Download GLS2014

30/10/2014
OLS
• Let X be a (N x K)matrix of k explanatory
variables, including a column of 1 for the
constant term, over N observations.
• Let y be a vector of N observations on the
dependent variable
• Let B be a vector of paramters
• Let e be a vector of N residual terms
• y = XB + e
OLS & GLS
Bo Sjö
2014
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Min sum of squares
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Properties of OLS
Min (ee’) => S(b) objective function.
S(b) = (y – XB)’(y – XB)
[S(b)/δb] = -2(X’y – X’XB)
Solving for B gives b (the estimate)
b = [X’X]-1 [X’y] = 0
b = B + [X’X]-1 [X’e]
The Gauss-Markov conditions:
yt = xt’βi + εt or matrix form y = x’ β + ε
εt is a random variable (a process)
E{εt} = 0
Correct specification (+No err in variable)
E{εt εt} = σ2
Homoscedasticity
E{εt εt±/-k} = 0 for all k ≠ t
No autocorrelation
E{εt | X} = 0
Weak exogeneity
Var {εt | X} = 0
We can add linearity, Normality in εt for inference
If these conditions are fulfilled OLS is Best Linear unbiased estimator BLUE,
and the estimated coefficients are ‘good estimates ‘ of the true parameters
of interest. Estimates will asymptotically have a normal distribution. (CLT)
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GLS and FGLS (or EGLS)
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Estimating the OLS
In you first course you learn how to use OLS and understand
the coefficients in a mutivariate linear regression model.
You focus on the problems of heteroscedasticity and
autocorrelation.
These problems can be analysed and ’solved’ with Generalized
Least squares GLS.
If you know and can estimate the heteroscedastity and
autocorrelation correctly, you can pre-wash you data to resore
the desired residual properties.
Since GLS assumes that we know the covariance matrix, it
must be replaced by a estimates, which leads to Feasible
Generalized Least Squares (FGLS) or Estimated Generalized
Least Squares (EGLS).
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• In the basic textbook, the estimated
parameters will be given by
B= (X’X)-1 (X’y)
• This requiers that the expected value (b) is
unbiased.
• Take expectations, substitute y with xB + e
E(b) = B + E{[X’X]-1 [X’e]}
• Since E(B) = B, it is constant.
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• What happend with the last term?
E(b) = B + E{[X’X]-1 [X’e]}
If the Gauss-Markov holds etc.
It will have an expected value of zero.
Taking plim show
1) That E[X’X]-1 can be viewed as a constant.
2) E[X’e] -p > 0, typpically because
E(X)E(e)=E(X) × 0 = 0, if independent
• The last term is always interesting
E{[X’X]-1 [X’e]}
It tells about bias – if non-zero
It gives the variance (and efficiency) as
We look at Var(b-B)
Also consistency, drive T to infinity and analyse
what happens with the estimates.
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Besides
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GLS Estimation
Heteroscedasticity and autocorrelation
Specification
System equation - exogeneity
Error in variable problems (measurement)?
What is y?
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You want E{ε’ ε} = σ2I
Where I is the identity matrix
But you get E{ε’ ε} = σ2 Ω
To get what you want find the inverse of Ω,
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Finding the inverse, means finding P such that P’P = Ω-1
Use P to construct new variables such that
Py = Px’ β + Pε
y* = x*‘ β + ε* where V{ε* } = σ2I
– such that Ω Ω-1 =I,
– Continuous, truncated, only positive values,
ordered variables, probability, time measure
variable (duration)?
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FGLS & GMM
• Since we cannot know the P matrix it must be
estimated -> Feasable GLS.
• Here you operate on the variables.
• But, suppose the problem is misspecification
then FGLS leads totally wrong. In time series
use dynamic specification (lags) instead.
• But you can also operate in the σ2 Ω
expression directly. This is GMM.
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