"""Data-driven dimension selection for nonparametric instrumental variables."""
from .lepski import npiv_j, npiv_jhat_max
[docs]
def npiv_choose_j(
y,
x,
w,
x_grid=None,
j_x_degree=3,
k_w_degree=4,
k_w_smooth=2,
knots="uniform",
basis="tensor",
x_min=None,
x_max=None,
w_min=None,
w_max=None,
grid_num=50,
boot_num=99,
check_is_fullrank=False,
seed=None,
):
r"""Select optimal B-spline dimensions.
Implements the full data-driven selection procedure from [1]_, combining a maximum
dimension selection step with a Lepski-style test. This procedure selects a
data-driven sieve dimension, :math:`\tilde{J}`, that is sup-norm rate-adaptive. This means
it adapts to unknown features of the data-generating process (e.g., smoothness of
:math:`h_0`, instrument strength) to achieve the minimax optimal convergence rate in sup-norm,
.. math::
\sup_x |\hat{h}_{\tilde{J}}(x) - h_0(x)|.
The procedure involves two main steps. First, determine a maximum feasible dimension, :math:`\hat{J}_{\max}`,
based on the sample size and an estimate of the sieve measure of ill-posedness, :math:`\hat{s}_J^{-1}`.
This step defines the search grid :math:`\hat{\mathcal{J}}` as
.. math::
\hat{J}_{\max} = \min \left\{ J \in \mathcal{T} : J \sqrt{\log J} \hat{s}_J^{-1} \leq c \sqrt{n} \right\}.
Second, use a Lepski-style method with a multiplier bootstrap to select the optimal dimension
:math:`\hat{J}` from the grid :math:`\hat{\mathcal{J}}`. This is done by comparing estimates across different
dimensions and selecting the smallest dimension that is not statistically different from estimates
at larger dimensions. The final choice is :math:`\tilde{J} = \min\{\hat{J}, \hat{J}_n\}`,
where :math:`\hat{J}_n` is a slightly smaller, more conservative dimension for stability.
.. math::
\hat{J} = \min \left\{ J \in \hat{\mathcal{J}} : \sup_{x, J_2 > J} \left|
\frac{\hat{h}_J(x) - \hat{h}_{J_2}(x)}{\hat{\sigma}_{J, J_2}(x)} \right|
\leq \theta_{1-\hat{\alpha}}^* \right\}.
Parameters
----------
y : ndarray
Dependent variable vector.
x : ndarray
Endogenous regressor matrix.
w : ndarray
Instrument matrix.
x_grid : ndarray, optional
Grid points for evaluation. If None, created automatically.
j_x_degree : int, default=3
Degree of B-spline basis for :math:`X`.
k_w_degree : int, default=4
Degree of B-spline basis for :math:`W`.
k_w_smooth : int, default=2
Smoothness parameter for :math:`K` selection.
knots : {"uniform", "quantiles"}, default="uniform"
Knot placement method.
basis : {"tensor", "additive", "glp"}, default="tensor"
Type of basis for multivariate :math:`X`:
- "tensor": Full tensor product of univariate bases
- "additive": Sum of univariate bases
- "glp": Generalized linear product (hierarchical)
x_min, x_max, w_min, w_max : float, optional
Range limits for basis construction.
grid_num : int, default=50
Number of grid points for evaluation.
boot_num : int, default=99
Number of bootstrap replications for confidence bands.
check_is_fullrank : bool, default=False
Whether to check if basis matrices have full rank.
seed : int, optional
Random seed for reproducibility.
Returns
-------
dict
Dictionary containing:
- **j_x_segments**: Selected number of segments for :math:`X`
- **k_w_segments**: Corresponding segments for :math:`W`
- **j_tilde**: Selected dimension
- **theta_star**: Bootstrap critical value
- **j_hat_max**: Maximum feasible dimension
- Additional diagnostic information
See Also
--------
npiv_jhat_max : Compute maximum feasible dimension
npiv_j : Lepski-style test for dimension selection
References
----------
.. [1] Chen, X., Christensen, T. M., & Kankanala, S. (2024).
Adaptive Estimation and Uniform Confidence Bands for Nonparametric
Structural Functions and Elasticities. https://arxiv.org/abs/2107.11869.
"""
# Step 1: Compute maximum feasible J and data-driven grid
tmp1 = npiv_jhat_max(
x=x,
w=w,
j_x_degree=j_x_degree,
k_w_degree=k_w_degree,
k_w_smooth=k_w_smooth,
knots=knots,
basis=basis,
x_min=x_min,
x_max=x_max,
w_min=w_min,
w_max=w_max,
)
j_x_segments_set = tmp1["j_x_segments_set"]
k_w_segments_set = tmp1["k_w_segments_set"]
alpha_hat = tmp1["alpha_hat"]
# Step 2: Run Lepski test with bootstrap
tmp2 = npiv_j(
y=y,
x=x,
w=w,
x_grid=x_grid,
j_x_degree=j_x_degree,
k_w_degree=k_w_degree,
j_x_segments_set=j_x_segments_set,
k_w_segments_set=k_w_segments_set,
knots=knots,
basis=basis,
x_min=x_min,
x_max=x_max,
w_min=w_min,
w_max=w_max,
grid_num=grid_num,
boot_num=boot_num,
alpha=alpha_hat,
check_is_fullrank=check_is_fullrank,
seed=seed,
)
return {
"j_hat_max": tmp1["j_hat_max"],
"j_hat_n": tmp2["j_hat_n"],
"j_hat": tmp2["j_hat"],
"j_tilde": tmp2["j_tilde"],
"j_x_seg": tmp2["j_x_seg"],
"k_w_seg": tmp2["k_w_seg"],
"j_x_segments_set": j_x_segments_set,
"k_w_segments_set": k_w_segments_set,
"theta_star": tmp2["theta_star"],
}
