Data Envelopment Analysis

Within couple of days I have been testing DEA modelling (Data Envelopment Analysis) with different R-package. Finally,…. I have found such a comprehensive way to calculate Malmquist indices.

Productivity 1991_2009_Forecast Cubic_Smoothing_Spline10y2 — Chart 1

My primary purpose is to show how to use nonparametric methods for measuring efficiency and productivity by using R-programs nonparaeff -package.

At this same time I will show you how to present Malmquist indices within googleVis -world map and finally I will introduse how to make forecast chart for productivity (Chart 1), effectiveness and technical effectiveness indices. In this work I used R-program forecast -library.

Used R-library documentation:
nonparaeff
googleVis
forecast

First of all I would like to thank Author Dong-hyun Oh for nice work with this nonparaeff-package.

In this working paper I will use example of faremalm2. Like nonpraeff -package documentation we calculate Malmquist productivity growth index of OECD countries
(productivity, technical efficiency and efficiency). As data source we have used Penn World Table (like original sources) with following version:
OECD Timeseries 1980-1990 version pwt5.6
OECD Timeseries 1990-2009 version pwt7.0
In pwt7.0 I cannot find capital stock data, so I downloaded it from here (http://www.ifw-kiel.de/forschung/datenbanken/netcap)

Productivity calculation 1980-1990 As output variables is used Total GDP of a country. This variables is calculated using GDP per capita (rgdpl)
and amount of total population (pop). As input variables is used Total labor force (gdp/rgdpwok) and Total capital stock (kapw * labor)

In second productivity computing calculation 1990-2009 I only use one input and one output variables (labor and gdp). This is because there is no
capital stock data available in pwt7.0 (please correct this If I missed something). I will re-calculate this period asap when I get capital data from all countries.

In third productivity computing 1992-2002 I use pwt7.0 and get capital stock data from different OECD datasource
(http://www.ifw-kiel.de/forschung/datenbanken/netcap). As input I used labor and capital variables and as output variables gdp.

When use faremalm2 function you will get the following outputs:
pc.c –> Productivity growth for each country
pc.y –> A trend of productivity growth of SELECTED countries (in this work we use OECD)

tc.c –> Technical efficiency for each country
tc.y –> A trend of technical efficiency growth of SELECTED countries

ec.c –> Efficiency change for each country
ec.y –> A trend of efficiency change of SELECTED countries

Useage of faremalm2

faremalm2(dat = NULL, noutput = 1, id = “id”, year = “year”)
dat –> The format of this data frame is data.frame(id, year, outputs, inputs).
noutput –> number of outputs
id –> column name for DMU:s
year –> column name for the time index

## Malmquist productivity growth index of OECD countries #First install data from OECD install.packages("pwt") # (installing a package can take a couple of minutes)

#Setting the library into use library(pwt) ## Use Penn World Table library(nonparaeff) # DEA modelling

More information about pwt data you can find there (http://pwt.econ.upenn.edu/php_site/pwt_index.php)

#my.dat <- pwt5.6 #used in 1980-1990 #my.dat <- pwt6.3 # my.dat <- pwt7.0 #used in 1990-2009 and 1992-2002 #head(my.dat) summary(my.dat) #my.dat$country

#capital stock data missing so we get it from there # http://www.ifw-kiel.de/forschung/datenbanken/netcap #picking up OECD countries my.oecd.ctry <- c("AUS", "AUT", "BEL", "CAN", "CHE", "DNK", "ESP", "FIN", "FRA", "GBR", "GER", "GRC", "IRL", "ISL", "ITA", "JPN", "KOR", "LUX", "MEX", "NLD", "NOR", "NZL", "PRT", "SWE", "TUR", "USA", "DEU")
#NOTE WPCODE USED IN OLDER DATASET #adding country code #my.dat <- my.dat[my.dat$wbcode %in% my.oecd.ctry,] #use this with pwt5.6 my.dat <- my.dat[my.dat$isocode %in% my.oecd.ctry,] #use this with pwt7.0 summary(my.dat)

#selecting appropriate years #my.dat <- my.dat[my.dat$year %in% 1980:1990,] #my.dat <- my.dat[my.dat$year %in% 1950:1992,] my.dat <- my.dat[my.dat$year %in% 1990:2009,] #my.dat <- my.dat[my.dat$year %in% 1992:2002,] summary(my.dat) my.dat

Note! I encounter problem (NA or other reason) thats why I do this out of box. my.dat is transformed into data.frame and after
that modified with Excel. This is only used in timeseries 1992-2002 productivity calculation.

