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[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (260.94,115.70) –
(269.68,114.41) –
(274.82,122.01) –
(266.14,123.08) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (346.72,155.93) –
(357.79,161.57) –
(362.18,170.82) –
(351.12,164.75) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (202.48,106.22) –
(210.55,103.10) –
(216.48,110.26) –
(208.46,113.21) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (283.72,121.40) –
(292.88,121.34) –
(297.78,129.28) –
(288.67,129.08) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (373.90,178.47) –
(386.41,187.98) –
(390.72,198.13) –
(378.19,188.08) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (224.64,107.46) –
(232.95,104.86) –
(238.58,112.19) –
(230.33,114.61) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (307.17,130.14) –
(316.90,131.76) –
(321.58,140.15) –
(311.89,138.21) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (247.00,110.02) –
(255.60,108.15) –
(260.94,115.70) –
(252.41,117.36) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (331.67,143.01) –
(342.22,146.96) –
(346.72,155.93) –
(336.20,151.60) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (188.31,102.43) –
(196.31, 99.04) –
(202.48,106.22) –
(194.53,109.46) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (269.68,114.41) –
(278.64,113.56) –
(283.72,121.40) –
(274.82,122.01) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (357.79,161.57) –
(369.51,168.74) –
(373.90,178.47) –
(362.18,170.82) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (210.55,103.10) –
(218.78,100.13) –
(224.64,107.46) –
(216.48,110.26) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (292.88,121.34) –
(302.33,121.91) –
(307.17,130.14) –
(297.78,129.28) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (232.95,104.86) –
(241.43,102.49) –
(247.00,110.02) –
(238.58,112.19) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (316.90,131.76) –
(327.03,134.28) –
(331.67,143.01) –
(321.58,140.15) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (255.60,108.15) –
(264.39,106.63) –
(269.68,114.41) –
(260.94,115.70) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (342.22,146.96) –
(353.30,152.18) –
(357.79,161.57) –
(346.72,155.93) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (196.31, 99.04) –
(204.45, 95.75) –
(210.55,103.10) –
(202.48,106.22) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (278.64,113.56) –
(287.85,113.23) –
(292.88,121.34) –
(283.72,121.40) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (369.51,168.74) –
(382.01,177.72) –
(386.41,187.98) –
(373.90,178.47) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (218.78,100.13) –
(227.16, 97.34) –
(232.95,104.86) –
(224.64,107.46) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (302.33,121.91) –
(312.10,123.22) –
(316.90,131.76) –
(307.17,130.14) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (241.43,102.49) –
(250.09,100.41) –
(255.60,108.15) –
(247.00,110.02) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (327.03,134.28) –
(337.60,137.85) –
(342.22,146.96) –
(331.67,143.01) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (264.39,106.63) –
(273.41,105.53) –
(278.64,113.56) –
(269.68,114.41) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (353.30,152.18) –
(365.03,158.88) –
(369.51,168.74) –
(357.79,161.57) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (204.45, 95.75) –
(212.74, 92.61) –
(218.78,100.13) –
(210.55,103.10) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (287.85,113.23) –
(297.35,113.51) –
(302.33,121.91) –
(292.88,121.34) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (227.16, 97.34) –
(235.70, 94.78) –
(241.43,102.49) –
(232.95,104.86) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (312.10,123.22) –
(322.26,125.39) –
(327.03,134.28) –
(316.90,131.76) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (250.09,100.41) –
(258.95, 98.66) –
(264.39,106.63) –
(255.60,108.15) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (337.60,137.85) –
(348.71,142.64) –
(353.30,152.18) –
(342.22,146.96) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (273.41,105.53) –
(282.68,104.94) –
(287.85,113.23) –
(278.64,113.56) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (365.03,158.88) –
(377.51,167.32) –
(382.01,177.72) –
(369.51,168.74) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (212.74, 92.61) –
(221.19, 89.63) –
(227.16, 97.34) –
(218.78,100.13) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (297.35,113.51) –
(307.17,114.50) –
(312.10,123.22) –
(302.33,121.91) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (235.70, 94.78) –
(244.42, 92.48) –
(250.09,100.41) –
(241.43,102.49) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (322.26,125.39) –
(332.87,128.57) –
(337.60,137.85) –
(327.03,134.28) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (258.95, 98.66) –
(268.03, 97.32) –
(273.41,105.53) –
(264.39,106.63) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (348.71,142.64) –
(360.44,148.87) –
(365.03,158.88) –
(353.30,152.18) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (282.68,104.94) –
(292.23,104.93) –
(297.35,113.51) –
(287.85,113.23) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (221.19, 89.63) –
(229.79, 86.86) –
(235.70, 94.78) –
(227.16, 97.34) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (307.17,114.50) –
