Pith. sign in

REVIEW 3 major objections 4 minor 55 references

High-Dimensional Spatial Autoregression with Latent Factors by Diversified Projections

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper's central claim is that a factor-augmented spatial autoregression (FSAR) can estimate hidden common shocks from high-dimensional responses and select the true covariates for every response component simultaneously with probability

desk verdict A genuinely useful new FSAR model for high-dimensional spatial data, but the gap between the theory (fixed M satisfying an eigenvalue condition) and the practice (data-dependent random-partition M) needs bridging before the main theorems can be taken at face value. read the letter →

arxiv 2509.00742 v1 pith:P2464SV4 submitted 2025-08-31 stat.ME

Jiaxin Shi , Xuening Zhu , Jing Zhou , Baichen Yu , Hansheng Wang This is my paper · ORCID
classification stat.ME MSC 62M3062H2562J07
keywords factor-augmentedspatialautoregressionmultivariatedatalatentfactormodeldiversifiedprojectionsSCADvariableselectionuniformconsistencyhigh-dimensionalresponsenetwork
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper is trying to establish that multivariate spatial autoregression remains feasible when the number of responses p diverges with the sample size, a regime where existing multivariate spatial models become computationally prohibitive. Its proposal, the FSAR model, imposes a fixed-dimensional factor structure on the p-dimensional error vector, cutting the parameter count from O(p^2) to O(p(d+q)). The main theoretical claims are that a diversified-projection step recovers the latent factors from componentwise residuals, that the factor-augmented likelihood estimator is √n-consistent and asymptotically normal, and that a SCAD penalty with a BIC-type tuning rule selects the true covariate set for every response component simultaneously with probability tending to one. A reader should care because the method turns an apparently intractable high-dimensional spatial estimation problem into p parallel low-dimensional regressions while keeping uniform statistical guarantees.

What carries the argument

The central object is the projection matrix M together with the rotated factor matrix H = M^T B^*/p. The diversified projections estimator bZ_i = M^T bε_i / p estimates Z_i H^T; Condition (C9)(2), which requires rank(H)=d and comparable eigenvalues for H^T H, is what guarantees the projection preserves the factor signal. The second load-bearing mechanism is the componentwise SCAD likelihood with the BIC penalty (log n)(log(pq))^{2/α}/n, which drives the uniform selection consistency. These two pieces convert one high-dimensional factor-plus-spatial problem into p small penalized regressions.

What would settle it

Simulate data with the true loading matrix B* nearly orthogonal to a fixed projection M (so H = M^T B*/p has a singular value near zero), then run the full three-step procedure; the theorem predicts the FMLE and SCAD selection should lose consistency, while any setup that still yields good estimates would contradict the role of Condition (C9)(2).

Watch

Extended reading notes

Core claim

Treating each response dimension as its own spatial autoregression and placing a shared factor model on the errors yields the FSAR model. The paper shows the latent factors can be learned by diversified projections: projecting the estimated residual matrix onto a well-chosen matrix M makes M^T ε_i/p concentrate around a rotated version of the latent factor Z_i. Once the estimated factors are inserted as covariates, each component is fit by maximum likelihood with a SCAD penalty. The decisive result is uniform selection consistency: under λ_n → 0 and √n λ_n / (log(pq))^{1/α} → ∞, the selected model equals the true model for every j = 1,...,p with probability going to one. The FMLE is √n-consi

Load-bearing premise

Everything rests on the analyst's choice of projection matrix M: if M is nearly orthogonal to the true factor loadings, the factor estimates never converge to the latent signals, and the subsequent likelihood and selection results collapse.

