Marina Adshade, Paul Corcuera, Giulia Lo Forte, Jane Platt
29 May 2024
This module uses the Penn World Tables which measure income, input, output, and productivity, covering 183 countries between 1950 and 2019. Before beginning this module, download this data in the specified Stata format.
In economics, we typically have data consisting of many units observed at a particular point in time. This is called cross-sectional data. There may be several different versions of the data set that are collected over time (monthly, annually, etc.), but each version includes an entirely different set of individuals.
For example, let’s consider a Canadian cross-sectional data set: General Social Survey Cycle 31: Family, 2017. In this data set, the first observation is a 55 year old married woman who lives in Alberta with two children. When the General Social Survey Cycle 25: Family, 2011 was collected six years earlier, there were probably similar women surveyed, but it is extremely unlikely that this exact same woman was included in that data set as well. Even if she was included, we would have no way to match her data over the two years of the survey.
Cross-sectional data allows us to explore variation between individuals at one point in time but does not allow us to explore variation over time for those same individuals.
Time-series data sets contain observations over several years for only one unit, such as country, state, province, etc. For example, measures of income, output, unemployment, and fertility for Canada from 1960 to 2020 would be considered time-series data. Time-series data allows us to explore variation over time for one individual unit (e.g. Canada), but does not allow us to explore variation between individual units (i.e. multiple countries) at any one point in time.
Panel data allows us to observe the same unit across multiple time periods. For example, the Penn World Tables is a panel data set that measures income, output, input, and productivity, covering 183 countries from 1950 to the near present. There are also microdata panel data sets that follow the same people over time. One example is the Canadian National Longitudinal Survey of Children and Youth (NLSCY), which followed the same children from 1994 to 2010, surveying them every two years as they progressed from childhood to adulthood.
Panel data sets allow us to answer questions that we cannot answer with time-series and cross-sectional data. They allow us to simultaneously explore variation over time for individual countries (for example) and variation between individuals at one point in time. This approach is extremely productive for two reasons:
In this sense, panel data sets allow us to answer empirical questions that cannot be answered with other types of data such as cross-sectional or time-series data.
Before we move forward exploring panel data sets in this module, we should understand the two main types of panel data:
We learned the techniques to create a balanced panel in Module 7. Essentially, all that is needed is to create a new data set that includes only the years for which there are no missing values.
The first step in any panel data analysis is to identify which variable is the panel variable and which variable is the time variable. The panel variable is the identifier of the units that are observed over time. The second step is indicating that information to Stata.
We are going to use the Penn World Data (discussed above) in this example. In that data set, the panel variable is either country or countrycode, and the time variable is year.
Variable Storage Display Value
name type format label Variable label
-------------------------------------------------------------------------------
country str34 %34s Country name
countrycode str3 %9s 3-letter ISO country code
year int %10.0g YearWhen the decribe command executed, did you see that the variable year is an integer (i.e. a number like 2020) and that country or countrycode are string variables (i.e. they are words like “Canada”)? Specifying the panel and time variables requires that both of the variables we are using are coded as numeric variables, and so our first step is to create a new numeric variable that represents the country variable.
To do this, we can use the encode command that we saw in Module 6.
We can see in our data editor that this command created a unique code for each country and saved it in a variable that we have named ccode. For example, in the data editor we can see that Canada was given the code 31 and Brazil was given the code 25.
Now we are able to proceed with specifying both our panel and time variables by using the command xtset. With this command, we first list the panel variable and then the time variable, followed by the interval of observation.
Panel variable: ccode (strongly balanced)
Time variable: year, 1950 to 2019
Delta: 1 yearWe can tell that we have done this correctly when the output indicates that the “Time variable” is “year”.
Within our panel data set, our use of this command above states that we observe countries (indicated by country codes) over many time periods that are separated into year groupings (delta = 1 year, meaning that each country has an observation for each year, specified by the yearly option). The option for periodicity of the observations is helpful. For instance, if we wanted each country to have an observation for every two years instead of every year, we would specify delta(2) as our periodicity option to xtset.
Always make sure to check the output of xtset carefully to see that the time variable and panel variable have been properly specified.
For now, we are going to focus on the skills we need to run our own panel data regressions. In section 15.6, there are more details about the econometrics of panel data regressions that may help with the understanding of these approaches. Please make sure you understand that theory before beginning your own research.
Now that we have specified the panel and time variables we are working with, we can begin to run regressions using our panel data. For panel data regressions we simply replace regress witht the command xtreg.
Let’s try this out by regressing the natural log of GDP per capita on the natural log of human capital. We have included the describe to help us understand the variables we are using in this exercise.
Variable Storage Display Value
name type format label Variable label
-------------------------------------------------------------------------------
rgdpe float %14.3g Expenditure-side real GDP at
chained PPPs (in mil. 2017US$)
pop double %10.0g Population (in millions)
hc float %9.0g * Human capital index, see note hc
(2,411 missing values generated)
(4,173 missing values generated)
Random-effects GLS regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 0.5907 avg = 59.6
Overall = 0.6006 max = 70
Wald chi2(1) = 8408.76
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.081454 .0226987 91.70 0.000 2.036965 2.125942
_cons | 7.344036 .0612318 119.94 0.000 7.224024 7.464048
-------------+----------------------------------------------------------------
sigma_u | .71051066
sigma_e | .3932119
rho | .76553536 (fraction of variance due to u_i)
------------------------------------------------------------------------------The coefficients in a panel regression are interpreted similarly to those in a basic OLS regression. Because we have taken the natural log of our variables, we can interpret the coefficient on each explanatory variable as being a \(\beta\) % increase in the dependent variable associated with a 1% increase in the explanatory variable.
Thus, in the regression results above, a 1% increase in human capital leads to a roughly 2% increase in real GDP per capita. That’s a huge effect, but then again this model is almost certainly misspecified due to omitted variable bias. Namely, we are likely missing a number of explanatory variables that explain variation in both GDP per capita and human capital, such as savings and population growth rates.
One thing we know is that GDP per capita can be impacted by the individual characteristics of a country that do not change much over time. For example, it is known that distance from the equator has an impact on the standard of living of a country; countries that are closer to the equator are generally poorer than those farther from it. This is a time-invariant characteristic that we might want to control for in our regression. Similarly, we know that GDP per capita could be similarly impacted in many countries by a shock at one point in time. For example, a worldwide global recession would affect the GDP per capita of all countries at a given time such that values of GDP per capita in this time period are uniformly different in all countries from values in other periods. That seems like a time-variant characteristic (time trend) that we might want to control for in our regression. Fortunately, with panel data regressions, we can account for these sources of endogeneity. Let’s look at how panel data helps us do this.
We refer to shocks that are invariant based on some variable (e.g. household level shocks that don’t vary with year or time-specific shocks that don’t vary with household) as fixed-effects. For instance, we can define household fixed-effects, time fixed-effects, and so on. Notice that this is an assumption on the error terms, and as such, when we include fixed-effects to our specification they become part of the model we assume to be true.
When we ran our regression of log real GDP per capita on log human capital from earlier, we were concerned about omitted variable bias and endogeneity. Specifically, we were concerned about distance from the equator positively impacting both human capital and real GDP per capita, in which case our measure of human capital would be correlated with our error term, preventing us from interpreting our regression result as causal. We are now able to add country fixed-effects to our regression to account for this and come closer to determining the pure effect of human capital on GDP growth. There are two ways to do this. Let’s look at the more obvious one first.
