Growth Curve Analysis of CO2 Emission Per Capita
Introduction
The co-movement and causal relationship between CO2 Emission and growth has been investigated by various studies. Some of these studies focuse on the Enviromental Kuznets Curve analysis and try to test validity of EKC, while other studies directly explore this relationship in terms of growth basis. There are also some studies that investigate this relationship by considering both growth and enviromental base.
There are significant number of studies that investigate EKC hypothesis and try to validate that. Selden and Song(1994) investigate inverted U relationship between pollution and economic development by using a cross national panel of data on emissions and they find that per capita emissions of pollutants indicates inverted U-relationship with per capita GDP.
Holtz-Eakin and Selden(1995) examine the relationship between economic development and carbon-dioxide emissions. Their study suggests that low income countries with high marginal propensity to emit have a tendency to continue emission growth.
Agras and Chapman(1999) try to examine the EKC by testing energy-income and CO2-income relationships. They find no evidence for the existence of EKC when they analyse current incomes for energy with price and trade variables.
Andreoni and Levinson(2001) investigate EKC by using a straight-forward static model. They find evidence that suggests the curve doesn’t depend on the dynamics of growth and other extarnalities.
Coondoo and Dinda(2002) examine the income-CO2 emission by Granger causality in the range of countries cross section. They find different results between income and emissions for different regions.
Galeotti et al.(2009) focuse on the question whether the Enviromental Kuznet’s Curve exists or not. Their results indicate that the EKC hypothesis remains a delicate concept.
Park and Lee(2011) examine the relationship between economic development and air pollution at the regional level for Korea. They find the potential existence of U-shaped and N-shaped curves while the results don’t indicate a dominant shape of EKC.
Shahpaz et al.(2014) investigate the existance of EKC in case of Tunisia by using annual time series data for the period of 1971-2010. The results of their sudy indicate that there is a long run relationship between economic growth, energy consumption, trade openness and CO2 emissions. They also concludes the existance of EKC curve.
There are also various studies that investigate the causality and co-movement relationship between energy and income on the basis of growth. This type of relationship is used by researchers to see economic development of a country and it’s energy dependency.
For instance, if there is a causality running from income to energy, this type of relationship indicates less-dependent economy, that means energy conservation policies has no or little effect on income(Jumbe,2004). If there exists causality running from energy consumption to income, this type of relationship indicates energy-dependent economy, that implies income may be negatively affected due to shortage of energy.(Lee, 2005). There may exist bi-directional causality between energy consumption and income.(Soytas and Sari, 2003).In the study of (Soytas and Sari, 2003), they found bi-directional causality runnig for Argentina. Altınay and Karagol (2004) find no causality between energy consumption and GDP by using data for the period 1950-2000 for Turkey.
Oh and Lee (2004) analysis the causal relationship between energy consumption and economic growth by using querterly data for the period 1981:1 to 2000:4 for South Korea. Oh and Lee (2004) find long-run undirectional
causality from GDP to energy consumption.
Pokharel (2007) investigates the relationship between energy consumption and economic growth by taking into account different econometric models for Nepal. The study of Pokharel (2007) shows the importance of energy end-use patterns and their implications on economic growth.
The study by Lise and Montfont (2007) find causality running undirectionally from GDP to energy consumption for Turkey by using data for the period 1970-2003.This implies that the role of energy consumption is relatively small on income.
Hu and Lin (2008) find non-linear long run equilibrium relationship between GDP and diaggregated energy consumption for Taiwan by using querterly data for the period of 1982:1 to 2006:4.
Constantini and Martini (2010) estimate the causal relationship between energy consumption and economic growth for a large sample of developed and developing countries by considering the data for the period 1960-2005. They find different results for different sectors.
Apergis and Payne (2010) examine the causal relationship between energy consumption and economic growth for South American countries by using data for the period 1980-2005, similarly. The evidence in their work confirm the hypothesis that the energy consumption has importance in the growth process for South America. There are also some recent studies that examine the relationship between energy and income on the multivariate basis.
Iriani(2006) investigates the causal relationship between gross domestic product and energy consumption in the six countries of the Gulf Cooperation Coincil. The results shows that there is a uni-directional causality running from GDP to energy consumption.
Keppler and Betaller(2010) try to analyse the relationship between electricity and gas price, carbon and CO2 emissions in European Union. They find statistically robust and theoretically coherent results.
Niu et. al(2011) investigate the causal relationship between energy consumption, GDP growth and carbon emissions for eight Asia-Pacific countries from 1971 to 2005. They find there is a long-run relationship between these variables.
Jayanthakumaran et al.(2012) compare China and India in terms of the relationship between relationships between growth, trade, energy use both in the short-run and in the long run. They find results that shows causal relationship runs from per capita income to CO2 emission. However, they can not find this type of causal relationship for India.
Andersson and Karpestam(2013) examine the short-term and the long-term determinants of energy intensity, carbon intensity and scale effects for eight developed economies and two emerging economies from 1973 to 2007.They find that there is a difference between the short-term and the long-term results. They conclude that the climate policies affect emmission more than the short-term.,
Menegaki and Ozturk(2013) investigate the causal relationship between economic growth and energy for 26 European countries. They use a multivariate panel framework over the period 1975–2009 that uses a two-way fixed effects model and including greenhouse gas emissions,capital,fossil energy consumption. The results of their study indicate that there is a bi-directional relationship growth and political stability, capital and fossil energy consumption.
