A Ridge-Regularized Jackknifed Anderson-Rubin Test

Figures & data

Table 1 Anderson and Rubin (1949) tests with many IVs: schematic comparison of main assumptions and results in the literature.

	kn	Z	ε
Kaffo and Wang (2017) and Anatolyev and Gospodinov (2011)	$k_n/n\to \tau ,0\le \tau <1,n\to \infty$	Z is fixed, obeys restrictive balanced design assumption	iid, $\mathbb{E}[{\epsilon }_i]=0,\mathbb{E}[{\epsilon }_i^4]<\infty$
Belloni et al. (2012)	$\hspace{0.17em}\log \hspace{0.17em}(\max (k_n,n))=o(n^{1/3}),n\to \infty$	Z is fixed	independent, $\mathbb{E}[{\epsilon }_i]=0,\mathbb{E}[{\epsilon }_i^3]<\infty$
Kapetanios, Khalaf, and Marcellino (2015)	Factor-dependent	Potentially dependent data, $\mathbb{E}[Z_{il}^4]<\infty ,l=1,\dots ,k_n,\mathbb{E}[{\epsilon }_i^4]<\infty$
Carrasco and Tchuente (2016)	$n\to \infty$	$Z_i{\epsilon }_i$ is iid, $\mathbb{E}[Z_i{\epsilon }_i]=0,\mathbb{E}[X_{ig}^2]<\infty$
Bun, Farbmacher, and Poldermans (2020)	$k_n/n\to \tau ,0\le \tau <1,n\to \infty$	$Z_i{\epsilon }_i$ is iid, Z obeys restrictive balanced design assumption, $\mathbb{E}[Z_i{\epsilon }_i]=0,\mathbb{E}[Z_i^8{\epsilon }_i^8]<\infty$
Crudu, Mellace, and Sándor (2020)	$k_n/n\to \tau ,0\le \tau <1,k_n\to \infty$ as $n\to \infty$	Z is fixed, $P_{ii}\le 1-\delta ,0<\delta <1$	independent, $\mathbb{E}[{\epsilon }_i]=0,\mathbb{E}[{\epsilon }_i^4]<\infty$
Mikusheva and Sun (2022)	$k_n/n\to \tau ,0\le \tau <1,k_n\to \infty$ as $n\to \infty$	Z is fixed, $P_{ii}\le 1-\delta ,0<\delta <1$	independent, $\mathbb{E}[{\epsilon }_i]=0,\mathbb{E}[{\epsilon }_i^6]<\infty$
RJAR	$r_n\to \infty$ as $n\to \infty$	Z is fixed, $\underset{n\to \infty }{\lim \inf }\frac{1}{r_n}{\sum }_{i=1}^n\sum _{j\ne i}{(P_{ij}^{{\gamma }_n})}^2>0$	independent, $\mathbb{E}[{\epsilon }_i]=0,\mathbb{E}[{\epsilon }_i^4]<\infty$

NOTE: n is the number of observations, k_n is the number of IVs, and g is the number of endogenous variables. Z is the $n\times k_n$ matrix of IVs. X is the n × g matrix of endogenous variables. ε is the $n\times 1$ vector of structural error terms.

$r_n:=\text{rank}(Z)$. See also (1).

$P:=Z{(Z\prime Z)}^{-1}Z\prime ,P^{{\gamma }_n}:=Z{(Z\prime Z+{\gamma }_n{I}_{k_n})}^{-1}Z\prime$ for ${\gamma }_n\ge 0$ if r_n = k_n and ${\gamma }_n\ge {\gamma }_{-}>0$ if r_n < k_n.

Fig. 1 PP Plots for Sparse IVs, homoscedastic errors, β = 1, ${\mu }^2=0,H_0:{\beta }_0=1$.

Fig. 2 Power curves for 30 IVs. Nominal test size of 5% indicated by the grey horizontal line. $H_0:{\beta }_0=1$.

Fig. 3 Power curves for 90 IVs. Nominal test size of 5% indicated by the grey horizontal line. $H_0:{\beta }_0=1$.

Fig. 4 Power curves for 190 IVs. Nominal test size of 5% indicated by the gray horizontal line. $H_0:{\beta }_0=1$.

Fig. 5 95% confidence sets for β_s for the application in (12) with k_n = 38 IVs. ${\max }_i{P}_{ii}=0.944$. ${\gamma }_n^{*}=0,r_n^{-1}{\sum }_{i=1}^n\sum _{j\ne i}{(P_{ij}^{{\gamma }_n^{*}})}^2=0.513$.

Fig. 6 95% confidence sets for β_s for the application in (12) with k_n = 342 IVs. ${\gamma }_n^{*}=5.299,r_n^{-1}{\sum }_{i=1}^n\sum _{j\ne i}{(P_{ij}^{{\gamma }_n^{*}})}^2=0.106$.

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