Approach 1: create a series of country dummy variables and include them in the regression. For example, we would have one dummy variable called “Canada” that would be equal to 1 if the country is Canada and 0 if not. We would have dummy variables for all but one of the countries in this data set to avoid perfect collinearity. Rather than defining all of these dummies manually and including them in our regress command, we can simply add i.varname into our regression. Stata will then manually create all of the country dummy variables for us.
Random-effects GLS regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 1.0000 avg = 59.6
Overall = 0.8991 max = 70
Wald chi2(145) = 75668.31
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.072537 .0228576 90.67 0.000 2.027737 2.117337
|
ccode |
ALB | -1.167585 .0801743 -14.56 0.000 -1.324724 -1.010446
ARE | 2.266366 .0796921 28.44 0.000 2.110172 2.422559
ARG | -.8561933 .0743902 -11.51 0.000 -1.001995 -.7103912
ARM | -1.485533 .093271 -15.93 0.000 -1.66834 -1.302725
AUS | -.0942302 .076069 -1.24 0.215 -.2433226 .0548622
AUT | -.1109968 .0753938 -1.47 0.141 -.2587659 .0367723
BDI | -1.520566 .0752978 -20.19 0.000 -1.668147 -1.372985
BEL | .0601811 .0749744 0.80 0.422 -.086766 .2071282
BEN | -.857801 .0750248 -11.43 0.000 -1.004847 -.7107551
BFA | -.8977982 .0753508 -11.91 0.000 -1.045483 -.7501132
BGD | -1.242003 .0751348 -16.53 0.000 -1.389265 -1.094742
BGR | -.7604451 .0809396 -9.40 0.000 -.9190838 -.6018064
BHR | .8210965 .0794591 10.33 0.000 .6653595 .9768335
BLZ | -1.343531 .080911 -16.61 0.000 -1.502113 -1.184948
BOL | -1.347422 .073696 -18.28 0.000 -1.491864 -1.202981
BRA | -.4142286 .0733045 -5.65 0.000 -.5579028 -.2705545
BRB | -.1897638 .0770783 -2.46 0.014 -.3408345 -.0386931
BRN | 1.503302 .0800975 18.77 0.000 1.346314 1.66029
BWA | -.8169213 .0759202 -10.76 0.000 -.9657221 -.6681205
CAF | -1.307198 .0752984 -17.36 0.000 -1.45478 -1.159616
CAN | -.0235712 .0759494 -0.31 0.756 -.1724293 .1252868
CHE | .1159736 .0763893 1.52 0.129 -.0337467 .2656938
CHL | -.6640075 .0747736 -8.88 0.000 -.8105611 -.5174539
CHN | -1.251941 .0738216 -16.96 0.000 -1.396628 -1.107253
CIV | -.5281156 .0753013 -7.01 0.000 -.6757034 -.3805278
CMR | -.9435662 .0754434 -12.51 0.000 -1.091432 -.7956999
COD | -1.136322 .0728257 -15.60 0.000 -1.279058 -.9935867
COG | -.7672164 .0755483 -10.16 0.000 -.9152883 -.6191445
COL | -.4126524 .0735289 -5.61 0.000 -.5567664 -.2685384
CRI | -.3754629 .0737845 -5.09 0.000 -.520078 -.2308479
CYP | -.0297054 .0740581 -0.40 0.688 -.1748567 .1154459
CZE | -.3349684 .0940311 -3.56 0.000 -.519266 -.1506708
DEU | -.3038158 .0760558 -3.99 0.000 -.4528823 -.1547492
DNK | -.0638862 .0757018 -0.84 0.399 -.2122589 .0844866
DOM | -.5496054 .0736276 -7.46 0.000 -.6939128 -.405298
DZA | .4004323 .0755001 5.30 0.000 .2524548 .5484097
ECU | -.6859267 .0738495 -9.29 0.000 -.830669 -.5411843
EGY | -1.028446 .0730225 -14.08 0.000 -1.171567 -.8853244
ESP | .0026079 .074305 0.04 0.972 -.1430272 .1482431
EST | -.5866867 .0936584 -6.26 0.000 -.7702539 -.4031196
ETH | -1.462973 .0753127 -19.43 0.000 -1.610583 -1.315362
FIN | -.0552752 .0751749 -0.74 0.462 -.2026153 .092065
FJI | -.7626918 .0763733 -9.99 0.000 -.9123807 -.613003
FRA | .1069984 .0748937 1.43 0.153 -.0397905 .2537872
GAB | .2872627 .0756826 3.80 0.000 .1389276 .4355979
GBR | -.2046964 .0758355 -2.70 0.007 -.3533312 -.0560616
GHA | -.7666579 .0743406 -10.31 0.000 -.9123628 -.620953
GMB | -.3783084 .0752947 -5.02 0.000 -.5258834 -.2307334
GRC | -.1195682 .0746092 -1.60 0.109 -.2657995 .0266631
GTM | -.3600827 .0729039 -4.94 0.000 -.5029717 -.2171937
GUY | -1.071105 .0800011 -13.39 0.000 -1.227904 -.9143052
HKG | .3536698 .077052 4.59 0.000 .2026506 .5046889
HND | -.9253552 .0731616 -12.65 0.000 -1.068749 -.781961
HRV | -.5280945 .0932547 -5.66 0.000 -.7108703 -.3453187
HTI | -1.153332 .0753401 -15.31 0.000 -1.300996 -1.005669
HUN | -.5205111 .0811291 -6.42 0.000 -.6795212 -.3615009
IDN | -1.026255 .0758132 -13.54 0.000 -1.174846 -.8776637
IND | -1.180681 .072924 -16.19 0.000 -1.323609 -1.037753
IRL | -.1043488 .0749147 -1.39 0.164 -.2511789 .0424813
IRN | .1575227 .0740786 2.13 0.033 .0123313 .302714
IRQ | -.0331739 .0789483 -0.42 0.674 -.1879098 .121562
ISL | .3615295 .0746757 4.84 0.000 .2151678 .5078912
ISR | -.2988862 .075613 -3.95 0.000 -.447085 -.1506874
ITA | .1210815 .0744401 1.63 0.104 -.0248184 .2669814
JAM | -.8681193 .0747782 -11.61 0.000 -1.014682 -.7215567
JOR | -.7779743 .0743328 -10.47 0.000 -.9236639 -.6322848
JPN | -.4223239 .0757315 -5.58 0.000 -.570755 -.2738929
KAZ | -.7565804 .0932048 -8.12 0.000 -.9392584 -.5739024
KEN | -1.179702 .0730714 -16.14 0.000 -1.32292 -1.036485
KGZ | -1.924722 .0931645 -20.66 0.000 -2.107321 -1.742123
KHM | -1.389133 .0787891 -17.63 0.000 -1.543556 -1.234709
KOR | -.8586359 .0754249 -11.38 0.000 -1.006466 -.7108059
KWT | 1.682594 .0793737 21.20 0.000 1.527025 1.838164
LAO | -1.250598 .0788503 -15.86 0.000 -1.405142 -1.096054
LBR | -1.564942 .0765941 -20.43 0.000 -1.715064 -1.41482
LKA | -1.229281 .0741029 -16.59 0.000 -1.37452 -1.084042
LSO | -1.406844 .0757763 -18.57 0.000 -1.555363 -1.258326
LTU | -.4579488 .0931202 -4.92 0.000 -.640461 -.2754365
LUX | .6663584 .0745599 8.94 0.000 .5202237 .8124932
LVA | -.4205205 .0929532 -4.52 0.000 -.6027054 -.2383357
MAC | .8761201 .0796854 10.99 0.000 .7199396 1.0323
MAR | -.3318478 .0728292 -4.56 0.000 -.4745903 -.1891052
MDA | -1.713512 .0930228 -18.42 0.000 -1.895834 -1.531191
MDG | -1.289452 .0753642 -17.11 0.000 -1.437163 -1.141741
MDV | -.1380271 .0790971 -1.75 0.081 -.2930545 .0170003
MEX | -.0596504 .0737608 -0.81 0.419 -.204219 .0849182
MLI | -1.217787 .0753184 -16.17 0.000 -1.365408 -1.070165
MLT | -.6047981 .0752351 -8.04 0.000 -.7522562 -.45734
