Transcript Why do we need econometrics?

Why do we need econometrics?
• If there are two points and we want
to know what relation describes that?
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Why do we need econometrics?
• But if there’s more than just two
points for two variables?
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Why do we need econometrics?
• How would we look for this line?
MINIMISING THE RESIDUALS!!!!
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What is an econometric model?
 Some things about reality are known…
– GDP per capita
– capital accumulation
– volume of trade
 … but the relations between them are
unknown
– correlation
– causality
 we need a tool to seek the latter using the former
 Costs? We need to simplify the reality
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An example of a model
• Suppose you wanted to see what is the
degree of gender discrimination in wages.
• Your model:
wages=f (gender and ???)
–
–
–
–
–
education
experience
profession
city/rural area
…
• We cannot consider everything because:
– no data
– model quality => STATISTICS
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Random versus deterministic
• What is a variable?
• What is a random variable?
– example: height of all the people in this room
• Can you ever get a deterministic number
from a random one?
• What is EXPECTED VALUE?
– for a deterministic variable
– for a random variable
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Are residuals form this graph
random or deterministic?
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An example of a model revisited
• Let’s go back to the example of gender
discrimination:
• We said the model was like this
wages
=
f (gender and ???)
• But now we know that in fact:
wages
=
constant +
coeff*education +
coeff*experience +
coeff*gender +
coeff*whatevereslewethinkof +
residuals
• We don’t know the coefficients => we seek
a method to find them!!!
• Residuals depend on how we choose the
coefficients and are unknown (random)
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Finding a method
• We want to minimise our „error”:
or
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Finding a method
• We can write each of the elements as :
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Finding a method
• What we have is:
– X – a matrix of exogenous (input) variables
(„knowns”)
– y - a vector of the endogenous (but still input)
variable (we think we know the results of the
random process)
– ɛ – unknown residuals that can be only estimated
using residuals from the model
– β – unknown parameters that we want to
estimate (output)
• What we need is:
– a model that will let us know β’s, with ɛ’s as
small as only possible
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Finding a method
• Let’s define:
• Where:
is a theoretical, fitted value of y’s
» e’s are only estimates of ɛ’s, but do
not have to be equal
» b’s are only estimats of β’s, but are
chosen such that, y and y hat are as
close as possible
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Finding a method
• We find the method for estimation by
minimising the residuals, but:
– There is a lot of them
– They can be very big (positive and negative) and
still add up to zero
=> we need to take squares (distances) and not
direct values
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Finding a method
• We look for the first order conditions for:
• So we differentiate and put equal to zero:
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Finding a method
• When it comes to matrices, multiplication is no
longer as straightforward (it matters what comes
first and you can’t divide)
• What you can is pre-multiply by an inverted matrix
• In order for a matrix to be invertible, it has to be
nonsingular (no row and no column is a linear
combination of the others)
• X’X is a matrix seems to meet these conditions 
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Finding a method
• We have an optimum, but we don’t know if
it’s a max or a min => need to find second
derivative and prove it’s positive to be sure
to have a minimum (so residuals as small
as possible)
• It is positive, so we have found what we
were looking for 
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Properties of OLS
1.
2.
3.
4.
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X’e=0
Fitted and actual values of y are on
average equal
Σe=0 (for a model with a constant)
There is nothing more systematic about y
than already explained by X (fitted y and
residuals are not correlated)
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Properties of OLS
• If a model has a constant…
• … and then
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Is OLS the best?
•
Can we be sure that OLS will always give us the
best possible estimator?
•
If assumptions are fulfilled, OLS is BLUE (meaning
Best Linear Unbiased Estimator)
Assumptions:
1. y=Xβ
2. X is deterministic and exogenous
3. E(ɛi)=0
4. Cov(ɛi,ɛj)=0
5. Var(ɛi)=σ2
What do we loose on linear and unbiased?
•
•
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Variance-covariance matrix
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What do we know about OLS
properties
• It is unbiased:
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What do we know about OLS
properties?
• The variance of the parameters is given by:
so we only need to find an estimator of σ,
but:
so…
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What do we know about OLS
properties?
…
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Why do we need the properties?
• How can we say that a model is good?
– We only know that among linear and unbiased we
have estimators of β that yield lowest errors)
• How can we say if one model is better than
other?
– So far we didn’t ask this question at all!
• How can we say AT ALL if a variable really is
correlated with another?
– So far we only considered setting up a model,
but in reality this is an implicit hypothesis and
needs to be tested!
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How good our model is?
•
We can ask how big are the residuals
when compared to the input values
TSS=ESS+RSS
with a constant
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How good our estimates are?
• We can test the values we have obtained
vis-a-vis a hypothesis that they are zero
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Preview of coming attractions
• Hypothesis testing
• Understanding the output of any statistical
package (or tables in papers you have to
read )
• Interpretation
• Prognosis
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