Cowan et. al (2014) investigate the causal relationship between electricity consumption, economic growth and CO2 emissions in the BRICS countries for the period 1980-2010. The results of their study indicate different results for different countries. So, they conclude that the energy policies can not be uniformly implemented for all of the countries in BRICS.
Data and Methodology
Growth Curve Analysis
Explanation of GCA
Growth curve analysis is a multilevel regression technique designed for analysis of time course or longitudinal data.Growth curve analysis provides a way to address those challenges by explicitly modeling change over time and quantifying both group-level and individual-level differences.(Mirman, 2014)
Structure of Growth Curve Model
Level 1
GCA analysis begin with setting up base model in which each variables in the model has growth paths that is generated by some parameters in the model. Then, the second level models will be generated by adding these parameters as group effect on the intercept and linear term of the model respectively.
At the first level, the base model is set up where Yij is given by time j and individual i.
So,
Yij = α0i + α1i * Timej + εij
At the base model, α1i is the growth rate of individual i and the residuals are assumed to be identically and normally distributed.
Level 2
GCA analysis continues with level 2 models which are generated by adding group effects(specified by level 1 model) on the intercept and linear term of the model respectively.
So, level 2 model where group effect is added on the intercept is as follows:
α0i = γ00 + γ0D * D + τ0i
And, level 2 model where group effect is added on the linear term of the model can be given as follows:
α1i = γ10 + γ1D * D + τ1i
In the second level models, γ0D represents the group effect of D on the intercept and γ1D represents the group effect of D on the linear term, τ0i represents unexplained variance in intercept and τ1i represents unexplained variance in slope.
1.3. Longest Common Subsequence
Explanation of LCS
LCS is used to find similar common patterns within symbolic data.It is adapted to find the common patterns of real valued sequences as well.(Imer and Ozkan, 2007)
It is to find the longest subsequence common to all sequences in a set of sequences, as well.
LCS function defined (Wikipedia)
Let two sequences be defined as follows: X = {X1,X2, ... , Xm} and Y = {Y1,Y2, ... , Yn}.
The prefixes of X are X1, 2, ..., m and the prefixes of Y are X1, 2, ..., n. Let LCS(Xi,Yj ) represent the set of longest common subsequence of prefixes Xi and Yj. This set of sequences is given by the following:
$$
L(y_i) = \begin{cases}
0 & \text{if } i=0 text{or } j=0 \\
LCS(X_{i-1},Y_{j-1}) + 1 & \text{if } x_{i}=y_{j}\\
longest(LCS(X_{i}+Y_{j-1}),LCS(X_{i-1}+Y_{j})) & \text{if } X_{i} \neq Y_{j}
\end{cases}
$$
LCS(Xi, Yj) = 0 if i = 0 or j = 0
LCS(Xi, Yj) = LCS(Xi − 1, Yj − 1) + 1 if xi=yj
LCS(Xi, Yj) = longest(LCS(Xi + Yj − 1),LCS(Xi − 1 + Yj) if Xi different from Yj
To find the longest subsequences common to Xi and Yj, compare the elements Xi and Yj. If they are equal, then the sequence LCS(Xi − 1, Yj) is extended by that element, Xi. If they are not equal, then the longer of the two sequences, LCS(Xi, Yj − 1), and LCS(Xi − 1, Yj), is retained. (If they are both the same length, but not identical, then both are retained.) Notice that the subscripts are reduced by 1 in these formulas. That can result in a subscript of 0. Since the sequence elements are defined to start at 1, it was necessary to add the requirement that the LCS is empty when a subscript is zero.
1.4 The Linear Panel Model
The basic linear panel models used in econometrics can be described through suitable restrictions of the following general model(Croissant and Millo, 2008):
Yit = αit + βit + εij
The basic linear panel models used in econometrics can be described through suitable restrictions of the following general model(Croissant and Millo, 2008):
Yit = αit + βitTxit + uit
where i = 1, ..., n is the individual(group,country,...) index, t = 1, ..., T is the time index and uit a random disturbance term of mean 0.
If the individual component is correlated with the regressors, then the ordinary least squares (OLS) estimator for β would be inconsistent, so it is customary to treat the μi as a further set of n parameters to be estimated, as if in the general model αit = αi for all t. This is called the fixed effects (also known as within or least squares dummy variables) model, usually estimated by OLS on transformed data, and gives consistent estimates for β, which can be seen as follows:
Yit = αi + βiTxit + μi + εit
If the individual-specific component μi is uncorrelated with the regressors, a situation which is usually termed random effects, the overall error uit also is, so the OLS estimator is consistent.