MMR | -1.47045 .0759577 -19.36 0.000 -1.619324 -1.321576
MNG | -1.550264 .0801544 -19.34 0.000 -1.707364 -1.393164
MOZ | -1.402513 .07531 -18.62 0.000 -1.550118 -1.254909
MRT | -.4266718 .0753664 -5.66 0.000 -.5743872 -.2789564
MUS | -.0791482 .073458 -1.08 0.281 -.2231232 .0648268
MWI | -1.68129 .0738461 -22.77 0.000 -1.826026 -1.536554
MYS | -.3608378 .0749055 -4.82 0.000 -.5076499 -.2140257
NAM | -.4384733 .0759446 -5.77 0.000 -.587322 -.2896246
NER | -.8025278 .0753428 -10.65 0.000 -.950197 -.6548585
NGA | -.8191766 .075341 -10.87 0.000 -.9668423 -.671511
NIC | -.4132997 .0731293 -5.65 0.000 -.5566304 -.2699689
NLD | .0442159 .0754439 0.59 0.558 -.1036515 .1920833
NOR | .0543472 .0758195 0.72 0.473 -.0942563 .2029506
NPL | -1.282033 .0753016 -17.03 0.000 -1.429622 -1.134445
NZL | -.3006866 .0759628 -3.96 0.000 -.4495709 -.1518022
PAK | -.7906198 .0728637 -10.85 0.000 -.93343 -.6478096
PAN | -.6960321 .0740998 -9.39 0.000 -.841265 -.5507993
PER | -.9140058 .0737523 -12.39 0.000 -1.058558 -.7694538
PHL | -1.208707 .0737143 -16.40 0.000 -1.353185 -1.06423
POL | -.665956 .0810286 -8.22 0.000 -.8247692 -.5071429
PRT | .3215888 .0732968 4.39 0.000 .1779298 .4652478
PRY | -.8082201 .0737547 -10.96 0.000 -.9527766 -.6636636
QAT | 1.716625 .079682 21.54 0.000 1.560451 1.872799
ROU | -.9929521 .077292 -12.85 0.000 -1.144442 -.8414625
RUS | -.5689844 .0933972 -6.09 0.000 -.7520394 -.3859293
RWA | -1.370132 .075312 -18.19 0.000 -1.517741 -1.222523
SAU | .9389691 .0795731 11.80 0.000 .7830087 1.09493
SDN | -.627489 .0786598 -7.98 0.000 -.7816594 -.4733187
SEN | -.41376 .0752945 -5.50 0.000 -.5613346 -.2661854
SGP | .4528528 .076504 5.92 0.000 .3029077 .6027979
SLE | -1.1959 .07559 -15.82 0.000 -1.344054 -1.047746
SLV | -1.483257 .0730552 -20.30 0.000 -1.626443 -1.340071
SRB | -.9042736 .0930796 -9.72 0.000 -1.086706 -.721841
SVK | -.5414365 .0938912 -5.77 0.000 -.7254599 -.3574131
SVN | -.2446127 .0936807 -2.61 0.009 -.4282236 -.0610018
SWE | .001472 .0755478 0.02 0.984 -.146599 .1495431
SWZ | -.2669642 .078989 -3.38 0.001 -.4217798 -.1121485
SYR | -1.112671 .0757991 -14.68 0.000 -1.261235 -.9641075
TGO | -1.182079 .075378 -15.68 0.000 -1.329817 -1.034341
THA | -.7933918 .0733773 -10.81 0.000 -.9372087 -.6495749
TJK | -2.386482 .0932714 -25.59 0.000 -2.569291 -2.203674
TTO | -.0485307 .0744453 -0.65 0.514 -.1944407 .0973794
TUN | -.2149054 .0755464 -2.84 0.004 -.3629737 -.0668372
TUR | .2710638 .0731255 3.71 0.000 .1277405 .4143872
TWN | -.0768687 .0742117 -1.04 0.300 -.2223209 .0685835
TZA | -1.298408 .0753746 -17.23 0.000 -1.44614 -1.150677
UGA | -1.635895 .0729136 -22.44 0.000 -1.778803 -1.492987
UKR | -1.223806 .0933087 -13.12 0.000 -1.406687 -1.040924
URY | -.2971627 .0740501 -4.01 0.000 -.4422983 -.1520272
USA | .0665248 .0762303 0.87 0.383 -.0828838 .2159333
VEN | -.105541 .0733484 -1.44 0.150 -.2493012 .0382191
VNM | -1.661203 .079402 -20.92 0.000 -1.816828 -1.505577
YEM | -.9614948 .0899015 -10.69 0.000 -1.137699 -.785291
ZAF | -.2013624 .0736927 -2.73 0.006 -.3457974 -.0569274
ZMB | -1.531063 .0744627 -20.56 0.000 -1.677007 -1.385119
ZWE | -1.047877 .0741844 -14.13 0.000 -1.193275 -.9024778
|
_cons | 7.91387 .0557836 141.87 0.000 7.804536 8.023204
-------------+----------------------------------------------------------------
sigma_u | 0
sigma_e | .3932119
rho | 0 (fraction of variance due to u_i)
------------------------------------------------------------------------------The problem with this approach is that we end up with a huge table containing the coefficients of every country dummy, none of which we care about. We are interested in the relationship between GDP and human capital, not the mean values of GDP for each country relative to the omitted one. Luckily for us, a well-known result is that controlling for fixed-effects is equivalent to adding multiple dummy variables. This leads us into the second approach to including fixed-effects in a regression.
Approach 2: We can alternatively apply fixed-effects to the regression by adding fe as an option on the regression.
Fixed-effects (within) regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 0.5907 avg = 59.6
Overall = 0.6006 max = 70
F(1, 8491) = 8221.37
corr(u_i, Xb) = 0.2961 Prob > F = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.072537 .0228576 90.67 0.000 2.02773 2.117343
_cons | 7.369692 .0159552 461.90 0.000 7.338416 7.400968
-------------+----------------------------------------------------------------
sigma_u | .73516756
sigma_e | .3932119
rho | .77755934 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(144, 8491) = 174.45 Prob > F = 0.0000We obtained the same coefficient and standard errors on our lnhc explanatory variable using both approaches!
One type of model we can also run is a random-effects model. The main difference between a random and fixed-effects model is that, with the random-effects model, differences across countries are assumed to be random. This allows us to treat time-invariant variables such as latitude as control variables. To run a random-effects model, just add re as an option in xtreg like below.
Random-effects GLS regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 0.5907 avg = 59.6
Overall = 0.6006 max = 70
Wald chi2(1) = 8408.76
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.081454 .0226987 91.70 0.000 2.036965 2.125942
_cons | 7.344036 .0612318 119.94 0.000 7.224024 7.464048
-------------+----------------------------------------------------------------
sigma_u | .71051066
sigma_e | .3932119
rho | .76553536 (fraction of variance due to u_i)
------------------------------------------------------------------------------As we can see, with this data and choice of variables, there is little difference in results between all of these models.
