Panel Data Analysis V¶
In this section we estimate a panoply of panel data models and try to determine which one is most appropriate for our data. We outline some tests — statistical and conceptual — that can be used to select from a set of panel data models.
Quick reminder¶
Let’s quickly remind ourselves of the key questions we need to ask before estimating panel data models:
How do your explanatory variables influence the outcome?
Is your statistical model specified correctly?
Let’s see how these questions map to the various panel data models we can estimate, and what tests we can run to help us select the most appropriate model (if it exists).
Defining our statistical model¶
Let’s return to our panel data on charities and define a statistical model for predicting a charity’s annual gross income as a function of its age, the scale of its charitable activities, where it is located, what type of charity it is, and the number of sources of income it has, and the share of its income provided by government.
How do your explanatory variables influence the outcome?¶
We are interested in a model that allows us to include observed time-invariant explanatory variables, as these are of substantive interest. For example, are local charities typically smaller than national or international organisations?
It is possible that the effect of the observed time-varying explanatory variables may differ depending on whether we consider them from a within or between perspective. For example, the effect of gaining an additional income source — say new funding from government — may be different for a change within an individual charity than a comparison of two charities.
Is your statistical model specified correctly?¶
We would be surprised if there wasn’t a correlation between the observed explanatory variables and the unobserved unit-specific effects. We only have six observed explanatory variables in the model, of which some do not vary much within charities (e.g., number of income sources), and some do not vary much between charities (e.g., a charity is either a social services organisation or not).
So before estimating models, we clearly want one that includes observed time-invariant explanatory variables and addresses the likely violation of independence of errors assumption.
Estimating models¶
Pooled OLS¶
Is the Pooled OLS model appropriate? That is, can we ignore the fact that charities likely differ in unobserved ways?
use "./data/charity-panel-analysis-2020-09-10.dta", clear
(Contains annual accounts of charities in E&W for financial years 2006-2017)
regress linc orgage localc west genchar nsources govern_share
est store pols
Source | SS df MS Number of obs = 23,826
-------------+---------------------------------- F(6, 23819) = 410.54
Model | 2225.8864 6 370.981066 Prob > F = 0.0000
Residual | 21523.6961 23,819 .903635591 R-squared = 0.0937
-------------+---------------------------------- Adj R-squared = 0.0935
Total | 23749.5825 23,825 .996834524 Root MSE = .9506
------------------------------------------------------------------------------
linc | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
orgage | .0036028 .00015 24.01 0.000 .0033087 .0038969
localc | -.3302434 .0130224 -25.36 0.000 -.3557682 -.3047187
west | .1121865 .0253139 4.43 0.000 .0625697 .1618033
genchar | -.3170303 .0139082 -22.79 0.000 -.3442913 -.2897693
nsources | .1053884 .0050963 20.68 0.000 .0953993 .1153774
govern_share | .000644 .0002035 3.16 0.002 .0002451 .0010429
_cons | 14.96317 .0236406 632.94 0.000 14.91683 15.00951
------------------------------------------------------------------------------
We can perform an autocorrelation test to check whether independence of errors assumption is violated:
*net sj 3-2 st0039
*net install st0039
xtserial linc orgage localc west genchar nsources govern_share
Wooldridge test for autocorrelation in panel data
H0: no first-order autocorrelation
F( 1, 2165) = 114.998
Prob > F = 0.0000
The results of the Wooldridge strongly suggest the error terms are correlated across observations. In practice this means that the values of these variables typically vary less within than across units. An obvious example would be the orgage variable:
l regno orgage in 1/11, clean
regno orgage
1. 200048 46
2. 200048 47
3. 200048 48
4. 200048 49
5. 200048 50
6. 200048 51
7. 200048 52
8. 200048 53
9. 200048 54
10. 200048 55
11. 200048 56
tabstat orgage, s(min max)
variable | min max
-------------+--------------------
orgage | 0 499
----------------------------------
xtsum orgage
Variable | Mean Std. Dev. Min Max | Observations
-----------------+--------------------------------------------+----------------
orgage overall | 39.20129 42.4661 0 499 | N = 23826
between | 42.35708 5 494 | n = 2166
within | 3.162344 34.20129 44.20129 | T = 11
Overall results suggest the average age of a charity 39.
Between results collapse data set down to one row per unit, hence slightly different figures to overall results. Min and Max now refer to average values.
Within results calculate differences between observed value for a unit in a given period and the unit’s mean value across all periods (and the global mean also, hence why results are counter-intuitive).
