EC3303 Econometrics I
Department of Economics, NUS
Spring, 2010
JongHoon Kim
Spring 2010 1EC3303
Spring 2010 EC3303 2
EC3303 Econometrics ILecturer: JongHoon KIM, AS2, 04‐40Class: Thu, 14.00 – 16.00, LT11
Tutorials: Mon – Fri, 10.00 – 12.00, AS4, 01‐17
Contact via emails please…
Starts on Week3 (Jan 25‐29)
Office Hours: Mon, 13.00 – 15.00
Assessment: Final Exam 60% + Continuous Assessment 40%Problem sets 20% + Midterm test 20%
2 ‐ 3 After the Midterm breakTextbook:Stock, J.H. and M.W. Watson (2006): Introduction to Econometrics, Second edition.
Boston: Pearson Addison Wesley. (HB 139 Sto 2006, CL, HSSML)
Supplementary reading:Wooldridge, J.M. (2005): Introductory Econometrics: A Modern Approach, Third edition. Gujarati, Damodar N. (2003): Basic Econometrics, Fourth edition. NY: McGraw-Hill.
Any statistics textbook…
Spring 2010 EC3303 3
Chapter I. Introduction
1. Overview - What is Econometrics? (S-W Ch1)
Reasoning and
conjecture
Global warming/Chinese economy in 10 yrs time?
Effectiveness of “caning” in SG penal system?
High time to buy a car or a HDB flat?
Global warming/Chinese economy in 10 yrs time?
Effectiveness of “caning” in SG penal system?
High time to buy a car or a HDB flat?
Observed and stylized facts
A medical case in SG 2008: An upsurge(150 or so over 5 months) of low bloodpressure shock cases 7 in coma, 4 death…?
A medical case in SG 2008: An upsurge(150 or so over 5 months) of low bloodpressure shock cases 7 in coma, 4 death…?
Theory (Model)
data + statistical tools/methods
Economics + Metric(Measure) = Econometrics
Definitive/quantitative questions with definitive/quantitative answers
Spring 2010 EC3303 4
Examples:
(a) Effect of reducing class size on elementary school education
test scorei class sizeistudent i
meaningful effect? How large?
pure(distinguishable) effect?
(b) Effect of cigarette taxes on reducing smoking
cigarette consumptioni cigarette sales priceiprice elasticity?
other factors?reverse “causality”?
How much can Apple price‐gouge SG customers on its 4G i‐phones?
i-phone salesi i-phone retail pricei
(c) Forecasting future inflation rates – SG’s inflation rate2010?
Benefits of the Casinos/Universal Theme Park at Sentosa/Marina Bay?
How many will survive EC3303 through to Final Exam?
Spring 2010 EC3303 5
(d) Explaining abrupt crime drop in 1990s in US (Levitt, Freakonomics…)
1. Innovative policing strategy2. Increased reliance on prisons3. Changes in crack and other drug markets4. Aging of the population5. Tough gun-control laws6. Strong economy7. Increased number of police8. All other explanations
(increased used of capital punishment, gun buybacks, and etc.)
Spring 2010 EC3303 6
(d) Explaining abrupt crime drop in 1990s in US (Levitt, Freakonomics…)
1. Innovative policing strategy
4. Aging of the population5. Tough gun-control laws6. Strong economy
8. All other explanations (increased used of capital punishment, gun buybacks, and etc.)
2. Increased reliance on prisons3. Changes in crack and other drug markets
7. Increased number of police
Legalization of abortion -1973, US Supreme Court Ruling on Roe v. Wade
(d’) Seeking determinants of crime rates (Levitt(1996))
crime ratet incarceration ratet
year t
other factors?
reverse “causality”?
Spring 2010 EC3303 7
(e) Understanding global warming (the effect of CO2 emission)
Vol NorPoleIcet CO2 emissiontreversed “causality”?
?
true scale of the effect?…“global warming hoax”??
