Panel data
Panel data has both me series and cross-sec onal dimension.
- Measure same collec on of objects over several periods examine how rela onship between objects
change over me. We need varia on both over me and in cross-sec on.
Explanatory variables could be:
- Firm-level variables (earnings growth of firm i in year t)
- Time-series variables (GDP growth)
- Cross-sec onal variables (industry dummies)
Panel data models / methods:
1. Pooled OLS
2. Seemingly unrelated regressions (SUR) SKIP, old fashioned
3. Fixed effects es mator
4. Random effects es mator
5. Fama-MacBeth es mator
Approaches differ primarily in how the constant and slope coefficients vary in the cross-sec on (i) and me (t).
Main focus on:
1. Pooled OLS
2. Fixed effects es mator
3. Fama-MacBeth es mator
Constant alpha and slope beta vary over me
1. Pooled OLS
Pool all informa on of i and t into one (so one big single regression). Assumes constant term and slope, coefficients
do not vary across i and t (alpha and beta are fixed). No heterogeneity in alpha and beta. This method is advisable
in small sample (both N and T small).
+ Simple
- Hard to reliably es mate a more general model
2. Fixed effects model
Slope coefficients the same across i (firms), but constant term allowed to differ across i. Adding a firm specific fixed
effect, stable over me and specific to firm (for example the sector a firm operates in). Can think of ai as capturing
all (omi ed) variables that affect yit cross-sec onally but do not vary over me. Fixed effects take out the average
differences across for example firms and provides the appropriate slope.
+ Control for unobservable differences across units (less omi ed variables)
- Cannot iden fy effects of variables that are constant over me, only variables that vary over me. (because of
absorp on in fixed effects)
One approach: incorporate N dummy variables (fixed effects are basically many dummies). This is a standard
regression via OLS.
- Not prac cal if N is large (causes many dummies) use within transforma on
, Within transforma on: Take the me-series mean of each en ty (firm):
Subtract this from the values of the variable Yi1,…,YiT.
Do this for all en es i and all explanatory variables. This regression does not have an intercept, since now the
dependent variable will have zero mean by construc on.
You can also have me-fixed effects (instead of en ty fixed i). For example: tax rate changes may influence y, but in
the same way for all firms i. Time-fixed effects can be done by adding T me dummies.
+ Control for any common me-series varia on
- Can no longer iden fy effect of variable that only varies over me (and not in cross-sec on)
Avoid too many dummies with within transforma on: subtract the cross-sec onal average at me t from each
observa on.
Also possible to allow for both en ty and me fixed effects within the same model.
3. Random effects model
Similar to fixed effects: addressing effects that are constant over me. But in random effects you do not include
firm varia on (it is assumed to be constant over me and random). Random variable: ei. This framework requires
the assump ons that the new cross-sec onal error term has:
zero mean and constant variance
is independent of the individual observa on error term vit
is independent of the explanatory variable
+ Not adding es mators (like dummies)
+ Can s ll es mate effect of variables constant over me
+ Works well for small samples
- Addi onal assump ons on the error term
Panel data has both me series and cross-sec onal dimension.
- Measure same collec on of objects over several periods examine how rela onship between objects
change over me. We need varia on both over me and in cross-sec on.
Explanatory variables could be:
- Firm-level variables (earnings growth of firm i in year t)
- Time-series variables (GDP growth)
- Cross-sec onal variables (industry dummies)
Panel data models / methods:
1. Pooled OLS
2. Seemingly unrelated regressions (SUR) SKIP, old fashioned
3. Fixed effects es mator
4. Random effects es mator
5. Fama-MacBeth es mator
Approaches differ primarily in how the constant and slope coefficients vary in the cross-sec on (i) and me (t).
Main focus on:
1. Pooled OLS
2. Fixed effects es mator
3. Fama-MacBeth es mator
Constant alpha and slope beta vary over me
1. Pooled OLS
Pool all informa on of i and t into one (so one big single regression). Assumes constant term and slope, coefficients
do not vary across i and t (alpha and beta are fixed). No heterogeneity in alpha and beta. This method is advisable
in small sample (both N and T small).
+ Simple
- Hard to reliably es mate a more general model
2. Fixed effects model
Slope coefficients the same across i (firms), but constant term allowed to differ across i. Adding a firm specific fixed
effect, stable over me and specific to firm (for example the sector a firm operates in). Can think of ai as capturing
all (omi ed) variables that affect yit cross-sec onally but do not vary over me. Fixed effects take out the average
differences across for example firms and provides the appropriate slope.
+ Control for unobservable differences across units (less omi ed variables)
- Cannot iden fy effects of variables that are constant over me, only variables that vary over me. (because of
absorp on in fixed effects)
One approach: incorporate N dummy variables (fixed effects are basically many dummies). This is a standard
regression via OLS.
- Not prac cal if N is large (causes many dummies) use within transforma on
, Within transforma on: Take the me-series mean of each en ty (firm):
Subtract this from the values of the variable Yi1,…,YiT.
Do this for all en es i and all explanatory variables. This regression does not have an intercept, since now the
dependent variable will have zero mean by construc on.
You can also have me-fixed effects (instead of en ty fixed i). For example: tax rate changes may influence y, but in
the same way for all firms i. Time-fixed effects can be done by adding T me dummies.
+ Control for any common me-series varia on
- Can no longer iden fy effect of variable that only varies over me (and not in cross-sec on)
Avoid too many dummies with within transforma on: subtract the cross-sec onal average at me t from each
observa on.
Also possible to allow for both en ty and me fixed effects within the same model.
3. Random effects model
Similar to fixed effects: addressing effects that are constant over me. But in random effects you do not include
firm varia on (it is assumed to be constant over me and random). Random variable: ei. This framework requires
the assump ons that the new cross-sec onal error term has:
zero mean and constant variance
is independent of the individual observa on error term vit
is independent of the explanatory variable
+ Not adding es mators (like dummies)
+ Can s ll es mate effect of variables constant over me
+ Works well for small samples
- Addi onal assump ons on the error term
