Tutorial 14: Working with grouped data
Johannes Karreth
RPOS 517, Day 14
This tutorial shows you:
- how to think about initial considerations when working with grouped data (in R)
- how to include fixed effects for groups in regression models
- how to estimate clustered standard errors
Note on copying & pasting code from the PDF version of this tutorial: Please note that you may run into trouble if you copy & paste code from the PDF version of this tutorial into your R script. When the PDF is created, some characters (for instance, quotation marks or indentations) are converted into non-text characters that R won’t recognize. To use code from this tutorial, please type it yourself into your R script or you may copy & paste code from the source file for this tutorial which is posted on my website.
Note on R functions discussed in this tutorial: I don’t discuss many functions in detail here and therefore I encourage you to look up the help files for these functions or search the web for them before you use them. This will help you understand the functions better. Each of these functions is well-documented either in its help file (which you can access in R by typing ?ifelse, for instance) or on the web. The Companion to Applied Regression (see our syllabus) also provides many detailed explanations.
As always, please note that this tutorial only accompanies the other materials for Day 14 (in this case, the readings in the syllabus) and that you need to have worked through the materials for that day before tackling this tutorial. More than on the other days of our seminar so far, these notes only scratch the surface of the issues arising with grouped data. I also strongly encourage you to self-study the theory behind fixed effects before using them in your own work. Two recent articles on the topic are worth a look; they include references to other canonical articles on fixed effects, so take a look - even though these articles go beyond the treatment of FEs in this tutorial:
- Bell, A. and Jones, K. (2015). “Explaining fixed effects: Random effects modeling of time-series cross-sectional and panel data.” Political Science Research and Methods 3(1): 133–153.
- Clark, T. S. and Linzer, D. A. (2015). “Should I use fixed or random effects?” Political Science Research and Methods 3(2): 399–408.
Both of these articles point you to textbook treatments of fixed effects in grouped data; you must consult at least one of these textbooks (e.g., Greene, Wooldridge) before using fixed effects in your work.
Grouped data and the OLS assumptions
In the tutorial for Day 7, you already encountered the two major types of grouped data in social science:
individuals nested in higher-level units
- Example: survey respondents in an international survey are “nested” in countries (500 respondents in the U.S., 500 in Canada, etc.)
observations from the same unit observed over at least 2 time periods (“time-series cross-sectional data”)
- Example: yearly country-level economic growth measures observed for 20 countries over 30 years
It should be clear from our previous discussions that both of these types of grouped data likely violate at least one OLS assumption: the assumption of independence of units. That is, the residual of observation \(k\) does not predict the residual of observation \(n\), or, in more descriptive terms, observations \(k\) and \(n\) have nothing in common that the regression model is not already accounting for. Grouped data likely violate this assumption because observations that are part of the same group likely share some characteristics that are difficult to model.
Similarly, grouped data may also be likely to produce heteroskedastic errors if groups exhibit different patterns of residuals. Think of the following two examples:
In studies of opinions toward policies, individuals in one country may all be framing their opinion in terms of some shared cultural experience germane to that country. Respondents from the same country are unlikely to be independent from each other.
Economic growth over time in the United States may be more volatile than in Sweden due to difficult-to-measure factors. Economic growth measures from the United States are not independent from each other.
A simple solution: group-specific intercepts
From Day 9 of our seminar, you recall that dummy variables can be used to assign separate intercepts to different groups. This means that each group receives a separate intercept for its (virtual) regression line, while the relationships between all other predictors are the same (i.e., parallel regression lines) for all groups. In a simple regression equation, we can include group dummies \(\alpha_j\) as follows, where groups are indexed by \(j\) and individual observations are indexed by \(i\):
\[ y_{ij} = \alpha + \beta x_i + \gamma_j + \varepsilon_i \]
If this equation represents the data-generating process, you can see in the figure below that simply fitting a regression with one intercept and one coefficient on \(x\) will yield an estimate of -6.1, but that coefficient estimate does not capture the relationship between \(x\) and \(y\):
summary(m1)##
## Call:
## lm(formula = y ~ x, data = sim.dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -26.2246 -7.9990 -0.0794 8.5255 28.0011
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -7.63022 0.48953 -15.59 <2e-16 ***
## x -6.08620 0.07762 -78.41 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.77 on 1998 degrees of freedom
## Multiple R-squared: 0.7547, Adjusted R-squared: 0.7546
## F-statistic: 6148 on 1 and 1998 DF, p-value: < 2.2e-16If you fit a model that includes dummy variables (\(\gamma_j\)) for each group, the slope of the regression line changes quite dramatically. In the figure below, each red line represents the regression line for each group. Note that although each group has its own intercept, all groups exhibit the same (positive) relationship between \(x\) and \(y\). You can see that not accounting for the groups will massively bias the estimate of \(\beta\); you can see this by comparing the slope of the black line and the red lines in the right figure. The black line is the slope from a regression that does not account for the groups. Once you do account for groups by estimating separate intercepts for each group \(j\), your estimate for \(\beta\) changes to 2.5.
