Chow test
Comparing coefficients across regressions is common. Chow test is one of them. If you’d like to compare coefficients of regressions for two subsets, that’s the original Chow test.
The idea is to interact the subset indicator with all the covariates or only the covariate you are interested (treatment). If you only interact the dummy with the treatment variable, then you are assuming all other covariates have the same effect across the two subsets. This may or may not be reasonable.
This post is inspired by Austin Nicholas (https://www.stata.com/statalist/archive/2009-11/msg01485.html). The case with overlapping samples is all from his code.
Let’s see a simple example:
est clear
sysuse nlsw88, clear
reg wage hours if south
est sto south
reg wage hours if !south
est sto nonsouth
suest south nonsouth
est sto suest
gen hours1=hours*(south==1)
gen hours2=hours*(south==0)
reg wage south hours?
est sto chow
test _b[hours1]-_b[hours2]=0
esttab south nonsouth suest chow, nogaps mti
(NLSW, 1988 extract)
Source | SS df MS Number of obs = 938
-------------+---------------------------------- F(1, 936) = 12.47
Model | 344.732583 1 344.732583 Prob > F = 0.0004
Residual | 25866.3404 936 27.634979 R-squared = 0.0132
-------------+---------------------------------- Adj R-squared = 0.0121
Total | 26211.0729 937 27.973397 Root MSE = 5.2569
------------------------------------------------------------------------------
wage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .0623497 .0176532 3.53 0.000 .0277053 .0969941
_cons | 4.520583 .6957145 6.50 0.000 3.155242 5.885923
------------------------------------------------------------------------------
Source | SS df MS Number of obs = 1,304
-------------+---------------------------------- F(1, 1302) = 55.93
Model | 1929.41943 1 1929.41943 Prob > F = 0.0000
Residual | 44919.1023 1,302 34.5000785 R-squared = 0.0412
-------------+---------------------------------- Adj R-squared = 0.0404
Total | 46848.5217 1,303 35.9543528 Root MSE = 5.8737
------------------------------------------------------------------------------
wage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .1107536 .01481 7.48 0.000 .0816995 .1398076
_cons | 4.357811 .564756 7.72 0.000 3.24988 5.465743
------------------------------------------------------------------------------
Simultaneous results for south, nonsouth Number of obs = 2,242
------------------------------------------------------------------------------
| Robust
| Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
south_mean |
hours | .0623497 .0174426 3.57 0.000 .0281629 .0965366
_cons | 4.520583 .6599379 6.85 0.000 3.227128 5.814037
-------------+----------------------------------------------------------------
south_lnvar |
_cons | 3.319082 .1435305 23.12 0.000 3.037768 3.600397
-------------+----------------------------------------------------------------
nonsouth_m~n |
hours | .1107536 .0134936 8.21 0.000 .0843066 .1372006
_cons | 4.357811 .4633326 9.41 0.000 3.449696 5.265927
-------------+----------------------------------------------------------------
nonsouth_l~r |
_cons | 3.540962 .1006119 35.19 0.000 3.343766 3.738157
------------------------------------------------------------------------------
(4 missing values generated)
(4 missing values generated)
Source | SS df MS Number of obs = 2,242
-------------+---------------------------------- F(3, 2238) = 36.91
Model | 3502.37892 3 1167.45964 Prob > F = 0.0000
Residual | 70785.4426 2,238 31.6288841 R-squared = 0.0471
-------------+---------------------------------- Adj R-squared = 0.0459
Total | 74287.8215 2,241 33.1494072 Root MSE = 5.624
------------------------------------------------------------------------------
wage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
south | .1627711 .9199871 0.18 0.860 -1.641346 1.966888
hours1 | .0623497 .0188858 3.30 0.001 .0253142 .0993852
hours2 | .1107536 .0141803 7.81 0.000 .0829456 .1385616
_cons | 4.357811 .5407453 8.06 0.000 3.297397 5.418226
------------------------------------------------------------------------------
( 1) hours1 - hours2 = 0
F( 1, 2238) = 4.20
Prob > F = 0.0405
----------------------------------------------------------------------------
(1) (2) (3) (4)
south nonsouth suest chow
----------------------------------------------------------------------------
main
hours 0.0623*** 0.111*** 0.0623***
(3.53) (7.48) (3.57)
south 0.163
(0.18)
hours1 0.0623***
(3.30)
hours2 0.111***
(7.81)
_cons 4.521*** 4.358*** 4.521*** 4.358***
(6.50) (7.72) (6.85) (8.06)
----------------------------------------------------------------------------
south_lnvar
_cons 3.319***
(23.12)
----------------------------------------------------------------------------
nonsouth_m~n
hours 0.111***
(8.21)
_cons 4.358***
(9.41)
----------------------------------------------------------------------------
nonsouth_l~r
_cons 3.541***
(35.19)
----------------------------------------------------------------------------
N 938 1304 2242 2242
----------------------------------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001In the above example, we are interested in the effect of hours on wage for south and non-south subsets. The Chow test is to use the entire sample which has both south and nonsouth data. Then use the interaction of south indicator and hours to find the effect of hours for south and nonsouth. By including these two subsamples in the same regression, we can test the equality of the two coefficients.
