Searching for the Robust Method to Estimate Total Factor Productivity

at Firm Level

Yin Heng Li Shigang Liu Di

SEBA Beijing Normal University Email: yheng@bnu.edu.cn

Motivation

Discuss the robust TFP estimation method at firm level, using competitive industry as an example.

What does TFP measure?

Evaluate the input-output efficiency Labor productivity cannot describe the true

efficiency at firm level The core of TFP estimation is dealing with

the substitution among input factors

The importance of TFP estimation

Productivity is not everything, but approximates everything in the long run. Krugman （ 1997 ）

The factors affecting TFP– Is the TFP of firms improved?

Misallocation of resources

- Can the economic environment promotes firms

with high TFP, and suppress or expel those

with low TFP?

The current situation of TFP estimation

Great differences exist even in researches appeared in the top journals

- Young (1995)’s estimation of the growth rate of TFP

in Hong Kong and Taiwan district in China is

between 2% and 3%, the growth rate of Korea is

1.7%

- Hsieh (1999) got 3% more than Young’s.

Structure

The measurement of data and variables Traditional methods Firms’ decision and structure estimation Value-added or gross output production

function Sample selection, function form and other

robust test summary

The measurement of data and variables

Panel construction– Goal : identify firms across years– Problems :

Different firms may share the same code Firms may change the code because of changing name

or structure etc.– Idea :

Make sure that firms with the same code is the same one;

Match firms with the combination of relatively stable information, such as name, head, telephone, etc.;

Correct the wrong matching.

The measurement of data and variables

The measurement of output– The real output– deflate gross output from three dimensions:

time(year), space(province), industry(two-digit) The measurement of input

– Construct the input deflator from time(year), space(province) and industry(two-digit), based on input-output table, like Brandt (2012).

The measurement of data and variables

The measurement of capital– Estimate nominal investment from the found year

with the data of original fixed capital– Deflate the nominal investment to get the real

investment– Get the real capital with perpetual inventory

method The measurement of labor

– Total average number of staff

The measurement of data and variables

The choice of industries– Two-digit industry 18: manufacture of clothing, shoes

and hats; two-digit industry 19: manufacture of leather, fur and feather

Data clearing – Delete the sample with non-positive output, capital,

labor and input – Delete the sample with less than 8 workers– Delete the sample with bigger value-added than

output

Table 1: Descriptive Statistics

Year Obs. Output Capital Labor Material

Mean Std. Mean Std. Mean Std. Mean Std.

1998 8795 29634.70 71628.43 4589.94 13460.20 331.38 787.72 19406.05 48162.89

1999 8482 31358.37 76530.40 4640.89 14266.12 331.24 638.53 20317.37 51764.68

2000 8872 33683.06 89291.19 4399.33 14301.95 332.19 663.23 21526.77 58845.14

2001 10269 33896.71 99122.08 4030.10 14122.47 320.76 616.40 21716.56 65105.94

2002 11488 34759.67 106787.20 3763.74 13890.40 315.26 592.85 22062.48 69204.64

2003 13219 37826.08 129803.20 3758.72 14743.61 321.98 642.17 23571.01 82674.73

2004 16210 36327.99 165626.10 3577.54 19389.14 316.30 616.25 21990.80 108535.00

2005 17549 43655.98 201137.20 3969.65 21885.35 319.08 657.42 26398.21 131528.40

2006 19260 47903.11 235153.90 4283.38 28367.65 313.16 676.45 28537.35 150848.60

2007 21314 52152.53 249305.10 4344.99 27960.65 301.58 671.94 30418.95 159284.20

Traditional methods

DEA (Data Envelopment Analysis) Index method Tradition parametric methods

– OLS– FE– BB

Traditional methods

DEA– Considering the heterogeneity of firms’ TFP– Get the TFP measurement from the input and

output data with linear programming, treating the production process as a black box

– It is a determinate method which can be sensitive to the random error or extreme values.

Traditional methods

Index method

– Free of function form– Need the given the parameter of return to scale– Without the consideration of random error– Based on the hypothesis that all inputs are static

input without adjustment cost

Traditional methods

Parametric method

– Based on the set that all firms in the same industries have the same elasticity of output of capital, labor and input

– Deal with random error

Traditional methods

OLS– Endogenous problem

FE– Neglect the change of TFP with time

AB/BB– ，– System GMM

Table 2：Traditional Methods

DEA INDEX OLS FE BB

Coef. Std. Coef. Std. Coef. Std.