(317.37,116.33) –
(322.26,125.39) –
(312.10,123.22) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (244.42, 92.48) –
(253.35, 90.50) –
(258.95, 98.66) –
(250.09,100.41) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (332.87,128.57) –
(344.00,132.94) –
(348.71,142.64) –
(337.60,137.85) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (268.03, 97.32) –
(277.36, 96.46) –
(282.68,104.94) –
(273.41,105.53) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (360.44,148.87) –
(372.92,156.78) –
(377.51,167.32) –
(365.03,158.88) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (292.23,104.93) –
(302.10,105.60) –
(307.17,114.50) –
(297.35,113.51) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (229.79, 86.86) –
(238.59, 84.35) –
(244.42, 92.48) –
(235.70, 94.78) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (317.37,116.33) –
(328.01,119.13) –
(332.87,128.57) –
(322.26,125.39) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (253.35, 90.50) –
(262.49, 88.91) –
(268.03, 97.32) –
(258.95, 98.66) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.00,132.94) –
(355.74,138.70) –
(360.44,148.87) –
(348.71,142.64) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (277.36, 96.46) –
(286.96, 96.15) –
(292.23,104.93) –
(282.68,104.94) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (302.10,105.60) –
(312.35,107.08) –
(317.37,116.33) –
(307.17,114.50) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (238.59, 84.35) –
(247.57, 82.13) –
(253.35, 90.50) –
(244.42, 92.48) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (328.01,119.13) –
(339.17,123.07) –
(344.00,132.94) –
(332.87,128.57) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (262.49, 88.91) –
(271.88, 87.77) –
(277.36, 96.46) –
(268.03, 97.32) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (355.74,138.70) –
(368.23,146.08) –
(372.92,156.78) –
(360.44,148.87) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (286.96, 96.15) –
(296.89, 96.50) –
(302.10,105.60) –
(292.23,104.93) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (312.35,107.08) –
(323.02,109.49) –
(328.01,119.13) –
(317.37,116.33) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (247.57, 82.13) –
(256.78, 80.29) –
(262.49, 88.91) –
(253.35, 90.50) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (339.17,123.07) –
(350.93,128.35) –
(355.74,138.70) –
(344.00,132.94) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (271.88, 87.77) –
(281.54, 87.17) –
(286.96, 96.15) –
(277.36, 96.46) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (296.89, 96.50) –
(307.18, 97.63) –
(312.35,107.08) –
(302.10,105.60) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (323.02,109.49) –
(334.21,113.01) –
(339.17,123.07) –
(328.01,119.13) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (256.78, 80.29) –
(266.23, 78.87) –
(271.88, 87.77) –
(262.49, 88.91) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (350.93,128.35) –
(363.42,135.22) –
(368.23,146.08) –
(355.74,138.70) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (281.54, 87.17) –
(291.52, 87.20) –
(296.89, 96.50) –
(286.96, 96.15) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (307.18, 97.63) –
(317.90, 99.66) –
(323.02,109.49) –
(312.35,107.08) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (334.21,113.01) –
(346.00,117.82) –
(350.93,128.35) –
(339.17,123.07) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (266.23, 78.87) –
(275.95, 77.97) –
(281.54, 87.17) –
(271.88, 87.77) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (291.52, 87.20) –
(301.86, 87.97) –
(307.18, 97.63) –
(296.89, 96.50) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (317.90, 99.66) –
(329.12,102.75) –
(334.21,113.01) –
(323.02,109.49) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (346.00,117.82) –
(358.49,124.16) –
(363.42,135.22) –
(350.93,128.35) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (275.95, 77.97) –
(285.99, 77.68) –
(291.52, 87.20) –
(281.54, 87.17) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (301.86, 87.97) –
(312.63, 89.62) –
(317.90, 99.66) –
(307.18, 97.63) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (329.12,102.75) –
(340.93,107.09) –
(346.00,117.82) –
(334.21,113.01) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (285.99, 77.68) –
(296.39, 78.09) –
(301.86, 87.97) –
(291.52, 87.20) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (312.63, 89.62) –
(323.88, 92.28) –
(329.12,102.75) –
(317.90, 99.66) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (340.93,107.09) –
(353.44,112.90) –
(358.49,124.16) –
(346.00,117.82) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (296.39, 78.09) –
(307.20, 79.35) –
(312.63, 89.62) –
(301.86, 87.97) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (323.88, 92.28) –
(335.72, 96.15) –
(340.93,107.09) –
(329.12,102.75) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (307.20, 79.35) –
(318.50, 81.58) –
(323.88, 92.28) –
(312.63, 89.62) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (335.72, 96.15) –
(348.25,101.43) –
(353.44,112.90) –
(340.93,107.09) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (318.50, 81.58) –
(330.37, 84.97) –
(335.72, 96.15) –
(323.88, 92.28) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (330.37, 84.97) –
(342.92, 89.72) –
(348.25,101.43) –
(335.72, 96.15) –