The other way to do this is to use Stata’s “suest” command. This command basically take the two regressions and the variance covariance structure; then a test of the difference between two coefficients can be done. However, “suest” does not work for some commands. In my opinion, using interaction can be more flexible.
A comparison with two different outcomes
We can also use the same idea to compare the effect of some treatment on two different outcomes, if we have the same set of covariates. We just need to “stack” the two outcomes and run a pooled regression with some interactions.
Here is an example.
est clear
sysuse nlsw88, clear
reg south wage hours
est sto south
reg smsa wage hours
est sto smsa
suest south smsa
est sto suest
preserve
gen Y1=south
gen Y2=smsa
gen id=_n
reshape long Y, i(id) j(subsample)
gen wage1=wage*(subsample==1)
gen wage2=wage*(subsample==2)
gen hours1=hours*(subsample==1)
gen hours2=hours*(subsample==2)
reg Y wage? hours? subsample
test _b[wage1]-_b[wage2]=0
est sto stacked
esttab south smsa suest stacked, nogaps mti
(NLSW, 1988 extract)
Source | SS df MS Number of obs = 2,242
-------------+---------------------------------- F(2, 2239) = 30.60
Model | 14.513685 2 7.25684251 Prob > F = 0.0000
Residual | 531.049205 2,239 .237181423 R-squared = 0.0266
-------------+---------------------------------- Adj R-squared = 0.0257
Total | 545.56289 2,241 .24344618 Root MSE = .48701
------------------------------------------------------------------------------
south | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
wage | -.0124053 .0018099 -6.85 0.000 -.0159545 -.008856
hours | .0047722 .0009916 4.81 0.000 .0028277 .0067167
_cons | .3372108 .0387354 8.71 0.000 .2612497 .4131718
------------------------------------------------------------------------------
Source | SS df MS Number of obs = 2,242
-------------+---------------------------------- F(2, 2239) = 36.17
Model | 14.6274602 2 7.31373012 Prob > F = 0.0000
Residual | 452.719551 2,239 .202197209 R-squared = 0.0313
-------------+---------------------------------- Adj R-squared = 0.0304
Total | 467.347012 2,241 .208543959 Root MSE = .44966
------------------------------------------------------------------------------
smsa | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
wage | .0139103 .0016711 8.32 0.000 .0106332 .0171873
hours | .0003663 .0009155 0.40 0.689 -.0014291 .0021616
_cons | .5820585 .0357648 16.27 0.000 .511923 .6521941
------------------------------------------------------------------------------
Simultaneous results for south, smsa Number of obs = 2,242
------------------------------------------------------------------------------
| Robust
| Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
south_mean |
wage | -.0124053 .0019287 -6.43 0.000 -.0161854 -.0086251
hours | .0047722 .0009825 4.86 0.000 .0028465 .0066978
_cons | .3372108 .0379918 8.88 0.000 .2627483 .4116733
-------------+----------------------------------------------------------------
south_lnvar |
_cons | -1.43893 .0096645 -148.89 0.000 -1.457872 -1.419988
-------------+----------------------------------------------------------------
smsa_mean |
wage | .0139103 .0018313 7.60 0.000 .010321 .0174996
hours | .0003663 .0009099 0.40 0.687 -.0014172 .0021497
_cons | .5820585 .0361037 16.12 0.000 .5112965 .6528205
-------------+----------------------------------------------------------------
smsa_lnvar |
_cons | -1.598512 .0195103 -81.93 0.000 -1.636751 -1.560272
------------------------------------------------------------------------------
(j = 1 2)
Data Wide -> Long
-----------------------------------------------------------------------------
Number of observations 2,246 -> 4,492
Number of variables 23 -> 23
j variable (2 values) -> subsample
xij variables:
Y1 Y2 -> Y
-----------------------------------------------------------------------------
(8 missing values generated)
(8 missing values generated)
Source | SS df MS Number of obs = 4,484
-------------+---------------------------------- F(5, 4478) = 109.69
Model | 120.488157 5 24.0976314 Prob > F = 0.0000
Residual | 983.768757 4,478 .219689316 R-squared = 0.1091
-------------+---------------------------------- Adj R-squared = 0.1081
Total | 1104.25691 4,483 .246320971 Root MSE = .46871
------------------------------------------------------------------------------
Y | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