This, however, will not always be the case. The test to determine if you should use the fixed-effects model (fe) or the random-effects model (re) is called the Hausman test.
To run this test in Stata, start by running a fixed-effects model and ask Stata to store the estimation results under then name “fixed”:
Fixed-effects (within) regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 0.5907 avg = 59.6
Overall = 0.6006 max = 70
F(1, 8491) = 8221.37
corr(u_i, Xb) = 0.2961 Prob > F = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.072537 .0228576 90.67 0.000 2.02773 2.117343
_cons | 7.369692 .0159552 461.90 0.000 7.338416 7.400968
-------------+----------------------------------------------------------------
sigma_u | .73516756
sigma_e | .3932119
rho | .77755934 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(144, 8491) = 174.45 Prob > F = 0.0000Next, run a random-effects model and again ask Stata to store the estimation results as “random”:
Random-effects GLS regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 0.5907 avg = 59.6
Overall = 0.6006 max = 70
Wald chi2(1) = 8408.76
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.081454 .0226987 91.70 0.000 2.036965 2.125942
_cons | 7.344036 .0612318 119.94 0.000 7.224024 7.464048
-------------+----------------------------------------------------------------
sigma_u | .71051066
sigma_e | .3932119
rho | .76553536 (fraction of variance due to u_i)
------------------------------------------------------------------------------Then, run the command for the Hausman test, which compares the two sets of estimates:
---- Coefficients ----
| (b) (B) (b-B) sqrt(diag(V_b-V_B))
| fixed random Difference Std. err.
-------------+----------------------------------------------------------------
lnhc | 2.072537 2.081454 -.0089169 .0026904
------------------------------------------------------------------------------
b = Consistent under H0 and Ha; obtained from xtreg.
B = Inconsistent under Ha, efficient under H0; obtained from xtreg.
Test of H0: Difference in coefficients not systematic
chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
= 10.98
Prob > chi2 = 0.0009As we can see, the results of this test suggest that we would reject the null hypothesis that the random-effects model is preferred, and thus we should adopt a fixed-effects model.
Let’s say we have run a panel data regression with fixed-effects, and we think that no more needs to be done to control for factors that are constant across our cross-sectional variables (i.e. countries) at any one point in time (i.e. years). However, for very long series (for example those over 20 years), we will want to check that time dummy variables are not also needed.
The Stata command testparm tests whether the coefficients on three or more variables are equal to zero. When used after a fixed-effects panel data regression that includes time dummies, testparm will tell us if the dummies are equal to 0. If they are equal to zero, then no time-fixed-effects are needed. If they are not, we will want to include them in all of our regressions.
As we have already learned, we can add i.year to include a new dummy variable for each year and include that in our regression. Now, let’s test to see if that is necessary in the fixed-effects regression by running the command for testparm.
Random-effects GLS regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.5673 min = 30
Between = 0.5271 avg = 59.6
Overall = 0.4653 max = 70
Wald chi2(70) = 11032.47
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | .9953885 .0533463 18.66 0.000 .8908317 1.099945
|
year |
1951 | .0178031 .0697244 0.26 0.798 -.1188543 .1544605
1952 | .0259721 .0694461 0.37 0.708 -.1101397 .1620839
1953 | .0262229 .0689145 0.38 0.704 -.108847 .1612928
1954 | .053819 .0679378 0.79 0.428 -.0793367 .1869747
1955 | .0964718 .0670622 1.44 0.150 -.0349678 .2279114
1956 | .1219016 .067064 1.82 0.069 -.0095415 .2533446
1957 | .1455531 .0670669 2.17 0.030 .0141045 .2770018
1958 | .1477829 .0670708 2.20 0.028 .0163265 .2792392
1959 | .185058 .0666733 2.78 0.006 .0543807 .3157352
1960 | .26242 .0621097 4.23 0.000 .1406873 .3841528
1961 | .2753844 .0620285 4.44 0.000 .1538108 .396958
1962 | .3057185 .0619506 4.93 0.000 .1842976 .4271395
1963 | .309277 .0619687 4.99 0.000 .1878206 .4307333
1964 | .3625061 .0618943 5.86 0.000 .2411955 .4838166
1965 | .3923993 .0619157 6.34 0.000 .2710469 .5137518
1966 | .4045928 .0619431 6.53 0.000 .2831867 .525999
1967 | .4182151 .0619724 6.75 0.000 .2967514 .5396788
1968 | .4384561 .062004 7.07 0.000 .3169305 .5599817
1969 | .4726771 .0620378 7.62 0.000 .3510853 .5942689
1970 | .5309956 .0603276 8.80 0.000 .4127557 .6492354
1971 | .549766 .0603772 9.11 0.000 .4314289 .668103
1972 | .5697978 .0604308 9.43 0.000 .4513557 .68824
1973 | .5952057 .0604877 9.84 0.000 .4766519 .7137595
1974 | .6207654 .0605473 10.25 0.000 .5020948 .739436
1975 | .6107479 .0606101 10.08 0.000 .4919542 .7295415
1976 | .6360785 .0606861 10.48 0.000 .517136 .755021
1977 | .6468602 .0607667 10.64 0.000 .5277597 .7659608
1978 | .6535494 .0608511 10.74 0.000 .5342833 .7728154
1979 | .6578622 .0609388 10.80 0.000 .5384243 .7773001
1980 | .6581445 .0610291 10.78 0.000 .5385296 .7777593
1981 | .6472568 .0611513 10.58 0.000 .5274026 .7671111
1982 | .6205573 .0612795 10.13 0.000 .5004517 .7406629
1983 | .6021466 .0614131 9.80 0.000 .4817792 .7225141
1984 | .6037983 .0615516 9.81 0.000 .4831594 .7244372
1985 | .5789695 .0616955 9.38 0.000 .4580485 .6998904
1986 | .5658215 .0618343 9.15 0.000 .4446285 .6870146
1987 | .571656 .0619775 9.22 0.000 .4501824 .6931296
1988 | .573049 .0621251 9.22 0.000 .451286 .6948119
1989 | .5724024 .0622043 9.20 0.000 .4504842 .6943205
1990 | .6285131 .061546 10.21 0.000 .5078852 .749141
1991 | .5972128 .0617097 9.68 0.000 .476264 .7181616
1992 | .5732003 .0618758 9.26 0.000 .451926 .6944746
1993 | .559798 .0620456 9.02 0.000 .4381909 .6814051
1994 | .5535301 .0622197 8.90 0.000 .4315816 .6754786
1995 | .5775349 .0623968 9.26 0.000 .4552394 .6998303
1996 | .6067848 .0625627 9.70 0.000 .4841642 .7294054