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 31.25,415.50) – (367.49,340.17);

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] (367.49,340.17) – (337.05, 29.85);

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] (367.49,340.17) – (474.64,447.33);

*(d) RKHS CAPCE

Nonparametric estimated surfaces in setting (B) (Mean, $N=10000$). X-axis is the value of  treatment variable ($X=x$), Y-axis is the value of  covariate ($W=w$), and Z-axis is the value of CAPCE.

§.§ Additional experiments: No interaction  between covariates  and unobserved confounders

In this section, we give additional experiments with no interaction  between covariates  and unobserved confounders.

SCM Settings. We consider the following two SCMs:  $W:=U+E_1, X:=Z+W+U+E_2$, and

\begin{eqnarray}
\label{eq-scm}
\left\{
\begin{array}{l}
Y:=10X^2+WX+X+W+50\, U+E_3\ \ \ \hspace*{\fill}\cdots\text{ (C)}\\Y:=\text{exp}(X)\, \text{exp}(W)+50\, U+E_3\ \ \ \hspace*{\fill}\cdots\text{ (D)}\\
\end{array}
\right..
\end{eqnarray}

The other settings of each estimator are the same as in setting (A) and (B).

The basic statistics of estimated coefficients by $100$ time simulations of
PTSLS and P-CAPCE in setting (C) are shown in Tables  <ref> and <ref>.
The MSE of each estimator in settings (C) and (D) are shown in Table  <ref>.
The results  show that the performance of the previous works PTSLS, NTSLS, Kernel IV is comparable with our proposed methods under the settings where the interaction between the covariates $W$ and unobserved confounders $U$ is absent.

Basic statistics of the P-CAPCE estimator over 1000 runs when $N=1000$ and $N=10000$ in setting (C).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -4.419     -38.895     -21.176

1st Qu.     0.896      0.351       19.614

Median      0.983      1.361       19.919

3rd Qu.     1.065      2.620       20.246

Max.        6.066      19.864      41.021

Mean        0.944      1.151       19.642

SD          0.811      5.535       4.884

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        0.522     -21.630     18.680

1st Qu.     0.976     0.158       19.901

Median      1.000     0.830       20.006

3rd Qu.     1.021     1.989       20.101

Max.        1.724     24.138      21.684

Mean        0.999     0.966       19.998

SD          0.106     4.851       0.305

Basic statistics of the PTSLS over 1000 runs when $N=1000$ and $N=10000$ in setting (C).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        0.350     -2.045     15.831

1st Qu.     0.957     -0.199     19.303

Median      1.021     1.009      19.731

3rd Qu.     1.101     1.921      19.953

Max.        1.456     5.165      20.260

Mean        1.029     0.997      19.474

SD          0.155     1.609      0.782

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        0.904     -1.251     19.501

1st Qu.     0.986     0.515      19.905

Median      1.003     0.962      19.963

3rd Qu.     1.021     1.407      20.019

Max.        1.074     3.148      20.120

Mean        1.003     0.939      19.939

SD          0.028     0.814      0.118

MSE of estimators in settings (C) and (D).

1c|MSE          PTSLS    NTSLS     Kernel IV     P-CAPCE     S-CAPCE     RKHS CAPCE

(C) $N=1000$      3.004     4.124     8.165      8.316    11.827     3.572

(C) $N=10000$     0.254     0.367     0.614     0.774     1.324     0.567
(D) $N=1000$      139.713     4.224      4.438     52.861      4.254      2.943

(D) $N=10000$     57.006      0.334     1.026      40.062      0.477       0.696

§.§ Additional Experiments: Weaker interaction between covariates  and unobserved confounders

In this section, we give additional experiments with weak interaction  between covariates  and unobserved confounders.