Editorial extensions

If this is right

  • Full multivariate spatial autoregression requires O(p^2) parameters; under FSAR the count drops to O(p(d+q)) and each response component can be fitted in parallel.
  • Factor-estimation error is not ignored: it appears in the FMLE's asymptotic covariance, so standard errors remain valid even though the factors are estimated rather than observed.
  • As long as the response dimension p is large relative to √n, latent common shocks can be separated from spatial spillover effects in estimation.
  • The SCAD-plus-BIC procedure recovers all p true covariate sets simultaneously, so model selection no longer has to be checked one outcome at a time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The factor-recovery step is agnostic about the sparse estimator used afterward, so swapping SCAD for another componentwise penalty should preserve the core argument and may allow post-selection inference.
  • Since Condition (C9)(2) is not verified by the data, a practical stability check is to repeat the analysis over several projection matrices and compare estimated factors and selected models; disagreement warns that the projection may be nearly orthogonal to the loadings.
  • The (log pq)^{2/α} factor in the BIC penalty implies that when p and q grow at different rates, the effective signal requirement tightens; applied users should monitor selection stability instead of trusting all p models simultaneously.
  • For small or slowly growing p, the consistency guarantee disappears; in that regime a dynamic factor or full multivariate spatial model is a safer alternative.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper proposes a factor-augmented spatial autoregressive (FSAR) model for high-dimensional multivariate response spatial data. The error vector in a componentwise SAR model is given a fixed-dimensional factor structure. Estimation proceeds in three steps: componentwise QMLE residuals, factor estimation via diversified projections (Fan and Liao 2022), and a per-component factor-augmented QMLE; SCAD penalization with a BIC criterion is used for variable selection. The main theoretical claims are uniform consistency of the componentwise estimator (Theorem 1), consistency of the projected factor estimator (Theorem 2), sqrt(n)-consistency and asymptotic normality of the FMLE (Theorem 3), and uniform selection consistency of SCAD and BIC (Theorems 4-5). Simulations on three network structures and a 287-city macroeconomic dataset illustrate the method.

Significance. If correct, the paper offers a computationally attractive route to MSAR with many responses: per-component parallel estimation, a parsimonious factor-augmented error structure, and a uniform selection guarantee stronger than usual pointwise model selection. The use of the diversified projections idea in a spatial-autoregression context is a sensible extension, and the authors are explicit about tail, network, and identification conditions. The provision of code on GitHub is a plus. However, the proof of the central asymptotic results is not contained in the arXiv submission, and the connection between the theoretical Condition (C9) on the projection matrix and the data-dependent random-partition implementation is unproved; these issues need to be resolved before the claims can be accepted.

major comments (3)
  1. [Section 2.3, 3.2; Theorems 2-4] Condition (C9)(2) requires rank(H)=d and lambda_min(H^T H) >> 1/p with H = M^T B*/p. In the theory M is pre-specified, but in the numerical implementation (Section 3.2 and 3.3) M is obtained from 10% of the data by a random partition. No theorem proves that this estimated M satisfies (C9)(2). For a generic bounded random projection, entries of H are O_p(p^{-1/2}) and eigenvalues of H^T H are O_p(1/p), violating the condition; the condition requires M to be strongly aligned with B*. If it fails, Theorem 2 fails and the FMLE likelihood (2.5) is misspecified, so Theorems 3 and 4 do not cover the implemented procedure. A lemma showing that the random-partition construction yields (C9)(2) with high probability, or a reformulated condition on the estimated M, is needed.
  2. [Eq. (2.7), Condition (C10); Appendix] The arXiv version states that all proofs are in Appendices A-C, but these appendices are not included. The expansion (2.7) is central to Theorem 3, and the matrices A_j and D_j in Condition (C10) are defined only in the omitted (A.30) and (A.33). Without these materials the covariance formula in Theorem 3, the identifiability condition, and the uniform selection proofs in Theorems 4-5 cannot be checked. The authors should include the appendices or point to a publicly accessible supplement.
  3. [Section 2.3 and Section 3.2] The random-partition method itself is not fully specified: it is described only as estimating M from 10% of the data. The reader cannot determine whether M is a PCA loading estimate, an eigenvalue-thresholded estimate, or some other construction. This matters because the required alignment with B* in (C9)(2) is precisely what must be demonstrated for the method actually used in the simulations and real-data analysis.
minor comments (4)
  1. [Condition (C6)] The definition of the induced norms contains an undefined 'm': 'max_{1<=i<=m}' should presumably be 'max_{1<=i<=n}'.
  2. [Theorem 3 display] The covariance matrices in Theorem 3 are hard to parse because of the block notation and the fact that Sigma_Qj is deferred to an omitted appendix. A clean partition of the parameter vector and a full definition of Sigma_Qj in the main text or a visible supplement would help.
  3. [Section 3.2, Table 1] The statement that 'a CP_j is randomly selected over 1<=j<=p' is not reproducible. Reporting the distribution of coverage probabilities across j, or at least the range and median, would be more informative.
  4. [Section 3.3] The factor dimension in the real-data example is selected by eigenvalue ratios without formal post-selection justification. A reference to Bai and Ng (2002) or Lam and Yao (2012) for the ratio-based rule would strengthen the presentation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained, with asymptotic results conditional on explicit identification conditions rather than on fitting the target quantities.

