1. Non Linear Functions in Panel Data Models-
Algorithm Using Stata
Presentation at the Stata User Group Meeting in Berlin
8. April 2005
Fred Ramb and Markus Reitzig

2. Overview
Theoretical models often are nonlinear e.g. in economics the well known q-model
Micro panel data
NLLS estimator is biased
Fixed effects estimator is efficient
Standard software package is not available

3. Overview Motivation Theory q Model Linearized Fixed Effects Estimator
Stata Code
Pooled Cross Sectional Nonlinear Least Squares
Panel Data Random Effects and Fixed Effects
Empirical Results
Summary

4. Theory (I): q Model Cobb-Douglas Production function
Logarithmic transformation
With for small x:

5. Theory (II): Algorithm
Logarithmic transformation
With for larger x, linearized version is estimated:
with:

6. Stata Code: NLLS program nlacc if "`1'" =="?" {
global S_1 "B_0 X1 C1 C2 C3 T1 T2 T3 T4 T5"
global B_0= 0
global X1= 0
global C1= 0
global C2= 0
global C3= 0
global T1= 0
global T2= 0
global T3= 0
global T4= 0
global T5= 0
exit }
replace `1'=$B_0 + ln(1+($X1*x2)) + $C1*control1+$C2*control2 + $C3*control3 /*
*/ $T1*dyear1 + $T3*dyear2 + $T3*dyear3 + $T4*dyear4 + $T5*dyear5
end
nl acc y

7. Stata Code: Fixed Effects
xtreg y x control1 control2 time*, fe
gen ex = _b[x]
gen y_s = y - (log(1+ex*x))
gen x_s = x / (1+ (ex*x))
log off
local i=1
while `i' < 50 {
quietly xtreg y_s x_s control1 control2 time*, fe
quietly replace ex = ex + _b[x_s]
quietly replace y_s = y - (log(1+ex*x))
quietly replace x_s = x2 / (1+ (ex*x))
local i = `i' + 1 }
log on
replace y_s = y - (log(1+ex*x)) + (ex*(x / (1+ (ex*x))))
xtreg y_s x_s control1 control2 time*, fe

8. Stata Code: Random Effects
xtreg y x control1 control2 time*, re
gen exr = _b[x]
gen y_sr = y - (log(1+exr*x))
gen x_sr = x / (1+ (exr*x)
log off
local i=1
while `i' < 50 {
quietly replace exr = exr + _b[x_sr]
quietly replace y_sr = y-(log(1+exr*x))
quietly replace x_sr = x / (1+ (exr*x))
quietly xtreg y_sr x_sr control1 control2 time*, re
local i = `i' + 1 }
log on
replace y_sr = y-(log(1+exr*x)) + (exr*(x / (1+ (exr*x))))
xtreg y_sr x_sr control1 control2 time*, re
xttest0

9. Stata Code: Hausman Test
xtreg y_s x_s control1 control2 time*, fe
est store fixed
xtreg y_sr x_sr control1 control2 time*, re
hausman fixed

10. Empirical Results: Dataset
Sources
Hoppenstedt: accounting data
Datastream: stock market data
Sample
Consolidated financial statements
Manufacturing sector
Estimation Sample
2,786 observations / 676 firms (Overall)
1,463 observations / 325 firms (HGB)
532 observations / 139 firms (IAS)
791 observations / 212 firms ( US-GAAP)

11. Empirical Results: Example 1

12. Empirical Results: Example 2

13. Summary
Pros
Different results by estimating pooled cross sectional nonlinear least squares on the one hand and by estimating fixed effects using the algorithm on the other hand.
Results using the algorithm are more efficient
Cons
Do-File is limited to the q-model
Heteroskedasticity? Robust estimation is necessary
Local or global maximum? Bootstrap is necessary
Dynamics? GMM is necessary

14. Thank you for your attention

15. Appendix (I)

17. Appendix (III)