SCM Settings. We consider the following two SCMs:  $W:=U+E_1, X:=Z+W+U+E_2$, and

\begin{eqnarray}
\label{eq-scm}
\left\{
\begin{array}{l}
Y:=10X^2+WX+X+W+10 (W^5+W^4+W^3+W^2)U+E_3\ \hfill\cdots\text{(E)}\\
Y:=\text{exp}(X)\text{exp}(W)+5 (W^5+W^4+W^3+W^2)U+E_3\hfill\cdots\text{(F)}
\end{array}
\right..
\end{eqnarray}

The other settings of each estimator are the same as in setting (A) and (B).

The basic statistics of estimated coefficients by $100$ time simulations of
PTSLS and P-CAPCE in setting (E) are shown in Tables  <ref> and <ref>.
The MSE of estimators in settings (E) and (F) are shown in Table  <ref>.
The results show that our methods are superior to the previous works PTSLS, NTSLS, Kernel IV while the performance differences are less than that in the settings (A) and (B) where the interaction  between covariates  and unobserved confounders are stronger.

Basic statistics of the P-CAPCE estimator over 1000 runs when $N=1000$ and $N=10000$ in setting (E).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -2.056     -9.914     -6.174

1st Qu.     -0.103     -0.589     11.002

Median      0.848      1.996      15.528

3rd Qu.     1.898      6.510      29.150

Max.        2.047      3.502      25.181

Mean        1.185      1.765      18.248

SD          2.416      5.991      14.165

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        0.057     -1.633     10.648

1st Qu.     0.635     -0.286     17.381

Median      1.193     1.187      22.993

3rd Qu.     1.743     2.306      23.704

Max.        2.047     3.502      25.181

Mean        1.160     0.979      20.531

SD          0.662     1.713      4.798

Basic statistics of the PTSLS over 1000 runs when $N=1000$ and $N=10000$ in setting (E).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -5.345     -1.064     -3.732

1st Qu.     -0.074     9.187      14.958

Median      1.195      10.862     20.450

3rd Qu.     2.667      13.322     24.988

Max.        5.766      34.620     37.757

Mean        1.195      10.862     20.450

SD          2.161      5.437      8.597

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -1.072     7.402      11.514

1st Qu.     0.394      9.613      17.370

Median      0.957      10.748     19.355

3rd Qu.     1.446      11.977     21.967

Max.        2.684      15.641     26.645

Mean        0.934      10.883     19.459

SD          0.778     1.687      3.478

MSE of estimators in settings (E) and (F).

1c|MSE          PTSLS    NTSLS     Kernel IV     P-CAPCE     S-CAPCE     RKHS CAPCE

(E) $N=1000$      170.152     129.132     113.733     29.408      17.181     31.267

(E) $N=10000$     64.645      55.104      65.055     3.592       4.527      3.171
(F) $N=1000$      141.678     28.73       24.125     101.902     13.84      14.806

(F) $N=10000$     61.569      4.323       3.197     42.422      1.788      1.731

§.§ Additional Experiments: More than one heterogeneity feature

In this section, we give additional experiments with more than one heterogeneity feature.
SCM Settings. We consider the following two SCMs:  $W_1:=U+E_1$, $W_2:=U+E_2$, $X:=Z+W+U+E_3$, and

\begin{eqnarray}
\label{eq-scm}
\left\{
\begin{array}{l}
Y:=10X^2+W_1X+X+W_1+10 (W_1^5+W_1^4+W_1^3+W_1^2)U+E_4\ \hfill\cdots\text{(E)}\\
Y:=\text{exp}(X)\text{exp}(W_1)+5 (W_1^5+W_1^4+W_1^3+W_1^2)U+E_4\hfill\cdots\text{(F)}
\end{array}
\right..
\end{eqnarray}

The other settings of each estimator are the same as in setting (A) and (B).

The basic statistics of estimated coefficients by $100$ time simulations of
PTSLS and P-CAPCE in setting (G) are shown in Tables  <ref> and <ref>.
The MSE of estimators in settings (G) and (H) are shown in Table  <ref>.