Packages, data and plotting
Installing and Loading Necessary Packages
In the first step , install and load packages.
library(knitr)
opts_chunk$set(highlight=TRUE, echo=TRUE, warning=FALSE, message=FALSE)
library(XML) #In order to read xml files, this package is loaded. Step 2.2
library(WDI) # In order to get data from WDI, this package is loaded. Step 2.2.3 library(ggplot2) # In order to plot WDI data, this package is loaded. Step 2.3
library(lme4) # In order to use Linear Mixed Models, this package is loaded. Step 4library(plm) # In order to use Panel Linear Model, this package is loaded. Step 5.library(googleVis) # In order to plot gVisNotionChart, this package is loaded. Step 2.3.1library(xlsx) # In order to read and write xlsx files, this package is loaded. Step 3.1 Obtaining the Data Set
Reading the OECD Country CodesIn order to get OECD countries list and their iso codes, following commands are run.
library(XML) #In order to read xml files, this package is loaded. Step 2.2
# Read web page that contains OECD Countries list
page <- readLines("http://en.wikipedia.org/wiki/Organisation_for_Economic_Co-operation_and_Development")
# Read OECD Countries list
oecd<-readHTMLList(page, which=2)
# Read web page that contains countries iso codes
page1<-readLines("http://en.wikipedia.org/wiki/ISO_3166-1_alpha-2")
# Read iso countries table from page
iso<-readHTMLTable(page1, which=3)In order to get OECD countries iso codes , following for loop is used
oecdcode<-as.character(1:length(oecd))
for(i in 1:length(oecd)){
for( j in 1:nrow(iso)){
if((oecd[i]==iso[j,2])==TRUE){
oecdcode[i]<-as.character(iso[j,1])
}
}
}
oecdcode[28]<-"KR" #South Korea's ISO code is KROECD Countries Growth Data
Get OECD countries' GDP(NY.GDP.MKTP.CD GDP), CO2 Emission(EN.ATM.CO2E.KT), Population(SP.POP.TOTL), Energy Consumption(EG.USE.COMM.KT.OE) and Gross capital Formation(NE.GDI.TOTL.CD) data between 1980-2013 from WDI Database.
inds <- c('NY.GDP.MKTP.CD','EN.ATM.CO2E.KT','NE.GDI.TOTL.CD','EG.USE.COMM.KT.OE',
'SP.POP.TOTL')
indnams <- c("GDP", "CO2 Emission","GPF","Energy Use","Population")
encar <- WDI(country=oecdcode, indicator=inds,
start=1980, format(Sys.Date(), "%Y"),extra=FALSE)Visualization of the Data Set
It is always good to start with Motion Chart like Hans Rosling style for getting as much information as possible out of this type of data set.
#### GoogleVisMotion Chart
library(googleVis)
op <- options(gvis.plot.tag='chart')
colnum <- match(inds, names(encar))
names(encar)[colnum] <- indnams
names(encar)[1] <- paste("Code")
names(encar)[2] <- paste("Country")
names(encar)[3] <- paste("Year")
names(encar)[4] <- paste("GDP")
names(encar)[5] <- paste("CO2")
names(encar)[6] <- paste("GPF")
names(encar)[7] <- paste("ENUSE")
names(encar)[8] <- paste("Population")
# Divide by population in order to get GDP per capita, CO2 Emission per capita, Energy use per capita
encar <- transform(encar, GDPPC = encar[,4] / encar[,8], CO2PC =encar[,5] / encar[,8], GPFPC = encar[,6] / encar[,8], ENUSEPC =encar[,7] / encar[,8])
suppressPackageStartupMessages(library(googleVis))
# Define GoogleVisMotionChart object
N <- gvisMotionChart(encar,
idvar="Country", timevar="Year",
xvar="CO2PC", yvar="GDPPC",
sizevar="Population",
options=list(width=700, height=600))
# In order to load gvisMotionChart, you may have to add your working directory to the trusted location in the global security settings of your Flash Player.
# print(N, "chart")
plot(N)ggplot of OECD countries CO2PC and GDPPC data
library(ggplot2)
ggplot(data=encar, aes(x=CO2PC, y=GDPPC,
color=Country))+
xlab("CO2 emissions (metric tons per capita)")+
ylab("GDP per capita, PPP (current international $)")+
geom_point() + geom_path() + theme_bw() + theme(legend.position="none")
Take log GDP, CO2 Emission, Gross capital formation and GDPPC
encar[,4:6]<-log(encar[,4:6])
encar[,9]<-log(encar[,9])ggplot of GDPPC, CO2PC and CO2PC/GDPPC by years
ggplot(encar, aes(Year, GDPPC),shape=Country) +
stat_summary(fun.data=mean_se, geom="pointrange") + theme_bw()
ggplot(encar, aes(Year, CO2PC),shape=Country) +
stat_summary(fun.data=mean_se, geom="pointrange") + theme_bw()
ggplot(encar, aes(Year, CO2PC/GDPPC),shape=Country) +
stat_summary(fun.data=mean_se, geom="pointrange") + theme_bw()
## Exploration of Data
Longest Common Subsequence for Trajectory Similarity
library(xlsx) # In order to read and write xlsx files, this package
# Read and convert data to matrix in order to use LCS algorithm
CO2PCGDPPC<-read.xlsx("CO2LGDP.xlsx", sheetIndex = 1)
CO2PCGDPPC<-data.frame(CO2PCGDPPC)
CO2PCGDPPC<-as.matrix(CO2PCGDPPC)
# LCSDistM(Imer and Ozkan, 2007) function is defined in order to compute LCS distance between countries
LCSDistM = function(dat) {
require(qualV)
nc=dim(dat)[2]
ld2=matrix(rep(0,nc*nc),nrow=nc)
nms=colnames(dat)
colnames(ld2)=nms
rownames(ld2)=nms
for (i in 1:nc) {
for (j in i:nc) {
lcs <- LCS(f.curve((1:length(na.omit(dat[,i]))),na.omit(dat[,i])),
f.curve((1:length(na.omit(dat[,j]))),na.omit(dat[,j]))) # too much noise
ld2[i,j]=lcs$QSI
}
}
ld2=1-ld2
hc.reslc=hclust(as.dist(t(ld2)))
plot(hc.reslc)
return(ld2)
}
# Call LCSDistM function
LCSCO2GDP<-LCSDistM(CO2PCGDPPC)
Visualization and Exploration Results
By the results of Cluster Dendrogram, we can conclude that there are similarities between some countries. We can seperate these cluster groups as follows:
- Korea Republic and United States
- Switzerland, Greece together with Portugal
- Denmark, New Zealand together with Turkey
- Belgium, Hungary together with Israel and Mexico
- Japan and Spain together with Luxemburg
- Netherlands, Austria together with Sweden
- Norway, France together with Italy
- Canada and United Kingdom
- Australia and Iceland
- Chile, Finland together with Ireland
- Germany and Poland
Czech Republic, Slovak Republic with Estonia and Slovenia
By results of LCS Cluster Dendrogram, we can conclude following results:Czech Republic and Slovak Republic together with Estonia and Slovenia have the highest similarities among these groups. Czech Republic and Slovak Republic are geographically very close and they were part of Austria- Hungary in the past. Estonia and Slovenia are geographically close similarly. They have similar behaviours when we evaluate them in terms of energy basis.