wage1 | -.0124053 .0017419 -7.12 0.000 -.0158202 -.0089903
wage2 | .0139103 .0017419 7.99 0.000 .0104953 .0173252
hours1 | .0047722 .0009543 5.00 0.000 .0029013 .0066431
hours2 | .0003663 .0009543 0.38 0.701 -.0015046 .0022372
subsample | .2448477 .0527214 4.64 0.000 .1414877 .3482078
_cons | .092363 .0833599 1.11 0.268 -.0710635 .2557896
------------------------------------------------------------------------------
( 1) wage1 - wage2 = 0
F( 1, 4478) = 114.12
Prob > F = 0.0000
----------------------------------------------------------------------------
(1) (2) (3) (4)
south smsa suest stacked
----------------------------------------------------------------------------
main
wage -0.0124*** 0.0139*** -0.0124***
(-6.85) (8.32) (-6.43)
hours 0.00477*** 0.000366 0.00477***
(4.81) (0.40) (4.86)
wage1 -0.0124***
(-7.12)
wage2 0.0139***
(7.99)
hours1 0.00477***
(5.00)
hours2 0.000366
(0.38)
subsample 0.245***
(4.64)
_cons 0.337*** 0.582*** 0.337*** 0.0924
(8.71) (16.27) (8.88) (1.11)
----------------------------------------------------------------------------
south_lnvar
_cons -1.439***
(-148.89)
----------------------------------------------------------------------------
smsa_mean
wage 0.0139***
(7.60)
hours 0.000366
(0.40)
_cons 0.582***
(16.12)
----------------------------------------------------------------------------
smsa_lnvar
_cons -1.599***
(-81.93)
----------------------------------------------------------------------------
N 2242 2242 2242 4484
----------------------------------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001In this example, we are interested in comparing the effect of wage on south vs. smsa (not interesting, but just as an example). What I did is to reshape it to long format, stacking south and smsa as “Y”. Then creat interaction of other covariates with subsample indicator. Then run the regression with Y on the interaction terms.
Overlapping samples
What if we’d like to compare coefficients for two overlapping subsamples? As I mentioned, Austin Nichols gave the following example:
est clear
sysuse nlsw88, clear
ta south smsa
reg wage hours if south
est sto south
reg wage hours if smsa
est sto smsa
suest south smsa
est sto suest
preserve
expand 2
bys idcode: g n=_n
keep if (n==1&south)|(n==2&smsa)
g hours1=hours*!(n==1&south)
g hours2=hours*!(n==2&smsa)
reg wage hours? n, cl(idcode)
est sto stacked
restore
esttab south smsa suest stacked, nogaps mti
(NLSW, 1988 extract)
Lives in | Lives in SMSA
the south | Not SMSA SMSA | Total
-----------+----------------------+----------
Not south | 308 996 | 1,304
South | 357 585 | 942
-----------+----------------------+----------
Total | 665 1,581 | 2,246
Source | SS df MS Number of obs = 938
-------------+---------------------------------- F(1, 936) = 12.47
Model | 344.732583 1 344.732583 Prob > F = 0.0004
Residual | 25866.3404 936 27.634979 R-squared = 0.0132
-------------+---------------------------------- Adj R-squared = 0.0121
Total | 26211.0729 937 27.973397 Root MSE = 5.2569
------------------------------------------------------------------------------
wage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .0623497 .0176532 3.53 0.000 .0277053 .0969941
_cons | 4.520583 .6957145 6.50 0.000 3.155242 5.885923
------------------------------------------------------------------------------
Source | SS df MS Number of obs = 1,578
-------------+---------------------------------- F(1, 1576) = 46.48
Model | 1594.14881 1 1594.14881 Prob > F = 0.0000
Residual | 54048.2539 1,576 34.2945773 R-squared = 0.0286
-------------+---------------------------------- Adj R-squared = 0.0280
Total | 55642.4027 1,577 35.2837049 Root MSE = 5.8562
------------------------------------------------------------------------------
wage | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .0953636 .0139872 6.82 0.000 .0679281 .1227991
_cons | 4.861519 .5443826 8.93 0.000 3.793729 5.929309
------------------------------------------------------------------------------
Simultaneous results for south, smsa Number of obs = 1,934
------------------------------------------------------------------------------
| Robust
| Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
south_mean |
hours | .0623497 .0174432 3.57 0.000 .0281617 .0965378
_cons | 4.520583 .6599613 6.85 0.000 3.227082 5.814083
-------------+----------------------------------------------------------------
south_lnvar |
_cons | 3.319082 .1435356 23.12 0.000 3.037758 3.600407