1997 | .6172768 .0627314 9.84 0.000 .4943255 .7402281
1998 | .6004642 .0629023 9.55 0.000 .4771778 .7237505
1999 | .6137021 .0630744 9.73 0.000 .4900785 .7373256
2000 | .6560094 .0632477 10.37 0.000 .5320462 .7799725
2001 | .6595905 .0634081 10.40 0.000 .5353129 .7838682
2002 | .6736759 .0635703 10.60 0.000 .5490804 .7982715
2003 | .6980967 .0637339 10.95 0.000 .5731806 .8230129
2004 | .7491733 .0638976 11.72 0.000 .6239364 .8744102
2005 | .8281019 .0640635 12.93 0.000 .7025398 .953664
2006 | .8763791 .0642322 13.64 0.000 .7504862 1.002272
2007 | .9197114 .0644033 14.28 0.000 .7934833 1.04594
2008 | .9621744 .0645753 14.90 0.000 .8356091 1.08874
2009 | .921703 .0647463 14.24 0.000 .7948026 1.048603
2010 | .9858718 .0649159 15.19 0.000 .858639 1.113105
2011 | 1.043475 .0651151 16.03 0.000 .9158513 1.171098
2012 | 1.060546 .0653156 16.24 0.000 .9325295 1.188562
2013 | 1.053693 .0655184 16.08 0.000 .9252792 1.182107
2014 | 1.056419 .065723 16.07 0.000 .9276043 1.185234
2015 | 1.023772 .0659305 15.53 0.000 .8945501 1.152993
2016 | 1.015305 .066141 15.35 0.000 .8856715 1.14494
2017 | 1.021399 .0663533 15.39 0.000 .8913495 1.151449
2018 | 1.035342 .0665694 15.55 0.000 .9048686 1.165816
2019 | 1.036413 .0667891 15.52 0.000 .905509 1.167317
|
_cons | 7.439635 .077874 95.53 0.000 7.287005 7.592265
-------------+----------------------------------------------------------------
sigma_u | .66502037
sigma_e | .36415315
rho | .76932191 (fraction of variance due to u_i)
------------------------------------------------------------------------------
( 1) 1951.year = 0
( 2) 1952.year = 0
( 3) 1953.year = 0
( 4) 1954.year = 0
( 5) 1955.year = 0
( 6) 1956.year = 0
( 7) 1957.year = 0
( 8) 1958.year = 0
( 9) 1959.year = 0
(10) 1960.year = 0
(11) 1961.year = 0
(12) 1962.year = 0
(13) 1963.year = 0
(14) 1964.year = 0
(15) 1965.year = 0
(16) 1966.year = 0
(17) 1967.year = 0
(18) 1968.year = 0
(19) 1969.year = 0
(20) 1970.year = 0
(21) 1971.year = 0
(22) 1972.year = 0
(23) 1973.year = 0
(24) 1974.year = 0
(25) 1975.year = 0
(26) 1976.year = 0
(27) 1977.year = 0
(28) 1978.year = 0
(29) 1979.year = 0
(30) 1980.year = 0
(31) 1981.year = 0
(32) 1982.year = 0
(33) 1983.year = 0
(34) 1984.year = 0
(35) 1985.year = 0
(36) 1986.year = 0
(37) 1987.year = 0
(38) 1988.year = 0
(39) 1989.year = 0
(40) 1990.year = 0
(41) 1991.year = 0
(42) 1992.year = 0
(43) 1993.year = 0
(44) 1994.year = 0
(45) 1995.year = 0
(46) 1996.year = 0
(47) 1997.year = 0
(48) 1998.year = 0
(49) 1999.year = 0
(50) 2000.year = 0
(51) 2001.year = 0
(52) 2002.year = 0
(53) 2003.year = 0
(54) 2004.year = 0
(55) 2005.year = 0
(56) 2006.year = 0
(57) 2007.year = 0
(58) 2008.year = 0
(59) 2009.year = 0
(60) 2010.year = 0
(61) 2011.year = 0
(62) 2012.year = 0
(63) 2013.year = 0
(64) 2014.year = 0
(65) 2015.year = 0
(66) 2016.year = 0
(67) 2017.year = 0
(68) 2018.year = 0
(69) 2019.year = 0
chi2( 69) = 1363.89
Prob > chi2 = 0.0000Stata runs a joint test to see if the coefficients on the dummies for all years are equal to 0. The null hypothesis on this test is that they are equal to zero. As the p-value is less than 0.05, we can reject the null hypothesis and will want to include the year dummies in our analysis.
Panel data also provides us with a new source of variation: variation over time. This means that we have access to a wide variety of variables we can include. For instance, we can create lags (variables in previous periods) and leads (variables in future periods). Once we have defined our panel data set using the xtset command (which we did earlier) we can create the lags using Lnumber.variable and the leads using Fnumber.variable.
For example, let’s create a new variable that lags the natural log of GDP per capita by one period.
If we wanted to lag this same variable ten periods, we would write it as such:
We can include lagged variables directly in our regression if we believe that past values of real GDP per capita influence current levels of real GDP per capita.
Fixed-effects (within) regression Number of obs = 7,208
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.9721 min = 20
Between = 0.9999 avg = 49.7
Overall = 0.9954 max = 60
F(62, 7001) = 3940.00
corr(u_i, Xb) = 0.8913 Prob > F = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lngdp |
L1. | .9902435 .0038061 260.17 0.000 .9827823 .9977047
L10. | -.0399322 .0036316 -11.00 0.000 -.0470513 -.0328131
|
lnhc | .0195021 .0149505 1.30 0.192 -.0098054 .0488096
|
year |
1961 | .0012136 .0150409 0.08 0.936 -.0282712 .0306984
1962 | .0089642 .0149834 0.60 0.550 -.0204077 .0383362
1963 | .0005079 .0148733 0.03 0.973 -.0286482 .0296639
1964 | .0307714 .0146698 2.10 0.036 .0020141 .0595287
1965 | .0185671 .0144896 1.28 0.200 -.009837 .0469711
1966 | .0053137 .0144952 0.37 0.714 -.0231014 .0337287
1967 | -.0038829 .0145012 -0.27 0.789 -.0323097 .0245438
1968 | -.0014469 .014507 -0.10 0.921 -.029885 .0269913
1969 | .0282496 .0143934 1.96 0.050 .0000341 .0564652
1970 | .0342442 .0135273 2.53 0.011 .0077266 .0607617
1971 | .0140959 .0135249 1.04 0.297 -.0124169 .0406088
1972 | .0213268 .0135197 1.58 0.115 -.005176 .0478295
1973 | .0250646 .0135374 1.85 0.064 -.0014729 .0516021
1974 | .0244523 .0135415 1.81 0.071 -.0020932 .0509977
1975 | -.0109177 .0135639 -0.80 0.421 -.037507 .0156716
1976 | .0264938 .0135787 1.95 0.051 -.0001245 .0531121
1977 | .0172719 .0136042 1.27 0.204 -.0093965 .0439403
1978 | .0148633 .0136293 1.09 0.276 -.0118544 .0415809
1979 | .0102008 .0136565 0.75 0.455 -.0165702 .0369718
1980 | .0085316 .0133178 0.64 0.522 -.0175754 .0346385
1981 | .001079 .013354 0.08 0.936 -.0250989 .0272569
1982 | -.0138718 .0133915 -1.04 0.300 -.0401233 .0123796
1983 | -.0047129 .0134308 -0.35 0.726 -.0310413 .0216154
1984 | .0162799 .0134749 1.21 0.227 -.0101349 .0426948
1985 | -.0104281 .013513 -0.77 0.440 -.0369177 .0160615
1986 | .0014367 .0135592 0.11 0.916 -.0251434 .0280168
1987 | .0209369 .0136041 1.54 0.124 -.0057313 .0476051
1988 | .0170562 .0136506 1.25 0.212 -.009703 .0438155
1989 | .0193004 .0136972 1.41 0.159 -.0075503 .0461512
1990 | .0242895 .0137437 1.77 0.077 -.0026522 .0512312
