1. Esman M. Nyamongo Central Bank of Kenya
Panel data analysis
Econometrics Course organized by the COMESA Monetary Institute (CMI) on 2-11 June 2014, KSMS Nairobi Kenya
Esman M. Nyamongo
Central Bank of Kenya

2. The random-effects model
Recall:
Random-effects model
(1) Differences in the vi are randomly distributed between units
(2) Values of vi (time invariant effects) are uncorrelated with the other regressors -[ substantive assumption]
Here vi is not a set of fixed parameters to be estimated, but as random variables;
Here vi, which was fixed in FE, has a random component (ui)
- thus random effects model

3. This equation can be reformulated as: Or
This randomness implies that the vi are distributed independently of and independently of Xit
With these assumptions we only have to estimate two parameters v and not N vis

4. That is the problem of too many parameters with FE leading to loss of DF can be avoided if are assumed random, i.e drawn from a given distribution with the following assumptions:
i.e independent of vit
The benefit of this approach is that you concede variation across the cross-sections, but don’t estimate ‘N-1’ of these variations
However, in this approach you introduce a more complicated variance structure and OLS is no longer appropriate

5. When the true model is the random effects model:
OLS will produce consistent estimates of B, but standard errors will be understated
OLS is not efficient compared to (feasible) generalised least squares
The solution is straight forward- use GLS
First derive an estimator of the variance- covariance matrix of the error term
Second, use this covariance structure in the estimation of B

6. The (homoscedastic) variance and the covariance of is given as:
OLS weights all observations equally but it is not ideal for error-components model
The (homoscedastic) variance and the covariance of is given as:
But the covariance for and is:
Meaning that we have serial correlation over time only between the disturbances of the same individual.

7. Based on the underlying assumptions of the RE estimator, the error variance-covariance matrix of the disturbance term of each individual cross-section unit given as:
Where 1T is Tx1 vector of ones

8. The variance-covariance matrix for the error term for the full NT observations of the stacked model can be stated as:
Where is a TxT matrix
Then we need the inverse above as:
Going through specific algebra

9. Let us assume the following:
and
JT is a T xT matrix of ones

10. The homoscedastic variance is given as:
for all i,t.
But serial correlation for the same i over t
i=j and s=t (same cross-section, same year)
i=j and (same cross section, different year)
In this case we need to use GLS- Generalized Least Squares to estimate
Which is an NTxNT matrix. The GLS method allows us to better make use of the more complicated error structure which has been introduced- this is an improvement over OLS

11. Here we use a trick; we know Replace JT by since And IT by Then:
In order to use GLS, we need an estimate of the appropriate variance-covariance matrix. How do we do this?- steps to get this matrix
Here we use a trick; we know
Replace JT by since
And IT by
Then:
Simplifies to:
‘averaging matrix’ ‘deviations’

12. Now we know:
Therefore:
Pre-multiplying using our new matrix:
where
Created a ‘weighted deviation’ transformation with K+1 parameters to estimate
But for this technique to work we need estimates of and to proceed

13. ……..suggestions
True residuals are not known, therefore we use estimated residuals:
Methods:
Wallace and Hussein (1969)- use OLS residuals
Amemiya (1971) use LSDV residuals
Swamy and Arora (1972) run ‘WITHIN’ regression and then ‘BETWEEN’ regression:
Nerlove (1971) estimate
Using from LSDV and from RSS from ‘WITHIN’
Details on Swamy and Arora (1972)…..

14. ….. Swamy and arora (19720 method
Run WITHIN regression and then BETWEEN regression
We know what WITHIN is but BETWEEN!
BETWEEN – takes the average of the N cross-sections and runs a regression on this.
Very easy- no boring transformations!
What next?
Then construct as a weighted average of the two methods as:
This allows us to estimate without having to transform the data and estimate the variance- covariance matrix:

15. RE- estimation results

16. Testing the validity of the RE
This is usually stated as:
Here the randomness implies that vi are distributed independently of Eit and

17. The hausman test
Tests if the random effects are uncorrelated with the explanatory variables
However, what the test does not do is to suggest what is the way forward if we reject the null.
It does not say we use the fixed effects

18. Hausman test results
P-value= >0.05 => fail to reject the null hypothesis that the model is correctly specified.

19. 2 way RE estimation results

20. Testing for random effects XX
The hypothesis:
The LM test is stated as:
under H0
You can try this at home!