Transcript Random Variable Discrete rv

Econometrics
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Lecture 1
• Syllabus
• Introduction of Econometrics:
Why we study econometrics?
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Introduction
• What is Econometrics?
• Econometrics consists of the application of mathematical
statistics to economic data to lend empirical support to
the models constructed by mathematical economics and
to obtain numerical results.
• Econometrics may be defined as the quantitative analysis
of actual economic phenomena based on the concurrent
development of theory and observation, related by
appropriate methods of inference.
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What is Econometrics?
Econometrics
Economics
Statistics
Mathematics
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Why do we study econometrics?
• Rare in economics (and many other areas without labs!) to have
experimental data
• Need to use nonexperimental, or observational data to make
inferences
• Important to be able to apply economic theory to real world data
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Why it is so important?
• An empirical analysis uses data to test a theory or to estimate a
relationship
• A formal economic model can be tested
• Theory may be ambiguous as to the effect of some policy change –
can use econometrics to evaluate the program
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The Question of Causality
• Simply establishing a relationship between variables is rarely sufficient
• Want to get the effect to be considered causal
• If we’ve truly controlled for enough other variables, then the
estimated effect can often be considered to be causal
• Can be difficult to establish causality
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Purpose of Econometrics
• Structural Analysis
• Policy Evaluation
• Economical Prediction
• Empirical Analysis
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Methodology of Econometrics
• 1. Statement of theory or hypothesis.
• 2. Specification of the mathematical model of the theory.
• 3. Specification of the statistical, or econometric model.
• 4. Obtaining the data.
• 5. Estimation of the parameters of the econometric model.
• 6. Hypothesis testing.
• 7. Forecasting or prediction.
• 8. Using the model for control or policy purposes.
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Example：Kynesian theory of consumption
•1. Statement of theory or hypothesis.
Keynes stated: The fundamental psychological law is
that men/women are disposed, as a rule and on
average, to increase their consumption as their
income increases, but not as much as the increase
in their income.

In short, Keynes postulated that the marginal
propensity to consume (MPC)边际消费倾向, the
rate of change of consumption for a unit change in
income, is greater than zero but less than 1
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2.Specification of the
mathematical model of the theory
• A mathematical economist might suggest the
following form of the Keynesian consumption function:
Y   0  1 X
0  1  1
Consumption
expenditure
Income
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3. Specification of the statistical,
or econometric model.
• To allow for the inexact relationships between economic
variables, the econometrician would modify the
deterministic consumption function as follows:
Y   0  1 X  u
U, known as disturbance, or error term
• This is called an econometric model.
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4. Obtaining the data.
year
Y
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
X
3081.5
3240.6
3407.6
3566.5
3708.7
3822.3
3972.7
4064.6
4132.2
4105.8
4219.8
4343.6
4486
4595.3
4714.1
4620.3
4803.7
5140.1
5323.5
5487.7
5649.5
5865.2
6062
6136.3
6079.4
6244.4
6389.6
6610.7
6742.1
6928.4
Sourse: Data on Y (Personal Consumption Expenditure) and X (Gross
Domestic Product),1982-1996) all in 1992 billions of dollars
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5. Estimation of the parameters
of the econometric model.
• reg y x
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Source |
SS
df
MS
Number of obs = 15
-------------+-----------------------------F( 1, 13) = 8144.59
Model | 3351406.23 1 3351406.23
Prob > F = 0.0000
Residual | 5349.35306 13 411.488697
R-squared = 0.9984
-------------+-----------------------------Adj R-squared = 0.9983
Total | 3356755.58 14 239768.256
Root MSE = 20.285
-----------------------------------------------------------------------------y | Coef.
Std. Err. t
P>|t| [95% Conf. Interval]
-------------+---------------------------------------------------------------x | .706408
.0078275 90.25 0.000 .6894978 .7233182
_cons | -184.0779 46.26183 -3.98 0.002 -284.0205 -84.13525
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6. Hypothesis testing.
As noted earlier, Keynes expected the
MPC to be positive but less than 1. In
our example we found it is about 0.70.
 Then, is 0.70 statistically less than 1?
If it is, it may support keynes’s theory.

Such confirmation or refutation of econometric theories on the basis of
sample evidence is based on a branch of statistical theory know as
statistical inference (hypothesis testing)
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7.Forecasting or prediction.
• To illustrate, suppose we want to predict the mean
consumption expenditure for 1997. The GDP value for
1997 was 7269.8 billion dollars. Putting this value on the
right-hand of the model, we obtain 4951.3 billion dollars.
• But the actual value of the consumption expenditure
reported in 1997 was 4913.5 billion dollars. The estimated
model thus overpredicted.
• The forecast error is about 37.82 billion dollars.
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Using the model for control or policy
purposes.
• This is on the opposite way of forecasting.
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Example: Returns to Education
A model of human capital investment implies
getting more education should lead to higher
earnings
• In the simplest case, this implies an equation
like
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Earnings   0  1education  u
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Example: (continued)
• The estimate of b1, is the return to education, but can it be considered
causal?
• While the error term, u, includes other factors affecting earnings, want
to control for as much as possible
• Some things are still unobserved, which can be problematic
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Types of Data Sets
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Distribution, Densities and Moments
Random Variable
Discrete r.v.: Binary data; Count data.
Probability Distribution

 p x   1
i 1
Continuous r.v.
i
Cumulative Distribution Function F ( x)  Pr( X  x)
Probability Density Function
:
:


 f ( x)dx   F ( x)dx  1
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Normal Distribution
• PDF:
1
f ( x) 
exp[( x   ) 2 / 2 2 ]
 2
1 2
 ( x)  (2x) exp( x )
2
 ( x )    ( y ) dy
1/ 2
• CDF:

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Monments of Random Variables
• Expectation/Population Mean:
• Discrete r.v:
• Continous r.v.:
• Monment:
m
E ( X )   p( xi ) xi
i 1

E ( X )   xf ( x)dx

mk ( X )   x k f ( x)dx
The expectation of a random variable is
often referred to as its first moment.
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Calculations
• (Mean) Expected value of :
•
• Variance of :
  M (0)
'
 2  E( X 2 )   2
 2  M '' (0)  [M ' (0)] 2
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