econometricians do.“
„Econometrics is the study of the application of statistical methods to
the analysis of economic phenomena.“
„Econometrics is based upon the development of statistical methods
for estimating economic relationships, testing economic theories, and
evaluating and implementing government and business policy.“
Wooldridge (2009).
The 1930s: foundation of Econometric Society (journal
Econometrica).
Ragnar Frisch (1933) explains in the first issue of Econometrica:
„. . . it is the unification of statistics, economic theory and
mathematics that constitutes econometrics.“
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 11 / 69
Introduction to econometrics
„Nobel Prize“ in economics
1969 – first „Nobel Prize“ in economics.
„for having developed and applied dynamic models for the analysis of
economic processes.“
Ragnar Frisch (1895–1973) Jan Tinbergen (1903–1994)
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 12 / 69
Introduction to econometrics
Who are econometricians?
Economists – utilizing economic theory to improve their empirical
analyses of the problems they address.
Mathematicians – formulating economic theory in ways that make it
appropriate for statistical testing.
Accountants – finding and collecting economic data and relating
theoretical economic variables to observable ones.
Applied statisticians – trying to estimate economic relationships or
predict economic events.
Theoretical statisticians – applying their skills to the development of
statistical techniques appropriate to the empirical problems
characterizing the science of economics.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 13 / 69
Introduction to econometrics
Categories of econometrics
Econometrics
Theoretical
Classical Bayesian
Applied
Classical Bayesian
Classical – mainstream.
Bayesian – rising importance and attractiveness in economic
applications.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 14 / 69
Introduction to econometrics
Why to use econometrics?
Econometrics shows us how to use data in a sensible ans systematic
manner to shed light on economic questions.
Testing whether financial markets are weak-form informationally
efficient.
Measuring and forecasting the volatility of bond returns.
Explaining the determinants of bond credit ratings used by the ratings
agencies.
Modelling long-term relationships between prices and exchange rates.
Forecasting the correlation between the stock indices of two countries.
Examining the effects of job training on worker productivity.
Estimating the effect of the minimum wage on unemployment.
Estimating the effect of law enforcement on city crime levels. Does the
presence of more police officers on the street deter crime? How much?
Investigating why some people choose to travel to work by car and
others choose to travel by public transport.
Estimating the effect of advertising on sales.
Estimating factors which determine consumer behaviour.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 15 / 69
Introduction to econometrics Economic model
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 16 / 69
Introduction to econometrics Economic model
Economic models – examples
Empirical analysis – using data to test theories or to estimate
relationships among variables of interest.
Economic model → to test economic theories.
Examples:
Economic model of crime.
Job training and worker productivity.
Formal economic model = a background for econometric analysis.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 17 / 69
Introduction to econometrics Economic model
Economic model of crime – introduction
Gary Becker – Nobel prize winner, model desribing individual’s
participation in crime (1968).
Gary S. Becker (*1930)
Utility maximization framework – costs and rewards of criminal
activities.
Model describing the amount of time spent in criminal activity as a
function of various factors.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 18 / 69
Introduction to econometrics Economic model
Economic model of crime
y = f (x1, x2, x3, x4, x5, x6, x7)
y = hours spent in criminal activities,
x1 = „wage“ for an hour spent in criminal activity,
x2 = hourly wage in legal employment,
x3 = income other than from crime or employment,
x4 = probability of getting caught,
x5 = probability of being convicted if caught,
x6 = expected sentence if convicted,
x7 = age.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 19 / 69
Introduction to econometrics Economic model
Economic model of crime – specification
Representative list of the main factors affecting a person’s decisionto
participate in crime.
Function f (·) not specified (depends on an underlying utility function,
rarely known).
Using economic theory to predict the effect that each variable would
have on = the basis for an econometric analysis of individual criminal
activity.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 20 / 69
Introduction to econometrics Economic model
Job training and worker productivity – introduction
Effectiveness of a publicly funded job training program (teaching
computer use in the manufacturing process, the twenty-week
programme, any worker may participate manufacturing worker may
participate, enrollment in all or part of the program is voluntary).
To examine the effects of job training on worker productivity
(represented by hourly wage).
Little need for formal economic theory.
Basic economic understanding sufficient → factors such as education,
experience, and training affect worker productivity.
Reasonable belief – workers are paid commensurate with their
productivity.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 21 / 69
Introduction to econometrics Economic model
Job training and worker productivity – model
wage = f (educ, exper, training)
wage = hourly wage,
educ = years of formal education,
exper = years of workforce experience,
training = weeks spent in job training.