1991 | -.0055244 .0137892 -0.40 0.689 -.0325553 .0215066
1992 | .0166306 .013828 1.20 0.229 -.0104765 .0437378
1993 | .0131152 .0138731 0.95 0.345 -.0140804 .0403107
1994 | .0209707 .0139261 1.51 0.132 -.0063287 .04827
1995 | .04204 .0139722 3.01 0.003 .0146504 .0694297
1996 | .0453148 .0140222 3.23 0.001 .0178271 .0728025
1997 | .0275372 .01408 1.96 0.051 -.0000638 .0551383
1998 | .0011516 .0141347 0.08 0.935 -.0265567 .0288599
1999 | .0320776 .0141664 2.26 0.024 .0043071 .0598481
2000 | .0578138 .0140784 4.11 0.000 .0302158 .0854118
2001 | .0177771 .0141252 1.26 0.208 -.0099125 .0454668
2002 | .0276489 .0141701 1.95 0.051 -.0001289 .0554267
2003 | .0378643 .014221 2.66 0.008 .0099869 .0657418
2004 | .0647254 .0142765 4.53 0.000 .0367391 .0927116
2005 | .0943584 .0143474 6.58 0.000 .0662333 .1224836
2006 | .0659652 .0144305 4.57 0.000 .037677 .0942534
2007 | .0622005 .0145032 4.29 0.000 .0337698 .0906311
2008 | .0613123 .0145712 4.21 0.000 .0327483 .0898764
2009 | -.0205425 .0146453 -1.40 0.161 -.0492517 .0081667
2010 | .0855195 .0146892 5.82 0.000 .0567243 .1143148
2011 | .0811768 .014778 5.49 0.000 .0522076 .1101461
2012 | .0419435 .0148695 2.82 0.005 .0127947 .0710923
2013 | .0193823 .014945 1.30 0.195 -.0099143 .048679
2014 | .031128 .0150153 2.07 0.038 .0016933 .0605626
2015 | -.0008208 .015101 -0.05 0.957 -.0304234 .0287818
2016 | .0252179 .0151697 1.66 0.096 -.0045193 .0549551
2017 | .0416232 .0152491 2.73 0.006 .0117304 .071516
2018 | .0515126 .0153355 3.36 0.001 .0214505 .0815748
2019 | .0374197 .0153921 2.43 0.015 .0072466 .0675928
|
_cons | .4137854 .0275731 15.01 0.000 .3597337 .4678371
-------------+----------------------------------------------------------------
sigma_u | .05404699
sigma_e | .08054093
rho | .3104913 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(144, 7001) = 2.65 Prob > F = 0.0000While we included lags from the previous period and 10 periods back as examples, we can use any period for our lags. In fact, including lag variables as controls for recent periods such as one lag back and two lags back is the most common choice for inclusion of past values of independent variables as controls.
Finally, these variables are useful if we are trying to measure the growth rate of a variable. Recall that the growth rate of a variable X is just equal to \(ln(X_{t}) - ln(X_{t-1})\) where the subscripts indicate time.
For example, if we want to now include the natural log of the population growth rate in our regression, we can create that new variable by taking the natural log of the population growth rate \(ln(pop_{t}) - ln(pop_{t-1})\)
Another variable that might also be useful is the natural log of the growth rate of GDP per capita.
Let’s put this all together in a regression and see what results we get:
Fixed-effects (within) regression Number of obs = 5,465
Group variable: ccode Number of groups = 140
R-squared: Obs per group:
Within = 0.0680 min = 2
Between = 0.0108 avg = 39.0
Overall = 0.0462 max = 60
F(71, 5254) = 5.40
corr(u_i, Xb) = -0.1746 Prob > F = 0.0000
------------------------------------------------------------------------------
dlngdp | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lngdp |
L1. | -.1853119 .04033 -4.59 0.000 -.2643755 -.1062483
|
lnhc | .2705899 .1966261 1.38 0.169 -.1148789 .6560587
lnn | -.0523217 .0265974 -1.97 0.049 -.1044636 -.0001798
|
year |
1952 | -.099967 .2305104 -0.43 0.665 -.5518633 .3519292
1953 | .0090347 .2217199 0.04 0.967 -.4256285 .443698
1954 | .1131285 .212706 0.53 0.595 -.3038636 .5301206
1955 | .0080017 .2107581 0.04 0.970 -.4051718 .4211752
1956 | -.4641057 .2093588 -2.22 0.027 -.8745361 -.0536754
1957 | -.1526395 .211047 -0.72 0.470 -.5663794 .2611005
1958 | -.6997954 .2277545 -3.07 0.002 -1.146289 -.253302
1959 | -.1232425 .2141894 -0.58 0.565 -.5431428 .2966578
1960 | .0462978 .2089085 0.22 0.825 -.3632497 .4558452
1961 | -.3480797 .2001159 -1.74 0.082 -.74039 .0442307
1962 | -.1151027 .19578 -0.59 0.557 -.4989128 .2687075
1963 | -.0732617 .1966156 -0.37 0.709 -.4587099 .3121865
1964 | .0327713 .1923909 0.17 0.865 -.3443948 .4099373
1965 | -.0570053 .1969998 -0.29 0.772 -.4432068 .3291962
1966 | -.5657967 .1954319 -2.90 0.004 -.9489244 -.182669
1967 | -.3328726 .1985421 -1.68 0.094 -.7220975 .0563524
1968 | -.1813929 .1940778 -0.93 0.350 -.5618661 .1990803
1969 | .0219682 .1936469 0.11 0.910 -.3576602 .4015966
1970 | .3018397 .1961577 1.54 0.124 -.082711 .6863905
1971 | -.2508719 .1916733 -1.31 0.191 -.6266313 .1248875
1972 | -.0877766 .1913399 -0.46 0.646 -.4628823 .2873291
1973 | .1639995 .1915518 0.86 0.392 -.2115216 .5395205
1974 | -.0058386 .1932006 -0.03 0.976 -.384592 .3729148
1975 | -.2887302 .2057281 -1.40 0.161 -.6920427 .1145823
1976 | .0649726 .1942182 0.33 0.738 -.3157758 .4457209
1977 | -.1821347 .1968885 -0.93 0.355 -.5681181 .2038487
1978 | -.1281384 .1971856 -0.65 0.516 -.5147042 .2584274
1979 | -.1026343 .1982133 -0.52 0.605 -.4912148 .2859461
1980 | -.2207011 .2022804 -1.09 0.275 -.6172548 .1758526
1981 | -.2144361 .2060877 -1.04 0.298 -.6184536 .1895815
1982 | -.8013063 .2191248 -3.66 0.000 -1.230882 -.3717306
1983 | -.2978487 .2101168 -1.42 0.156 -.7097649 .1140675
1984 | -.1921488 .2037307 -0.94 0.346 -.5915457 .207248
1985 | -.6192931 .2121557 -2.92 0.004 -1.035206 -.2033798
1986 | -.0247539 .2050512 -0.12 0.904 -.4267395 .3772316
1987 | -.0836047 .2049267 -0.41 0.683 -.4853462 .3181368
1988 | -.1773129 .2033501 -0.87 0.383 -.5759636 .2213378
1989 | -.216709 .2061752 -1.05 0.293 -.6208982 .1874801
1990 | -.1029481 .2037307 -0.51 0.613 -.5023449 .2964487
1991 | -.2113553 .2114437 -1.00 0.318 -.6258728 .2031622
1992 | -.2797653 .2095338 -1.34 0.182 -.6905387 .1310081
1993 | -.3005134 .2071802 -1.45 0.147 -.7066727 .1056458
1994 | .0027848 .2056116 0.01 0.989 -.4002994 .405869
1995 | -.0109579 .2042036 -0.05 0.957 -.4112818 .389366
1996 | .1573324 .2062262 0.76 0.446 -.2469568 .5616215
1997 | -.2668611 .2057516 -1.30 0.195 -.6702198 .1364975
1998 | -.2182675 .2151946 -1.01 0.310 -.6401383 .2036034
1999 | -.0502273 .2116084 -0.24 0.812 -.4650676 .364613
