malmquist {Benchmarking}

Malmquist index for firms in a panel

Description

Estimate Malmquist index for firms in a panel data set. The data set does not need to be balanced.

Usage

malmquist(X, Y, ID, TIME, RTS = "vrs", ORIENTATION = "in", SAMEREF=FALSE, 
    SLACK = FALSE, DUAL = FALSE, DIRECT = NULL, param = NULL, 
    TRANSPOSE = FALSE, FAST = TRUE, LP = FALSE, CONTROL = NULL, LPK = NULL)

Arguments

X

Inputs of firms in many periods, a (T*K) x m matrix of observations of K firms with m outputs (firm x input) in at the most T periods.

Y

Outputs of firms in many periods, a (T*K) x n matrix of observations of K0 firms with n outputs (firm x input) in at the most T periods.

ID

Identifier for the firms in rows of X and Y.

TIME

Array with period number for each row in the input maxtrix X and output matrixY

RTS

Returns to scale assumption as in dea.

ORIENTATION

Input efficiency "in" (1), output efficiency "out" (2), and graph efficiency "graph" (3) as in dea.

SAMEREF

Use the same units for reference technology when comparing two periods. This is not restricted to same units in several timpe periods, but only to pairwise periods comparisons for Malmquist. Default is to use available and possible differnt units in pairwise periods.

SLACK

See dea.

DUAL

See dea.

DIRECT

See dea.

param

See dea.

TRANSPOSE

See dea.

FAST

See dea.

LP

See dea.

CONTROL

See dea.

LPK

See dea.

Details

Malmquist uses malmq for the calculations of the necessary efficiencies, and the returned indices are as in malmq. The data must be a long data set with regards to TIME and ID; se the example below.

Note that the calculated index are index comparing a period and the previous period. To compare the development over time the indices must be turned into a chain index as shown in the example below.

Value

m

Malmquist indicies, an array of length T*K in the order of ID and TIME, i.e. the order of the rows of X.

tc

Technical change indices, an array of length T*K.

ec

Efficiency indices, an array of length T*K.

id

Index for firms as ID

time

Index for time as TIME

e00

The efficiencies for period 0 with reference technology from period 0.

e10

The efficiencies for period 1 with reference technology from period 0.

e11

The efficiencies for period 1 with reference technology from period 1.

e01

The efficiencies for period 0 with reference technology from period 1.

Note

The lagged values e11 are not necessary equal to values of e00 as the reference technology for the two periods could be generated by different units, if the units in different time periods are not the same.

Author(s)

Peter Bogetoft and Lars Otto [e-mail]

References

Peter Bogetoft and Lars Otto; Benchmarking with DEA, SFA, and R; Springer 2011

See Also

dea, malmq

Examples

 x0 <- matrix(c(10, 28, 30, 60),ncol=1)
 y0 <- matrix(c(5, 7, 10, 15),ncol=1)
 x1 <- matrix(c(12, 26, 16, 60 ),ncol=1)
 y1 <- matrix(c(6, 8, 9, 15 ),ncol=1)
 x2 <- matrix(c(13, 26, 15, 60 ),ncol=1)
 y2 <- matrix(c(7, 9, 10, 15 ),ncol=1)
 
 dea.plot(x0, y0, RTS="vrs", txt=TRUE)
 dea.plot(x1, y1, RTS="vrs", add=TRUE, col="red")
 dea.plot(x2, y2, RTS="vrs", add=TRUE, col="blue")
 points(x1, y1, col="red", pch=16)
 # points(x2, y2, col="blue", pch=17)
 text(x1, y1, 1:dim(x1)[1], col="red", adj=-1)
 text(x2, y2, 1:dim(x1)[1], col="blue", adj=-1)
 legend("bottomright", legend=c("Period 0", "Period 1", "Period 2"),
    col=c("black", "red", "blue"), lty=1, pch=c(1,16, 17),  bty="n")
 
 X <- rbind(x0, x1, x2)
 Y <- rbind(y0, y1, y2)
 # Make ID and TIME variables one way or another
 ID <- rep(1:dim(x1)[1], 3)
 # TIME <- c(rep(0,dim(x1)[1]), rep(1,dim(x1)[1]), rep(2,dim(x1)[1]))
 TIME <- gl(3, dim(x1)[1], labels=0:2)
 # This is how the data for Malmquist must look like
 data.frame(TIME, ID, X, Y)
 mq <- malmquist(X,Y, ID, TIME=TIME) 
 data.frame(TIME, ID, X, Y, mq$e00, mq$e01, mq$e10, mq$e11, mq$m, mq$tc)[order(ID, TIME),]
 
 # How to make the Malmquist indices to a chain index
 # Make data.frame with indices
 DM <- data.frame(TIME, ID, m=mq$m, tc=mq$tc, ec=mq$ec)
 # Set missing index for first period to 1, the base
 DM[DM$TIME==0, c("m","tc", "ec")] <- 1
 # Make chain index of the individual indices
 AD <- aggregate(cbind(m=DM$m), by=list(ID=DM$ID), cumprod)
 # Compare chain index to original index
 data.frame(ID, TIME, m=c(AD$m), DM$m)