full rationale

The paper's central claims (Theorems 1-5) are derived from standard likelihood and penalized-likelihood objectives: CMLE in (2.4), FMLE in (2.5), and SCAD in Section 2.5. The latent factors are estimated from first-step residuals via the diversified projections identity M^T ε_i/p = H Z_i + M^T ω_i/p; Theorem 2 then shows bZ estimates ZH^T under Condition (C9), which is an explicit non-orthogonality/identifiability condition, not an assumption that already contains the theorem's conclusion. The √n-consistency of the initial CMLE is imported from the external result of Lee (2004), and the paper proves its own uniform version as Theorem 1. Theorems 3 and 4 explicitly account for the generated-regressor error through Eq. (2.7), so the spatial coefficients are not fitted inputs relabeled as predictions. The BIC/SCAD selection results are proved from the likelihood and standard separation/eigenvalue conditions (C10)-(C11); no fitted constant is renamed as a prediction. There is no load-bearing self-citation: Fan and Liao (2022) and Lee (2004) are external published results, and the authors' own prior work is used only as background or comparison. The paper's stated limitations—Condition (C9) assumes a pre-specified projection M non-orthogonal to B*, while Sections 3.2 and 3.3 use a data-dependent random-partition M without a matching theorem, and Appendices A-C are omitted from the arXiv version—are verifiability/correctness gaps, not circular reductions. No equation in the paper equates the target parameter to the estimator or defines the estimand in terms of the fitted values.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on the paper's regularity conditions (C1)-(C11), most of which are standard in the factor and SAR literature. No ad hoc fitted constants are introduced into the theoretical analysis; the SCAD tuning parameter is selected by a BIC criterion, and the projection matrix is obtained by a random partition (data-driven). The latent factors are a standard statistical device, not a new physical entity.

assumptions (6)
  • domain assumption Condition (C1): errors and factors are sub-Weibull(α) with α ∈ (0,2], independent of X
    Needed for uniform convergence over p components; assumed on the data generating process.
  • domain assumption Condition (C4): √n/p → 0 and log p = O(n^{αγ})
    Ensures p diverges fast enough for consistent factor estimation via diversified projections; restricts the growth rate of response dimension.
  • domain assumption Condition (C9): the projection matrix M satisfies rank(H)=d, λ_min(H^T H) ≫ 1/p and λ_max(H^T H) ≤ C λ_min(H^T H)
    Load-bearing for factor estimability; if M is orthogonal to B*, the projected factors carry no information.
  • domain assumption Condition (C10): row sums of |τ_jk| are O(1) and spectral conditions on A_j, D_j
    Sparsity of the idiosyncratic covariance is required for the asymptotic distribution of the FMLE.
  • domain assumption Condition (C11): strict separation of risk between true and underfitted models
    Needed for BIC selection consistency; a standard condition in model selection theory.
  • standard math |ρ_j| < 1 and S_j = I_n - ρ_j W invertible
    Standard in SAR literature; assumed in Condition (C6).

how reviews work

0 comments
Cite this review

Pith. "Pith review of High-Dimensional Spatial Autoregression with Latent Factors by Diversified Projections." pith.science (2026). https://pith.science/paper/P2464SV4

@misc{pith2026250900742,
  author       = {Pith},
  title        = {Pith review of: High-Dimensional Spatial Autoregression with Latent Factors by Diversified Projections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P2464SV4}},
  note         = {Machine review of arXiv:2509.00742}
}
read the original abstract