2000 | -.0443905 .2062698 -0.22 0.830 -.448765 .3599839
2001 | -.3561773 .218427 -1.63 0.103 -.7843851 .0720305
2002 | -.1815304 .2177039 -0.83 0.404 -.6083205 .2452598
2003 | -.1755016 .2099487 -0.84 0.403 -.5870883 .2360851
2004 | .07888 .2080334 0.38 0.705 -.3289518 .4867119
2005 | .428666 .2079812 2.06 0.039 .0209364 .8363957
2006 | .0705911 .2110394 0.33 0.738 -.3431339 .4843161
2007 | .2116385 .2104657 1.01 0.315 -.2009617 .6242387
2008 | .2391556 .2145273 1.11 0.265 -.1814071 .6597184
2009 | .0991337 .2403729 0.41 0.680 -.3720971 .5703644
2010 | .3560309 .2123908 1.68 0.094 -.0603433 .7724051
2011 | .3664368 .2138722 1.71 0.087 -.0528416 .7857153
2012 | -.1996652 .2189568 -0.91 0.362 -.6289115 .2295812
2013 | -.4467897 .2319267 -1.93 0.054 -.9014624 .0078829
2014 | -.237489 .2273998 -1.04 0.296 -.6832871 .2083092
2015 | -.1815811 .2342816 -0.78 0.438 -.6408703 .2777081
2016 | -.6170238 .2272141 -2.72 0.007 -1.062458 -.1715897
2017 | -.3791262 .2221476 -1.71 0.088 -.8146279 .0563755
2018 | -.5515301 .2236345 -2.47 0.014 -.9899467 -.1131135
2019 | -.7027819 .2269077 -3.10 0.002 -1.147615 -.2579485
|
_cons | -1.977118 .3548434 -5.57 0.000 -2.672758 -1.281477
-------------+----------------------------------------------------------------
sigma_u | .41161403
sigma_e | .99999749
rho | .14488033 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(139, 5254) = 4.47 Prob > F = 0.0000While there are the typical concerns with interpreting the coefficients of regressions (i.e. multicollinearity, inferring causality), there are some topics which require special treatment when working with panel data.
As always, when running regressions, we must consider whether our residuals are heteroskedastic (not constant for all values of \(X\)). To test our panel data regression for heteroskedasticity in the residuals, we need to calculate a modified Wald statistic. Fortunately, there is a Stata package available for installation that will make this test very easy for us to conduct. To install this package into your version of Stata, simply type:
checking xttest3 consistency and verifying not already installed...
all files already exist and are up to date.Let’s now test this with our original regression, the regression of log real GDP per capita on log human capital with the inclusion of fixed-effects.
Fixed-effects (within) regression Number of obs = 8,637
Group variable: ccode Number of groups = 145
R-squared: Obs per group:
Within = 0.4919 min = 30
Between = 0.5907 avg = 59.6
Overall = 0.6006 max = 70
F(1, 8491) = 8221.37
corr(u_i, Xb) = 0.2961 Prob > F = 0.0000
------------------------------------------------------------------------------
lngdp | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.072537 .0228576 90.67 0.000 2.02773 2.117343
_cons | 7.369692 .0159552 461.90 0.000 7.338416 7.400968
-------------+----------------------------------------------------------------
sigma_u | .73516756
sigma_e | .3932119
rho | .77755934 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(144, 8491) = 174.45 Prob > F = 0.0000
Modified Wald test for groupwise heteroskedasticity
in fixed effect regression model
H0: sigma(i)^2 = sigma^2 for all i
chi2 (145) = 75913.77
Prob > chi2 = 0.0000
The null hypothesis is homoskedasticity (or constant variance of the error term). From the output above, we can see that we reject the null hypothesis and conclude that the residuals in this regression are heteroskedastic.
The best method for dealing with heteroskedasticity in panel data regression is by using generalized least squares, or GLS. There are a number of techniques to estimate GLS equations in Stata, but the recommended approach is the Prais-Winsten method.
This is easily implemented by replacing the command xtreg with xtpcse and including the option het.
Linear regression, heteroskedastic panels corrected standard errors
Group variable: ccode Number of obs = 8,637
Time variable: year Number of groups = 145
Panels: heteroskedastic (unbalanced) Obs per group:
Autocorrelation: no autocorrelation min = 30
avg = 59.565517
max = 70
Estimated covariances = 145 R-squared = 0.6006
Estimated autocorrelations = 0 Wald chi2(1) = 16264.57
Estimated coefficients = 2 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Het-corrected
lngdp | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 2.652185 .0207961 127.53 0.000 2.611425 2.692944
_cons | 6.979568 .0162396 429.79 0.000 6.947738 7.011397
------------------------------------------------------------------------------In time-series setups where we only observe a single unit over time (no cross-sectional dimension) we might be worried that a linear regression model like
\[ Y_t = \alpha + \beta X_t + \varepsilon_t \]
can have errors that not only are heteroskedastic (i.e. that depend on observables \(X_t\)) but can also be correlated across time. For instance, if \(Y_t\) was income, then \(\varepsilon_t\) may represent income shocks (including transitory and permanent components). The permanent income shocks are, by definition, very persistent over time. This would mean that \(\varepsilon_{t-1}\) affects (and thus is correlated with) shocks in the next period \(\varepsilon_t\). This problem is called serial correlation or autocorrelation, and if it exists, the assumptions of the regression model (i.e. unbiasedness, consistency, etc.) are violated. This can take the form of regressions where a variable is correlated with lagged versions of the same variable.
To test our panel data regression for serial correlation, we need to run a Woolridge test. Fortunately, there are multiple packages in Stata available for installation that make this test automatic to conduct. Run the command below to see some of these packages.
Search of official help files, FAQs, Examples, and Stata Journals
-----------------------------------------------------------------
FAQ . . . . Testing for panel-level heteroskedasticity and autocorrelation
. . . . . . . . . . . . . . . . . . . . . . . . V. Wiggins and B. Poi
6/13 How do I test for panel-level heteroskedasticity
and autocorrelation?
