Econometric
Definition
Econometric means economic measurement. Econometric is social science in
which tools of mathematics, statistics and economics theory apply to analyze the
economic phenomena. It is the application of mathematics, statistics and
economics.
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Methodology of Econometrics
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How we study the econometric to analyze the economic theory. What is needed
is a methodology, i.e. a step-by-step procedure. The details are given below.
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1. Statement of the theory / Economic Problem
2. Convert this into mathematical model.
3. Transform into statistical model.
4. Data requirement
5. Estimation
6. Hypothesis testing
7. Forecasting/Prediction
8. Use for Policy recommendation.
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Statement of the theory / Economic Problem
Y = F(X)
Where
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A theory should have a prediction. In statistics and econometrics, we also speak
of hypothesis. One example is the marginal propensity to consume (MPC)
proposed by Keynes.
Y =Consumption and X = Income
Specification of the Mathematical Model
This is where the algebra enters. We need to use mathematical skills to produce
an equation. Assume a theory predicting that income increases the consumption.
In economic terms, there is positive relation ship b/w income and consumption.
The equation is:
,
Where Y is the variable for consumption and
is a constant and
is the
coefficient of income, and X is a measurement of income. We also call
intercept and
a slope coefficient.
Normally, we would expect both
and
to be positive.
Specification of the Econometric Model
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Here, we assume that the mathematical model is correct but we need to account
for the fact that it may not be so. We add an error term, u to the equation above.
It is also called a random (stochastic) variable. It represents other nonquantifiable or unknown factors that affect Y. It also represents mis
measurements that may have entered the data. The econometric equation is:
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The error term is assumed to follow some sort of statistical distribution. This will
be important later on.
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Obtain Data
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We need data for the variables above. This can be obtained from government
statistics agencies and other sources. A lot of data can also be collected on the
Internet in these days. But we need to learn the art of finding appropriate data
from the ever increasing huge loads of data
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Estimation of the model
Here, we quantify
and , i.e. we obtain numerical estimates. This is done by
statistical technique called regression analysis.
Hypothesis Testing
Now we go back to the part where we had economic theory. The prediction was
that income is good for the consumption. Does the econometric model support
this hypothesis. What we do here is called statistical inference (hypothesis
testing). Technically speaking, the
coefficient should be greater than 0.
Forecasting
If the hypothesis testing was positive, i.e. the theory was concluded to be correct,
we forecast the values of the wage by predicting the values of education. For
example, how much would someone earn for an additional year of schooling? If
the X variable is the years of schooling, the coefficient gives the answer to the
question.
Use for Policy Recommendation
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Lastly, if the theory seems to make sense and the econometric model was not
refuted on the basis of the hypothesis test, we can go on to use the theory for
policy recommendation. If your theory was really good, then maybe you will earn
the Nobel Prize of Economics.
Simple regression model
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Derivation of Ordinary Least Squares Estimators
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y i = β 0 + β 1 xi + u i where y i is a dependent variable, x i is an independent right-hand side
(RHS) variable, u i is the error term (unobservable), β 0 and β 1 are coefficients. The
ordinary least squares procedure minimizes the error sum of squares (SSE). The
minimization problem is given as follows:
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ei = Y – Ŷ
^
^
Minimize SSE ( β 0, β 1)
∂SSE
n
n
^
^

2
2
ˆ
u
=
y
−
−
β
β

∑
∑
i
i
0
1 xi 

i =1
i =1 
n
^
^


= 2 ∑  y i − β 0 − β 1 x i  (− 1 ) = 0

i =1 
(1 - a)
n
^
^


= 2 ∑  y i − β 0 − β 1 xi  (− x i ) = 0

i =1 
(1 - b)
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F .O.C.
=
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Ŷ=Y–Ŷ
^
∂ β0
∂SSE
^
∂ β1
Step 1: Derive the OLS estimate of β 0 ( βˆ 0 )
n
^
^


y
−
−
β
β

∑
i
0
1 xi  = 0

i =1 
n
n
i =1
i =1
n
^
^
∑ y i − ∑ β 0 − ∑ β 1 xi = 0
n
i =1
n
^
i =1
_
^
re-arrange
^
∑ y i − n β 0 − ∑ β 1 xi = 0
i =1
divide equation (1-a) by -2
^
_
y − β 0 − β1 x = 0
divide both sides by n.
re-arrange
_
^
_
^
^
solve the OLS estimate β 0
β 0 = y − β1 x
Step 2: Derive the OLS estimate of β 1 , ( βˆ1 )
^
i =1
n
n
^
i
+ β 1 ∑ xi
0
i =1
n
^
n
^
∑x y = β ∑x + β ∑x
i =1
i
i
0
i
i =1
1
i =1
2
i
re-arrange equation (1-a)
(2)
re-arrange equation (1-b)
(3)
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n
∑ y = nβ
Now multiply equation (2) by the sum of xi and equation (3) by n. Subsequently,
n
n
^
n
^
^
n
2
n
^
n∑ xi y i = n β 0 ∑ xi + n β 1 ∑ xi2
i =1
i =1
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i =1
Subtract equation (4) from equation (5).
(5)
n

n ∑ x i y i − ∑ x i ∑ y i = n β 1 ∑ x − β 1 ∑ x i 
i =1
i =1
i =1
i =1
 i =1 
Solving equation (6) yields the OLS estimate of β 1 ( βˆ1 ) :
n
n
n
^
n
^
n
i =1
n∑ x
n
n ∑ xi y i − ∑ x i ∑ y i
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^
Step 3: β 1 =
n
i =1
i =1
i =1
n
n

n∑ xi2 − ∑ xi 
i =1
 i =1 
2
2
(6)
n
i =1
n
i =1
n
^
n ∑ xi y i − ∑ x i ∑ y i
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β1 =
2
i
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n
(4)
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n

x
y
=
n
x
+
β
β
∑
i ∑ i
0∑ i
1 ∑ x i 
i =1
i =1
i =1
 i =1 
n


− ∑ x i 
 i =1 
2
i
n
=
i =1
n
(7)
2
_
_
n∑ ( xi − x)( y i − y )
i =1
n
_
n∑ ( xi − x) 2
=
Cov ( x, y )
Var ( x)
i =1
The numerator of equation (7) can be re-written as follows:
n
_
_
n
_
n
_
n
n
n
n
n
i =1
i =1
i =1
i =1
_ _
n∑ ( xi − x )( yi − y ) = n (∑ xi yi ) − n x ( ∑ yi ) − n y ( ∑ xi ) − n 2 x y
i =1
i =1
n
n
i =1
n
i =1
n
i =1
n
i =1
n
i =1
i =1
i =1
i =1
= n (∑ xi yi ) − ∑ yi ∑ xi − ∑ yi ∑ xi + ∑ yi ∑ xi
= n (∑ xi yi ) − ∑ yi ∑ xi
The denominator of equation (7) can be re-written as follows:
(8)
n
_
_2
= n ∑ x − 2 n x ( ∑ xi ) + n x
i =1
2
i
n
= n ∑x
i =1
n
= n ∑x
i =1
2
2
i


 n

− 2  ∑ xi  +  ∑ xi 
 i =1

 i =1

2
i
 n

−  ∑ xi 
 i =1

n
2
2
(9)
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i =1
2
fre
i =1
n
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n∑ ( xi − x)
2
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_
ww
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