Forum: open-discussion

RE: Questions about residuals and efficiencies in the frontier package [ Reply ]
By: W Miao on 2013-05-15 08:48

(Continued from my response on May 13, I revised my question number 4 as below. Thanks!)

4.
In my case, x1, x2 are log(inputs), and y=log(output). Let me consider “sfa1”, the time-invariant efficiency case in this question.
In response to your reply ” 2. Yes (but please note that U is assumed to be zero in the OLS estimation).”
Do you mean:
sfa1$mleParam stores the estimation results with U,
i.e. , the estimation results for
y=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V-U
with V~N(0, \sigma^2_V), U~| N(0, \sigma^2_U)| , and x1, x2, y, V depend on i and t, U depends on only i, and this is the model for calculating technical efficiencies,

while sfa1$olsParam stores the estimation results without U, i.e.
y=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V, where
V~N(0, \sigma^2_V), and x1, x2, y, V depend on i and t?

RE: Questions about residuals and efficiencies in the frontier package [ Reply ]
By: W Miao on 2013-05-13 08:24

[forum:39567]

Dear Prof Henningsen,

Thank you very much for your patience and your kind reply.

sfa1 <- frontierQuad( yName = "y",
xNames = c("x1","x2"),
data = mydata)

sfa2 <- frontierQuad( yName = "y",
xNames = c("x1","x2"),
data = mydata,
timeEffect=TRUE)
a2<-efficiencies(sfa2)

> sfa1

Call:
frontierQuad frontierQuad( yName = "y", xNames = c("x1","x2"), data = mydata)

Maximum likelihood estimates
a_0 a_1 a_2 b_1_1 b_1_2
-21.7295 3.1600 1.4485 -0.1260 -0.1640
b_2_2 sigmaSq gamma
0.1751 0.3385 0.7649
> sfa2

Call:
frontierQuad( yName = "y", xNames = c("x1","x2"),
data = mydata, timeEffect=TRUE)

Maximum likelihood estimates
a_0 a_1 a_2 b_1_1
-12.377355 1.699485 1.650747 -0.079181
b_1_2 b_2_2 sigmaSq gamma
-0.073640 0.007892 0.176129 0.603941
time
0.056823

4.
In my case, x1, x2 are log(inputs), and y=log(output). Let me consider “sfa1”, the time-invariant efficiency case in this question.
In response to your reply ” 2. Yes (but please note that U is assumed to be zero in the OLS estimation).”
Do you mean:
When we estimate the above reported coefficients and the residuals, we estimate the model
y=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V, where
V~N(0, \sigma^2_V), and x1, x2, y, V depend on i and t, but
when we attempt to calculate the technical efficiencies, we estimate the model
y=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V-U
with V~N(0, \sigma^2_V), U~| N(0, \sigma^2_U)| , and x1, x2, y, V depend on i and t, U depends on only i?
Why not just report the parameter estimation of the second model with -U?

5. With “timeEffect=TRUE”, a2<-efficiencies(sfa2) provides estimates of time-varying technical efficiencies for each firm.
y=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V-U
In this case, x1, x2, y, V, U all depend on i and t.
The technical efficiencies are given by exp[-exp(-\eta (t-T))U_i], where U_i is independent of t. I believe that the estimated parameter “time” in the second model is exactly minus \eta here. Am I right?

6. For the first case WITHOUT “timeEffect=TRUE”, does the computation of efficiencies involve the estimation of the model
y=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V-U, where x1, x2, y, V depend on i and t, while U depends only on i? Confronting the efficiency estimates in the two cases, I find that the efficiencies for firm i in the time-invariant case is somehow close to, but not exactly, the mean of time-varying efficiencies of firm i over all the time periods. There seems to be no simple relationship between the efficiencies in the two cases, like efficiency(i, time-invariant)= mean[efficiencies(time-varying, i, t=1, …..T)]. Am I right?

Thank you very much,

Miao

RE: Questions about residuals and efficiencies in the frontier package [ Reply ]
By: Arne Henningsen on 2013-05-12 21:15

[forum:39565]

Dear Miao

I suggest that you use the residuals() method for obtaining the residuals of the frontier model (estimated by the maximum likelihood method).

The model specification in your post is almost correct. You just need to replace "ln(y)" by "y".

1. Please read the documentation of sfa() and the documentation of residuals.frontier(). You can use argument "asInData" of residuals.frontier() to select the format of the returned residuals.

2. Yes (but please note that U is assumed to be zero in the OLS estimation).

3. The efficiencies() methods returns the expected / estimated technical efficiencies, i.e. E[ exp(-U) ]. Whether U (and hence exp(-U)) remains constant over time or might change over time depends on your model specification (e.g. argument "timeEffect"). Please read the working paper:

Coelli, T.J. (1996): A Guide to FRONTIER Version 4.1: A Computer
Program for Stochastic Frontier Production and Cost Function Estimation, Working Paper No. 7/96, Centre for Efficiency and Productivity Analysis (CEPA), Department of Econometrics, University of New England, Armidale, Australia,
https://absalon.itslearning.com/data/ku/103018/publications/coelli96.pdf

... and please do not forget to cite the "frontier" package in your publications :-)

Best regards,
Arne

Questions about residuals and efficiencies in the frontier package [ Reply ]
By: W Miao on 2013-05-10 08:12

[forum:39564]

Hi,

I use a wrapper function of "sfa" and the function "efficiencies" to find out the efficiencies.

sfa1 <- frontierQuad( yName = "y",
xNames = c("x1","x2"),
data = mydata)

a<-efficiencies(sfa1)

res1<-sfa1$olsResid
res2<-sfa1$resid

I believe that the model in my case is
ln(y)=b0+b1*x1+b2*x2+(b11*(x1^2)+b22*(x2^2))/2+b12*x1*x2+V-U.
(If I am wrong , please correct me)

1. Are the residuals sfa1$olsResid and sfa1$resid are basically the same in content except that the former is listed straightly in a column, while the later is in a matrix form?

2. Do the two residual refer to V-U ?

3. Does the function "efficiencies" provide the technical efficiencies, which is given by exp(-U)? Since U might change with time for the same firm, does the technical efficiency for a specific firm equal exp(-mean(U)), where the mean is taken over different periods for the same firm?

Thanks,

Miao