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  • Quantile instrumental‐variables (IV) estimators

    • Inverse quantile regression (IQR)

    • Smoothed estimating equations

  • IQR estimator

    • Confidence intervals robust to weak instruments

    • Graphical convergence diagnostics

  • Simultaneous estimation over quantiles

  • Visualization of effects over quantiles

  • Specialized tests

    • IV not affecting the outcome

    • Equality of endogenous effects across quantiles

    • Effects greater than zero across quantiles

    • IV being exogenous instead of endogenous

When we want to study the effects of covariates on different quantiles of the outcome, we use quantile regression. But what if we suspect that a covariate is endogenous? The new ivqregress command models quantiles of the outcome and, at the same time, controls for problems that arise from endogeneity.


When we use linear regression, we model the mean of the outcome. Yet, sometimes, we would like to study features of the outcome distribution other than the mean. For example, a policymaker may want to learn how participation in a 401(k) retirement plan would affect the lower-level, median, and upper-level conditional quantiles of net wealth.

ivqregress estimates parameters at quantiles of the outcome distribution and accounts for endogeneity problems that arise for reasons such as self-selection, omission of a relevant variable, or measurement error. For example, participation in the 401(k) program may be endogenous because the people who do and do not participate may have different saving preferences, which will affect net wealth growth.

When quantile regression matters

Suppose we have a simple model \(E(y|x) = \beta_0 + x\beta_1\), where \(y\) is the outcome variable and \(x\) is a covariate. \(x\) takes values in \(\{0, 1, 2, 3, 4, 5, 6\}\). By definition, \(\beta_1\) fully characterizes the effects of increasing one unit of \(x\) on the conditional mean of outcome \(y\); that is, \(\beta_1 = E(y|x=a + 1) - E(y|x=a)\). Below, we consider two scenarios of the data-generating process.

1. Location shifted only. The probability density function of the outcome conditional on \(x=a+1\), \(f(y|x=a+1)\), is only location shifted relative to \(f(y|x=a)\). In this case, \(\beta_1\) summarizes the effect of \(x\) not only on the conditional mean but also on each conditional quantile of \(y\). This case is illustrated in the left panel of Figure 1.

2. Location shifted and rescaled. The probability density function of the outcome conditional on \(x=a+1\), \(f(y|x=a+1)\), is both location shifted and rescaled relative to \(f(y|x=a)\). In this case, \(\beta_1\) summarizes the effect of \(x\) only on the conditional mean but not on the conditional quantiles of \(y\). This case is illustrated in the right panel of Figure 1.

Figure 1

In the left panel, we see that each conditional density is parallel relative to the others; only the location has been shifted. In this case, \(\beta_1\) captures the shift in both conditional mean and any other conditional quantiles of the outcome. As a result, running a linear regression provides as much information about \(\beta_1\) as a quantile regression.

In contrast, in the right panel, the conditional density for each level of \(x\) has a different location and a different shape. Thus, \(\beta_1\) can summarize the shifts in conditional mean, which generally differ from the shifts in conditional quantiles. Quantile regression becomes necessary to learn about the effects of \(x\) on the conditional quantiles of the outcome.

Let's see it work

We want to estimate the effect of 401(k) participation (p401k) on different conditional quantiles of net financial assets (assets). We use data reported by Chernozhukov and Hansen (2004). These data are from a sample of households in the 1990 Survey of Income and Program Participation (SIPP). For the head of household, we have data on income (income), age (age), number of people in the family (familysize), marital status (married), participation in IRA (ira), participation in pension benefit (pension), home ownership (ownhome), and years of education (educ).

We suspect 401(k) participation is endogenous because it may depend on unobserved factors such as saving preference that also impact financial assets. We will use 401(k) eligibility (e401k) as an instrument for 401(k) participation.

We use the IQR estimator (ivqregress iqr) to estimate the effect of 401(k) participation on the conditional median (the default) of the net financial assets.