98-02

k - - - - 0.0431 0.0017 0.0320 0.0032 0.0632 0.0247

l - - - - 0.0738 0.0024 0.1057 0.0048 0.2333 0.0497

m - - - - 0.8678 0.0018 0.7490 0.0033 0.8643 0.0363

03-07

k - - - - 0.0492 0.0013 0.0599 0.0023 0.1282 0.0215

l - - - - 0.1191 0.0020 0.1527 0.0044 0.0106 0.0349

m - - - - 0.8158 0.0017 0.6969 0.0027 0.9518 0.0204

Year 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07

Growth % 4.2136 6.8645 2.6128 3.2202 1.5407 2.7222 2.3363 4.2434 0.4844 0.1696

Ratio 90/10 4.1528 5.4557 2.2865 2.5604 1.4636 1.5532 1.7671 1.8278 1.8370 1.7388

95/5 6.9894 9.1132 3.0946 3.7640 1.7207 1.8629 2.1345 2.2355 2.2774 2.1942

Obs. 16780 29704 16780 29704 16780 29704 16780 29704 16780 29704

Firms’ decision and structure estimation

The more information of firm’s action and decision we use, the more robust and accurate result we can get.

Tradition methods neglect the information of firm’s action and decision structure.

Firms’ decision and structure estimation

Data generating process at firm level– Firms choose input and output to maximize the

profit based on the observed productivity

– Where is planed output, the real output is

Firms’ decision and structure estimation

The decision structure of firm’s factor input: dynamic and static– Two adjustment frictions make the firm’s input

decision dynamic: Adjustment cost, such as the cost of installment, test

and dismantle Adjustment lag, because the factor used now is decided

at the former period

Firms’ decision and structure estimation

The decision structure of firm’s dynamic input: take capital as an example

1 1 1

, , max , ,

1, , , , ,

1

it

it it it it it it itR

it it it it it it it

V K J K J C R

E V K J K J R

Firms’ decision and structure estimation

The decision structure of firm’s static input : materials

max , , , ,it

it it it t it it it it MtM

K J PQ K M M P

Firms’ decision and structure estimation

The decision structure of firm’s labor input ( may change with industry)– Treated as dynamic if the adjustment cost cannot

be neglected Adjustment cost : training cost when employing new

staff and the cost of layoff Adjustment lag : new staff can only get to work after the

training – Treated as static if the adjustment cost can be

neglected

Firms’ decision and structure estimation

Model – C-D production function– Hicks-neutral techniques– Static labor input

Firms’ decision and structure estimation

Olley & Pakes （ 1996 ）– Get productivity from the investment function , and then take it into the production function– Step 1. get with nonparametric method, and then

the productivity can be expressed as = ；– Step 2. let productivity follows the Markov

process,get the estimation of with the moment condition

Firms’ decision and structure estimation

Levinsohn & Petrin (2003)– A great loss of investment information – Use materials as proxy variables:

,it it itm m k ,it it itk m

Firms’ decision and structure estimation

Bond & Söderbom (2005)and Ackerberg et al. (2006): Collinearity problem

– Robinson (1988): “The variables in the parametric part cannot be predicted by those in the nonparametric part in the sense of OLS.”

– Newey et al. (1999): There should exist no function between parametric part and nonparametric part in semi-parametric model.

, ;it it itl l k ,it it itm m k

Firms’ decision and structure estimation

Ackerberg et al.(2006)– Capital is decided before TFP – Labor decision is before materials

,it it itl l k

, ,it it it itm m l k

Firms’ decision and structure estimation

– Step 1. the production function is , get 、 with nonparametric method, and the productivity is

– Step 2.the productivity follows Markov process, get the other parameters with the moment condition

Firms’ decision and structure estimation

The idea of the new structural estimation of TFP at firm level– Review the index method about estimating static

input Solow (1957) ； Caves et al. (1982) ； Hall (1989)

– Separate the estimation of static input and dynamic input

Gandhi et al. (2011)