summary(m2)##
## Call:
## lm(formula = y ~ x + factor(group), data = sim.dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.1255 -1.3554 -0.0137 1.3575 6.7746
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.79171 0.14563 -67.23 <2e-16 ***
## x 2.49291 0.03891 64.07 <2e-16 ***
## factor(group)b -10.14603 0.20337 -49.89 <2e-16 ***
## factor(group)c -20.09365 0.21400 -93.90 <2e-16 ***
## factor(group)d -29.96795 0.22937 -130.65 <2e-16 ***
## factor(group)e -39.91740 0.25551 -156.23 <2e-16 ***
## factor(group)f -49.82431 0.27905 -178.55 <2e-16 ***
## factor(group)g -59.98434 0.30932 -193.93 <2e-16 ***
## factor(group)h -69.93534 0.34039 -205.46 <2e-16 ***
## factor(group)i -79.83151 0.37202 -214.59 <2e-16 ***
## factor(group)j -89.76941 0.40167 -223.49 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.991 on 1989 degrees of freedom
## Multiple R-squared: 0.9916, Adjusted R-squared: 0.9916
## F-statistic: 2.361e+04 on 10 and 1989 DF, p-value: < 2.2e-16Fitting these separate intercepts for groups is commonly known as estimating fixed effects for groups. Informally, fixed effects have some advantages:
- FEs account for any group-level sources of variation in the outcome you’re trying to explain - sources of variation that are not captured in your regression otherwise.
- FEs allow you to estimate the “base level” of your outcome of interest for each group. That is, you can make statements such as “on average, economic growth is higher in the United States than it is in Sweden.”
- FEs are also popular as a tool for causal identification under certain circumstances (see., e.g, chapter 5 in Mostly Harmless Econometrics by Angrist and Pischke).
In this example, you can see that using FEs for groups now provides an estimate (\(\beta\)) for comparing the relationship between \(x\) and \(y\) within groups and not between groups. This is important (more on this below).
FEs in nested data: a survey example
To illustrate the advantages of fixed effects in nested data (e.g., survey respondents nested in different geographic units), we’ll use a cleaned-up and reduced version of the Cooperative Study of Electoral Systems. I provided these data on my website. We’ll use these data to estimate a model of satisfaction with democracy. The variables in this dataset are:
| Variable | Description |
|---|---|
satisdem |
Index of satisfaction with democracy, from 1 (dissatisfied) to 4 (satisfied) |
married |
Married |
employed |
Full-time employed |
union |
Union member |
voted |
Respondent voted in last election |
female |
Female respondent |
age |
Age |
education |
Education |
lrself |
Left/right self-placement |
cname |
Name of the respondent’s country and year of the last election |
To obtain a list of the countries represented in the survey, we can use the table() command:
cses_dat <- rio::import("http://www.jkarreth.net/files/cses3_reduced.csv")
table(cses_dat$cname)##
## AUSTRALIA (2007) AUSTRIA (2008) CANADA (2008)
## 1168 1004 1731
## CZECH REPUBLIC (2010) DENMARK (2007) ESTONIA (2011)
## 1510 1273 770
## FINLAND (2011) FRANCE (2007) GERMANY (2009)
## 1178 1940 1784
## GREECE (2009) ICELAND (2009) IRELAND (2007)
## 923 1094 674
## ISRAEL (2006) JAPAN (2007) NETHERLANDS (2010)
## 1065 1098 1897
## NEW ZEALAND (2008) NORWAY (2009) POLAND (2007)
## 803 1700 1788
## PORTUGAL (2009) SLOVAKIA (2010) SLOVENIA (2008)
## 1024 984 682
## SOUTH KOREA (2008) SPAIN (2008) SWEDEN (2006)
## 745 936 1019
## SWITZERLAND (2007) TAIWAN (2008) UNITED STATES (2008)
## 2945 1024 1830
## URUGUAY (2009)
## 812For this tutorial, I’ll treat the outcome - satisfaction with democracy, a 4-point scale - as continuous. Estimating a model with individual predictors “pools” all respondents and returns the following results:
sd.pooled <- lm(satisdem ~ married + employed + union + voted +