Other factors might generally affect the wage rate × model captures
the essence of the problem.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 22 / 69
Introduction to econometrics Econometric model
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 23 / 69
Introduction to econometrics Econometric model
Economic and econometric model
Specification of economic model ⇒ econometric model.
Specification of relationships among variables, f (·).
Solving the problem with „unobservable“ variables, e.g.:
„wage“ that an individual can earn from criminal activity – well defined
variable × difficult to observe;
probability of being arrested – for a given individual canot be obtained
× observable arrest statistics → „proxy“ variable.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 24 / 69
Introduction to econometrics Econometric model
Econometric model of crime
crime = α + β1wagem + β2othinc + β3freqarr + β4freqconv
+ β5avgsen + β6age +
crime = some measure of the frequency of criminal activity,,
wagem = wage that can be earned in legal employment,
othinc = income from other sources (assets etc.),
freqarr = frequency of arrests for prior infractions
(aprroximated probability of arrest),
freqconv = frequency of conviction,
avgsen = average sentence length after conviction,
age = age.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 25 / 69
Introduction to econometrics Econometric model
Econometric model of crime – explanation
Choice of variables determined by the economic theory as well as data
considerations.
error term: = unobserved factors („wage“ for criminal activity,
moral character, family background) and errors in measuring things
(criminal activity, probability of arrest).
parameters: α, β1, . . . , β6 = directions and strengths of the
relationship between explained variable, crime, and explaining
variables, factors used to determine crime in model.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 26 / 69
Introduction to econometrics Econometric model
Econometric model of job training and worker productivity
wage = α + β1educ + β2exper + β3training +
wage = hourly wage,
educ = years of formal education,
exper = years of workforce experience,
training = weeks spent in job training.
= factors such as quality of education, family background and other
factors influencing person’s wage.
β3 = parameter measuring effects of job training on wage.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 27 / 69
Introduction to econometrics Econometric model
Other issues
Econometric analysis begins by specifying an econometric models
without consideration of the details of the model’s creation (careful
derivation might be very often difficult.
Possible use of economic reasoning and common sense as guides for
choosing the variables.
This approach loses some of the richness of economic analysis ×
commonly and effectively applied by careful researchers.
Stating hypothesis in terms of the unknown parameters (e.g. wage
that can be earned in legal employment, wagem, has no effect on
criminal behaviour ⇔ β1 = 0).
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 28 / 69
Introduction to econometrics Econometric model
Econometric modelling
1a. Economic or financial theory (previous studies)
1b. Formulation of an estimable theoretical model
2. Collection of data
3. Model estimation
4. Is the model statistically adequate?
No Yes
Reformulate model 5. Interpret model
6. Use for analysis
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 29 / 69
Introduction to econometrics Econometric model
Steps in econometric modelling
1 General statement of the problem – formulation of a theoretical
model, or intuition from economic theory that two or more variables
should be related to one another in a certain way.
2 Collection of data – external or internal (our own surveys) sources.
3 Choice of estimation method – single equation or multiple equation
technique?.
4 Statistical evaluation of the model – What assumptions were required
to estimate the parameters of the model optimally? Were these
assumptions satisfied by the data or the model? Does the model
adequately describe the data?
5 Evaluation of the model from a theoretical perspective – Are the
parameter estimates of the sizes and signs that the theory or
suggested?.
6 Use of model – testing the theory, formulating forecasts or suggested
courses of action (e.g. „if inflation and GDP rise, buy stocks in sector
X“), as an input to government policy.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 30 / 69
Working with data
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 31 / 69
Working with data Types of economic data
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 32 / 69
Working with data Types of economic data
Times series
Data collected at specific points in time.
Observations on variable at time (for an individual – country, firm,
household, etc.).
Yt t = 1, . . . , T
Observations are not independent (working with times series often
requires special tools).
Many frequencies.
Seasonal adjustment – if necessary.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 33 / 69
Working with data Types of economic data
Time series – example
Tabulka: Minimal wage, unemployment and other data for Puerto Rico
Obsno Year Avgmin Avgcov Unemp GNP
1 1950 0.20 20.1 15.4 878.7
2 1951 0.21 20.7 16.0 925.0
3 1952 0.23 22.6 14.8 1015.9
...
...
...
...
...
...