#uploading my.dat #my.dat_df <- as.data.frame(my.dat) #out #write.table(my.dat_df, file = "c:/data/my.dat_df.txt", sep = "\t", col.names = NA, qmethod = "double")

Capital stock data inserted in this phase. source: http://www.ifw-kiel.de/forschung/datenbanken/netcap

#...and in both source
 my.dat  <- read.table(".../mydat1990_2009.csv",  header=TRUE, 
sep=";", na.strings="NA", dec=",", strip.white=TRUE)
my.dat <- read.table("http://ekqvist.goeuropeinfo.com/rbloggerqvist/
data/oecd/mydat1990_2009.csv", header=TRUE, sep=";", na.strings="NA", 
dec=",", strip.white=TRUE)

#INSERTING some variables to my.dat (ie. INPUT and OUTPUT -variables calculation) my.dat$rgdpl <- as.numeric(my.dat$rgdpl) ## GDP per capita my.dat$pop <- as.numeric(my.dat$pop) ## total population (1000) my.dat$rgdpwok <- as.numeric(my.dat$rgdpwok) ## GDP per labor

my.dat$gdp <- my.dat$rgdpl * my.dat$pop ## Total GDP of a country my.dat$labor <- with(my.dat, gdp/rgdpwok) ## Total labor force

my.dat$capital <- with(my.dat, kapw * labor) ## Total capital stock           MISSING pwt7 note: this is used in pwt5.6
 #my.dat$capital <- as.numeric(my.dat$capital)  
## Total capital stock  ADDED FROM DIFF. SOURCE now used in time 
series 1993-2002

my.dat$kapw <- as.numeric(my.dat$kapw) ## Capital stock per labor         MISSING pwt7 note: this is used in pwt5.6
 #my.dat$kapw <- with(my.dat, my.dat$capital/my.dat$labor) ## 
Capital stock per labor now used in time series 1993-2002

#variable used in 1990-2009 Malmquist productivity calculation

#1980-1990
#1990-2009
#1992-2002

#these variables used in timeseries 1980-1990 and 1992-2002 #oecd <- my.dat[, c("country", "wbcode", "year", "gdp", "labor", "capital")] #used in time series 1980-1990 oecd <- my.dat[, c("country", "isocode", "year", "gdp", "labor")] # 1990-2009 #oecd <- my.dat[, c("country", "isocode", "year", "gdp", "labor", "capital")] #used in time series 1992-2002

summary(oecd) head(oecd)
#removing NA-rows oecd <- oecd[!is.na(oecd$capital),] summary(oecd)

#Now calcuatel productivity (pc), effiency (ec) and technical efficiency (tc) #huom. tämä on tehty eri aikasarjoille ja hyvä niin koska mahdollistaa eri aikajaksojen tarkastelun library(nonparaeff) # DEA modelling re.oecd <- faremalm2(dat = oecd, noutput = 1, id = "isocode", year = "year") #re.oecd <- faremalm2(dat = oecd, noutput = 1, id = "wbcode", year = "year")

summary(re.oecd)

###################################################### #ISOCODE USED IN #1990-2009 and 1992-2002 #note:

## productivity growth for each country pc.c <- tapply(re.oecd$pc, re.oecd$isocode, geometric.mean) #pc.c <- tapply(re.oecd$pc, re.oecd$wbcode, geometric.mean) ## a trend of productivity growth of SELECTED countries pc.y <- tapply(re.oecd$pc, re.oecd$year, geometric.mean)