female + age + education + lrself + I(lrself^2), data = cses_dat)
summary(sd.pooled)##
## Call:
## lm(formula = satisdem ~ married + employed + union + voted +
## female + age + education + lrself + I(lrself^2), data = cses_dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9506 -0.6481 0.2277 0.3564 1.9264
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.9640444 0.0270099 72.716 < 2e-16 ***
## married -0.0196254 0.0087782 -2.236 0.025378 *
## employed 0.0413796 0.0092576 4.470 7.85e-06 ***
## union 0.0705390 0.0104848 6.728 1.75e-11 ***
## voted 0.2682937 0.0118711 22.601 < 2e-16 ***
## female -0.0057596 0.0082341 -0.699 0.484256
## age 0.0009801 0.0002677 3.662 0.000251 ***
## education 0.0319880 0.0024815 12.891 < 2e-16 ***
## lrself 0.0785386 0.0060129 13.062 < 2e-16 ***
## I(lrself^2) -0.0052712 0.0005484 -9.612 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7631 on 34853 degrees of freedom
## (538 observations deleted due to missingness)
## Multiple R-squared: 0.03281, Adjusted R-squared: 0.03256
## F-statistic: 131.4 on 9 and 34853 DF, p-value: < 2.2e-16Sidenote: the “effects” package you encountered last week can also be used for OLS:
library(effects)
plot(allEffects(sd.pooled))
As you saw above, the countries in this survey are quite diverse. Would it make sense to consider respondents from the same country independent? And would you be worried about pooling respondents from a new democracy such as Slovakia and an established democracy such as the United States? Fixed effects can partially address these concerns. I am including FEs for countries (\(\gamma_j\)) by adding factor(cname) to the regression equation.
sd.fe <- lm(satisdem ~ married + employed + union + voted +
female + age + education + lrself + I(lrself^2) + factor(cname),
data = cses_dat)
summary(sd.fe)##
## Call:
## lm(formula = satisdem ~ married + employed + union + voted +
## female + age + education + lrself + I(lrself^2) + factor(cname),
## data = cses_dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.32681 -0.34898 0.04048 0.45081 2.24753
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.6234390 0.0337276 77.783 < 2e-16 ***
## married 0.0117220 0.0080508 1.456 0.145400
## employed -0.0002075 0.0085399 -0.024 0.980618
## union -0.0033486 0.0108239 -0.309 0.757043
## voted 0.1687882 0.0112828 14.960 < 2e-16 ***
## female -0.0230978 0.0075162 -3.073 0.002120 **
## age -0.0008120 0.0002492 -3.258 0.001123 **
## education 0.0237596 0.0024210 9.814 < 2e-16 ***
## lrself 0.0660353 0.0055609 11.875 < 2e-16 ***
## I(lrself^2) -0.0040384 0.0005085 -7.942 2.05e-15 ***
## factor(cname)AUSTRIA (2008) -0.2475184 0.0300769 -8.230 < 2e-16 ***
## factor(cname)CANADA (2008) -0.2258618 0.0266192 -8.485 < 2e-16 ***
## factor(cname)CZECH REPUBLIC (2010) -0.7069347 0.0274955 -25.711 < 2e-16 ***
## factor(cname)DENMARK (2007) 0.1488733 0.0288080 5.168 2.38e-07 ***
## factor(cname)ESTONIA (2011) -0.6335524 0.0325568 -19.460 < 2e-16 ***
## factor(cname)FINLAND (2011) -0.2700551 0.0290803 -9.287 < 2e-16 ***
## factor(cname)FRANCE (2007) -0.3603989 0.0259710 -13.877 < 2e-16 ***
## factor(cname)GERMANY (2009) -0.4899690 0.0265025 -18.488 < 