Basic statistics of the P-CAPCE estimator over 1000 runs when $N=1000$ and $N=10000$ in setting (E).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -2.056     -9.914     -6.174

1st Qu.     -0.103     -0.589     11.002

Median      0.848      1.996      15.528

3rd Qu.     1.898      6.510      29.150

Max.        2.047      3.502      25.181

Mean        1.185      1.765      18.248

SD          2.416      5.991      14.165

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        0.057     -1.633     10.648

1st Qu.     0.635     -0.286     17.381

Median      1.193     1.187      22.993

3rd Qu.     1.743     2.306      23.704

Max.        2.047     3.502      25.181

Mean        1.160     0.979      20.531

SD          0.662     1.713      4.798

Basic statistics of the PTSLS over 1000 runs when $N=1000$ and $N=10000$ in setting (E).

$N=1000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -5.345     -1.064     -3.732

1st Qu.     -0.074     9.187      14.958

Median      1.195      10.862     20.450

3rd Qu.     2.667      13.322     24.988

Max.        5.766      34.620     37.757

Mean        1.195      10.862     20.450

SD          2.161      5.437      8.597

$N=10000$          $1$    $W$    $X$

True Coeff.     1     1     20

Min.        -1.072     7.402      11.514

1st Qu.     0.394      9.613      17.370

Median      0.957      10.748     19.355

3rd Qu.     1.446      11.977     21.967

Max.        2.684      15.641     26.645

Mean        0.934      10.883     19.459

SD          0.778     1.687      3.478

MSE of estimators in settings (E) and (F).

1c|MSE          PTSLS    NTSLS     Kernel IV     P-CAPCE     S-CAPCE     RKHS CAPCE

(E) $N=1000$      170.152     129.132     113.733     29.408      17.181     31.267

(E) $N=10000$     64.645      55.104      65.055     3.592       4.527      3.171
(F) $N=1000$      141.678     28.73       24.125     101.902     13.84      14.806

(F) $N=10000$     61.569      4.323       3.197     42.422      1.788      1.731

§.§ Additional information on the application

We present detailed settings of the application in Section 6.
We applied  P-CAPCE and PTSLS.
We learn the expected values of basis functions by the nonlinear model,

We use terms $\{1,W,W^2,X,XW,XW^2\}$ for P-CAPCE and $\{1,W,W^2,X,XW,XW^2,X^2,X^2W,X^2W^2\}$ for PTSLS, and let $z_0=8$.
We estimate CAPCE via differentiating estimated $\mathbb{E}[Y_{x}|{w}]$ for PTSLS.

Regularize parameter is determined by test error from $\{1,10^{-1},10^{-2},10^{-3},\ldots\}$.

Setting of PTSLS estimator. We learn $\mathbb{E}[Y|Z=z], \mathbb{E}[X^pW^q|Z=z]$ for $p=0,1,2$ and $q=0,1,2$ by the nonlinear model,

We consider the following terms, $\{1,W,W^2,X,XW,XW^2,X^2,X^2W,X^2W^2\}$ to build model of $\mathbb{E}[Y_{x}|{W}={w}]$.

Regularize value is determined by test error from $\{1000,100,10,1,10^{-1}\}$.
We estimate CAPCE via differentiating estimated $\mathbb{E}[Y_{x}|{W}={w}]$.

The basic bootstrapping statistical properties of the P-CAPCE and PTSLS estimators are shown in Tables <ref> and <ref>. The predicted CAPCE  values  are shown in Tables <ref> and <ref>. The estimated CAPCE surfaces  are shown in Fig. <ref>.

Basic statistics of the test error of the CAPCE estimator over 1000 runs for regularization value; the bold number is the smallest.

$\lambda$     1     0.1     0.01

Min.       8457869       6021324       NA

1st Qu.        14552258      13155384      NA

Median     17036292      16429917      NA

3rd Qu.        20301844      19727542      NA

Max.       29788501      28455256      NA

Mean       17657884      16589425      NA

SD         4662390.0     5035935.0     NA

Basic statistics of the P-CAPCE estimator over 1000 bootstrapping.

Terms        $1$    $W$    $W^2$    $X$   $WX$   $W^2X$

Min.        -0.00267     -0.00003     -0.00126     -0.00006     -0.00084     -0.00237

1st Qu.     -0.00061     0.00002      0.00510      -0.00001     -0.00029     -0.00099

Median      -0.00006     0.00004      0.00904      -0.00001     -0.00016     -0.00053

3rd Qu.     0.00058      0.00007      0.01331      0.00000      -0.00008     -0.00017

Max.        0.00226      0.00018      0.02786      0.00004      0.00059      0.00068

Mean        -0.00003     0.00005      0.00949      -0.00001     -0.00018     -0.00056

SD          0.00090      0.00004      0.00602      0.00001      0.00018      0.00062

Basic statistics of the PTSLS estimator over 1000 bootstrapping.