Canadia and United Kingdom have similar behaviours. These countries were dominion of British Empire in the past. They are commonwealth countries, and they have some other similaties, as well.
Germany and Poland form another group. Again, these countries are very close geographically. Similar growth path behaviours can be observed when looking at these countries CO2 Emission per capita and GDP per capita growth paths.
Another interesting cluster group contains France and Norway. Despite of being not close geographically, having some dissimalities growth path of CO2 Emission per capita and growth path of GDP per capita have very similar trends.
Despite of having some dissimilarities, Greece and Switzerland form another group. These countries have also similar behaviors when we look at their CO2 Emission per capita and GDP per capita movements.
Another interesting group consists of Japan and Spain together with Luzemburg. These countries economic growth charts have very similar patterns while lookign at their economic growth chart.
USA and Korea Republic form another group as well. They have similar economic growth path, especially after then 2000.
Modelling: Growth Curve Analysis
Level 1 and level 2 models are given below where CO2PC:GDPPC is group effect that is added to the intercept and linear term of the model respectively:
** 4.1.1 Base model **
GDPPCij = α0i + α1i * Yearj + εij
** 4.1.2 Level 2 models **
α0i = γ00 + γ0CO2PC : GDPPC * CO2PC : GDPPC + τ0i
α1i = γ10 + γ1CO2PC : GDPPC * CO2PC : GDPPC + τ1i
library(lme4) # In order to use Linear Mixed Models, this package is loaded. Step 4
library(plm) # In order to use Panel Linear Model, this package is loaded. Step 5.
# First, we set up a base model m.base which has a random effect, but no fixed effect of CO2PC:GDPPC
m.base <- lmer(GDPPC ~ 1 + Year+ (1+Year |Country),
data=encar, REML=FALSE)
# Then, we add the CO2PC:GDPPC effect on the intercept with a new model m.0
m.0 <- lmer(GDPPC ~ 1 + Year + I(CO2PC/GDPPC) + (1 + Year | Country),
data=encar, REML=FALSE)
# Finally, we add the effect of CO2PC:GDPPC on the linear term with a new model m.1
m.1 <- lmer(GDPPC ~ 1 + Year + I(CO2PC/GDPPC) + Year:I(CO2PC/GDPPC) +
(1 + Year | Country), data=encar, REML=FALSE)Summary of models
summary(m.base)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: GDPPC ~ 1 + Year + (1 + Year | Country)
## Data: encar
##
## AIC BIC logLik deviance df.resid
## 42.5 72.5 -15.3 30.5 1076
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -3.224 -0.672 0.013 0.685 3.156
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Country (Intercept) 9.96e-02 0.3155
## Year 3.99e-08 0.0002 1.00
## Residual 5.02e-02 0.2241
## Number of obs: 1082, groups: Country, 34
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -1.08e+02 1.46e+00 -74.0
## Year 5.90e-02 7.33e-04 80.6
##
## Correlation of Fixed Effects:
## (Intr)
## Year -0.996
summary(m.0)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: GDPPC ~ 1 + Year + I(CO2PC/GDPPC) + (1 + Year | Country)
## Data: encar
##
## AIC BIC logLik deviance df.resid
## -25.0 9.3 19.5 -39.0 982
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.8072 -0.6691 -0.0144 0.6794 3.0359
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Country (Intercept) 5.67e-01 7.53e-01
## Year 1.14e-09 3.38e-05 -1.00
## Residual 4.63e-02 2.15e-01
## Number of obs: 989, groups: Country, 34
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -1.09e+02 1.63e+00 -67.0
## Year 5.95e-02 8.08e-04 73.6
## I(CO2PC/GDPPC) 8.59e+01 5.46e+01 1.6
##
## Correlation of Fixed Effects:
## (Intr) Year
## Year -0.997
## I(CO2PC/GDP -0.228 0.198
In the level 1 model in which the group effect(CO2PC:GDPPC) is added to the intercept. Fixed effect intercept value − 1.091 * e + 0.2 refers to the intercept of the model, 0.06160 refers to the Year group's slope, 4979.35138 refers to CO2PC:GDPPC group's intercept relative to Year group's intercept.