2e-16 ***
## factor(cname)GREECE (2009) -1.0509429 0.0308224 -34.097 < 2e-16 ***
## factor(cname)ICELAND (2009) -0.8102822 0.0300474 -26.967 < 2e-16 ***
## factor(cname)IRELAND (2007) -0.1718445 0.0338612 -5.075 3.90e-07 ***
## factor(cname)ISRAEL (2006) -0.9488008 0.0298130 -31.825 < 2e-16 ***
## factor(cname)JAPAN (2007) -0.6195115 0.0294470 -21.038 < 2e-16 ***
## factor(cname)NETHERLANDS (2010) -0.2640142 0.0260388 -10.139 < 2e-16 ***
## factor(cname)NEW ZEALAND (2008) -0.3921531 0.0321194 -12.209 < 2e-16 ***
## factor(cname)NORWAY (2009) -0.0091268 0.0265973 -0.343 0.731489
## factor(cname)POLAND (2007) -0.7727139 0.0269114 -28.713 < 2e-16 ***
## factor(cname)PORTUGAL (2009) -0.8151904 0.0300565 -27.122 < 2e-16 ***
## factor(cname)SLOVAKIA (2010) -0.7804179 0.0303292 -25.732 < 2e-16 ***
## factor(cname)SLOVENIA (2008) -0.8120247 0.0338397 -23.996 < 2e-16 ***
## factor(cname)SOUTH KOREA (2008) -0.7337151 0.0331257 -22.149 < 2e-16 ***
## factor(cname)SPAIN (2008) -0.1404658 0.0310338 -4.526 6.02e-06 ***
## factor(cname)SWEDEN (2006) -0.1101505 0.0301230 -3.657 0.000256 ***
## factor(cname)SWITZERLAND (2007) -0.1161815 0.0244739 -4.747 2.07e-06 ***
## factor(cname)TAIWAN (2008) -0.5626945 0.0299643 -18.779 < 2e-16 ***
## factor(cname)UNITED STATES (2008) 0.0151381 0.0277483 0.546 0.585379
## factor(cname)URUGUAY (2009) -0.0910952 0.0320575 -2.842 0.004491 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6946 on 34826 degrees of freedom
## (538 observations deleted due to missingness)
## Multiple R-squared: 0.1992, Adjusted R-squared: 0.1983
## F-statistic: 240.6 on 36 and 34826 DF, p-value: < 2.2e-16You can visualize the fixed effects using the dotchart() function you learned last week. First, you have to calculate the actual country intercepts. Recall that using dummy variables based on a categorical variable will result in the omission of the coefficient for one group (the baseline category). In our example, you may notice that the first country-wave (Australia 2007) is not represented in the regression output for the fixed effects model. When calculating country fixed effects (i.e., country intercepts), you need to take this into account. As before, you can think of the overall intercept (\(\alpha\)) as the intercept for an observation where all other covariates are 0. In this case, this is Australia (2007). The next country’s intercept (Austria 2008) is \(\alpha + \gamma_{\text{Austria}}\), i.e. 2.623439 \(+\) -0.2475184 = 2.3759206.
fe.dat <- data.frame(fe = coef(sd.fe)[c(1, grep("cname", names(coef(sd.fe))))],
cname = unique(cses_dat$cname))
mean <- fe.dat[1, ]$fe
fe.dat[-1, ]$fe <- fe.dat[-1, ]$fe + mean
fe.dat <- fe.dat[order(fe.dat$fe), ]
dotchart(x = fe.dat$fe, labels = fe.dat$cname, cex = 0.75,
pch = 19, xlab = "Satisfaction with democracy")
Using fixed effects here helps you account for unobserved sources of variation at the country-level. While this may be desirable, it also limits you: the fixed effects now account for all country-level sources of explanation for satisfaction with democracy. That is, if you would like to estimate the relationship between country-level covariates (such as the state of the economy, the integrity of the democratic process, etc.) and individual-level satisfaction with democracy, you cannot do so while fixed effects are part of the model. This should be logical: a dummy variable for each country and any country-level predictor will be perfectly collinear and the model is not identified.