Firms’ decision and structure estimation

new structural estimation of TFP– Get the following formula according to the optimal

condition of static input

– the Hicks-neutral technique allows

Where is the share of materials to nominal output– Get 、、 with nonparametric regression, and in

the situation of C-D production function, the mean of is

Firms’ decision and structure estimation

– If labor is static input, then get with the method above, if not, get the estimation of at the next step

– The productivity follows Markov process same as OP/LP/ACF, and get with the moment condition

Firms’ decision and structure estimation

New structural estimation of TFP– Step 1. estimate the parameter of static input

following the idea of index method– Step 2. estimate the parameter of dynamic input

following the idea of structural estimation– The advantages

Avoid the assumptions in the proxy variables method such as the reversible proxy function and the measurement error

Make full use of firms’ decision Solve the endogenous problem and the collinearity

problem

Table 3：Structural Estimation for Aggregate Output

OP LP ACF NEW-S1 NEW-S2

Coef. Std. Coef. Std. Coef. Std. Coef. Std. Coef. Std.

k 0.0459 0.0005 0.0084 0.0037 0.0115 0.0021 0.0537 0.0014 0.0621 0.0014

l 0.0936 0.0015 0.0773 0.0010 0.0501 0.0054 0.1358 0.0021 0.1090 0.0002

m 0.8309 0.0012 0.9144 0.0102 0.9372 0.0092 0.7302 0.0002 0.7302 0.0002

Year 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07

Growth% 0.3810 0.5745 0.2169 0.1836 0.2574 0.3691 1.4941 3.9874 1.5763 4.0379

Ratio 90/10 1.0562 1.0514 1.1133 1.0822 1.1816 1.1270 1.4517 1.4990 1.4676 1.5141

95/5 1.0856 1.0809 1.2142 1.1271 1.3234 1.2080 1.6544 1.7231 1.6803 1.7427

Obs. 67483 97196 97196 97196 97196

Gross output or Value-added?

Gross output (sales) is the real observable variable by firms who experience the production and management process, while value-added is just a statistical concept.

Value-added can be proper only if the theoretic definition is agreed with empirical measurement, which needs the following assumptions

Assumption 1. Labor and capital produce value-added following , and combine with materials according to to form output

Gross output or Value-added?

Gross output or Value-added?

The core in TFP estimation is to control the substitution among factors– Make the following choices to maximize profit

Labor intensive Capital intensive Outsource and material intensive

– Value-added production function only consider the substitution between labor and capital and neglect the efficiency from materials

 

Gross output or Value-added?

Gross output or Value-added?

The result misusing value-added

– New endogenous problem appears because is put into the error term

– TFP heterogeneity will be exaggerated because the heterogeneity coming from materials is put into TFP difference

Table 4：Structural Estimation for Value-added

OP LP ACF NEW-S

Coef. Std. Coef. Std. Coef. Std. Coef. Std.

k 0.3417 0.0028 0.2691 0.0054 0.1689 0.0115 0.3869 0.0044

l 0.4959 0.0040 0.2169 0.0024 0.0909 0.0133 0.0930 0.0001

Year 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07

Growth% 0.9916 1.3299 4.4379 9.4516 5.8774 12.0461 11.4778 17.9319

Ratio 90/10 1.6245 1.5912 5.1236 5.1989 6.3752 6.6307 8.8184 7.1292

95/5 1.9829 1.9762 9.0538 8.6154 12.0695 11.7247 19.7805 14.1120

Obs. 67489 97196 97196 96920

Sample selection, function form and other robust test

Sample selection problem– There is a great number of entry and exit in the

data, and we can only observe the existed ones The structure estimation method don’t have to

deal with sample selection problem because of the proxy of in the first step

Sample selection, function form and other robust test

We can only observe the existed samples with, and ,so there is endogenous problem in the second step

How to deal with it?– Rules of entry and exit ：– Conditional expectation ：

Sample selection, function form and other robust test

– The probability that a firm i stay in period t

– Get – Put into the conditional expectation of productivity

Table 5：Structural Estimation for Aggregate Output:

Sample Selection Considered

OP LP ACF NEW-S1 NEW-S2

Coef. Std. Coef. Std. Coef. Std. Coef. Std. Coef. Std.