summary(m.1)## Linear mixed model fit by maximum likelihood ['lmerMod']
## Formula: GDPPC ~ 1 + Year + I(CO2PC/GDPPC) + Year:I(CO2PC/GDPPC) + (1 +
## Year | Country)
## Data: encar
##
## AIC BIC logLik deviance df.resid
## -18.0 21.2 17.0 -34.0 981
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.7795 -0.6644 -0.0105 0.6660 3.0747
##
## Random effects:
## Groups Name Variance Std.Dev. Corr
## Country (Intercept) 5.35e-02 0.231364
## Year 6.02e-08 0.000245 1.00
## Residual 4.64e-02 0.215368
## Number of obs: 989, groups: Country, 34
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -1.14e+02 3.76e+00 -30.2
## Year 6.17e-02 1.89e-03 32.6
## I(CO2PC/GDPPC) 5.14e+03 3.85e+03 1.3
## Year:I(CO2PC/GDPPC) -2.55e+00 1.94e+00 -1.3
##
## Correlation of Fixed Effects:
## (Intr) Year I(CO2P
## Year -0.999
## I(CO2PC/GDP -0.903 0.905
## Y:I(CO2PC/G 0.902 -0.904 -1.000
In the level 1 model in which the group effect(CO2PC:GDPPC) is added to the linear term. Fixed effect intercept value -113.43157 refers to the intercept of the model , 0.06160 refers to the Year group's slope and 4979.35138 refers to CO2PC:GDPPC group's intercept relative to Year group's intercept and -2.46834 refers to CO2PC:GDPPC group's slope relative to Year group's slope.
Fixed and Random effects of models
m.base fixed effect
fixef(m.base)## (Intercept) Year
## -108.26744 0.05904
So, the intercept of the fixed effect of the base model is -108.25905357 and Year group's slope is 0.05903105.
m.base random effect
# Intercept and slope of the random effect model on each country are given below
ranef(m.base)## $Country
## (Intercept) Year
## Australia 0.182803 1.157e-04
## Austria 0.215556 1.364e-04
## Belgium 0.196035 1.241e-04
## Canada 0.205055 1.298e-04
## Chile -0.537751 -3.404e-04
## Czech Republic -0.370071 -2.342e-04
## Denmark 0.309967 1.962e-04
## Estonia -0.487165 -3.084e-04
## Finland 0.225720 1.429e-04
## France 0.192698 1.220e-04
## Germany 0.204747 1.296e-04
## Greece -0.095304 -6.032e-05
## Hungary -0.480312 -3.040e-04
## Iceland 0.283887 1.797e-04
## Ireland 0.130870 8.283e-05
## Israel -0.005012 -3.173e-06
## Italy 0.121422 7.685e-05
## Japan 0.274964 1.740e-04
## Korea, Rep. -0.264000 -1.671e-04
## Luxembourg 0.462609 2.928e-04
## Mexico -0.499436 -3.161e-04
## Netherlands 0.215955 1.367e-04
## New Zealand 0.038774 2.454e-05
## Norway 0.398610 2.523e-04
## Poland -0.572471 -3.623e-04
## Portugal -0.187112 -1.184e-04
## Slovak Republic -0.516729 -3.271e-04
## Slovenia -0.170109 -1.077e-04
## Spain -0.018012 -1.140e-05
## Sweden 0.290676 1.840e-04
## Switzerland 0.412818 2.613e-04
## Turkey -0.631763 -3.999e-04
## United Kingdom 0.159408 1.009e-04
## United States 0.312675 1.979e-04
Dotplot of random effect terms is given below:
# Save and print random effect terms on each country
re1 <- ranef(m.base, condVar=TRUE, whichel = "Country")
print(re1)## $Country
## (Intercept) Year
## Australia 0.182803 1.157e-04
## Austria 0.215556 1.364e-04
## Belgium 0.196035 1.241e-04
## Canada 0.205055 1.298e-04
## Chile -0.537751 -3.404e-04
## Czech Republic -0.370071 -2.342e-04
## Denmark 0.309967 1.962e-04
## Estonia -0.487165 -3.084e-04
## Finland 0.225720 1.429e-04
## France 0.192698 1.220e-04
## Germany 0.204747 1.296e-04
## Greece -0.095304 -6.032e-05
## Hungary -0.480312 -3.040e-04
## Iceland 0.283887 1.797e-04
## Ireland 0.130870 8.283e-05
## Israel -0.005012 -3.173e-06
## Italy 0.121422 7.685e-05
## Japan 0.274964 1.740e-04
## Korea, Rep. -0.264000 -1.671e-04
## Luxembourg 0.462609 2.928e-04
## Mexico -0.499436 -3.161e-04
## Netherlands 0.215955 1.367e-04
## New Zealand 0.038774 2.454e-05
## Norway 0.398610 2.523e-04
## Poland -0.572471 -3.623e-04
## Portugal -0.187112 -1.184e-04
## Slovak Republic -0.516729 -3.271e-04
## Slovenia -0.170109 -1.077e-04
## Spain -0.018012 -1.140e-05
## Sweden 0.290676 1.840e-04
## Switzerland 0.412818 2.613e-04
## Turkey -0.631763 -3.999e-04
## United Kingdom 0.159408 1.009e-04
## United States 0.312675 1.979e-04
##
## with conditional variances for "Country"
require(lattice)
# Dotplot of re1
dotplot(re1)## $Country
So, the highest random effect on intercept is observed on Luxemburg, and the least random effect on intercept is observed on Turkey.