See for yourself: here, I include a country-level predictor for economic growth before the election, gdpgrowth:
sd.fe2 <- lm(satisdem ~ married + employed + union + voted +
female + age + education + lrself + I(lrself^2) + factor(cname) +
gdpgrowth,
data = cses_dat)
summary(sd.fe2)##
## Call:
## lm(formula = satisdem ~ married + employed + union + voted +
## female + age + education + lrself + I(lrself^2) + factor(cname) +
## gdpgrowth, data = cses_dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.32681 -0.34898 0.04048 0.45081 2.24753
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.6234390 0.0337276 77.783 < 2e-16 ***
## married 0.0117220 0.0080508 1.456 0.145400
## employed -0.0002075 0.0085399 -0.024 0.980618
## union -0.0033486 0.0108239 -0.309 0.757043
## voted 0.1687882 0.0112828 14.960 < 2e-16 ***
## female -0.0230978 0.0075162 -3.073 0.002120 **
## age -0.0008120 0.0002492 -3.258 0.001123 **
## education 0.0237596 0.0024210 9.814 < 2e-16 ***
## lrself 0.0660353 0.0055609 11.875 < 2e-16 ***
## I(lrself^2) -0.0040384 0.0005085 -7.942 2.05e-15 ***
## factor(cname)AUSTRIA (2008) -0.2475184 0.0300769 -8.230 < 2e-16 ***
## factor(cname)CANADA (2008) -0.2258618 0.0266192 -8.485 < 2e-16 ***
## factor(cname)CZECH REPUBLIC (2010) -0.7069347 0.0274955 -25.711 < 2e-16 ***
## factor(cname)DENMARK (2007) 0.1488733 0.0288080 5.168 2.38e-07 ***
## factor(cname)ESTONIA (2011) -0.6335524 0.0325568 -19.460 < 2e-16 ***
## factor(cname)FINLAND (2011) -0.2700551 0.0290803 -9.287 < 2e-16 ***
## factor(cname)FRANCE (2007) -0.3603989 0.0259710 -13.877 < 2e-16 ***
## factor(cname)GERMANY (2009) -0.4899690 0.0265025 -18.488 < 2e-16 ***
## factor(cname)GREECE (2009) -1.0509429 0.0308224 -34.097 < 2e-16 ***
## factor(cname)ICELAND (2009) -0.8102822 0.0300474 -26.967 < 2e-16 ***
## factor(cname)IRELAND (2007) -0.1718445 0.0338612 -5.075 3.90e-07 ***
## factor(cname)ISRAEL (2006) -0.9488008 0.0298130 -31.825 < 2e-16 ***
## factor(cname)JAPAN (2007) -0.6195115 0.0294470 -21.038 < 2e-16 ***
## factor(cname)NETHERLANDS (2010) -0.2640142 0.0260388 -10.139 < 2e-16 ***
## factor(cname)NEW ZEALAND (2008) -0.3921531 0.0321194 -12.209 < 2e-16 ***
## factor(cname)NORWAY (2009) -0.0091268 0.0265973 -0.343 0.731489
## factor(cname)POLAND (2007) -0.7727139 0.0269114 -28.713 < 2e-16 ***
## factor(cname)PORTUGAL (2009) -0.8151904 0.0300565 -27.122 < 2e-16 ***
## factor(cname)SLOVAKIA (2010) -0.7804179 0.0303292 -25.732 < 2e-16 ***
## factor(cname)SLOVENIA (2008) -0.8120247 0.0338397 -23.996 < 2e-16 ***
## factor(cname)SOUTH KOREA (2008) -0.7337151 0.0331257 -22.149 < 2e-16 ***
## factor(cname)SPAIN (2008) -0.1404658 0.0310338 -4.526 6.02e-06 ***
## factor(cname)SWEDEN (2006) -0.1101505 0.0301230 -3.657 0.000256 ***
## factor(cname)SWITZERLAND (2007) -0.1161815 0.0244739 -4.747 2.07e-06 ***
## factor(cname)TAIWAN (2008) -0.5626945 0.0299643 -18.779 < 2e-16 ***
## factor(cname)UNITED STATES (2008) 0.0151381 0.0277483 0.546 0.585379
## factor(cname)URUGUAY (2009) -0.0910952 0.0320575 -2.842 0.004491 **
## gdpgrowth NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.6946 on 34826 degrees of freedom
## (538 observations deleted due to missingness)
## Multiple R-squared: 0.1992, Adjusted R-squared: 0.1983
## F-statistic: 240.6 on 36 and 34826 DF, p-value: < 2.2e-16The coefficient for GDP growth cannot be estimated because GDP growth does not vary for all respondents within one country.
To reiterate, using FEs in this example allows you to use regression to explain variation in satisfaction with democracy within countries: why are some people within one country more satisfied with democracy than others? The coefficients of the FE model provide these inferences: more educated respondents report higher satisfaction with democracy, etc.
Note: in both the pooled and FE specifications, you assume that there is one “true” relationships between predictors (e.g., education) and satisfaction with democracy, and that your model can approximate this one true relationship.
FEs in TSCS data: a political economy example
You can also think of time-series cross-sectional data as nested: observations are “nested” in time. As briefly discussed above, it may be desirable or necessary to account for unobserved sources of variation between groups by using fixed effects. An example comes from Geoffrey Garrett’s 1998 book Partisan Politics in the Global Economy. These data are discussed in section 5.2 of a recent article by Nathaniel Beck and Jonathan Katz, “Modeling Dynamics in Time-Series–Cross-Section Political Economy Data”, in the Annual Review of Political Science (Vol. 14: pp. 331-352). One of the chapters in Garrett’s book asks left-leaning governments in OECD countries are associated with lower economic growth rates. To answer this question, Garrett regresses economic growth rates on a measure of leftist participation in government (the percentage of cabinet posts held by members of Left parties), an indicator of the global economic climate (OECD demand), an indicator for oil dependency of a country, and an indicator for the institutionalization of centralized wage bargaining or corporatism. Garrett’s argument suggests that leftist governments should oversee less economic growth only when wage bargaining is not centralized, therefore he includes an interaction term of leftist government participation and corporatism.