## technical efficiency for each country
tc.c <- tapply(re.oecd$tc, re.oecd$isocode, geometric.mean)
#tc.c <- tapply(re.oecd$tc, re.oecd$wbcode, geometric.mean)

## a trend of technical efficiency growth of SELECTED countries
tc.y <- tapply(re.oecd$tc, re.oecd$year, geometric.mean)

## efficiency change for each country
ec.c <- tapply(re.oecd$ec, re.oecd$isocode, geometric.mean)
#ec.c <- tapply(re.oecd$ec, re.oecd$wbcode, geometric.mean)
## a trend of efficiency change of SELECTED countries
ec.y <- tapply(re.oecd$ec, re.oecd$year, geometric.mean)
######################################################
pc.c

#1980-1990, 1990-2009 #country -variable(we use it in google vis plotting) ## productivity growth for each country pc.c <- tapply(re.oecd$pc, re.oecd$country, geometric.mean)

## technical efficiency for each country
tc.c <- tapply(re.oecd$tc, re.oecd$country, geometric.mean)

## efficiency change for each country
ec.c <- tapply(re.oecd$ec, re.oecd$country, geometric.mean)

#missing time series 1992-2002 TURKEY, KOREA, LUXEMBOURG AND MEKSIKO capital stock data #VUOSITTAINEN startyear-endyear KESKIARVO TEHOKKUUSKERTOIMESSA (MALMQUIST - FARREL) #ec: efficiency change #TEHOKKUUSMUUTOS ec.c_df <- as.data.frame(ec.c) summary(ec.c_df) ec.c_df #tc: technical change #TEKNISEN TEHOKKUUDEN MUUTOS tc.c_df <- as.data.frame(tc.c)

#pc: productivity change #TUOTTAVUUDEN MUUTOS
pc.c_df <- as.data.frame(pc.c)

#LITTLE TRICK #ec: efficiency change #TEHOKKUUSMUUTOS write.table(ec.c_df, file = "c:/data/ec.c_df.txt", sep = "\t", col.names = NA, qmethod = "double") ec.c_df2 <- read.table("c:/data/ec.c_df.txt", header=TRUE, sep="\t", na.strings="NA", dec=".", strip.white=TRUE) names(ec.c_df2)<- c("Country", "Effiency")

#tc: technical change #TEKNISEN TEHOKKUUDEN MUUTOS write.table(tc.c_df, file = "c:/data/tc.c_df.txt", sep = "\t", col.names = NA, qmethod = "double") tc.c_df2 <- read.table("c:/data/tc.c_df.txt", header=TRUE, sep="\t", na.strings="NA", dec=".", strip.white=TRUE) names(tc.c_df2)<- c("Country", "Effiency")

#pc: productivity change #TUOTTAVUUDEN MUUTOS
write.table(pc.c_df, file = “c:/data/pc.c_df.txt”, sep = “\t”, col.names = NA, qmethod = “double”)
pc.c_df2 <- read.table(“c:/data/pc.c_df.txt”, header=TRUE, sep=”\t”, na.strings=”NA”, dec=”.”, strip.white=TRUE)
names(pc.c_df2)<- c(“Country”, “Effiency”)

#REMOVING NA-VALUES ec.c_df2 <- ec.c_df2[!is.na(ec.c_df2$Effiency),] tc.c_df2 <- tc.c_df2[!is.na(tc.c_df2$Effiency),] pc.c_df2 <- pc.c_df2[!is.na(pc.c_df2$Effiency),] summary(ec.c_df2)

#just cecking country names ec.c_df2$Country

#replace some country name with new one…
ec.c_df3 <- gsub(“United States of America”, “United States”, ec.c_df2$Country)
ec.c_df3 <- gsub(“Germany, West”, “Germany”, ec.c_df3)
ec.c_df3 <- gsub(“Korea, Republic”, “South Korea”, ec.c_df3)

tc.c_df3 <- gsub(“United States of America”, “United States”, tc.c_df2$Country)
tc.c_df3 <- gsub(“Germany, West”, “Germany”, tc.c_df3)
tc.c_df3 <- gsub(“Korea, Republic”, “South Korea”, tc.c_df3)

pc.c_df3 <- gsub(“United States of America”, “United States”, pc.c_df2$Country)
pc.c_df3 <- gsub(“Germany, West”, “Germany”, pc.c_df3)
pc.c_df3 <- gsub(“Korea, Republic”, “South Korea”, pc.c_df3)