http://www.stata.com/support/faqs/statistics/panel-level-
heteroskedasticity-and-autocorrelation/
SJ-3-2 st0039 . . Testing for serial correlation in linear panel-data models
(help xtserial if installed) . . . . . . . . . . . . . D. M. Drukker
Q2/03 SJ 3(2):168--177
test for serial correlation in random- or fixed-effects
one-way models that can be applied under general conditions
Search of web resources from Stata and other users
--------------------------------------------------
(contacting http://www.stata.com)
5 packages found (Stata Journal listed first)
---------------------------------------------
st0592 from http://www.stata-journal.com/software/sj20-1
SJ20-1 st0592. Jochmans (2019) test for serial ... / Jochmans (2019) test
for serial correlation in / panel-data models / by Koen Jochmans,
University of Cambridge, / Cambridge, UK / Vincenzo Verardi, Universite de
Namur, Namur, / Belgium / Support: kj345cam.ac.uk, vverardiunamur.be /
st0039 from http://www.stata-journal.com/software/sj3-2
SJ3-2 st0039. Testing for serial correlation in linear ... / Testing for
serial correlation in linear panel-data models / by David M. Drukker,
Stata Corporation / Support: ddrukker@stata.com / After installation,
type help xtserial
xtserial from http://www.stata.com/users/ddrukker
xtserial tests for serial correlation in linear panel-data models /
xtserial implements a test for serial correlation in the idiosyncratic /
errors of a linear panel-data model discussed by Wooldridge (2002). /
Drukker (2003) presents simulation evidence that this test has good size /
abar from http://fmwww.bc.edu/RePEc/bocode/a
'ABAR': module to perform Arellano-Bond test for autocorrelation / abar
performs the Arellano-Bond (1991) test for / autocorrelation. The test was
originally proposed for a / particular linear Generalized Method of
Moments dynamic panel / data estimator, but is quite general in its
xtserialpm from http://fmwww.bc.edu/RePEc/bocode/x
'XTSERIALPM': module to perform a portmanteau test for serial correlation
in panel data / xtserialpm performs the portmanteau test developed in
Jochmans / (2019). The procedure tests for serial correlation in the
errors / of a linear panel model after estimation of the regression /
(end of search)We can choose any one of these packages and follow the (brief) instructions to install it. Once it’s installed, we can conduct the Woolridge test for autocorrelation below.
The null hypothesis is that there is no serial correlation between residuals. From the output, we can see that we reject the null hypothesis and conclude the variables are correlated with lagged versions of themselves. One method for dealing with this is by using the same Prais-Winsten method to estimate a GLS equation. This is easily implemented by replacing the command xtreg with xtpcse and including the option corr(ar1).
note: estimates of rho outside [-1,1] bounded to be in the range [-1,1].
Prais–Winsten regression, heteroskedastic panels corrected standard errors
Group variable: ccode Number of obs = 8,637
Time variable: year Number of groups = 145
Panels: heteroskedastic (unbalanced) Obs per group:
Autocorrelation: common AR(1) min = 30
avg = 59.565517
max = 70
Estimated covariances = 145 R-squared = 0.7778
Estimated autocorrelations = 1 Wald chi2(1) = 890.10
Estimated coefficients = 2 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Het-corrected
lngdp | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
lnhc | 1.982825 .0664606 29.83 0.000 1.852565 2.113085
_cons | 7.426831 .0594964 124.83 0.000 7.31022 7.543441
-------------+----------------------------------------------------------------
rho | .9832443
------------------------------------------------------------------------------Note that we have continued to use the het option to account for heteroskedasticity in our standard errors. We can also see that our results have not drifted significantly from what they were originally when running our first, most simple regression of log GDP per capita on log human capital.
testparm to test both the need for year fixed-effects and, in the example we have been using here, country fixed-effects. Now that we have used encode to create a new country variable that is numeric, we can include country dummies simply by including i.ccode into our regression.
In the regressions that we have been running in this example, we have found that the level of human capital is correlated with the level of GDP per capita. But have we proven that having high human capital causes countries to be wealthier? Or is is possible that wealthier countries can afford to invest in human capital? This is known as the issue of reverse causality, and arises when our independent variable determines our dependent variable.
The Granger Causality test allows use to unpack some of the causality in these regressions. While understanding how this test works is beyond the scope of this notebook, we can look at an example using this data.
The first thing we need to do is ensure that our panel is balanced. In the Penn World Tables, there are no missing values for real GDP and for population, but there are missing values for human capital. We can balance our panel by simply dropping all of the observations that do not include that measure.
Next, we can run the test that is provided by Stata for Granger Causality: xtgcause. We need to install this package before we begin using the same approach you used with xtserial above.
Now let’s test the causality between GDP and human capital!
Dumitrescu & Hurlin (2012) Granger non-causality test results:
--------------------------------------------------------------
Lag order: 1
W-bar = 4.9331
Z-bar = 33.4892 (p-value = 0.0000)
Z-bar tilde = 31.4641 (p-value = 0.0000)
--------------------------------------------------------------
H0: lnhc does not Granger-cause lngdp.
H1: lnhc does Granger-cause lngdp for at least one panel (ccode).From our results, we can reject the null hypothesis that high levels of wealth in countries causes higher levels of human capital. The evidence seems to suggest that high human capital causes countries to be wealthier.
Please speak to your instructor, supervisor, or TA if you need help with this test.
In typical cross-sectional settings, it is hard to defend the selection on observables assumption (otherwise known as conditional independence). However, panel data allows us to control for unobserved time-invariant heterogeneity.
Consider the following example. Household income \(y_{jt}\) at time \(t\) can be split into two components:
\[ y_{jt} = e_{jt} + \Psi_{j} \]
where \(\Psi_{j}\) is a measure of unobserved household-level determinants of income, such as social programs targeted towards certain households.
Consider what happens when we compute each \(j\) household’s average income, average value of \(e\), and average value of \(\Psi\) across time \(t\) in the data:
\[ \bar{y}_{J}= \frac{1}{\sum_{j,t} \mathbf{1}\{ j = J \} } \sum_{j,t} y_{jt} \mathbf{1}\{ j = J \} \] \[ \bar{e}_{J}= \frac{1}{\sum_{j,t} \mathbf{1}\{ j = J \} } \sum_{j,t} e_{jt} \mathbf{1}\{ j = J \} \] \[ \bar{\Psi}_{J} = \Psi_{J} \]
Notice that the mean of \(\Psi_{j}\) does not change over time for a fixed household \(j\). Hence, we can subtract the two household level means from the original equation to get:
\[ y_{jt} - \bar{y}_{j} = e_{jt} - \bar{e}_{j} + \underbrace{ \Psi_{j} - \bar{\Psi}_{j} }_\text{equals zero!} \]
Therefore, we are able to get rid of the unobserved heterogeneity in household determinants of income via “de-meaning”! This is called a within-group or fixed-effects transformation. If we believe these types of unobserved errors/shocks are creating endogeneity, we can get rid of them using this powerful trick. In some cases, we may alternatively choose to do a first-difference transformation of our regression specification. This entails subtracting the regression in one period not from it’s expectation across time, but from the regression in the previous period. In this case, time-invariant characteristics are similarly removed from the regression since they are constant across all periods \(t\).
In this module, we’ve learned how to address linear regression in the case where we have access to two dimensions: cross-sectional variation and time variation. The usefulness of time variation is that it allows us to control for time-invariant components of the error term which may be causing endogeneity. We also investigated different ways for addressing problems such as heteroskedasticity and autocorrelation in our standard errors when working specifically with panel data. In the next module, we will cover a popular research design method: difference-in-differences.
| Command | Function |
|---|---|
xtset panelvar timevar, interval |
It tells Stata that we are working with panel data, as well as which variables are our panel variable, time variable, and what at what interval the data was recorded. |
xtreg depvar indepvar |
It runs a panel regression. We can add options to this, such as fe for fixed-effects, and re for random-effects. |
hausman model1 model2 |
It performs the Hausman test on model1 and model2 to determine which more accurately models our data. |
testparm i.varname |
It evaluates whether multiple coefficients are equal to zero. |
Lnumber.variable |
It creates a lagged variable. |
Fnumber.variable |
It creates a lead variable. |
xttest3 |
It calculates a modified Wald statistic to test for heteroskedasticity. |
xtpcse depvar indepvar, het |
It calculates a GLS regression to deal with heteroskedasticity, following the Prais-Winsten method. We can add corr(ar1) to account for serial correlation. |
xtserial depvar indepvar |
It conducts a Woolridge test for autocorrelation. |
xtgcause depvar indepvar |
It conducts a Granger Causality test for reverse causality. |