. webuse assets2
(Excerpt from Chernozhukov and Hansen (2004))

. ivqregress iqr assets (i.p401k = i.e401k) income age familysize
  i.married i.ira i.pension i.ownhome educ

Initial grid:
Quantile = 0.50: .........10.........20.........30

Adaptive grid:
Quantile = 0.50: .........10.........20.........30

IV median regression                                     Number of obs =   9,913
Estimator: Inverse quantile regression                   Wald chi2(9)  = 1289.75
                                                         Prob > chi2   =  0.0000

assets Coefficient std. err. z P>|z| [95% conf. interval]
Yes 5313.397 573.2818 9.27 0.000 4189.786 6437.009
income .1577512 .0124889 12.63 0.000 .1332735 .1822289
age 99.96526 8.561923 11.68 0.000 83.1842 116.7463
familysize -197.8251 54.36773 -3.64 0.000 -304.3838 -91.26627
Married -1359.124 227.3366 -5.98 0.000 -1804.696 -913.5528
Yes 22629.61 1022.706 22.13 0.000 20625.15 24634.08
Receives .. -693.8347 210.6176 -3.29 0.001 -1106.638 -281.0317
Yes -30.29657 154.7265 -0.20 0.845 -333.555 272.9618
educ -96.43983 32.09465 -3.00 0.003 -159.3442 -33.53547
_cons -4998.673 570.1315 -8.77 0.000 -6116.11 -3881.236
Endogenous: 1.p401k Exogenous: income age familysize 1.married 1.ira 1.pension 1.ownhome educ 1.e401k

The coefficient for p401k is 5,313. This means participation in a 401(k) would increase the median net financial assets by $5,313, conditional on other covariates, relative to a scenario where no one participates.

After ivqregress iqr, we can use estat dualci to obtain the dual confidence interval (CI) that is robust to weak instruments for the coefficient on the endogenous variables.

. estat dualci

Dual confidence interval                                    Number of obs = 9,913
Robust Dual
assets Coefficient std. err. z P>|z| [95% conf. interval]
Yes 5313.397 573.2818 9.27 0.000 3683.916 7304.986

The dual CI is usually wider than the regular CI; it provides more robust inference if the instruments are weak. Here the dual 95% CI is [3684, 7305], which is wider than the regular 95% CI [4190, 6437].

We have estimated the 401(k) participation (p401k) treatment effect on the conditional median of net financial assets (assets). However, from the policy designer's point of view, we may be more interested in estimating the treatment effect of p401k on other conditional quantiles of assets.

This time, we specify ivqregress smooth to use the smoothed estimating equations estimator to fit the model at different quantiles. In particular, we specify the quantile(10(10)90) option to fit the IVQR model at the 10th, 20th, . . . , 90th quantiles.

. ivqregress smooth assets (i.p401k = i.e401k) income age familysize      
  i.married i.ira i.pension i.ownhome educ, quantile(10(10)90)

Fitting smoothed IV quantile regression:

Quantile = .1:
Step 1: Bandwidth = 1327.0069    GMM criterion Q(b) = 9.224e-11
Step 2: Bandwidth = 1311.3131    GMM criterion Q(b) = 1.995e-10

(output omitted)

IV quantile regression                                    Number of obs =   9,913
Estimator: Smoothed estimating equations                  Wald chi2(81) = 4932.84
                                                          Prob > chi2   =  0.0000