(f) And many, many more interesting issues awaiting…
“H1N1 Flu pandemic hoax(scam)”,“Renewal of the contract hosting the F1 race in SG”,
Spring 2010 EC3303 8
Econometrics = Economics + Metrictheory data
Sources of Data:
(controlled) experiment
observation
Typical Economic Dataset:
cross‐sectional data
time series data
multiple entities at a given point in time
Individuals (person, firm,…), localities (city, states,…),…
a single entity over multiple peroids in time
panel data multiple entities over multiple peroids in time (longitudinal data)
“Devils are in the detail(s).”“Data is the least deceiving window toward truth.”(provided you know how to tease them without bungling)
“Why? …Why?... Why?”
Spring 2010 EC3303 9
2. Review of Probabilities (S-W Ch2) why do we care?
2.1 Probability Space
the (imaginary) collection of the whole “outcomes” in life
Probability Space (Sample Space), ΩΩ
a specific happening(realization) Outcome, ω
a collection of certain outcomes Event, E
a subset of Ω (basic unit to assign probability!)
Examples: i) the event E of tossing a coin to “head”
ii) the event E of finishing today’s lecture at 3.35pm sharp
iii) the event E of STI index “gaining” tomorrow
Events are the subsets resulting from introducing division(“partition”) of Ω.Here, for example, E and Ec.
ω.
E
Spring 2010 EC3303 10
More than two events occurring when…(i) a partition w/ multiple cuts (into mutually exclusive events)
“disjoint”
Ω
An example: Rolling a dice into{1,2}, {3,4}, {5,6}
Ω
An example: … ?
Probalibity: A relative measure of the likelihoods of events, satisfying
(a) 0 ≤ P(E) ≤ 1,(b) P(Ω) = 1 for any E Ω, and(c) P(E1 E2 ) = P(E1) + P (E2) + , for disjoint E1, E2,
P( i=1Ei) ∑i=1P(Ei)∞ ∞
E1 E2 E3 F1 F2 F3 Fn…
Spring 2010 EC3303 11
More than two events occurring when…(i) a partition w/ multiple cuts (into mutually exclusive events)
“disjoint”
Ω
An example: Rolling a dice into{1,2}, {3,4}, {5,6}
Ω
An example: … ?
Probalibity: A relative measure of the likelihoods of events, satisfying
(a) 0 ≤ P(E) ≤ 1,(b) P(Ω) = 1 for any E Ω, and(c) P(E1 E2 ) = P(E1) + P (E2) + , for disjoint E1, E2,
P( i=1Ei) ∑i=1P(Ei)∞ ∞
E F P(E) ≤ P(F),P(Ec) = 1 – P(E),P(E F) = P(E) +P(F) – P(E∩F),…
E1 E2 E3 F1 F2 F3 Fn…
Spring 2010 EC3303 12
(ii) multiple (overlapping) partitions (each w/ multiple cuts)
Ω
E1 E2
F1
F2
P(E1∩F1), P(E1∩F2), P(E2∩F1), P(E2∩F2) Joint probability
Spring 2010 EC3303 13
(ii) multiple (overlapping) partitions (each w/ multiple cuts)
Ω
E1 E2
F1
F2
P(E1∩F1), P(E1∩F2), P(E2∩F1), P(E2∩F2) Joint probability
P(E1) and P(E2) (or likewise, P(F1), P(F2) )Marginal probability
Spring 2010 EC3303 14
(ii) multiple (overlapping) partitions (each w/ multiple cuts)
Ω
E1 E2
F1
F2
P(E1∩F1), P(E1∩F2), P(E2∩F1), P(E2∩F2) Joint probability
P(E1) and P(E2) (or likewise, P(F1), P(F2) )Marginal probability
P(E1|F1) =
Conditional probabilityP(E1∩F1)P(F1)
(likewise, P(E2|F1), P(E1|F2), P(E2|F2))From these…
Statistical Independence P(E1) = P(E1|F1)betn E1 and F1
“how likely E1 to happen is oblivious of F1”
( P(E1∩F1) = P(E1)P(F1))
P(E2) = P(E2|F1), P(E1) = P(E1|F2), P(E2) = P(E2|F2)Statistical Independencebetn the two paritions
In general, with much finer partitions…?