37 1986 3.35 58.1 18.9 4281.6
38 1987 3.35 58.2 16.8 4496.7
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 34 / 69
Working with data Types of economic data
Time series – example (comments)
Data from Wooldridge (2009), available in gretl (prminwge.gdt).
Originally: Castillo-Freeman and Freeman (1992) – effects of minimal
wage in the Puerto Rico.
Obsno = observation;
Year = rok;
Avgmin = average (hourly) minimal wage for the year;
Avgcov = average coverage rate;
(% of workers covered by the minimum wage)
Unemp = unemployment rate;
GNP = gross national product.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 35 / 69
Working with data Types of economic data
Cross-sectional data
Sample of individuals, households, firms, cities or a variety of other
units taken at a given point in time.
Yi i = 1, . . . , N
Assumptions – obtained by random sampling from underlying
population (may be a problem).
Pooled cross sections – different, randomly sampled individuals at
different time periods (effective to analyse structural breaks in time.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 36 / 69
Working with data Types of economic data
Cross-sectional data – example 1
Tabulka: Cross-sectional data on wage and other individual characteristics.
Obsno Wage Educ Exper Female Married
1 3.10 11 2 1 0
2 3.24 12 22 1 1
3 3.00 11 2 0 0
4 6.00 8 44 0 1
5 5.30 12 7 0 1
...
...
...
...
...
...
525 11.56 16 5 0 1
526 3.50 14 5 1 0
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 37 / 69
Working with data Types of economic data
Cross-sectional data – example 1 (comments)
Data from Wooldridge (2009), available in gretl (wage1.gdt).
Abbreviated form on 526 working individuals for the year 1976.
Obsno = observation;
Wage = hourly wage;
Educ = years of education;
Exper = years of potential labor force experience;
Female = an indicator for gender;
Married = martial status.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 38 / 69
Working with data Types of economic data
Cross-sectional data – example 2
Tabulka: Data set on economic growth and country characteristics
Obsno. Country Gpcrgdp Govcons60 Second60
1 Argentina 0.89 9 32
2 Austria 3.32 16 50
3 Belgium 2.56 13 69
4 Bolivia 1.24 18 12
...
...
...
...
...
61 Zimbabwe 2.30 17 6
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 39 / 69
Working with data Types of economic data
Cross-sectional data – example 2 (comments)
Data from Wooldridge (2009).
Originally: De Long a Summers (1991) – the study of cross-country
growth rates.
Obsno = observation;
Country = country;
Gpcrgdp = growth in real per capita GDP from 1960 to 1985;
Govcons60 = government consumption as a percentage of GDP;
Second60 = adult secondary education rates.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 40 / 69
Working with data Types of economic data
Pooled cross-sectional data – example
Tabulka: Two Years of Housing Prices
Obsno. Year Hprice Proptax Sqrft Bdrms
1 1993 85500 42 1600 3
2 1993 67300 36 1440 3
3 1993 134000 38 2000 4
...
...
...
...
...
...
250 1993 243600 41 2600 4
251 1995 65000 16 1250 2
252 1995 182400 20 2200 4
...
...
...
...
...
...
520 1995 57200 16 1100 2
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 41 / 69
Working with data Types of economic data
Pooled cross-sectional data – example (comments)
Data from Wooldridge (2009).
Data on houses sold in 1993 (250 observations) and 1995 (270
observations).
Obsno = observation;
Year = years;
Hprice = house price;
Proptax = property tax;
Sqrft = lot size in square feet;
Bdrms = number of bedrooms.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 42 / 69
Working with data Types of economic data
Panel data
Time series and corss-sectional component.
Data on the same individuals (countries, cities, firms, etc.) over a
given time period.
Yit i = 1, . . . , N t = 1, . . . T
Some difficulties to collect them.
To estimate „individual effects“.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 43 / 69
Working with data Types of economic data
Panel data – example
Tabulka: A Two-Year Panel Data Set on City Crime Statistics
Obsno. Město Rok Murders Population Unem Police
1 1 1986 5 350000 8.7 440
2 1 1990 8 359200 7.2 471
3 2 1986 2 64300 5.4 75
4 2 1990 1 65100 5.5 75
...
...
...
...
...
...
...
297 149 1986 10 260700 9.6 286
298 149 1990 6 245000 9.8 334
299 150 1986 25 543000 4.3 520
300 150 1990 32 546200 5.2 493
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 44 / 69
Working with data Types of economic data
Panel data – example (comments)
Data from Wooldridge (2009).