fixef(m.0)## (Intercept) Year I(CO2PC/GDPPC)
## -109.17712 0.05946 85.87451
So, the intercept of the fixed effect of the base model is -109.11010321, Year group's slope is 0.05942099 and 87.34863755 refers to CO2PC:GDPPC group's intercept relative to Year group's intercept.
m.0 random effect
# Intercepts and slopes of the random effect model on each country are given below
ranef(m.0)## $Country
## (Intercept) Year
## Australia 0.361590 -1.624e-05
## Austria 0.554658 -2.491e-05
## Belgium 0.477833 -2.146e-05
## Canada 0.443120 -1.990e-05
## Chile -1.316527 5.912e-05
## Czech Republic -0.962225 4.321e-05
## Denmark 0.770571 -3.460e-05
## Estonia -1.312406 5.894e-05
## Finland 0.555040 -2.493e-05
## France 0.516084 -2.318e-05
## Germany 0.489359 -2.198e-05
## Greece -0.220070 9.883e-06
## Hungary -1.195123 5.367e-05
## Iceland 0.748944 -3.363e-05
## Ireland 0.315771 -1.418e-05
## Israel -0.009398 4.220e-07
## Italy 0.331602 -1.489e-05
## Japan 0.706314 -3.172e-05
## Korea, Rep. -0.664424 2.984e-05
## Luxembourg 1.015859 -4.562e-05
## Mexico -1.193285 5.359e-05
## Netherlands 0.529639 -2.378e-05
## New Zealand 0.105088 -4.719e-06
## Norway 0.984403 -4.421e-05
## Poland -1.476847 6.632e-05
## Portugal -0.428193 1.923e-05
## Slovak Republic -1.070662 4.808e-05
## Slovenia -0.424098 1.905e-05
## Spain -0.018281 8.210e-07
## Sweden 0.757881 -3.403e-05
## Switzerland 1.061030 -4.765e-05
## Turkey -1.541461 6.922e-05
## United Kingdom 0.404197 -1.815e-05
## United States 0.704019 -3.162e-05
Dotplot of random effect terms is given below:
# Save and print random effect terms on each country
re2 <- ranef(m.0, condVar=TRUE, whichel = "Country")
print(re2)## $Country
## (Intercept) Year
## Australia 0.361590 -1.624e-05
## Austria 0.554658 -2.491e-05
## Belgium 0.477833 -2.146e-05
## Canada 0.443120 -1.990e-05
## Chile -1.316527 5.912e-05
## Czech Republic -0.962225 4.321e-05
## Denmark 0.770571 -3.460e-05
## Estonia -1.312406 5.894e-05
## Finland 0.555040 -2.493e-05
## France 0.516084 -2.318e-05
## Germany 0.489359 -2.198e-05
## Greece -0.220070 9.883e-06
## Hungary -1.195123 5.367e-05
## Iceland 0.748944 -3.363e-05
## Ireland 0.315771 -1.418e-05
## Israel -0.009398 4.220e-07
## Italy 0.331602 -1.489e-05
## Japan 0.706314 -3.172e-05
## Korea, Rep. -0.664424 2.984e-05
## Luxembourg 1.015859 -4.562e-05
## Mexico -1.193285 5.359e-05
## Netherlands 0.529639 -2.378e-05
## New Zealand 0.105088 -4.719e-06
## Norway 0.984403 -4.421e-05
## Poland -1.476847 6.632e-05
## Portugal -0.428193 1.923e-05
## Slovak Republic -1.070662 4.808e-05
## Slovenia -0.424098 1.905e-05
## Spain -0.018281 8.210e-07
## Sweden 0.757881 -3.403e-05
## Switzerland 1.061030 -4.765e-05
## Turkey -1.541461 6.922e-05
## United Kingdom 0.404197 -1.815e-05
## United States 0.704019 -3.162e-05
##
## with conditional variances for "Country"
# Dotplot of re2
dotplot(re2)## $Country
So, the highest random effect on intercept is observed on Switzerland, and the least random effect on intercept is observed on Turkey.
m.1 fixed effect
fixef(m.1)## (Intercept) Year I(CO2PC/GDPPC)
## -113.55102 0.06166 5136.64719
## Year:I(CO2PC/GDPPC)
## -2.54759
So, the intercept of the fixed effect of the base model is -113.43157107, Year group's slope is 0.06160238, 4979.35138535 refers to CO2PC:GDPPC group's intercept relative to Year group's intercept and -2.46834253 refers to CO2PC:GDPPC group's slope relative to Year group's slope.