#as a dataframe MUUNNETAAN DATA FRAMEKSI
ec.c_df3 <- as.data.frame(ec.c_df3)
tc.c_df3 <- as.data.frame(tc.c_df3)
pc.c_df3 <- as.data.frame(pc.c_df3)

#check total number of rows KATSOTAAN RIVINUMEROINNIT MERGELLE
summary(ec.c_df2)
nrow(ec.c_df2)
nrow(ec.c_df3)

summary(tc.c_df2)
nrow(tc.c_df2)
nrow(tc.c_df3)

summary(pc.c_df2)
nrow(pc.c_df2)
nrow(pc.c_df3)

#used in time series #1980-1990 #1990-2009 #giving row id ...NUMEROIDDAAN ID MERGEÄ VARTEN ec.c_df2$id1=c(1:26) ec.c_df3$id1=c(1:26) tc.c_df2$id1=c(1:26) tc.c_df3$id1=c(1:26) pc.c_df2$id1=c(1:26) pc.c_df3$id1=c(1:26)
#...and for time series #1991-2002 #giving row id ...NUMEROIDDAAN ID MERGEÄ VARTEN ec.c_df2$id1=c(1:22) ec.c_df3$id1=c(1:22) tc.c_df2$id1=c(1:22) tc.c_df3$id1=c(1:22) pc.c_df2$id1=c(1:22) pc.c_df3$id1=c(1:22)
#merge data table and renamed country by id ..... ec.c_df4 <-merge(ec.c_df2, ec.c_df3, by.x="id1", by.y="id1", all = TRUE) tc.c_df4 <-merge(tc.c_df2, tc.c_df3, by.x="id1", by.y="id1", all = TRUE) pc.c_df4 <-merge(pc.c_df2, pc.c_df3, by.x="id1", by.y="id1", all = TRUE)

#check the data head(ec.c_df4) head(tc.c_df4) head(pc.c_df4) summary(ec.c_df4)

#selecting columns to the gvisGeoMapping ..... ec.c_df5 <- data.frame(ec.c_df4$ec.c_df3, ec.c_df4$Effiency ) tc.c_df5 <- data.frame(tc.c_df4$tc.c_df3, tc.c_df4$Effiency ) pc.c_df5 <- data.frame(pc.c_df4$pc.c_df3, pc.c_df4$Effiency )

#check the data head(ec.c_df5) head(tc.c_df5) head(pc.c_df5)

summary(ec.c_df5)
summary(tc.c_df5)
summary(pc.c_df5)

#naming trick names(ec.c_df5)<- c("Country", "Effiency change in OECD 1980-1990") names(tc.c_df5)<- c("Country", "Technical Effiency change in OECD 1980-1990") names(pc.c_df5)<- c("Country", "Productivity change in OECD 1980-1990")

#naming time series 1991-2009 names(ec.c_df5)<- c("Country", "Effiency change in OECD 1991-2009") names(tc.c_df5)<- c("Country", "Technical Effiency change in OECD 1991-2009") names(pc.c_df5)<- c("Country", "Productivity change in OECD 1991-2009")

#1991-2002 names(ec.c_df5)<- c("Country", "Effiency change in OECD 1991-2002") names(tc.c_df5)<- c("Country", "Technical Effiency change in OECD 1991-2002") names(pc.c_df5)<- c("Country", "Productivity change in OECD 1991-2002")

head(ec.c_df5) head(tc.c_df5) head(pc.c_df5)

#set library
library(googleVis)

