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1 MADE
Why do we need econometrics?
• If there are two points and we want to know what relation describes that?
X
Y
2 MADE
Why do we need econometrics?
• But if there’s more than just two points for two variables?
3 MADE
Why do we need econometrics?
• How would we look for this line?
MINIMISING THE RESIDUALS!!!!
4 MADE
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
5 MADE
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
6 MADE
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
7 MADE
Are residuals form this graph random or deterministic?
8 MADE
An example of a model revisited
• Let’s go back to the example of gender discrimination:
• We said the model was like thiswages = 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)
9 MADE
Finding a method
• We want to minimise our „error”:
or
10 MADE
Finding a method
• We can write each of the elements as :
11 MADE
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
12 MADE
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
13 MADE
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
14 MADE
Finding a method
• We look for the first order conditions for:
• So we differentiate and put equal to zero:
15 MADE
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
16 MADE
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
17 MADE
Properties of OLS
1. X’e=02. Fitted and actual values of y are on
average equal3. Σe=0 (for a model with a constant)4. There is nothing more systematic about
y than already explained by X (fitted y and residuals are not correlated)
18 MADE
Properties of OLS
• If a model has a constant…
• … and then
19 MADE
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=Xβ2. X is deterministic and exogenous3. E(ɛi)=0
4. Cov(ɛi,ɛj)=0
5. Var(ɛi)=σ2
• What do we loose on linear and unbiased?
20 MADE
Variance-covariance matrix
21 MADE
What do we know about OLS properties
• It is unbiased:
22 MADE
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…
23 MADE
What do we know about OLS properties?
…
24 MADE
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!
25 MADE
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
26 MADE
How good our estimates are?
• We can test the values we have obtained vis-a-vis a hypothesis that they are zero
27 MADE
Preview of coming attractions
• Hypothesis testing
• Understanding the output of any statistical package (or tables in papers you have to read )
• Interpretation
• Prognosis