Replication data are available from Jonathan Katz’s dataverse page for the 2011 article, as a tab-delimited file under the name “garrett1998.tab”. You can download the file from there. The relevant variables in it are:
| Variable | Description |
|---|---|
year |
Year of the observation |
country |
Country (as Correlates of War country codes) |
gdp |
Growth rate of GDP |
gdpl |
Growth rate of GDP (lagged from previous year) |
leftlab |
% of cabinet members from Left parties (transformed) |
corp |
Corporatism index |
demand |
Overall OECD GDP growth, weighted for each country by its trade with the other OECD nations |
oild |
Oil dependence of the economy |
First, I load the data and assign country names to the country codes. For this, I use the “countrycode” package.
garrett_dat <- rio::import("http://www.jkarreth.net/files/garrett1998.tab", format = "tsv")
summary(garrett_dat)## country year unem infl
## Min. : 2.0 Min. :1966 Min. : 0.6848 Min. :-0.700
## 1st Qu.:210.0 1st Qu.:1972 1st Qu.: 2.0992 1st Qu.: 3.700
## Median :282.5 Median :1978 Median : 4.5000 Median : 5.900
## Mean :287.4 Mean :1978 Mean : 4.9939 Mean : 6.688
## 3rd Qu.:380.0 3rd Qu.:1984 3rd Qu.: 7.3000 3rd Qu.: 9.075
## Max. :740.0 Max. :1990 Max. :13.0000 Max. :24.500
## gdp uneml infll gdpl
## Min. :-4.300 Min. : 0.600 Min. :-0.700 Min. :-4.300
## 1st Qu.: 1.877 1st Qu.: 2.000 1st Qu.: 3.800 1st Qu.: 2.000
## Median : 3.200 Median : 3.900 Median : 5.900 Median : 3.300
## Mean : 3.254 Mean : 4.832 Mean : 6.683 Mean : 3.336
## 3rd Qu.: 4.700 3rd Qu.: 7.100 3rd Qu.: 8.900 3rd Qu.: 4.795
## Max. :12.800 Max. :13.000 Max. :24.500 Max. :12.800
## trade capmob oild Icc_2
## Min. : 9.623 Min. :0.0000 Min. :-0.117810 Min. :0.00000
## 1st Qu.: 41.419 1st Qu.:0.0000 1st Qu.: 0.003662 1st Qu.:0.00000
## Median : 52.624 Median :1.0000 Median : 0.013674 Median :0.00000
## Mean : 57.076 Mean :0.8914 Mean : 0.015280 Mean :0.07143
## 3rd Qu.: 71.847 3rd Qu.:1.0000 3rd Qu.: 0.030052 3rd Qu.:0.00000
## Max. :146.020 Max. :4.0000 Max. : 0.083460 Max. :1.00000
## Icc_3 Icc_4 Icc_5 Icc_6
## Min. :0.00000 Min. :0.00000 Min. :0.00000 Min. :0.00000
## 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000
## Median :0.00000 Median :0.00000 Median :0.00000 Median :0.00000
## Mean :0.07143 Mean :0.07143 Mean :0.07143 Mean :0.07143
## 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000
## Max. :1.00000 Max. :1.00000 Max. :1.00000 Max. :1.00000
## Icc_7 Icc_8 Icc_9 Icc_10
## Min. :0.00000 Min. :0.00000 Min. :0.00000 Min. :0.00000
## 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000
## Median :0.00000 Median :0.00000 Median :0.00000 Median :0.00000
## Mean :0.07143 Mean :0.07143 Mean :0.07143 Mean :0.07143
## 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000
## Max. :1.00000 Max. :1.00000 Max. :1.00000 Max. :1.00000
## Icc_11 Icc_12 Icc_13 Icc_14
## Min. :0.00000 Min. :0.00000 Min. :0.00000 Min. :0.00000
## 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000 1st Qu.:0.00000
## Median :0.00000 Median :0.00000 Median :0.00000 Median :0.00000
## Mean :0.07143 Mean :0.07143 Mean :0.07143 Mean :0.07143
## 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000 3rd Qu.:0.00000
## Max. :1.00000 Max. :1.00000 Max. :1.00000 Max. :1.00000
## per6673 per7479 per8084 per8690 corp
## Min. :0.00 Min. :0.00 Min. :0.0 Min. :0.0 Min. :0.415
## 1st Qu.:0.00 1st Qu.:0.00 1st Qu.:0.0 1st Qu.:0.0 1st Qu.:2.189
## Median :0.00 Median :0.00 Median :0.0 Median :0.0 Median :3.230
## Mean :0.32 Mean :0.24 Mean :0.2 Mean :0.2 Mean :3.016
## 3rd Qu.:1.00 3rd Qu.:0.00 3rd Qu.:0.0 3rd Qu.:0.0 3rd Qu.:3.794
## Max. :1.00 Max. :1.00 Max. :1.0 Max. :1.0 Max. :4.820
## leftlab clint demand
## Min. :0.05066 Min. : 0.085 Min. :-39.17
## 1st Qu.:1.14301 1st Qu.: 2.542 1st Qu.: 92.59
## Median :2.17484 Median : 6.396 Median :177.52
## Mean :2.02770 Mean : 6.763 Mean :187.42
## 3rd Qu.:2.80958 3rd Qu.:10.015 3rd Qu.:257.06
## Max. :3.57084 Max. :17.162 Max. :644.79# install.packages("countrycode")
library(countrycode)
garrett_dat$cname <- countrycode(sourcevar = garrett_dat$country,
origin = "cown",
destination = "country.name")
table(garrett_dat$cname)##
## Austria Belgium
## 25 25
## Canada Denmark
## 25 25
## Federal Republic of Germany Finland
## 25 25
## France Italy
## 25 25
## Japan Netherlands
## 25 25
## Norway Sweden
## 25 25
## United Kingdom United States
## 25 25garrett_dat$cname <- ifelse(garrett_dat$cname == "Federal Republic of Germany",
"Germany",
garrett_dat$cname)
table(garrett_dat$cname)##
## Austria Belgium Canada Denmark Finland
## 25 25 25 25 25
## France Germany Italy Japan Netherlands
## 25 25 25 25 25
## Norway Sweden United Kingdom United States
## 25 25 25 25To capture Garrett’s argument about that the impact of leftist government participation on economic growth is conditional on the degree of corporatism, I include an interaction term between the two variables in the model below. This model pools all observations and assumes that each observation is independent from each other (once we’ve accounted for temporal autocorrelation by including a lag of the outcome variable). In other words, there is nothing inherently different about the United States in 1980 compared to Sweden in 1980, once we’ve controlled for economic growth in the previous year, the share of leftists in the cabinet, the level of corporatism, oil dependence, and the global economic climate.
gdp.pooled <- lm(gdp ~ gdpl + oild + demand + leftlab * corp, data = garrett_dat)
summary(gdp.pooled)##
## Call:
## lm(formula = gdp ~ gdpl + oild + demand + leftlab * corp, data = garrett_dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.1171 -1.2909 -0.0205 1.3530 6.6253
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.7514261 0.7356121 5.100 5.64e-07 ***
## gdpl 0.3696199 0.0478508 7.724 1.25e-13 ***
## oild -9.3066684 4.2953481 -2.167 0.03095 *
## demand 0.0052377 0.0009239 5.669 3.04e-08 ***
## leftlab -1.1000959 0.3588064 -3.066 0.00234 **
## corp -0.8592037 0.2836856 -3.029 0.00264 **
## leftlab:corp 0.3329478 0.1151210 2.892 0.00407 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.099 on 343 degrees of freedom
## Multiple R-squared: 0.2683, Adjusted R-squared: 0.2555
## F-statistic: 20.96 on 6 and 343 DF, p-value: < 2.2e-16Because the original paper contains an interaction term, we can plot the marginal effects:
library(devtools)
source_url("https://raw.githubusercontent.com/jkarreth/JKmisc/master/ggintfun.R")
ggintfun(gdp.pooled, varnames = c("leftlab", "corp"),
varlabs = c("% cabinet members from Left", "Corporatism"),
title = FALSE, rug = TRUE,
twoways = FALSE)
However, you may suspect that the control variables listed above do not capture all potential country-level characteristics that might influence economic growth. To account for all between-country variation, you can include fixed effects and re-estimate the model.