Terms        $1$    $W$    $W^2$    $X$   $WX$   $W^2X$

Min.        -0.00082     -0.03024     -0.00319     -0.01294     -0.47307     -0.00180

1st Qu.     0.00000      0.00023      0.00729      0.00005      0.00238      -0.00126

Median      0.00007      0.00259      0.00996      0.00134      0.06028      -0.00108

3rd Qu.     0.00012      0.00499      0.01341      0.00260      0.10789      -0.00089

Max.        0.00051      0.01539      0.04302      0.00773      0.30066      0.00046

Mean        0.00004      0.00191      0.01083      0.00102      0.04417      -0.00103

SD          0.00017      0.00617      0.00660      0.00299      0.11830      0.00039

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(238.18,227.44) –
(250.94,249.58) –
(229.96,265.46) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (272.03,233.60) –
(293.24,217.54) –
(304.77,237.40) –
(284.02,254.52) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (127.96,235.57) –
(148.06,222.73) –
(162.05,246.94) –
(141.77,260.43) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.50,220.30) –
(346.22,203.20) –
(355.89,220.39) –
(335.94,238.53) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (182.65,233.24) –
(203.60,219.30) –
(217.03,242.34) –
(196.12,257.07) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (375.70,202.38) –
(395.36,184.49) –
(402.58,198.80) –
(383.96,217.66) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (238.18,227.44) –
(259.57,212.37) –
(272.03,233.60) –
(250.94,249.58) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (293.24,217.54) –
(314.57,201.38) –
(325.50,220.30) –
(304.77,237.40) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (148.06,222.73) –
(168.60,209.63) –
(182.65,233.24) –
(162.05,246.94) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (346.22,203.20) –
(366.92,186.12) –
(375.70,202.38) –
(355.89,220.39) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.60,219.30) –
(224.90,205.13) –
(238.18,227.44) –
(217.03,242.34) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (259.57,212.37) –
(281.21,197.12) –
(293.24,217.54) –
(272.03,233.60) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (113.41,211.29) –
(133.37,198.86) –
(148.06,222.73) –
(127.96,235.57) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (314.57,201.38) –
(336.01,185.14) –
(346.22,203.20) –
(325.50,220.30) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (168.60,209.63) –
(189.58,196.24) –
(203.60,219.30) –
(182.65,233.24) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (366.92,186.12) –
(387.61,169.05) –
(395.36,184.49) –
(375.70,202.38) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (224.90,205.13) –
(246.56,190.73) –
(259.57,212.37) –
(238.18,227.44) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (281.21,197.12) –
(303.09,181.71) –
(314.57,201.38) –
(293.24,217.54) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (133.37,198.86) –
(153.84,186.11) –
(168.60,209.63) –
(148.06,222.73) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (336.01,185.14) –
(357.58,168.81) –
(366.92,186.12) –
(346.22,203.20) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (189.58,196.24) –
(211.02,182.56) –
(224.90,205.13) –
(203.60,219.30) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (246.56,190.73) –
(268.59,176.08) –
(281.21,197.12) –
(259.57,212.37) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] ( 97.97,187.43) –
(117.81,175.17) –
(133.37,198.86) –
(113.41,211.29) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (303.09,181.71) –
(325.23,166.11) –
(336.01,185.14) –
(314.57,201.38) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (153.84,186.11) –
(174.85,173.03) –
(189.58,196.24) –
(168.60,209.63) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (357.58,168.81) –
(379.26,152.38) –
(387.61,169.05) –
(366.92,186.12) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (211.02,182.56) –
(232.92,168.58) –
(246.56,190.73) –
(224.90,205.13) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (268.59,176.08) –
(291.00,161.17) –
(303.09,181.71) –
(281.21,197.12) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (117.81,175.17) –
(138.24,162.53) –
(153.84,186.11) –
(133.37,198.86) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.23,166.11) –
(347.62,150.33) –
(357.58,168.81) –
(336.01,185.14) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (174.85,173.03) –
(196.41,159.60) –
(211.02,182.56) –
(189.58,196.24) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (232.92,168.58) –
(255.32,154.29) –
(268.59,176.08) –
(246.56,190.73) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (291.00,161.17) –
(313.79,146.01) –
(325.23,166.11) –
(303.09,181.71) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (138.24,162.53) –
(159.29,149.52) –
(174.85,173.03) –
(153.84,186.11) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (347.62,150.33) –
(370.27,134.37) –
(379.26,152.38) –
(357.58,168.81) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (196.41,159.60) –
(218.56,145.81) –
(232.92,168.58) –
(211.02,182.56) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (255.32,154.29) –
(278.22,139.68) –
(291.00,161.17) –
(268.59,176.08) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (313.79,146.01) –
(336.98,130.59) –
(347.62,150.33) –
(325.23,166.11) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (159.29,149.52) –
(180.98,136.11) –
(196.41,159.60) –
(174.85,173.03) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (218.56,145.81) –
(241.30,131.64) –
(255.32,154.29) –
(232.92,168.58) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (278.22,139.68) –
(301.64,124.73) –
(313.79,146.01) –
(291.00,161.17) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (336.98,130.59) –
(360.58,114.89) –
(370.27,134.37) –
(347.62,150.33) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (180.98,136.11) –
(203.35,122.27) –
(218.56,145.81) –
(196.41,159.60) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (241.30,131.64) –
(264.67,117.09) –
(278.22,139.68) –
(255.32,154.29) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (301.64,124.73) –
(325.60,109.44) –
(336.98,130.59) –
(313.79,146.01) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.35,122.27) –
(226.43,108.00) –
(241.30,131.64) –
(218.56,145.81) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (264.67,117.09) –
(288.70,102.13) –
(301.64,124.73) –
(278.22,139.68) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.60,109.44) –
(350.12, 93.79) –
(360.58,114.89) –
(336.98,130.59) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (226.43,108.00) –
(250.26, 93.27) –
(264.67,117.09) –
(241.30,131.64) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (288.70,102.13) –
(313.40, 86.74) –
(325.60,109.44) –
(301.64,124.73) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (250.26, 93.27) –
(274.86, 78.05) –
(288.70,102.13) –
(264.67,117.09) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (313.40, 86.74) –
(338.81, 70.91) –
(350.12, 93.79) –
(325.60,109.44) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (274.86, 78.05) –
(300.28, 62.33) –
(313.40, 86.74) –
(288.70,102.13) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (300.28, 62.33) –
(326.57, 46.08) –
(338.81, 70.91) –
(313.40, 86.74) –