-------------+----------------------------------------------------------------
smsa_mean |
hours | .0953636 .0132806 7.18 0.000 .069334 .1213931
_cons | 4.861519 .4842914 10.04 0.000 3.912325 5.810713
-------------+----------------------------------------------------------------
smsa_lnvar |
_cons | 3.534987 .0910825 38.81 0.000 3.356469 3.713506
------------------------------------------------------------------------------
(2,246 observations created)
(1,969 observations deleted)
(7 missing values generated)
(7 missing values generated)
Linear regression Number of obs = 2,516
F(3, 1933) = 40.81
Prob > F = 0.0000
R-squared = 0.0399
Root MSE = 5.6403
(Std. err. adjusted for 1,934 clusters in idcode)
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours1 | .0953636 .0132886 7.18 0.000 .0693022 .121425
hours2 | .0623497 .0174536 3.57 0.000 .0281198 .0965796
n | .3409364 .6124145 0.56 0.578 -.8601261 1.541999
_cons | 4.179646 1.177889 3.55 0.000 1.869579 6.489713
------------------------------------------------------------------------------
----------------------------------------------------------------------------
(1) (2) (3) (4)
south smsa suest stacked
----------------------------------------------------------------------------
main
hours 0.0623*** 0.0954*** 0.0623***
(3.53) (6.82) (3.57)
hours1 0.0954***
(7.18)
hours2 0.0623***
(3.57)
n 0.341
(0.56)
_cons 4.521*** 4.862*** 4.521*** 4.180***
(6.50) (8.93) (6.85) (3.55)
----------------------------------------------------------------------------
south_lnvar
_cons 3.319***
(23.12)
----------------------------------------------------------------------------
smsa_mean
hours 0.0954***
(7.18)
_cons 4.862***
(10.04)
----------------------------------------------------------------------------
smsa_lnvar
_cons 3.535***
(38.81)
----------------------------------------------------------------------------
N 938 1578 1934 2516
----------------------------------------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001IV regression
What about IV regression?
sysuse nlsw88, clear
ivregress 2sls wage (hours=union) if south
ivregress 2sls wage (hours=union) if !south
gen hours1=hours*(south==1)
gen hours2=hours*(south==0)
gen union1=union*(south==1)
gen union2=union*(south==0)
ivregress 2sls wage south (hours? = union?)already preserved
r(621);
(NLSW, 1988 extract)
Instrumental-variables 2SLS regression Number of obs = 798
Wald chi2(1) = 2.61
Prob > chi2 = 0.1060
R-squared = .
Root MSE = 9.5807
------------------------------------------------------------------------------
wage | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .97819 .6050822 1.62 0.106 -.2077492 2.164129
_cons | -30.96026 23.32846 -1.33 0.184 -76.6832 14.76268
------------------------------------------------------------------------------
Endogenous: hours
Exogenous: union
Instrumental-variables 2SLS regression Number of obs = 1,079
Wald chi2(1) = 6.03
Prob > chi2 = 0.0140
R-squared = .
Root MSE = 7.0372
------------------------------------------------------------------------------
wage | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .6255843 .2546731 2.46 0.014 .1264342 1.124735
_cons | -14.9159 9.401508 -1.59 0.113 -33.34252 3.510714
------------------------------------------------------------------------------
Endogenous: hours
Exogenous: union
(4 missing values generated)
(4 missing values generated)
(368 missing values generated)
(368 missing values generated)
Instrumental-variables 2SLS regression Number of obs = 1,877
Wald chi2(3) = 21.75
Prob > chi2 = 0.0001
R-squared = .
Root MSE = 8.2154
------------------------------------------------------------------------------
wage | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
hours1 | .97819 .5188511 1.89 0.059 -.0387394 1.995119
hours2 | .6255843 .297311 2.10 0.035 .0428654 1.208303
south | -16.04435 22.81705 -0.70 0.482 -60.76495 28.67625
_cons | -14.9159 10.97553 -1.36 0.174 -36.42754 6.595735
------------------------------------------------------------------------------
Endogenous: hours1 hours2
Exogenous: south union1 union2We can see same Chow kind of test works, with IV regression, if we have the right interaction terms.
IV with fixed effects
However, when doing with a fixed effet IV, I seem to have difficulties. In this example, I use “reghdfe” to do an IV regression with fixed effect. We can also use “xtivreg2”, but “ivreghdfe” is supposed to be faster.