gdp.fe <- lm(gdp ~ gdpl + oild + demand + leftlab * corp + factor(cname), data = garrett_dat)
summary(gdp.fe)##
## Call:
## lm(formula = gdp ~ gdpl + oild + demand + leftlab * corp + factor(cname),
## data = garrett_dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9246 -1.1679 -0.0681 1.1914 5.4880
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.212508 2.826633 0.075 0.94012
## gdpl 0.250525 0.048714 5.143 4.65e-07 ***
## oild -18.415379 5.898115 -3.122 0.00195 **
## demand 0.008451 0.001088 7.767 1.02e-13 ***
## leftlab -0.887424 0.434177 -2.044 0.04175 *
## corp -0.416717 0.642045 -0.649 0.51676
## factor(cname)Belgium 0.406168 1.173264 0.346 0.72942
## factor(cname)Canada 2.079316 1.772377 1.173 0.24157
## factor(cname)Denmark 0.482628 0.798703 0.604 0.54608
## factor(cname)Finland 1.696598 0.879336 1.929 0.05454 .
## factor(cname)France 2.726026 2.344243 1.163 0.24573
## factor(cname)Germany 1.472264 0.974398 1.511 0.13176
## factor(cname)Italy 2.135334 1.145047 1.865 0.06309 .
## factor(cname)Japan 4.840057 1.739499 2.782 0.00571 **
## factor(cname)Netherlands 0.554957 1.640841 0.338 0.73542
## factor(cname)Norway -0.078887 0.975611 -0.081 0.93560
## factor(cname)Sweden -0.021682 0.637137 -0.034 0.97287
## factor(cname)United Kingdom 1.018206 1.429937 0.712 0.47693
## factor(cname)United States 2.602201 1.843101 1.412 0.15893
## leftlab:corp 0.375131 0.155862 2.407 0.01664 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.966 on 330 degrees of freedom
## Multiple R-squared: 0.3821, Adjusted R-squared: 0.3466
## F-statistic: 10.74 on 19 and 330 DF, p-value: < 2.2e-16You will notice that the \(R^2\) value has increased (which should make sense). You can now make the statement that all between country-level variation has been taken up by the fixed effects. This has one important implication that’s often overlooked. In this model with fixed effects, the coefficient estimates on leftist governments, corporatism, global demand, and oil dependence now take on a fundamentally different meaning: they express the relationship between a within-country change on \(x\) and \(y\). That is, you can not interpret the coefficient on oil dependency as “countries that depend more on oil exhibit less economic growth compared to other less oil-dependent countries.” Rather, these coefficients mean that “as a country becomes more dependent on oil, it exhibits lower economic growth compared to the same country when it was less dependent on oil.” This is an important distinction for when it comes to testing hypotheses about between-country variation: fixed effects do not allow this interpretation.
Lastly, you can plot the marginal effects again:
ggintfun(gdp.fe, varnames = c("leftlab", "corp"),
varlabs = c("% cabinet members from Left", "Corporatism"),
title = FALSE, rug = TRUE,
twoways = FALSE)
You will notice that once between-country variance is taken up by the inclusion of fixed effects, the following interpretation emerges: when a country moves toward more centralized wage bargaining, leftist cabinet participation is associated with more economic growth, while leftist cabinet participation has no impact on economic growth in the same country with lower levels of centralized wage bargaining.
Fixed effects: some caveats
As you saw in the discussion of the two examples above, fixed effects can be useful to account for unobserved confounders at the group level in grouped data. However, using fixed effects in regression has consequences and comes with some caveats. Some of these are:
- The interpretation of regression coefficients changes fundamentally and does not allow for the testing of hypotheses about differences between groups (see the Garrett example)
- Fixed effects take up all the variation between groups. This means that any covariates that are constant at the group level (e.g., the geographic size of a country) cannot be estimated in fixed-effects models
- Fixed effects estimate residual sources of between-group variation, but do not help you in explaining these sources of variation.
- Other important caveats can be found in the articles by Clark and Linzer and Bell and Jones and the references therein.
Managing grouped data
Grouped data will, more likely than not, require some data management, for instance when merging data or calculating group averages. You can achieve nearly all of these tasks using the “dplyr” package that you encountered in the tutorial for Day 7. Revisit that tutorial for information on how to reshape grouped data, use time-series operators, and other related skills.
Further reading
This discussion barely scratches the surface of grouped data and fixed effects, and does not provide any theoretical foundation for their use. Please see the recommendations above for further and important background on this topic. For further reading, you should consult the sources in the syllabus. An excellent introduction to grouped data (among other things) is:
- Gelman,A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. New York: Cambridge University Press.