Spring 2010 EC3303 15
How useful is it?
An example (from a German biostatistics text):A recently found contagious (and deadly) disease!You were tested positive (and diagnosed as so). Am I really infected? The prob.? The test’s known to detect 99 out of 100 true infected cases
98 out of 100 true uninfected cases
There is 1/1000 chance of getting infected.
Ω
E1 E2
F1
F2
tested positive tested negative
infected
not‐infeced
P(F1|E1) ?
= P(E1|F1)
= P(E2|F2)
= P(F1)
If in a population of 100,000…
99+1= 10099 1
99,900 = 1,998 + 97,902
1,998 97,902P(F1|E1) = =
P(E1∩F1)P(E1)
992,097
0.047
Spring 2010 EC3303 16
2.2 Random Variables and Probability Distributions
A “random variable”, Y “a numerical summary of a random outcome” (S-W)??a collection of (possibly infinitely many) numbers, which
takes on(“realizes to”) one of them when a certain event happens
A “partition” a random variable
ΩE Ec
Y
1if E happens
0if Ec happens
Y Bernoulli(p)where p = P(E)
A “partition” a random variable
Y
y1with F1
… ynwith Fn
…
Rolling a dice n = 6Daily SGD vs. USD n = ∞
ΩF1 F2F3 Fn…
Spring 2010 EC3303 17
The “probability distribution of Y,” PY
the list(ing) of all probabilities attached to all possible outcomes of Y( the wholesome of all probabilities of the events induced by Y)( the full knowledge of Pr{a ≤ Y ≤ b} for any a,b)
An example: Y Bernoulli(p) with p = P(E) Y 0 1p1 – p
P(Ec) P(E)finite(or countably many) values(“events”)Discrete random variable
(Discrete distn of a r.v.)
uncountably infinitely many values(“events”)Continuous random variable(Continuous distn of a r.v.)
Expressing/Describing a prob. distribution (of a r.v. Y) :
(i) tabulation feasible only in finite cases!(ii) p.m.f.(probability mass function) “pointwise probability (expression)”
p.d.f.(probability density function)(iii) c.d.f.(cumulative distribution function)“range‐wise probability (expression)”
Spring 2010 EC3303 18
Expressing/Describing a prob. distribution (of a r.v. Y) :
p.m.f.(probability mass function)“pointwise probability (expression)”
fY(x) = Pr{Y = x}For each possible value x of Y
0 1
1 – pp
only for discrete Y!
p.d.f.(probability density function)
“continuous version of pointwise probability”
μ
fY(x) (≠ Pr{Y = x})For each possible value x of Y
The height of pdf ≠ prob. why?
pmf of Bernoulli(p)
pdf of N(μ,1)
Spring 2010 EC3303 19
Expressing/Describing a prob. distribution (of a r.v. Y) :
p.m.f.(probability mass function)“pointwise probability (expression)”
fY(x) = Pr{Y = x}For each possible value x of Y
0 1
1 – pp
only for discrete Y!
p.d.f.(probability density function)
“continuous version of pointwise probability”
μ
fY(x) (≠ Pr{Y = x})For each possible value x of Y
The height of pdf ≠ prob. why?