sysuse nlsw88, clear
gen hours1=hours*(south==1)
gen hours2=hours*(south==0)
gen union1=union*(south==1)
gen union2=union*(south==0)
ivreghdfe wage (hours=union) if south, a(race) cluster(race)
ivreghdfe wage (hours=union) if !south, a(race) cluster(race)
ivreghdfe wage south (hours? = union?) , a(race) cluster(race)
already preserved
r(621);
(NLSW, 1988 extract)
(4 missing values generated)
(4 missing values generated)
(368 missing values generated)
(368 missing values generated)
(MWFE estimator converged in 1 iterations)
IV (2SLS) estimation
--------------------
Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on race
Number of clusters (race) = 3 Number of obs = 798
F( 1, 2) = 69.28
Prob > F = 0.0141
Total (centered) SS = 12186.8806 Centered R2 = -6.4527
Total (uncentered) SS = 12186.8806 Uncentered R2 = -6.4527
Residual SS = 90825.44631 Root MSE = 10.68
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | 1.107099 .133012 8.32 0.014 .5347947 1.679404
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 1.756
Chi-sq(1) P-val = 0.1852
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): 3.233
(Kleibergen-Paap rk Wald F statistic): 13.977
Stock-Yogo weak ID test critical values: 10% maximal IV size 16.38
15% maximal IV size 8.96
20% maximal IV size 6.66
25% maximal IV size 5.53
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments): 0.000
(equation exactly identified)
------------------------------------------------------------------------------
Instrumented: hours
Excluded instruments: union
Partialled-out: _cons
nb: total SS, model F and R2s are after partialling-out;
any small-sample adjustments include partialled-out
variables in regressor count K
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
race | 3 3 0 *|
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
(MWFE estimator converged in 1 iterations)
IV (2SLS) estimation
--------------------
Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on race
Number of clusters (race) = 3 Number of obs = 1079
F( 1, 2) = 1.41
Prob > F = 0.3564
Total (centered) SS = 19086.22115 Centered R2 = -2.3302
Total (uncentered) SS = 19086.22115 Uncentered R2 = -2.3302
Residual SS = 63560.97528 Root MSE = 7.682
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .7023322 .5905772 1.19 0.356 -1.838717 3.243381
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 0.789
Chi-sq(1) P-val = 0.3745
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): 5.501
(Kleibergen-Paap rk Wald F statistic): 3.772
Stock-Yogo weak ID test critical values: 10% maximal IV size 16.38
15% maximal IV size 8.96
20% maximal IV size 6.66
25% maximal IV size 5.53
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments): 0.000
(equation exactly identified)
------------------------------------------------------------------------------
Instrumented: hours
Excluded instruments: union
Partialled-out: _cons
nb: total SS, model F and R2s are after partialling-out;
any small-sample adjustments include partialled-out
variables in regressor count K
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
race | 3 3 0 *|
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
(MWFE estimator converged in 1 iterations)
IV (2SLS) estimation
--------------------
Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on race
Number of clusters (race) = 3 Number of obs = 1877
F( 3, 2) = 1036.24
Prob > F = 0.0010
Total (centered) SS = 32142.319 Centered R2 = -3.9041
Total (uncentered) SS = 32142.319 Uncentered R2 = -3.9041
Residual SS = 157627.5718 Root MSE = 9.174
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours1 | 1.142036 .0650507 17.56 0.003 .8621454 1.421927
hours2 | .6880691 .5265185 1.31 0.321 -1.577357 2.953495
south | -19.74773 22.10811 -0.89 0.466 -114.8712 75.37577
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 1.793
Chi-sq(1) P-val = 0.1806
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): 3.584
(Kleibergen-Paap rk Wald F statistic): 8.484
Stock-Yogo weak ID test critical values: 10% maximal IV size 7.03
15% maximal IV size 4.58
20% maximal IV size 3.95
25% maximal IV size 3.63
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Warning: estimated covariance matrix of moment conditions not of full rank.
overidentification statistic not reported, and standard errors and
model tests should be interpreted with caution.
Possible causes:
number of clusters insufficient to calculate robust covariance matrix
singleton dummy variable (dummy with one 1 and N-1 0s or vice versa)
partial option may address problem.
------------------------------------------------------------------------------
Instrumented: hours1 hours2
Included instruments: south
Excluded instruments: union1 union2
Partialled-out: _cons
nb: total SS, model F and R2s are after partialling-out;
any small-sample adjustments include partialled-out
variables in regressor count K
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
race | 3 3 0 *|
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computationWe can see I failed to replicate the first two regressions in the third regression. Why? Because we’ll need the fixed effect to be interacted with the subsample indicator to make it right.
