1. Esman M. Nyamongo Central Bank of Kenya
Panel data analysis
Econometrics Course organized by the COMESA Monetary Institute (CMI) on 9-13 February 2015, Kampala, Uganda
Esman M. Nyamongo
Central Bank of Kenya

2. panel Data- where we are coming from
There are 3 types of data
Cross sectional data
Time series data
Panel data
The cross-section and time series are the primary building blocks of Panel

3. Time series data
A time series is a set of observations on the values that a variable takes at different times
Such data may be collected at regular time intervals
Minutely and Hourly- collected literally continuously ( the so-called real time quote)
Daily- e.g., Financial time series-Stock prices, exchange rates; weather reports- rainfall, temperature
Weekly – e.g., money supply
Monthly- e.g., consumer price index
Quarterly- e.g., GDP
Semi-annually- e.g., Fiscal data
Annually- e.g., Fiscal data
Quinquennially ( every 5 years)- e.g., manufacturing survey
Decennially- (every 10 years)- e.g., population census data

4. Illustration of time series

5. Consumption and Income (annual)
The model setup:
Where t= time series

6. Cross-sectional data
Cross-section data is data on one or more variables collected at a particular point in time in time
Survey data- questionnaire is designed to capture all variables a research is looking for.
Macro data relating to different economic entities : countries, banks at a particular point in time. E.g
Other data

7. Summary data from income survey

8. Cross-country data
The model set up
Where i= cross -section

9. Panel data
Panel data is a combination of both time and cross-section data
Specialized type is the longitudinal or micropanel data where a cross-sectional unit (say, individual, family, firm) is surveyed over time.
Surveying same individual over time is able to provide useful information on the dynamics of individual/household/firm behavior

10. Example

11. Pooled data

12. Some Advantages of panel data
Controlling for individual heterogeneity. Panel data suggests that individuals, firms, states or countries are heterogeneous.
Panel data give more informative data, more variability, less collinearity among the variables, more degrees of freedom and more efficiency
Panel data are better able to study the dynamics of adjustment.
Panel data are better able to identify and measure effects that are simply not detectable in pure cross- section or pure time-series data.
Consider a country where labor participation rate is 50%!
it is possible each individual has 50% chance of participation
50% of individuals work 100% of the time and the other 50% work 0% of the time yielding 50% participation.

13. Panel data model set up Time series + cross-section
Where i= cross section; t= time series
How then do we estimate this model?
Many approaches

14. Part i

15. a. pooled estimator
Emphasizes the joint estimation of coefficients- ignores panel structure of the data
where y= dependent variable,
Xs= regressors
for all i and t. i.e
For a given cross section, observations are serially uncorrelated
Across cross-sections and time, the errors are homoscedastic

16. But how realistic is it to ignore the panel structure of the data?
These assumptions- classical assumptions therefore suggest we estimate the equation by OLS
Pooling- increases degrees of freedom, potentially lowering standard errors on the coefficients.
This involves stacking cross-sections in the data set
This form assumes same intercept and same slope for all coefficients
But how realistic is it to ignore the panel structure of the data?

17. Short practical session
Preparation of data for use in panel estimation
Estimation of pooled model using Eviews

18. Pooled regression results
Same slope and intercept is assumed
Homogeneity of the cross-section units

19. b. Error component model

20. B. The error-components model
Recall
In this model u may display 3 different schemes:
(i) consists of 3 individual shocks, each assumed to be independent of each other:
is cross section invariant shock- time effect
is time invariant shock- individual effect
is the error term with usual properties- uncorrelated with Xit

21. Scheme (i) is the TWO-WAY-ERROR COMPONENT MODEL
(ii) to yield:
Cross-section fixed effects model
(iii) to yield:
Time-fixed effects model.
Schemes (ii) and (iii) are referred to as ONE-AWAY ERROR COMPONENT MODEL
Scheme (i) is the TWO-WAY-ERROR COMPONENT MODEL

22. i. One way error component models
Follows from the restrictions imposed by the pooled model i.e joint intercept and slope
for i=1,2….N
for t=1,2….T
One-way error component model allows cross-section heterogeneity in the error term.
Error term (uit) becomes the sum of an individual specific effect ( vi- time invariant) and a ‘well behaved’ disturbance ( )

23. In this formulation we have:
The first part varies over cross-section units but is constant across time
The second part varies unsystematically (independently) across time and individuals

24. Estimation method
Two ways to estimate a regression model with error terms that are assumed to consist of several error components:
Fixed-effects-
(1) the constant/intercept in each equation is a separate parameter
(2) values of vi are potentially correlated with the other regressors
Random-effects –
(1) differences in the vi are randomly distributed between units
(2) values of vi are uncorrelated with the other regressors

25. 1. Fixed effects model Recall: Fixed-effects model-
(1) each equation constant is a separate parameter
(2) values of vi are potentially correlated with the other regressors

26. The main question is whether Xit is correlated with uit:
If no, then we have a seemingly unrelated regression
If yes, then we have a multi-equation system with common coefficients and endogenous regressors
Then how do we account for this endogeneity-
In time series we use instrumental variable estimation methods (2sls,3sls etc)
However, in panel we can handle this, under certain assumptions, without using instruments
How?