a b
Rather, Pr{a ≤ Y ≤ b} = ∫a fY(x)dxb
pmf of Bernoulli(p)
pdf of N(μ,1)
Spring 2010 EC3303 20
c.d.f.(cumulative distribution function)
“range‐wise probability (expression)”
FY(x) = Pr{Y ≤ x}
For each possible value x of Y
= ∑y ≤ x fY(x) (= fY(x)+ fY(x – 1)+ )discrete Y case
| |0 1
1 – p
p
1
= ∫–∞ fY(x)dyx
continuous Y case1
less intuitive than pmf /pdf, butmore convenience b/c always well‐defined
Spring 2010 EC3303 21
Expectations/Moments (of a r.v. Y)
Often, we focus only on certain characteristics of the distn PY, e.g.,“the middle value of all Y outcomes”,“the most likely value of Y”,“how scattered the range of all Y propable values are”,…
the mean of Y (= the expected value of Y)A measure of the center(ing) (counting in “prob”) of the distn PY
EY = y1fY(y1) + y2fY(y2) + + ykfY(yk) discrete Y with k outcomesprob.’s as proper weights
= ∑y yfY(y) ( ∫–∞ yfY(y)dy continuous version)∞
0 1
1 – pp
μ
μY =
μY
Spring 2010 EC3303 22
Expectations/Moments (of a r.v. Y)
Often, we focus only on certain characteristics of the distn PY, e.g.,“the middle value of all Y outcomes”,“the most likely value of Y”,“how scattered the range of all Y propable values are”,…
the mean of Y (= the expected value of Y)A measure of the center(ing) (counting in “prob”) of the distn PY
EY = y1fY(y1) + y2fY(y2) + + ykfY(yk) discrete Y with k outcomesprob.’s as proper weights
= ∑y yfY(y) ( ∫–∞ yfY(y)dy continuous version)∞
μY =
μY
the variance of Y (= the expected value of the“squared-deviations of Y from μY”)A measure of the dispersion (counting in “prob”) of the distn PY
Var(Y) = (y1 – μY)2fY(y1) + + (yk – μY)2fY(yk) discrete Y with k outcomes
= ∑y (y – μY)2fY(y) ( ∫–∞ (y – μY)2fY(y)dy continuous version)∞
σ²Y =
σ²Y
= E(Y– μY)2
Spring 2010 EC3303 23
Expectations/Moments (of a r.v. Y)
the mean of Y (= the expected value of Y)A measure of the center(ing) (counting in “prob”) of the distn PY
EY = y1fY(y1) + y2fY(y2) + + ykfY(yk) discrete Y with k outcomesprob.’s as proper weights
= ∑y yfY(y) ( ∫–∞ yfY(y)dy continuous version)∞
μY =
μY
the variance of Y (= the expected value of the“squared-deviations of Y from μY”)A measure of the dispersion (counting in “prob”) of the distn PY
Var(Y) = (y1 – μY)2fY(y1) + + (yk – μY)2fY(yk) discrete Y with k outcomes
= ∑y (y – μY)2fY(y) ( ∫–∞ (y – μY)2fY(y)dy continuous version)∞
σ²Y =
σ²Y
μ
√Var(Y)σY =
the standard deviation of Y
= E(Y– μY)2
Spring 2010 EC3303 24
Recall! (Important properties of μY and σ²Y ): Given a r.v. Y with μY and σ²YIf X = aY + b for any non-random numbers a, b,(a) EX = E(aY + b) = aEY + b = aμY + b(b) Var(X) = Var(aY + b) = a² Var(Y) = a²σ²YOther useful (higher) moments (of a r.v. Y)
the skewness of Y A measure of the asymmetry(inclination) of the distn PY
E(Y– μY)3
σ3Y
the kurtosis of Y A measure of the tail thinckness of the distn PY
E(Y– μY)4
σ4Y
Spring 2010 EC3303 25
Recall! (Important properties of μY and σ²Y ): Given a r.v. Y with μY and σ²YIf X = aY + b for any non-random numbers a, b,(a) EX = E(aY + b) = aEY + b = aμY + b(b) Var(X) = Var(aY + b) = a² Var(Y) = a²σ²YOther useful (higher) moments (of a r.v. Y)
the skewness of Y A measure of the asymmetry(inclination) of the distn PY
E(Y– μY)3
σ3Y
the kurtosis of Y A measure of the tail thinckness of the distn PY
E(Y– μY)4
σ4Y
Spring 2010 EC3303 26
2.3 Multiple RandomVariables (“More than one r.v.?”)
Remember! A “random variable”, Y
A “partition” a random variable
ΩE Ec
Y
1if E happens
0if Ec happens
Y Bernoulli(p)where p = P(E)
A “partition” a random variable
Y
y1with F1
… ynwith Fn
…
Rolling a dice n = 6Daily SGD vs. USD n = ∞
ΩF1 F2F3 Fn…