# Intercepts and slopes of the random effect model on each country are given below
ranef(m.1)## $Country
## (Intercept) Year
## Australia 0.113255 1.201e-04
## Austria 0.160383 1.701e-04
## Belgium 0.140409 1.489e-04
## Canada 0.135959 1.442e-04
## Chile -0.390222 -4.139e-04
## Czech Republic -0.274992 -2.917e-04
## Denmark 0.224838 2.385e-04
## Estonia -0.375531 -3.983e-04
## Finland 0.163447 1.734e-04
## France 0.147164 1.561e-04
## Germany 0.143228 1.519e-04
## Greece -0.065427 -6.940e-05
## Hungary -0.352213 -3.736e-04
## Iceland 0.216172 2.293e-04
## Ireland 0.092601 9.822e-05
## Israel -0.002925 -3.102e-06
## Italy 0.094777 1.005e-04
## Japan 0.205699 2.182e-04
## Korea, Rep. -0.193798 -2.056e-04
## Luxembourg 0.307776 3.264e-04
## Mexico -0.354046 -3.755e-04
## Netherlands 0.155501 1.649e-04
## New Zealand 0.029095 3.086e-05
## Norway 0.286493 3.039e-04
## Poland -0.430279 -4.564e-04
## Portugal -0.128851 -1.367e-04
## Slovak Republic -0.313893 -3.329e-04
## Slovenia -0.125501 -1.331e-04
## Spain -0.007924 -8.404e-06
## Sweden 0.217309 2.305e-04
## Switzerland 0.305592 3.241e-04
## Turkey -0.456018 -4.837e-04
## United Kingdom 0.117423 1.245e-04
## United States 0.214496 2.275e-04
Dotplot of random effect terms is given below:
# Save and print random effect terms on each country
re3 <- ranef(m.1, condVar=TRUE, whichel = "Country")
print(re2)## $Country
## (Intercept) Year
## Australia 0.361590 -1.624e-05
## Austria 0.554658 -2.491e-05
## Belgium 0.477833 -2.146e-05
## Canada 0.443120 -1.990e-05
## Chile -1.316527 5.912e-05
## Czech Republic -0.962225 4.321e-05
## Denmark 0.770571 -3.460e-05
## Estonia -1.312406 5.894e-05
## Finland 0.555040 -2.493e-05
## France 0.516084 -2.318e-05
## Germany 0.489359 -2.198e-05
## Greece -0.220070 9.883e-06
## Hungary -1.195123 5.367e-05
## Iceland 0.748944 -3.363e-05
## Ireland 0.315771 -1.418e-05
## Israel -0.009398 4.220e-07
## Italy 0.331602 -1.489e-05
## Japan 0.706314 -3.172e-05
## Korea, Rep. -0.664424 2.984e-05
## Luxembourg 1.015859 -4.562e-05
## Mexico -1.193285 5.359e-05
## Netherlands 0.529639 -2.378e-05
## New Zealand 0.105088 -4.719e-06
## Norway 0.984403 -4.421e-05
## Poland -1.476847 6.632e-05
## Portugal -0.428193 1.923e-05
## Slovak Republic -1.070662 4.808e-05
## Slovenia -0.424098 1.905e-05
## Spain -0.018281 8.210e-07
## Sweden 0.757881 -3.403e-05
## Switzerland 1.061030 -4.765e-05
## Turkey -1.541461 6.922e-05
## United Kingdom 0.404197 -1.815e-05
## United States 0.704019 -3.162e-05
##
## with conditional variances for "Country"
# Dotplot of re3
dotplot(re3)## $Country
So, the highest random effect on intercept is observed on Luxemburg, and the least random effect on intercept is observed on Turkey.
ANOVA test for Models
Now, we can compare these models by using ANOVA test. The results of this test will give us the group effect of CO2PC:GDPPC.
# Results of ANOVA test can be seen below:
anova(m.base, m.0, m.1)## Data: encar
## Models:
## m.base: GDPPC ~ 1 + Year + (1 + Year | Country)
## m.0: GDPPC ~ 1 + Year + I(CO2PC/GDPPC) + (1 + Year | Country)
## m.1: GDPPC ~ 1 + Year + I(CO2PC/GDPPC) + Year:I(CO2PC/GDPPC) + (1 +
## m.1: Year | Country)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## m.base 6 42.5 72.5 -15.3 30.5
## m.0 7 -25.0 9.3 19.5 -39.0 69.5 1 <2e-16 ***
## m.1 8 -18.0 21.2 17.0 -34.0 0.0 1 1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The results of ANOVA test show that adding a fixed effect CO2PC:GDPPC on the intercept improves the model fit significantly, while adding CO2PC:GDPPC on the slope doesn’t improve the model fit.
Following models are employed in order to see fixed and random effects of CO2PC:GDPPC within econometrics panel data modeling.
Fixed effect modelencar.fe <- plm(GDPPC ~ 1 +Year+I(CO2PC/GDPPC), data = encar,index = "Country", model = "within")Results of CO2PC:GDPPC fixed effect on within model can be given as follows:
# Summary of fixed effects:
summary(encar.fe)## Oneway (individual) effect Within Model
##
## Call:
## plm(formula = GDPPC ~ 1 + Year + I(CO2PC/GDPPC), data = encar,
## model = "within", index = "Country")
##
## Unbalanced Panel: n=34, T=18-31, N=989
##
## Residuals :
## Min. 1st Qu. Median 3rd Qu. Max.
## -0.60600 -0.14400 -0.00304 0.14600 0.65900
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## Year 5.94e-02 8.08e-04 73.54 <2e-16 ***
## I(CO2PC/GDPPC) 6.72e+01 5.55e+01 1.21 0.23
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 303
## Residual Sum of Squares: 44
## R-Squared : 0.855
## Adj. R-Squared : 0.824
## F-statistic: 2800.86 on 2 and 953 DF, p-value: <2e-16
Results of Oneway (individual) effect Within Model indicate that the coefficient Year is significant, while the coefficient CO2PC:GDPPC is not significant.R-Square of model is 0.855. In the model, the estimate of coefficient year 5.94 * e − 0.2 indicates how much GDPPC changes, on avarage per country, when year increases by one unit. Similarly, the estimate of coefficient 6.72 * e + 0.1 indicates how much GDPPC changes, on avarage per country, when CO2PC:GDPPC increases by one unit.