#1980-1990 #me/230712 #1 #KMEAN VALUE OF MALMQUIST INDICES 1980-1990 Efficency change Geo1=gvisGeoMap(ec.c_df5, locationvar="Country", numvar="Effiency change in OECD 1980-1990", options=list (height=450, width=800, dataMode='regions')) #plot(Geo1) cat(Geo1$html$chart, file=".../oecd_mean_eff_1980_1990_malmquist2.html")

#2 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES 1980-1990 Efficency change Geo2=gvisGeoMap(tc.c_df5, locationvar="Country", numvar="Technical Effiency change in OECD 1980-1990", options=list(height=450, width=800, dataMode='regions')) #plot(Geo2) cat(Geo2$html$chart, file=".../oecd_mean_tech_eff_1980_1990_malmquist2.html")

#3 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES 1980-1990 Efficency change Geo3=gvisGeoMap(pc.c_df5, locationvar="Country", numvar="Productivity change in OECD 1980-1990", options=list(height=450, width=800, dataMode='regions')) #plot(Geo3) cat(Geo3$html$chart, file=".../oecd_mean_prod_1980_1990_malmquist2.html")

#1991-2009 (capital var dropped) #me/240712 #1 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES Geo1=gvisGeoMap(ec.c_df5, locationvar="Country", numvar="Effiency change in OECD 1991-2009", options=list (height=450, width=800, dataMode='regions')) plot(Geo1) cat(Geo1$html$chart, file=".../oecd_mean_eff_1991_2009_malmquist2.html")

#2 #VALUE OF MALMQUIST INDICES Geo2=gvisGeoMap(tc.c_df5, locationvar="Country", numvar="Technical Effiency change in OECD 1991-2009", options=list(height=450, width=800, dataMode='regions')) plot(Geo2) cat(Geo2$html$chart, file=".../oecd_mean_tech_eff_1991_2009_malmquist2.html")

#3 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES Geo3=gvisGeoMap(pc.c_df5, locationvar="Country", numvar="Productivity change in OECD 1991-2009", options=list(height=450, width=800, dataMode='regions')) plot(Geo3) cat(Geo3$html$chart, file=".../oecd_mean_prod_1991_2009_malmquist2.html")

#1991-2002 (capital as input) #me/240712 #1 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES Geo1=gvisGeoMap(ec.c_df5, locationvar="Country", numvar="Effiency change in OECD 1991-2002", options=list (height=450, width=800, dataMode='regions')) plot(Geo1) cat(Geo1$html$chart, file=".../oecd_mean_eff_1991_2002_malmquist2.html")

#2 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES Geo2=gvisGeoMap(tc.c_df5, locationvar="Country", numvar="Technical Effiency change in OECD 1991-2002", options=list(height=450, width=800, dataMode='regions')) plot(Geo2) cat(Geo2$html$chart, file=".../oecd_mean_tech_eff_1991_2002_malmquist2.html")

#3 #KESKIARVO MEAN VALUE OF MALMQUIST INDICES Geo3=gvisGeoMap(pc.c_df5, locationvar="Country", numvar="Productivity change in OECD 1991-2002", options=list(height=450, width=800, dataMode='regions')) plot(Geo3) cat(Geo3$html$chart, file=".../oecd_mean_prod_1991_2002_malmquist2.html")

World Map (producitivity, efficiency)

The cubic smoothing spline model

Now we plot productivity, effectiveness and technical effectiveness mean values historical trend and forecast by using
The cubic smoothing spline model. It is equivalent to an ARIMA(0,2,2) model.