Here is another try:
sysuse nlsw88, clear
gen hours1=hours*(south==1)
gen hours2=hours*(south==0)
gen union1=union*(south==1)
gen union2=union*(south==0)
gen race1=race*(south==1)
gen race2=race*(south==0)
ivreghdfe wage (hours=union) if south, a(race) cluster(race)
ivreghdfe wage (hours=union) if !south, a(race) cluster(race)
ivregress 2sls wage south (hours? = union?) i.race?, cluster(race)
already preserved
r(621);
(NLSW, 1988 extract)
(4 missing values generated)
(4 missing values generated)
(368 missing values generated)
(368 missing values generated)
(MWFE estimator converged in 1 iterations)
IV (2SLS) estimation
--------------------
Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on race
Number of clusters (race) = 3 Number of obs = 798
F( 1, 2) = 69.28
Prob > F = 0.0141
Total (centered) SS = 12186.8806 Centered R2 = -6.4527
Total (uncentered) SS = 12186.8806 Uncentered R2 = -6.4527
Residual SS = 90825.44631 Root MSE = 10.68
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | 1.107099 .133012 8.32 0.014 .5347947 1.679404
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 1.756
Chi-sq(1) P-val = 0.1852
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): 3.233
(Kleibergen-Paap rk Wald F statistic): 13.977
Stock-Yogo weak ID test critical values: 10% maximal IV size 16.38
15% maximal IV size 8.96
20% maximal IV size 6.66
25% maximal IV size 5.53
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments): 0.000
(equation exactly identified)
------------------------------------------------------------------------------
Instrumented: hours
Excluded instruments: union
Partialled-out: _cons
nb: total SS, model F and R2s are after partialling-out;
any small-sample adjustments include partialled-out
variables in regressor count K
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
race | 3 3 0 *|
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
(MWFE estimator converged in 1 iterations)
IV (2SLS) estimation
--------------------
Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on race
Number of clusters (race) = 3 Number of obs = 1079
F( 1, 2) = 1.41
Prob > F = 0.3564
Total (centered) SS = 19086.22115 Centered R2 = -2.3302
Total (uncentered) SS = 19086.22115 Uncentered R2 = -2.3302
Residual SS = 63560.97528 Root MSE = 7.682
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours | .7023322 .5905772 1.19 0.356 -1.838717 3.243381
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): 0.789
Chi-sq(1) P-val = 0.3745
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): 5.501
(Kleibergen-Paap rk Wald F statistic): 3.772
Stock-Yogo weak ID test critical values: 10% maximal IV size 16.38
15% maximal IV size 8.96
20% maximal IV size 6.66
25% maximal IV size 5.53
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments): 0.000
(equation exactly identified)
------------------------------------------------------------------------------
Instrumented: hours
Excluded instruments: union
Partialled-out: _cons
nb: total SS, model F and R2s are after partialling-out;
any small-sample adjustments include partialled-out
variables in regressor count K
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
race | 3 3 0 *|
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
note: 3.race1 omitted because of collinearity.
note: 3.race2 omitted because of collinearity.
Instrumental-variables 2SLS regression Number of obs = 1,877
Wald chi2(7) = 101696.08
Prob > chi2 = 0.0000
R-squared = .
Root MSE = 9.0693
(Std. err. adjusted for 3 clusters in race)
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
hours1 | 1.107099 .1085357 10.20 0.000 .8943732 1.319825
hours2 | .7023322 .4819806 1.46 0.145 -.2423324 1.646997
south | -21.52299 22.35225 -0.96 0.336 -65.33259 22.28662
|
race1 |
1 | 3.054296 .1451752 21.04 0.000 2.769758 3.338834
2 | 1.987268 .176538 11.26 0.000 1.64126 2.333277
3 | 0 (omitted)
|
race2 |
1 | -.4816127 .434723 -1.11 0.268 -1.333654 .3704287
2 | -2.065527 .7694573 -2.68 0.007 -3.573635 -.5574181
3 | 0 (omitted)
|
_cons | -17.01633 18.01689 -0.94 0.345 -52.32879 18.29614
------------------------------------------------------------------------------
Endogenous: hours1 hours2
Exogenous: south 1.race1 2.race1 1.race2 2.race2 union1 union2This works. Basically we use dummies which are interactions of subsample indicator and the fixed effect dummies. This would not work if we have a lot of fixed effect units.
But we can trick “reghdfe” to use a two way fixed effect option:
This way we can do a test to see whether hours effect differs between these two samples.
sysuse nlsw88, clear
gen hours1=hours*(south==1)
gen hours2=hours*(south==0)
gen union1=union*(south==1)
gen union2=union*(south==0)
gen race1=race*(south==1)
gen race2=race*(south==0)
ivreghdfe wage south (hours? = union?) , a(race1 race2) cluster(race)
test _b[hours1]-_b[hours2]=0already preserved
r(621);
(NLSW, 1988 extract)
(4 missing values generated)
(4 missing values generated)
(368 missing values generated)
(368 missing values generated)
note: variable #4 is probably collinear with the fixed effects (all partialled-
> out values are close to zero; tol = 1.0e-09)
(MWFE estimator converged in 2 iterations)
warning: -ranktest- error in calculating underidentification test statistics;
may be caused by collinearities
warning: -ranktest- error in calculating weak identification test statistics;
may be caused by collinearities
IV (2SLS) estimation
--------------------
Estimates efficient for homoskedasticity only
Statistics robust to heteroskedasticity and clustering on race
Number of clusters (race) = 3 Number of obs = 1877
F( 2, 2) = 13010.62
Prob > F = 0.0001
Total (centered) SS = 31273.10174 Centered R2 = -3.9367
Total (uncentered) SS = 31273.10174 Uncentered R2 = -3.9367
Residual SS = 154386.4216 Root MSE = 9.089
------------------------------------------------------------------------------
| Robust
wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
hours1 | 1.107099 .1331773 8.31 0.014 .5340838 1.680115
hours2 | .7023322 .5914076 1.19 0.357 -1.84229 3.246954
south | 0 (omitted)
------------------------------------------------------------------------------
Underidentification test (Kleibergen-Paap rk LM statistic): .
