Dear All,
If valid instruments are not available, we can use Lewbel's (2018, 2012) heteroskedasticity-based constructed instrument. This has also been shown to be applicable in the case of binary endogenous variables.
It constructs valid instruments on the basis of information contained in the heteroskedasticity of the error term of the structural equation. The estimator gain efficiency, if we also include an external instrument. This what is available in my data, but which alone does not constitute a valid instrument, as shown via:
Estimating the Lewbel approach in a panel fixed effects model with clustered standard erros, I write:
Now I am wondering whether the excluded instrument can also be the fitted probit estimate of the first stage "y1_hat". This is a bit tricky in interpretation but yields a higher F-Test than the alternative with z1 (results are fairly similar)
If valid instruments are not available, we can use Lewbel's (2018, 2012) heteroskedasticity-based constructed instrument. This has also been shown to be applicable in the case of binary endogenous variables.
It constructs valid instruments on the basis of information contained in the heteroskedasticity of the error term of the structural equation. The estimator gain efficiency, if we also include an external instrument. This what is available in my data, but which alone does not constitute a valid instrument, as shown via:
Code:
xtlogit y1 z1 x, fe vce(boot)
predict y1_hat
xtivreg2 y2 x (y1 = y1_hat), fe cluster(hhid) first endog(I)
FIXED EFFECTS ESTIMATION
------------------------
Number of groups = 198 Obs per group: min = 3
avg = 3.0
max = 3
First-stage regressions
-----------------------
FIXED EFFECTS ESTIMATION
------------------------
Number of groups = 198 Obs per group: min = 3
avg = 3.0
max = 3
First-stage regression of y1:
Statistics robust to heteroskedasticity and clustering on hhid
Number of obs = 594
Number of clusters (hhid) = 198
------------------------------------------------------------------------------
| Robust
y1 | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
y1_hat | -.2426688 .2516973 -0.96 0.336 -.7375261 .2521885
x1 | .2264221 .0475699 4.76 0.000 .1328958 .3199484
x2 | -.1929559 .0392403 -4.92 0.000 -.2701054 -.1158063
x3 | -.0001582 .0001217 -1.30 0.194 -.0003975 .000081
x4 | .0008101 .0001883 4.30 0.000 .0004398 .0011803
x5 | -.0160819 .0660399 -0.24 0.808 -.1459217 .113758
x6 | -.0001422 .0001727 -0.82 0.411 -.0004819 .0001974
------------------------------------------------------------------------------
F test of excluded instruments:
F( 1, 197) = 0.93
Prob > F = 0.3362
Sanderson-Windmeijer multivariate F test of excluded instruments:
F( 1, 197) = 0.93
Prob > F = 0.3362
Summary results for first-stage regressions
-------------------------------------------
(Underid) (Weak id)
Variable | F( 1, 197) P-val | SW Chi-sq( 1) P-val | SW F( 1, 197)
y1 | 0.93 0.3362 | 0.94 0.3313 | 0.93
NB: first-stage test statistics cluster-robust
Stock-Yogo weak ID F test critical values for single endogenous regressor:
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 i.i.d. errors only.
Underidentification test
Ho: matrix of reduced form coefficients has rank=K1-1 (underidentified)
Ha: matrix has rank=K1 (identified)
Kleibergen-Paap rk LM statistic Chi-sq(1)=0.77 P-val=0.3788
Weak identification test
Ho: equation is weakly identified
Cragg-Donald Wald F statistic 1.00
Kleibergen-Paap Wald rk F statistic 0.93
Stock-Yogo weak ID test critical values for K1=1 and L1=1:
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.
Weak-instrument-robust inference
Tests of joint significance of endogenous regressors B1 in main equation
Ho: B1=0 and orthogonality conditions are valid
Anderson-Rubin Wald test F(1,197)= 1.41 P-val=0.2363
Anderson-Rubin Wald test Chi-sq(1)= 1.43 P-val=0.2313
Stock-Wright LM S statistic Chi-sq(1)= 2.83 P-val=0.0922
Estimating the Lewbel approach in a panel fixed effects model with clustered standard erros, I write:
Code:
ivreg2h y2 X (y1 = z), fe cluster(hhid) first
Comment