assets Coefficient std. err. z P>|z| [95% conf. interval]
Yes 3191.667 486.2193 6.56 0.000 2238.695 4144.639
income .0318585 .0123707 2.58 0.010 .0076124 .0561046
age 128.9268 15.42632 8.36 0.000 98.69178 159.1618
familysize -329.8374 125.4774 -2.63 0.009 -575.7687 -83.90615
Married -1480.013 386.4611 -3.83 0.000 -2237.463 -722.5635
Yes 7914.049 342.9506 23.08 0.000 7241.878 8586.22
Receives .. -5.356704 334.9869 -0.02 0.987 -661.919 651.2056
Yes 1043.279 308.722 3.38 0.001 438.1945 1648.363
educ -289.8807 53.06713 -5.46 0.000 -393.8904 -185.8711
_cons -7631.313 1214.725 -6.28 0.000 -10012.13 -5250.496
(output omitted)
Yes 15525.23 3035.965 5.11 0.000 9574.848 21475.61
income .8311508 .0574108 14.48 0.000 .7186277 .9436738
age 486.9876 51.61654 9.43 0.000 385.821 588.1541
familysize -586.2617 193.5936 -3.03 0.002 -965.6983 -206.8252
Married -3877.165 781.2296 -4.96 0.000 -5408.347 -2345.983
Yes 67888.86 4902.106 13.85 0.000 58280.91 77496.81
Receives .. -4829.506 898.9147 -5.37 0.000 -6591.346 -3067.665
Yes 715.6272 722.8727 0.99 0.322 -701.1773 2132.432
educ 14.5293 110.8781 0.13 0.896 -202.7878 231.8464
_cons -19953.21 2326.698 -8.58 0.000 -24513.45 -15392.96
Endogenous: 1.p401k Exogenous: income age familysize 1.married 1.ira 1.pension 1.ownhome educ 1.e401k

The results show the estimates for the effect of 401(k) participation on each conditional quantile of the asset. The coefficient interpretation is similar to before, except we are looking at different conditional quantiles. For example, for quantile q90, the estimate for the coefficient on p401k is 15,525. Thus, 401(k) participation would increase net financial assets' 90% conditional quantile by $15,525.

In addition to looking at the exact numerical estimates from the coefficient table, we can use estat coefplot to visualize the p401k's treatment effect from the lower to the upper quantile.

. estat coefplot

Figure 2

The dots in the plot show the point estimates of p401k's treatment effect on different conditional quantiles of assets, and the gray bound shows the 95% pointwise CI. We see an upward trend of p401k's treatment effect. At lower-level quantiles such as the 10th, 20th, 30th, and 40th quantiles, the treatment effect is relatively flat. However, the treatment effect increases in the upper-level quantiles. The red line shows the two-stage least-squares estimates, which can be used as a benchmark.

We can use estat endogeffects to test the following hypotheses regarding the endogenous covariate:

  • No effect : The 401(k) participation does not affect net financial assets for all the estimated quantiles.

  • Constant effect : The 401(k) participation’s treatment effect is constant for all the estimated quantiles.

  • Dominance : The 401(k) participation is unambiguously positive for all the estimated quantiles; that is, the coefficient values are strictly positive.

  • Exogeneity : The 401(k) participation is exogenous.

We use estat endogeffects to show the Kolmogorov–Smirnov statistic and the 95% critical value for each hypothesis. We can reject the null hypothesis if the test statistic is greater than the critical value; otherwise, we cannot reject the null hypothesis. We specify the rseed() option to make the results reproducible because the critical values are generated from a bootstrap sample.

. estat endogeffects, rseed(12345671)

Tests for endogenous effects           Replications = 100
Null hypothesis KS statistic 95% critical value
No effect 11.507 2.593
Constant effect 5.351 2.391
Dominance 0.000 2.556
Exogeneity 4.195 2.526
Note: If the KS statistic < critical value, there is insufficient evidence to reject the null hypothesis. (KS = Kolmogorov—Smirnov)

We find that 401(k) participation has some effect, treatment is not constant across different quantiles, and 401(k) participation is endogenous. The test for dominance indicates that 401(k) participation is unambiguously beneficial for all the estimated quantiles of assets.

The test results are consistent with the coefficient plot produced by estat coefplot, where we saw that the treatment effects are positive (dominance and no effect hypotheses) and upward trended (constant effect hypothesis).


Chernozhukov, V., and C. Hansen. 2004. The effects of 401(k) participation on the wealth distribution: An instrumental quantile regression analysis. Review of Economics and Statistics 86: 735–751.

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Read more about instrumental-variables quantile regression in the Stata Base Reference Manual; see [R] ivqregress.

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