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] ( 29.85,347.74) – (359.21,206.62);

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] (359.21,206.62) – (323.72, 32.04);

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] (359.21,206.62) – (476.04,412.52);

*(a) P-CAPCE

[use as bounding box,fill=fillColor,fill opacity=0.00] (0,0) rectangle (505.89,505.89);

[clip] ( 12.00, 12.00) rectangle (493.89,493.89);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (238.82,291.75) – (414.63,221.47);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (414.63,221.47) – (476.04,412.52);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (476.04,412.52) – (227.06,473.85);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (227.06,473.85) – (238.82,291.75);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (111.78,154.95) – (238.82,291.75);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (227.06,473.85) – ( 29.85,347.74);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 29.85,347.74) – (111.78,154.95);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (111.78,154.95) – (323.72, 32.04);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (323.72, 32.04) – (414.63,221.47);

[clip] (  0.00,  0.00) rectangle (505.89,505.89);

[text=drawColor,rotate=-30.57,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.00] at (198.73, 76.47) Years of Education;

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (108.48,145.83) – (319.40, 21.26);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (312.64, 23.80) –
(319.40, 21.26) –
(313.91, 25.96);

[text=drawColor,rotate= 65.04,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.00] at (390.39,125.68) IQ;

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (342.27, 49.40) – (412.65,200.58);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] (410.78,193.60) –
(412.65,200.58) –

[text=drawColor,rotate=-65.54,anchor=base,inner sep=0pt, outer sep=0pt, scale=  1.00] at ( 50.92,225.54) CAPCE;

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 92.84,171.81) – ( 31.53,306.59);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round] ( 35.62,300.63) –
( 31.53,306.59) –
( 33.34,299.59);

[clip] ( 12.00, 12.00) rectangle (493.89,493.89);