Two-year panel data set on crime and related statistics for 150 cities
in the United States (1986 and 1990).
Obsno = observation;
City = city;
Year = year;
Murders = number of murders;
Population = population number;
Unem = unemployment rate;
Police = number of policemen.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 45 / 69
Working with data Types of economic data
Data transformation
Depends on purposes of economic analysis.
Difference: ∆Yt = Yt − Yt−1.
Growth rate (% change):
%∆Yt =
Yt
Yt−1
− 1 =
Yt − Yt−1
Yt−1
(×100[%])
Diference of logarithms (to approximate growth rate):
%∆Yt ≈ ln (Yt) − ln (Yt−1) (×100[%])
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 46 / 69
Working with data Working with data – graphical methods
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 47 / 69
Working with data Working with data – graphical methods
Time series graphs
100
150
200
250
300
350
400
450
1940 1950 1960 1970 1980 1990 2000
Směnnýkurz(£/$)
Rok
Obrázek: Time series plot of UK pound/US dollar exchange rate.Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 48 / 69
Working with data Working with data – graphical methods
Histograms
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5000 10000 15000
Relativníčetnost
HDP na obyvatele (v USD)
celkový počet zemí = 90
Obrázek: Histogram of GDP per capita for 90 countries.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 49 / 69
Working with data Working with data – graphical methods
Absolute and relative frequencies
Tabulka: Frequency table for GDP per capita data.
Interval (USD)
Frequency
Absolute Relative
0-2000 33 36.67 %
2001-4000 22 24.44 %
4001-6000 7 7.78 %
6001-8000 3 3.33 %
8001-10000 4 4.44 %
10001-12000 2 2.22 %
12001-14000 9 10.00 %
14001-16000 6 6.67 %
16001-18000 4 4.44 %
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 50 / 69
Working with data Working with data – graphical methods
Scatter plots (XY plots)
0
1
2
3
4
5
6
0 500 1000 1500 2000 2500 3000
Průměrnýročníúbyteklesníplochy(%)
Počet obyvatel na tisíc hektarů rozlohy
Nikaragua
Obrázek: XY plot of population density against deforestation.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 51 / 69
Working with data Descriptive statistics and correlation
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 52 / 69
Working with data Descriptive statistics and correlation
Sample mean and variance
Random sample: Y1, . . . , YN.
Sample mean:
Y =
N
i=1 Yi
N
Sample standard deviation:
sY =
N
i=1 Yi − Y
2
N − 1
Sample variance: s2
Y .
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 53 / 69
Working with data Descriptive statistics and correlation
Histogram and bell-shaped distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 4
Hustotapravděpodobnosti(relativníčetnost)
X
X
N(0.0030101,0.99163)
Test statistic for normality:
Chi-squared(2) = 2.482 pvalue = 0.28904
Obrázek: Histogram for a bell-shaped distribution.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 54 / 69
Working with data Descriptive statistics and correlation
Expected values and variances
Random variable Y .
Expected value and population mean: E(Y ) ≡ µ
E(Y ) =
N
i=1
yi p(yi ) discrete random variable, sample space{y1, . . . yN}
E(Y ) =
∞
−∞
yf (y)dy continuous random variable
Population variance: var(Y ) = σ2
var(Y ) = E[(Y − µ)2
] = E(Y 2
) − µ2
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 55 / 69
Working with data Descriptive statistics and correlation
Example – expected return on the stock market
Probability 70% (0.7) – stable markets (return 1%); probability 10%
– falling markets (return −10%); probability 20% – rising markets
(return 5%).
p(yi ) = Pr(Y = yi ).
E(Y ) = p (0.05) 0.05 + p (0.01) 0.01 + p (−0.10) (−0.10)
= 0.20 × 0.05 + 0.70 × 0.01 + 0.10 × (−0.10)
= 0.007.
Expected return on the stock next month is 0.7% (i.e. a bit less than
1%).
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 56 / 69
Working with data Descriptive statistics and correlation
Example – expected ucertainty of stock returns
Risk of expected returns → Y 2: (0.05)2 = 0.0025, (0.01)2 = 0.0001 a
(−0.10)2 = 0.01.
E(Y 2
) = p (0.0025) × 0.0025 + p (0.0001) × 0.0001
+ p (0.01) × 0.01
= 0.20 × 0.0025 + 0.70 × 0.0001 + 0.10 × 0.01
= 0.00157.
var(Y ) = E Y 2
− [E(Y )]2
= 0.00157 − (0.007)2
= 0.001521.