############################################################### library(forecast) #library(ggplot2) #1980-1990 #1990-2009 #1991-2002 ##################################################################### #1 #1980-1990 ##################################################################### #PRODUCTIVITY pc.y fcast_pc <- splinef(pc.y,h=10, fan=T) #NOTE confidence leve 50, 99 plot(fcast_pc, main="Productivity, forecast from Cubic Smoothing Spline", ylab="Malmquist indices - Productivity", xlab="Year(0=1980, 10=1990, 20=2000)")

# example – output graph to jpeg file
jpeg(“…/Productivity 1980_1990_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_pc, main=”Productivity, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Productivity”, xlab=”Year(0=1980, 10=1990, 20=2000)”)
dev.off()

#TECHNICAL EFFICIENCY
fcast_tc <- splinef(tc.y,h=10, fan=T)
plot(fcast_tc, main=”Technical Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Technical Efficiency”, xlab=”Year(0=1980, 10=1990, 20=2000)”)

# example – output graph to jpeg file
jpeg(“…/Technical efficiency 1980_1990_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_tc, main=”Technical Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Technical Efficiency”, xlab=”Year(0=1980, 10=1990, 20=2000)”)
dev.off()

#EFFICIENCY
fcast_ec <- splinef(ec.y,h=10, fan=T)
plot(fcast_ec, main=”Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Efficiency”, xlab=”Year(0=1980, 10=1990, 20=2000)”)

# example – output graph to jpeg file
jpeg(“G:/data/home/2012/marko/blogi_rbloggerqvist/tekstiaihiot/40/Efficiency 1980_1990_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_ec, main=”Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Efficiency”, xlab=”Year(0=1980, 10=1990, 20=2000)”)
dev.off()

summary(fcast_pc)
summary(fcast_tc)
summary(fcast_ec)

#####################################################################
#2
#1991-2009
#####################################################################
#PRODUCTIVITY
pc.y
fcast_pc <- splinef(pc.y,h=10, fan=T) #NOTE confidence leve 50, 99
plot(fcast_pc, main=”Productivity, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Productivity”, xlab=”Year(0=1990, 20=2010)”)

# example – output graph to jpeg file
jpeg(“…/Productivity 1991_2009_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_pc, main=”Productivity, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Productivity”, xlab=”Year(0=1990, 20=2010)”)
dev.off()

# example – output graph to jpeg file
jpeg(“G:/data/home/2012/marko/blogi_rbloggerqvist/tekstiaihiot/40/Technical efficiency 1991_2009_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_tc, main=”Technical Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Technical Efficiency”, xlab=”Year(0=1990, 20=2010)”)
dev.off()

# example – output graph to jpeg file
jpeg(“…/Efficiency 1991_2009_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_ec, main=”Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Efficiency”, xlab=”Year(0=1990, 20=2010)”)
dev.off()

summary(fcast_pc)
summary(fcast_tc)
summary(fcast_ec)
######################################################################3
#1991-2002 (capital included)
#
#####################################################################
#PRODUCTIVITY
fcast_pc <- splinef(pc.y,h=10, fan=T) #NOTE confidence leve 50, 99
plot(fcast_pc, main=”Productivity, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Productivity”, xlab=”Year(1=1991, 10=2000, 20=2012)”)

# example – output graph to jpeg file
jpeg(“…/Productivity 1991_2002_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_pc, main=”Productivity, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Productivity”, xlab=”Year(1=1991, 10=2000, 20=2012)”)
dev.off()

# example – output graph to jpeg file
jpeg(“…/Technical efficiency 1991_2002_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_tc, main=”Technical Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Technical Efficiency”, xlab=”Year(1=1991, 10=2000, 20=2012)”)
dev.off()

# example – output graph to jpeg file
jpeg(“…/Efficiency 1991_2002_Forecast Cubic_Smoothing_Spline10y2.jpg”)
plot(fcast_ec, main=”Efficiency, forecast from Cubic Smoothing Spline”, ylab=”Malmquist indices – Efficiency”, xlab=”Year(1=1991, 10=2000, 20=2012)”)
dev.off()

summary(fcast_pc)
summary(fcast_tc)
summary(fcast_ec)
#####################################################################

Ok, that’s it,
Have fun with Linear Programming for the Malmquist Productivity Growth Index and its wide range of applications…

Marko

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Category: Forecasting

Linear Programming for the Malmquist Productivity Growth Index

Data Envelopment Analysis

Useage of faremalm2

World Map (producitivity, efficiency)

1980-1990

1991-2002

1991-2009

The cubic smoothing spline model