We can examine fixed effects of CO2PC:GDPPC on each country:
# Summary of fixed effects on each country:
summary(fixef(encar.fe))## Estimate Std. Error t-value Pr(>|t|)
## Australia -108.75 1.63 -66.5 <2e-16 ***
## Austria -108.59 1.62 -67.0 <2e-16 ***
## Belgium -108.66 1.63 -66.8 <2e-16 ***
## Canada -108.68 1.63 -66.5 <2e-16 ***
## Chile -110.31 1.62 -68.2 <2e-16 ***
## Czech Republic -109.97 1.63 -67.3 <2e-16 ***
## Denmark -108.39 1.62 -66.7 <2e-16 ***
## Estonia -110.29 1.64 -67.4 <2e-16 ***
## Finland -108.59 1.63 -66.8 <2e-16 ***
## France -108.63 1.62 -67.0 <2e-16 ***
## Germany -108.65 1.63 -66.7 <2e-16 ***
## Greece -109.30 1.62 -67.4 <2e-16 ***
## Hungary -110.19 1.62 -67.9 <2e-16 ***
## Iceland -108.42 1.62 -66.9 <2e-16 ***
## Ireland -108.81 1.62 -67.0 <2e-16 ***
## Israel -109.11 1.62 -67.2 <2e-16 ***
## Italy -108.80 1.62 -67.1 <2e-16 ***
## Japan -108.45 1.62 -66.8 <2e-16 ***
## Korea, Rep. -109.71 1.62 -67.6 <2e-16 ***
## Luxembourg -108.15 1.64 -65.9 <2e-16 ***
## Mexico -110.20 1.62 -68.1 <2e-16 ***
## Netherlands -108.61 1.63 -66.8 <2e-16 ***
## New Zealand -109.01 1.62 -67.2 <2e-16 ***
## Norway -108.20 1.62 -66.7 <2e-16 ***
## Poland -110.44 1.63 -67.8 <2e-16 ***
## Portugal -109.50 1.62 -67.7 <2e-16 ***
## Slovak Republic -110.08 1.63 -67.6 <2e-16 ***
## Slovenia -109.49 1.63 -67.3 <2e-16 ***
## Spain -109.12 1.62 -67.3 <2e-16 ***
## Sweden -108.41 1.62 -66.9 <2e-16 ***
## Switzerland -108.14 1.62 -66.8 <2e-16 ***
## Turkey -110.52 1.62 -68.4 <2e-16 ***
## United Kingdom -108.73 1.62 -67.0 <2e-16 ***
## United States -108.44 1.64 -66.2 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Therefore, fixed effect of CO2PC:GDPPC on each country is significant in our model.
encar.re <- plm(GDPPC ~ 1 +Year+I(CO2PC/GDPPC), data = encar,index = "Country", model = "random")# Summary of random effects:
summary(encar.re)## Oneway (individual) effect Random Effect Model
## (Swamy-Arora's transformation)
##
## Call:
## plm(formula = GDPPC ~ 1 + Year + I(CO2PC/GDPPC), data = encar,
## model = "random", index = "Country")
##
## Unbalanced Panel: n=34, T=18-31, N=989
##
## Effects:
## var std.dev share
## idiosyncratic 0.0462 0.2149 0.1
## individual 0.4055 0.6368 0.9
## theta :
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.921 0.939 0.939 0.938 0.939 0.939
##
## Residuals :
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.626 -0.140 0.005 0.002 0.151 0.565
##
## Coefficients :
## Estimate Std. Error t-value Pr(>|t|)
## (Intercept) -1.09e+02 1.64e+00 -66.75 <2e-16 ***
## Year 5.95e-02 8.12e-04 73.20 <2e-16 ***
## I(CO2PC/GDPPC) 8.91e+01 5.47e+01 1.63 0.1
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Total Sum of Squares: 306
## Residual Sum of Squares: 46.1
## R-Squared : 0.849
## Adj. R-Squared : 0.847
## F-statistic: 2777.99 on 2 and 986 DF, p-value: <2e-16
Results of Oneway (individual) effect Random Effect Model indicate that the coefficient Year has a significant influence on GDPPC, while the coefficient CO2PC:GDPPC doesn't have a significant influence on GDPPC. R-Square of model is 0.8493. In the model, p-value 2 * e − 16 shows that our model is correct.
In order to compare fixed and random effect model, the Hausman test is employed
phtest(encar.fe, encar.re)##
## Hausman Test
##
## data: GDPPC ~ 1 + Year + I(CO2PC/GDPPC)
## chisq = 4.926, df = 2, p-value = 0.08517
## alternative hypothesis: one model is inconsistent
Since p value is greater than 0.05, random effect model is better fit than fixed effect model.
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