Chi-sq(.) P-val = .
------------------------------------------------------------------------------
Weak identification test (Cragg-Donald Wald F statistic): .
(Kleibergen-Paap rk Wald F statistic): .
Stock-Yogo weak ID test critical values: 10% maximal IV size 7.03
15% maximal IV size 4.58
20% maximal IV size 3.95
25% maximal IV size 3.63
Source: Stock-Yogo (2005). Reproduced by permission.
NB: Critical values are for Cragg-Donald F statistic and i.i.d. errors.
------------------------------------------------------------------------------
Warning: estimated covariance matrix of moment conditions not of full rank.
overidentification statistic not reported, and standard errors and
model tests should be interpreted with caution.
Possible causes:
number of clusters insufficient to calculate robust covariance matrix
singleton dummy variable (dummy with one 1 and N-1 0s or vice versa)
partial option may address problem.
------------------------------------------------------------------------------
Collinearities detected among instruments: 1 instrument(s) dropped
Instrumented: hours1 hours2
Included instruments: south
Excluded instruments: union1 union2
Partialled-out: _cons
nb: total SS, model F and R2s are after partialling-out;
any small-sample adjustments include partialled-out
variables in regressor count K
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
race1 | 4 0 4 |
race2 | 4 2 2 |
-----------------------------------------------------+
( 1) hours1 - hours2 = 0
F( 1, 2) = 0.31
Prob > F = 0.6325Chow test with different covariates
What if we want to compare coefficients across equations with different covariates? We can still do Chow test.
Say you have \[ Y_1 = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon_1\] \[ Y_2 = \gamma_0 + \gamma_1 X_1 + \epsilon_2\]
The way you can think of the second equation is that you still have \(X_2\) but it is just a constant, which means it will go into the constant term. \[ Y_2 = \gamma_0 + \gamma_1 X_1 + \gamma_2 C + \epsilon_2\]
So we can just replace \(X_2\) with a constant in the sample for \(Y_2\) and still do Chow test.
sysuse nlsw88, clear
reg south wage hours tenure
est sto south
reg smsa wage hours
est sto smsa
suest south smsa
est sto suest
preserve
gen Y1=south
gen Y2=smsa
gen id=_n
reshape long Y, i(id) j(subsample)
gen wage1=wage*(subsample==1)
gen wage2=wage*(subsample==2)
gen hours1=hours*(subsample==1)
gen hours2=hours*(subsample==2)
replace tenure=1 if subsample==2
gen tenure1= tenure*(subsample==1)
gen tenure2= tenure*(subsample==2)
reg Y wage? hours? tenure? subsample
est sto chow
test _b[hours1]-_b[hours2]=0
esttab suest chow, nogaps mti(NLSW, 1988 extract)
Source | SS df MS Number of obs = 2,227
-------------+---------------------------------- F(3, 2223) = 20.30
Model | 14.4507388 3 4.81691294 Prob > F = 0.0000
Residual | 527.507052 2,223 .23729512 R-squared = 0.0267
-------------+---------------------------------- Adj R-squared = 0.0254
Total | 541.957791 2,226 .243467112 Root MSE = .48713
------------------------------------------------------------------------------
south | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
wage | -.0119584 .0018359 -6.51 0.000 -.0155586 -.0083583
hours | .004828 .0010072 4.79 0.000 .0028528 .0068031
tenure | -.0024032 .0019221 -1.25 0.211 -.0061726 .0013661
_cons | .346361 .0392121 8.83 0.000 .2694648 .4232573
------------------------------------------------------------------------------
Source | SS df MS Number of obs = 2,242
-------------+---------------------------------- F(2, 2239) = 36.17
Model | 14.6274602 2 7.31373012 Prob > F = 0.0000
Residual | 452.719551 2,239 .202197209 R-squared = 0.0313
-------------+---------------------------------- Adj R-squared = 0.0304
Total | 467.347012 2,241 .208543959 Root MSE = .44966
------------------------------------------------------------------------------
smsa | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
wage | .0139103 .0016711 8.32 0.000 .0106332 .0171873
hours | .0003663 .0009155 0.40 0.689 -.0014291 .0021616
_cons | .5820585 .0357648 16.27 0.000 .511923 .6521941
------------------------------------------------------------------------------
Simultaneous results for south, smsa Number of obs = 2,242
------------------------------------------------------------------------------
| Robust
| Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