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (216.33,425.54) –
(237.57,399.08) –
(249.60,420.28) –
(228.54,450.88) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (237.57,399.08) –
(258.16,373.41) –
(269.80,390.94) –
(249.60,420.28) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (203.92,400.48) –
(225.22,377.62) –
(237.57,399.08) –
(216.33,425.54) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (258.16,373.41) –
(278.16,348.50) –
(289.18,362.79) –
(269.80,390.94) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (225.22,377.62) –
(246.11,355.19) –
(258.16,373.41) –
(237.57,399.08) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (278.16,348.50) –
(297.57,324.31) –
(307.79,335.75) –
(289.18,362.79) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (191.26,375.64) –
(212.54,355.85) –
(225.22,377.62) –
(203.92,400.48) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (246.11,355.19) –
(266.61,333.19) –
(278.16,348.50) –
(258.16,373.41) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (297.57,324.31) –
(316.43,300.81) –
(325.69,309.75) –
(307.79,335.75) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (212.54,355.85) –
(233.62,336.25) –
(246.11,355.19) –
(225.22,377.62) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (266.61,333.19) –
(286.72,311.61) –
(297.57,324.31) –
(278.16,348.50) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (316.43,300.81) –
(334.76,277.98) –
(342.90,284.75) –
(325.69,309.75) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (178.31,350.95) –
(199.49,333.75) –
(212.54,355.85) –
(191.26,375.64) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (233.62,336.25) –
(254.52,316.82) –
(266.61,333.19) –
(246.11,355.19) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (286.72,311.61) –
(306.46,290.42) –
(316.43,300.81) –
(297.57,324.31) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (334.76,277.98) –
(352.58,255.77) –
(359.47,260.68) –
(342.90,284.75) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (199.49,333.75) –
(220.68,316.54) –
(233.62,336.25) –
(212.54,355.85) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (254.52,316.82) –
(275.23,297.56) –
(286.72,311.61) –
(266.61,333.19) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (306.46,290.42) –
(325.83,269.63) –
(334.76,277.98) –
(316.43,300.81) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (165.03,326.36) –
(186.04,311.26) –
(199.49,333.75) –
(178.31,350.95) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (352.58,255.77) –
(369.91,234.18) –
(375.43,237.49) –
(359.47,260.68) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (220.68,316.54) –
(241.87,299.32) –
(254.52,316.82) –
(233.62,336.25) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (275.23,297.56) –
(295.75,278.48) –
(306.46,290.42) –
(286.72,311.61) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (325.83,269.63) –
(344.86,249.21) –
(352.58,255.77) –
(334.76,277.98) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (186.04,311.26) –
(207.25,296.04) –
(220.68,316.54) –
(199.49,333.75) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (369.91,234.18) –
(386.77,213.17) –
(390.82,215.13) –
(375.43,237.49) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (241.87,299.32) –
(263.07,282.10) –
(275.23,297.56) –
(254.52,316.82) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (295.75,278.48) –
(316.09,259.56) –
(325.83,269.63) –
(306.46,290.42) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (151.36,301.78) –
(172.17,288.35) –
(186.04,311.26) –
(165.03,326.36) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (344.86,249.21) –
(363.53,229.16) –
(369.91,234.18) –
(352.58,255.77) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (207.25,296.04) –
(228.65,280.67) –
(241.87,299.32) –
(220.68,316.54) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (386.77,213.17) –
(403.18,192.71) –
(405.66,193.57) –
(390.82,215.13) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (263.07,282.10) –
(284.28,264.87) –
(295.75,278.48) –
(275.23,297.56) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (316.09,259.56) –
(336.25,240.81) –
(344.86,249.21) –
(325.83,269.63) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (172.17,288.35) –
(193.33,274.70) –
(207.25,296.04) –
(186.04,311.26) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (363.53,229.16) –
(381.88,209.48) –
(386.77,213.17) –
(369.91,234.18) –

[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (228.65,280.67) –
(250.24,265.17) –
(263.07,282.10) –
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[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (329.98,136.55) –
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[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (317.00,113.65) –
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[draw=drawColor,line width= 0.4pt,line join=round,line cap=round,fill=fillColor] (275.09,102.31) –
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[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] (359.21,206.62) – (323.72, 32.04);

[draw=drawColor,line width= 0.4pt,dash pattern=on 1pt off 3pt ,line join=round,line cap=round] (359.21,206.62) – (476.04,412.52);

*(b) PTSLS

Bootstrap mean surface of each estimator. X-axis is years of education, Y-axis is IQ, and Z-axis is CAPCE.

Basic statistics of the test error of the PTSLS estimator over 1000 runs for regularization value; the bold number is the smallest.

$\lambda$    1000    100      10     1

Min.       708704433      718265667      707217598     NA

1st Qu.        774156768      752119583      764372413     NA