Square root of variance → 0.039 (std. dev.) Rightarrow expected
return 0.7% with uncertainty ±3.9%.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 57 / 69
Working with data Descriptive statistics and correlation
Correlation
Relationship between X a Y .
Correlation (correlation coefficient):
r =
N
i=1 Yi − Y Xi − X
N
i=1 Yi − Y
2 N
i=1 Xi − X
2
.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 58 / 69
Working with data Descriptive statistics and correlation
Properties of correlation
1 r ∈ −1, 1 .
2 r > 0 . . . positive correlation; r < 0 . . . negative correlation; r = 0
. . . no correlation; r = 1 or r = −1 . . . perfect correlation.
3 rXY = rYX .
4 rXX = 1.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 59 / 69
Working with data Descriptive statistics and correlation
Correlation – possible interpretations
De Vaus (2002) – Analyzing Social Science Data: 50 Key Problems in
Data Analysis.
Take with reserve!
Correlation coefficient (abs. value) Interpretation
0.01 – 0.09 trivial, no correlation
0.10 – 0.29 low
0.30 – 0.49 mild
0.50 – 0.69 substantial, strong
0.70 – 0.89 very strong
0.90 – 1.00 almost perfect
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 60 / 69
Working with data Descriptive statistics and correlation
Example – houses prices
Data set hprice.gdt, Koop (2008)
Data relating to N = 546 houses sold in Windsoru, Kanadě, in the
summer of 1987.
Y = the sales price of the house,
X = the size of its lot in square feet ⇒ rXY = 0.54
1 Houses with large lots tend to be worth more than thos with small lots.
2 There is a positive relationship between lot size and sales price.
3 The variation in lot size accounts for 29 % (i.e. 0.542
= 0.29) of the
variability in house prices.
Z = number of bedrooms ⇒ rYZ = 0.37: houses with more bedrooms
tend to be worth more than houses with fewer bedrooms.
rXZ = 0.15 ⇒ houses with larger lots tend to have more bedrooms
(small correlation ⇒ quite weak link between lot size and number of
bedrooms).
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 61 / 69
Working with data Descriptive statistics and correlation
Correlation and causality
Correlation does neccessarily imply causality.
It may be the case that an underlyind third variable is responsible for
correlation.
Direct (no interventing variable) and undirect correlation.
Use common sense or a convincingeconomic theory to establish
causality.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 62 / 69
Working with data Descriptive statistics and correlation
Correlation and XY plot – example 1
20
40
60
80
100
120
140
160
180
200
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
Cena(tis.kanadskýchdolarů)
Rozloha (čtvereční stopy)
Obrázek: XY plot of lot size against house price.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 63 / 69
Working with data Descriptive statistics and correlation
Correlation and XY plot – example 2
-2
-1
0
1
2
-4 -2 0 2 4
Y
X
Obrázek: XY plot of two perfectly correlated variables.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 64 / 69
Working with data Descriptive statistics and correlation
Correlation and XY plot – example 3
-2
-1
0
1
2
-3 -2 -1 0 1 2 3
Y
X
Obrázek: XY plot of uncorrelated variables.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 65 / 69
Working with data Descriptive statistics and correlation
Correlation matrix
Correlation between several variables.
X Y Z
X 1.000
Y 0.318 1.000
Z -0.131 0.097 1.000
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 66 / 69
Working with data Descriptive statistics and correlation
Population correlations and covariances
Example: portfolio risk; investment over the summer months in the
shares of two companies – an umbrella manufacturer and an ice
cream maker.
Overall portfolio might be much less risky than the individual stocks.
Population covariance:
cov(X, Y ) = E(XY ) − E(X)E(Y )
Population correlation:
corr(X, Y ) =
cov(X, Y )
var(X)var(Y )
Sample statistics → (consistent) estimates of population statistics.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 67 / 69
Exercises
Content
1 Course organization
2 Introduction to econometrics
Economic model
Econometric model
3 Working with data
Types of economic data
Working with data – graphical methods
Descriptive statistics and correlation
4 Exercises
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 68 / 69
Exercises
Exercises
Koop (2008): chapter 1.
Hill et al. (2008): B.9, B.12, B.13, C.3, C.5, C.6.
Introduction to econometrics (INEC) I. Introduction and working with data Autumn 2011 69 / 69