k 0.0457 0.0005 0.0259 0.0103 0.0304 0.0118 0.0343 0.0014 0.0341 0.0015

l 0.0936 0.0015 0.0773 0.0010 -0.0277 0.0380 0.1065 0.0028 0.1090 0.0002

m 0.8309 0.0012 0.7968 0.0783 0.8170 0.0665 0.7302 0.0002 0.7302 0.0002

Year 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07

Growth% 0.3848 0.5796 0.9931 1.8084 1.4416 2.4926 1.7720 4.4306 1.7608 4.4184

Ratio 90/10 1.0562 1.0514 1.2926 1.3153 1.5180 1.5157 1.5034 1.5482 1.4996 1.5456

95/5 1.0858 1.0810 1.4062 1.4371 1.7527 1.7450 1.7337 1.7862 1.7296 1.7821

Obs. 67483 97196 97196 97196 97196

Sample selection, function form and other robust test

Trans-log production function

Cobb-Douglas production function is a special situation of trans-log production function.

Table 6：Sensitivity Analysis

BASE CHECK1 CHECK 2 CHECK 3 CHECK 4

Coef. Std. Coef. Std. Coef. Std. Coef. Std. Coef. Std.

k 0.0717 0.0054 0.0771 0.0047 0.0587 0.0088 0.0821 0.0043 0.0752 0.0062

l 0.2944 0.0113 0.2679 0.0000 0.2588 0.0120 0.2752 0.0086 0.1887 0.0094

m 0.3925 0.0010 0.3925 0.0010 0.4049 0.0010 0.3984 0.0013 0.4719 0.0013

kk 0.0089 0.0004 0.0091 0.0004 0.0104 0.0006 0.0080 0.0003 0.0075 0.0004

ll 0.0274 0.0012 0.0307 0.0000 0.0261 0.0011 0.0290 0.0010 0.0263 0.0008

mm 0.0392 0.0001 0.0392 0.0001 0.0390 0.0001 0.0387 0.0001 0.0426 0.0001

kl 0.0035 0.0010 0.0019 0.0000 0.0053 0.0012 0.0038 0.0008 0.0033 0.0009

lm -0.0499 0.0002 -0.0499 0.0002 -0.0464 0.0002 -0.0494 0.0002 -0.0516 0.0002

mk -0.0185 0.0001 -0.0185 0.0001 -0.0222 0.0001 -0.0182 0.0001 -0.0188 0.0001

t - - - - - - 0.0342 0.0005

Year 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07 98-02 03-07

Growth% 1.0973 3.3210 1.0765 3.3036 0.5199 2.3967 1.4325 2.6047 0.8027 2.5936

Ratio 90/10 1.4631 1.4616 1.4631 1.4616 1.4167 1.4232 1.6414 1.3594 1.4308 1.3966

95/5 1.6683 1.6467 1.6683 1.6467 1.5899 1.5791 1.9345 1.5050 1.6283 1.5619

Obs. 97196 97196 97196 97196 97196

Summary

The problems of tradition methods– DEA method tries to measure TFP by construct a

set of substitution of factors by linear programming, but determinate method cannot get the robust estimation with the data at firm level, because the measurement error cannot be neglected.

Summary

– Index method is also not satisfactory because all the inputs are assumed to be static and the parameter of return to scale should be given.

– Traditional methods, such as FE,IV and dynamic panel, will not get the robust result because the disturbance should be given before the estimation.

Summary

Structural estimation method, which is becoming the most potential approach, tries to open the black box of the firms’ production process by making full use of the information of their behavior and decision-making. – Olley and Pakes (1996), Levinsohn and Petrin

(2003),Ackerberg et al.(2006) all face the “collineraity” problem.

– The new structural estimation, which combines the structural estimation with the traditional index method, may get the most robust estimation of TFP at firm level.

Summary

The definition of variables affects the robustness of TFP estimation

-measuring firms’ output with value-added will

exaggerate TFP heterogeneity seriously Sample selection and the production function

form also affect the TFP estimation

Summary

The most robust estimation of TFP for clothing and leather industry in China

Summary

Unsolved problem:– The use of proxy variable in structural method

and the index method need new foundation if firms have market power.

– More information is needed to separate the effect of demand and price from TFP

Thank you!