The presence of serial (auto) correlation suggests we cannot ignore the panel component of the data. However, that does not mean we need to estimate a panel model. We could use the regress, cluster() approach to relax the assumption that the error terms are independent and uncorrelated with the explanatory variables.
regress linc orgage localc west genchar nsources govern_share, cluster(regno)
Linear regression Number of obs = 23,826
F(6, 2165) = 39.07
Prob > F = 0.0000
R-squared = 0.0937
Root MSE = .9506
(Std. Err. adjusted for 2,166 clusters in regno)
------------------------------------------------------------------------------
| Robust
linc | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
orgage | .0036028 .0005828 6.18 0.000 .0024599 .0047458
localc | -.3302434 .0451017 -7.32 0.000 -.4186907 -.2417962
west | .1121865 .0896089 1.25 0.211 -.0635419 .2879149
genchar | -.3170303 .0455036 -6.97 0.000 -.4062657 -.2277949
nsources | .1053884 .013709 7.69 0.000 .0785042 .1322725
govern_share | .000644 .0006234 1.03 0.302 -.0005785 .0018665
_cons | 14.96317 .0686493 217.97 0.000 14.82854 15.09779
------------------------------------------------------------------------------
We no longer have underestimated standard errors, resulting in more more accurate t tests of the coefficients (note some variables are no longer statistically significant). However we may still want to control for unit-specific differences in the outcome — that is, is some of the variation in the outcome explained by unobserved heterogeneity?
We can check whether a Random Effects model is preferred over Pooled OLS by conducting a Breusch and Pagan Lagrangian multiplier test.
quietly xtreg linc orgage localc west genchar nsources govern_share, re
xttest0
Breusch and Pagan Lagrangian multiplier test for random effects
linc[regno,t] = Xb + u[regno] + e[regno,t]
Estimated results:
| Var sd = sqrt(Var)
---------+-----------------------------
linc | .9968345 .998416
e | .0795807 .2821005
u | .8226523 .9070018
Test: Var(u) = 0
chibar2(01) = 98352.33
Prob > chibar2 = 0.0000
Rejection of the null hypothesis suggests that there is a panel effect on the outcome, and that a Random Effects model is preferred over Pooled OLS.
Fixed Effects or Random Effects?¶
For most repeated contacts data sets, it would be erroneous to ignore the panel component of the data, even after controlling for autocorrelation of the error terms.
We then have a choice between Fixed Effects and Random Effects. (We ignore the Between Effects model as it is rarely insightful on its own, and is captured by the Random Effects model anyway.)
Hausman Test
The Hausman test checks whether the coefficients of the Random Effects model are consistent — that is, equivalent to those from the Fixed Effects model.
Failure to reject the null hypothesis (they are equivalent) provides evidence in favour of the Random Effects model, otherwise the Fixed Effects model is considered more appropriate.
quietly xtreg linc orgage localc west genchar nsources govern_share, fe
est store fixed
quietly xtreg linc orgage localc west genchar nsources govern_share, re
est store random
hausman fixed random
---- Coefficients ----
| (b) (B) (b-B) sqrt(diag(V_b-V_B))
| fixed random Difference S.E.
-------------+----------------------------------------------------------------
orgage | .0069072 .005022 .0018852 .000448
nsources | .0289886 .030451 -.0014624 .0004286
govern_share | .0010325 .001024 8.44e-06 .0000196
------------------------------------------------------------------------------
b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
Test: Ho: difference in coefficients not systematic
chi2(3) = (b-B)'[(V_b-V_B)^(-1)](b-B)
= 53.31
Prob>chi2 = 0.0000
In our example, it appears that the coefficients from the Random Effects model are inconsistent and thus the Fixed Effects model should be preferred.
Often you’ll find that the Hausman test favours the Fixed Effects model but this isn’t definitive proof that it is more appropriate.
Guidance on selecting an appropriate model¶
Confusing and conflicting advice is found throughout the statistical literature (Gelman and Hill, 2007).
In quantitative social science there is probably more support for Random Effects lately. Clark et al. (2010) state that Fixed Effects has its advantages but it limits the type of research questions that can be addressed.
Random Effects has qualities close to Fixed Effects where rich data are available i.e., where lots of observed time-varying explanatory variables are captured (Gayle and Lambert, 2018).
Selecting a model should first-and-foremost draw on theoretical insight on the relationship between the explanatory variables and the outcome.
Undertake the Hausman test but don’t be bound by it (Gayle and Lambert, 2018).
Estimate theoretically plausible statistical models and carefully compare their results.
Summary¶
QUESTION
Which model of charity income would you choose and why?
Based on all of the guidance and the results of the statistical tests, I selected the Random Effects model.