south_mean |
wage | -.0119584 .001953 -6.12 0.000 -.0157862 -.0081307
hours | .004828 .0009971 4.84 0.000 .0028737 .0067822
tenure | -.0024032 .0018928 -1.27 0.204 -.0061131 .0013066
_cons | .346361 .038473 9.00 0.000 .2709554 .4217667
-------------+----------------------------------------------------------------
south_lnvar |
_cons | -1.438451 .0096359 -149.28 0.000 -1.457337 -1.419565
-------------+----------------------------------------------------------------
smsa_mean |
wage | .0139103 .0018313 7.60 0.000 .010321 .0174996
hours | .0003663 .0009099 0.40 0.687 -.0014172 .0021497
_cons | .5820585 .0361037 16.12 0.000 .5112965 .6528205
-------------+----------------------------------------------------------------
smsa_lnvar |
_cons | -1.598512 .0195103 -81.93 0.000 -1.636751 -1.560272
------------------------------------------------------------------------------
(j = 1 2)
Data Wide -> Long
-----------------------------------------------------------------------------
Number of observations 2,246 -> 4,492
Number of variables 23 -> 23
j variable (2 values) -> subsample
xij variables:
Y1 Y2 -> Y
-----------------------------------------------------------------------------
(8 missing values generated)
(8 missing values generated)
(2,222 real changes made)
(15 missing values generated)
(15 missing values generated)
note: subsample omitted because of collinearity.
Source | SS df MS Number of obs = 4,469
-------------+---------------------------------- F(6, 4462) = 91.07
Model | 120.039676 6 20.0066126 Prob > F = 0.0000
Residual | 980.226603 4,462 .219683237 R-squared = 0.1091
-------------+---------------------------------- Adj R-squared = 0.1079
Total | 1100.26628 4,468 .246254762 Root MSE = .4687
------------------------------------------------------------------------------
Y | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
wage1 | -.0119584 .0017664 -6.77 0.000 -.0154215 -.0084954
wage2 | .0139103 .0017418 7.99 0.000 .0104954 .0173252
hours1 | .004828 .0009691 4.98 0.000 .002928 .0067279
hours2 | .0003663 .0009543 0.38 0.701 -.0015046 .0022371
tenure1 | -.0024032 .0018494 -1.30 0.194 -.006029 .0012225
tenure2 | .2356975 .0530397 4.44 0.000 .1317134 .3396816
subsample | 0 (omitted)
_cons | .346361 .0377289 9.18 0.000 .2723936 .4203285
------------------------------------------------------------------------------
( 1) hours1 - hours2 = 0
F( 1, 4462) = 10.76
Prob > F = 0.0010
--------------------------------------------
(1) (2)
suest chow
--------------------------------------------
main
wage -0.0120***
(-6.12)
hours 0.00483***
(4.84)
tenure -0.00240
(-1.27)
wage1 -0.0120***
(-6.77)
wage2 0.0139***
(7.99)
hours1 0.00483***
(4.98)
hours2 0.000366
(0.38)
tenure1 -0.00240
(-1.30)
tenure2 0.236***
(4.44)
subsample 0
(.)
_cons 0.346*** 0.346***
(9.00) (9.18)
--------------------------------------------
south_lnvar
_cons -1.438***
(-149.28)
--------------------------------------------
smsa_mean
wage 0.0139***
(7.60)
hours 0.000366
(0.40)
_cons 0.582***
(16.12)
--------------------------------------------
smsa_lnvar
_cons -1.599***
(-81.93)
--------------------------------------------
N 2242 4469
--------------------------------------------
t statistics in parentheses
* p<0.05, ** p<0.01, *** p<0.001We can see Chow type of “stacking” method generates the same result as “suest” in terms of point estimates. For standard errors you can use “robust” or “cluster” option in “reg” command.
Conclusion
For testing cross equations hypotheses, we can use “suest” or Chow type “stacking” method. Sometimes people use “sureg”. I prefer not use “sureg”. It is a GLS estimator for all the equations together. It relies on assumptions of the error term of the whole system. It assumes homoscedasticity for example. If it’s true, then GLS is more efficient; if not, then biased. If the equations have the same covariates, then it returns the same coefficient estimates as the single equation estimates. If different covariates, then different estimates as the single equation estimates.
The nice thing about Chow test is that it is very flexible, and it does not rely on Stata’s internal functions. You can do it with R or other programs. And it works for complicated models too, usually, as long as the single equation works. “suest” needs the results stored before hand, which may not be available even within stata.