Example 3LC1. Linear model: US production function (48 states in 9 regions)

Baltagi, Song and Jung (2001) estimated a Cobb Douglas production function for the US productivity data of Munnell (1990) for the period 1970-1986. The data set (productivity.dat) contains information by year, state and region on various capital and labour inputs. The same data were used by Rabe-Hesketh and Skrondal (2005, exercise 7.3). The data are for 48 US states (level 2) and 9 regions of the US (level 3). We estimate a 3-level model of the logarithm of private capital stock, using some of the state-level measures of unemployment and various publicly funded activities as covariates.

References

Baltagi, B. H., S. H. Song, and B. C. Jung, (2001), Journal of Econometrics 101: 357–381.

Munnell, A., (1990), Why has productivity growth declined? Productivity and public investment. New England Economic Review. Jan. / Feb.:3–22.

Rabe-Hesketh, S., and Skrondal, A., (2005), Multilevel and Longitudinal Modelling using Stata, Stata Press, Stata Corp, College Station, Texas.

Data description

Number of observations = 816

Number of level 2 cases (‘state’ = states of U.S.) = 48

Number of level 3 cases (‘region’ = region of U.S.) = 9

Number of variables (columns): 17

The variables are:

state =    states 1-48

region  =    regions 1-9

year  =     years 1970-1986

public   =      public capital stock

hwy   =   log(highway component of public capital stock)

water   =  log(water component of public capital stock)

other    =     log(bldg/other component of  public capital stock)

private =  log(private capital stock)

gsp  =    log(gross state product)

emp  =   log(non-agriculture payrolls)

unemp  =   state unemployment rate

The first few lines of productivity.dat look like:

Example

We will start by estimating a linear model (without random effects) for the logarithm of private capital stock, private with the covariates hwy, water, other and unemp.

We will then allow for both the state random effect (state) and for the region random effect (region), using mass 96 for both level 2 and level 3.

Both these random effects are significant and significant, sclev2= 0.29718 (0.46362E-02),  sclev3=0.25728 (0.59759E-02), and sclev2> sclev3.

Sabre commands

out productivity.log

data state region year public hwy water other private gsp emp unemp

case first=state second=region

yvar private

family g

constant cons

mass first=96 second=96

alpha 1

ari a

fit hwy water other unemp cons

dis m

dis e

stop

Sabre log file

<S> data state region year public hwy water other private gsp emp unemp

816 observations in dataset

<S> case first=state second=region

<S> yvar private

<S> family g

<S> constant cons

<S> mass first=96 second=96

<S> alpha 1

<S> ari a

<S> fit hwy water other unemp cons

Initial Homogeneous Fit:

Iteration       Log. lik.       Difference

__________________________________________

1          -92.779046

Iteration       Log. lik.         Step      End-points     Orthogonality

length    0          1      criterion

________________________________________________________________________

1           103.83069        1.0000    fixed  fixed       10.740

2           164.91413        1.0000    fixed  fixed      orthogonal

3           223.61434        1.0000    fixed  fixed       42.534

4           398.66698        0.2500    fixed  fixed       31.955

5           566.38688        1.0000    fixed  fixed       115.99

6           570.93147        0.5000    fixed  fixed       246.80

7           620.84031        1.0000    fixed  fixed       4.2063

8           623.61855        0.0313    fixed  fixed       2.0443

9           644.67990        0.0625    fixed  fixed       1.0336

10           650.52590        0.0156    fixed  fixed       6.8368

11           656.76716        0.2500    fixed  fixed       190.53

12           818.89009        1.0000    fixed  fixed      orthogonal

13           819.90069        1.0000    fixed  fixed       30.958

14           821.07209        0.5000    fixed  fixed       16.374

15           821.12792        0.0156    fixed  fixed       4.1429

16           821.43748        0.0313    fixed  fixed      orthogonal

17           823.01496        1.0000    fixed  fixed      orthogonal

18           824.22349        1.0000    fixed  fixed      orthogonal

19           825.35265        1.0000    fixed  fixed      orthogonal

20           826.26399        1.0000    fixed  fixed      orthogonal

21           827.09599        1.0000    fixed  fixed       11.708

22           827.21836        0.0078    fixed  fixed      orthogonal

23           827.86688        1.0000    fixed  fixed       1.9105

24           827.88532        0.0020    fixed  fixed       5.2971

25           828.22338        0.0313    fixed  fixed      orthogonal

26           830.19284        1.0000    fixed  fixed       13.645

27           830.88480        0.5000    fixed  fixed       21.926

28           831.07719        0.0625    fixed  fixed      orthogonal

29           833.75857        1.0000    fixed  fixed       6.6293

30           837.85297        0.1250    fixed  fixed       14.817

31           838.95540        0.0313    fixed  fixed       1.7552

32           839.49882        0.0313    fixed  fixed       23.321

33           839.69707        0.0625    fixed  fixed       541.01

34           845.39686        1.0000    fixed  fixed      orthogonal

35           845.68906        1.0000    fixed  fixed       47.137

36           845.75796        1.0000    fixed  fixed       1.8814

37           847.99216        0.0156    fixed  fixed       83.612

38           849.76283        0.1250    fixed  fixed      orthogonal

39           850.45014        1.0000    fixed  fixed       21.772

40           851.83832        0.0625    fixed  fixed       102.61

41           852.42080        0.0625    fixed  fixed       3.2700

42           852.50899        0.0078    fixed  fixed       67.357

43           854.46818        0.1250    fixed  fixed       7242.5

44           855.63389        1.0000    fixed  fixed       1092.9

45           856.11039        0.0625    fixed  fixed       20.617

46           856.21877        0.0039    fixed  fixed       12.124

47           856.25543        0.0039    fixed  fixed       6.6368

48           857.06500        0.0020    fixed  fixed       21.259

49           857.35161        0.0078    fixed  fixed       992.65

50           857.61633        1.0000    fixed  fixed       2620.9

51           859.23405        0.5000    fixed  fixed       1816.3

52           859.84857        1.0000    fixed  fixed       3010.6

53           859.92138        1.0000    fixed  fixed       11616.

54           859.92154        1.0000    fixed  fixed       2188.8

55           859.92154        1.0000    fixed  fixed

<S> dis m

X-vars            Y-var             Case-var

________________________________________________

cons              private           state

hwy                                 region

water

other

unemp

Univariate model

Standard linear

Gaussian random effects

Number of observations             =     816

Number of level 2 cases            =      48

Number of level 3 cases            =       9

X-var df           =     5

Sigma df           =     1

Scale df           =     2

Log likelihood =      859.92154     on     808 residual degrees of freedom

<S> dis e

Parameter              Estimate         Std. Err.

___________________________________________________

cons                    1.5157          0.53379E-01

hwy                    0.35939          0.13771E-01

water                  0.56907          0.12396E-01

other                  0.18537          0.13000E-01

unemp                  0.21874E-02      0.13306E-02

sigma                  0.71069E-01      0.17798E-02

sclev2                 0.29718          0.46362E-02

sclev3                 0.25728          0.59759E-02

<S> stop