27. Some approaches There are 3 approaches to doing this in panel:
The least squares dummy variables (LSDV) estimator
The within-group estimator
The first difference estimator
Discussed in turns

28. 1. Least Squares Dummy Variables (LSDV)
LSDV method applies OLS to levels with group specific dummies added to the list of regressors.
Explains why the estimator is called the Least- Squares Dummy Variables (LSDV) estimator
Consider the general model:
Stack the observations over t, to obtain:

29. The pooled regression is then:
Where is a Kronecker product
Since 1T is a regressor alongside Xit then we expect:

30. Too many parameters (especially with large N!
This model is appealing , but consider the number of parameters to be estimated!
K+1+(N-1)= k+N
K= parameters for the original X-regressors
1= parameter for the intercept
N-1= parameters for cross-section fixed effects (omitted x-section captured by intercept)
Too many parameters (especially with large N!
N-1
K+1

31. There are many parameters (Bs and Ds) being estimated.
But can be estimated using OLS following the Frisch-Waugh-Lovell (FWL) theorem on partitioned regressions:
Digression on FWL and partitioned regressions

32. Based on this result it can be shown that the fixed effects estimator is the partitioned OLS estimator of in pooled regression.
Where
It then follows that:
Therefore:

33. Whence it follows that:
This is basically a pooled OLS estimator on transformed data

34. Least Squares Dummy Variable est.

35. 2. Within, Q, Estimation
Still assumes individual effects although we no longer directly estimate them.
We demean the data- wipe out the individual effects- to estimate only B
How do we wipe out the individual effects?
We define a Q matrix . Where Q is defined such that:
where

36. Consider a simple regression:
The trick is to remove the fixed effects, vi.
How?
Step 1: Average over time t for each i.

37. Step 2: The transformed regression:
Then, stack by observation for t= 1, …., T, resulting in:

38. Degrees of freedom of FE estimator
The within-group fixed-effects estimator is pooled OLS on the transformed regression that has been stacked by observations:
Degrees of freedom of FE estimator
nT-k-n=n(T-1)-k
Why- lose 1 degree of freedom in each fixed effect estimated. There are k Bs to be estimated as well

39. General case:
Stack an individual i’s observations for t=1,…..,T, giving: Or
i= 1 , 2, …., n
Where 1T is a (Tx1) vector of ones

40. The fixed-effects estimator is applied to a T- equation system transformed from the original system above.
The matrix used for the transformation is the so-called annihilator associated with 1T:
where Q is a T x T matrix:

41. The matrices QT and PT are such that:
What does this mean- the QT and PT matrices takes you back to the transformed y

42. The transformed error-components model is then:
The pooled regression is stated as:
sss

43. The fixed-effects estimator is again obtained by pooled OLS on the transformed system:
Why?
QT is idempotent i.e QTQT=QT
In other words:

44. is an NTx1 vector with ‘stacked’ deviations
P is the matrix that averages across time for each individual cross section.
Thus pre-multiplying this regression by Q obtains deviations from means WITHIN each cross- section
is an NTx1 vector with ‘stacked’ deviations
The OLS estimator is therefore:
Demeaning the data will not change the estimates for .
Similar to running a regression with the line of best fit passing through the origin

45. Thus the ‘WITHIN’ model becomes a simple regression:
Individual effects can be solved ( not estimated). But we need the following assumption:
And solving:

46. Fixed Effects: Within estimation
Notice the fixed effects are not estimated.
Computed instead.

47. Within- For clickers The computed FE are indicated.
But what are these FE?

48. Disadvantages of the method
Demeaning the data means X-regressors which are themselves dummy variables cannot be used (sex, religion, etc)

49. Which one do we work with?- Pooled or FE
Both coefficients are positive and significant. Therefore satisfy some theory!
Which one do we choose?

50. Testing the joint validity of fixed effects
The null hypothesis
Alternative
HA: Not all equal to 0
We test the null hypothesis of no individual effects within applied Chow or F-test, combining the residual sum of squares for the regression both with constraints (under the null) and without (under alternative).
The recipe:
RSS- OLS on pooled model ( constant intercept)
URSS- OLS on LSDV

51. The F- statistics is stated as:
If N is ‘large enough’ can use ‘WITHIN’ estimation instead of LSDV for the RSS.
The decision rule:
P-value < 0.05 we reject the null hypothesis => FE are not redundant
P-value> 0.05, we fail to reject the null hypothesis => FE are redundant, suggesting pooled model is valid

52. Part ii

53. Ii Two way error component model
Recall:
i=1,….,N; t=1, …..,T
Where = unobservable individual effect; =unobservable time effect; stochastic disturbance
= selector matrix of ones and zeros- individual effects
= time dummies

54. 1. Estimating fixed effects
Here we still assume and are fixed parameters to be estimated and
Estimation with LSDV requires the estimation of {(N-1)+(T-1)} dummies
This can introduce rather severe loss of degrees of freedom
once again to avoid this problem we perform ‘ WITHIN’ transformation (similar to one-way model). Now, however, we must demean across both dimensions

55. Here we work with the following:
Where:JN= matrix of ones of dimension N
deviations across i
deviations across t
Transforming with Q sweeps out the time and individual effects

56. Here we have a simple regression (one-x regressor):
We now need 2 constraints to capture individual and time effects
Then we can compute the intercepts using
Again as in one way model we cannot use time-invariant or individual-invariant (dummy regressors) as Q wipes them out.

57. Two way fixed effects ESTIMATION RESULTS

58. For clickers
How do we interpret these result

59. Practical session

60. Supplementary information

61. Panel data set

62. Longitudinal/Micropanel data

63. Pooled regression
Same results

64. 1 Kronecker product

65. LSDV- For clickers

66. 2. Fwl and partitioned regressions
Consider the partitioned regression equation:
Where K=k1+k2
The least-squares estimators for B1 and B2 can be expressed as:
Where

67. 3. Annihilator matrix The residual maker and the Hat Matrix
Some useful matrices:
We know
Meaning:
Where is called the residual maker since it makes residuals out of y.
Matrix M is idempotent if [M2=MM=M]

68. M is a square matrix and idempotent- show
The matrix has the following properties as well:
The hat matrix (H)- makes y hat out of y.
where