We study one particular type of multivariate spatial autoregression (MSAR) model with diverging dimensions in both responses and covariates. This makes the usual MSAR models no longer applicable due to the high computational cost. To address this issue, we propose a factor-augmented spatial autoregression (FSAR) model. FSAR is a special case of MSAR but with a novel factor structure imposed on the high-dimensional random error vector. The latent factors of FSAR are assumed to be of a fixed dimension. Therefore, they can be estimated consistently by the diversified projections method \citep{fan2022learning}, as long as the dimension of the multivariate response is diverging. Once the fixed-dimensional latent factors are consistently estimated, they are then fed back into the original SAR model and serve as exogenous covariates. This leads to a novel FSAR model. Thereafter, different components of the high-dimensional response can be modeled separately. To handle the high-dimensional feature, a smoothly clipped absolute deviation (SCAD) type penalized estimator is developed for each response component. We show theoretically that the resulting SCAD estimator is uniformly selection consistent, as long as the tuning parameter is selected appropriately. For practical selection of the tuning parameter, a novel BIC method is developed. Extensive numerical studies are conducted to demonstrate the finite sample performance of the proposed method.

Figures

Figures reproduced from arXiv: 2509.00742 by the authors.

Figure 1
Figure 1. The log(MaxErrc) values for CMLE ρbj,cmle with d = 3. Different panels correspond to different network structures: DIM (the left), SBM (the middle), and LSM (the right). For a given panel, different groups correspond to different feature dimensions with p = 50, 100 and 200, respectively. For a given group, the lighter the color of the box is, the larger the sample size is. Then, we obtain the mean error as MErrf = (… view at source ↗
Figure 2
Figure 2. The CM values with q = 20 and d = 3. Different panels correspond to different network structures: DIM (the left), SBM (the middle), and LSM (the right). For a given panel, different lines correspond to different feature dimensions with p = 50 (solid), 100 (dashed) and 200 (dotted), respectively. S(j),T , which is in line with our theoretical findings in Theorem 5. 3.3. A Real Data Example To demonstrate the practica… view at source ↗
Figure 3
Figure 3. The histogram of the CMLE ρbj,cmles (the left panel) and the histogram of the estimated SE (the right panel). weight matrix W = (wi1i2 ) ∈ R n×n is then constructed based on geographical locations (Lee and Yu, 2010). We next fit an FSAR model to this dataset. Note that each node (i.e., city) is spatially connected with others through W. The associated spatial correlations ρj s reflect the spatial spillover effects a… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: The boxplots of the CMLE ρbj,cmles for four categories. From left to right: Tertiary (tertiary industry), Population (labor and population development), Fiscal (fiscal and financial resources), and Public Services (infrastructure and public services). indicators, measu…
Figure 5
Figure 5. Figure 5: The top 30 estimated eigenvalues of Eb (the left panel) and the top 30 eigen￾value ratios (the right panel). one-dimensional factor structure. This finding is not totally surprising, as these 50 indicators are all macroeconomics related. Therefore, they are heavily cor…
Figure 6
Figure 6. Figure 6: The scatter plot of the FMLE ρbfmle and the associated CMLE ρbcmle (the left panel). The boxplots of their standard errors (the right panel). 4. CONCLUDING REMARKS In this work, we study the problem of spatial autoregressive modeling for network data with high-dimensio…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 52 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in " " * FUNCTION format....

  3. [3]

    (1988), Spatial econometrics: methods and models, vol

    Anselin, L. (1988), Spatial econometrics: methods and models, vol. 4, Springer Science and Business Media

    work page 1988

  4. [4]

    (2020), Local fiscal multipliers and fiscal spillovers in the USA, IMF Economic Review, 68, 195--229

    Auerbach, A., Gorodnichenko, Y., and Murphy, D. (2020), Local fiscal multipliers and fiscal spillovers in the USA, IMF Economic Review, 68, 195--229

    work page 2020

  5. [5]

    (2012), Statistical analysis of factor models of high dimension, The Annals of Statistics, 40, 436

    Bai, J. (2012), Statistical analysis of factor models of high dimension, The Annals of Statistics, 40, 436

    work page 2012

  6. [6]

    and Li, K

    Bai, J. and Li, K. (2021), Dynamic spatial panel data models with common shocks, Journal of Econometrics, 224, 134--160

    work page 2021

  7. [7]

    and Ng, S

    Bai, J. and Ng, S. (2002), Determining the number of factors in approximate factor models, Econometrica, 70, 191--221

    work page 2002

  8. [8]

    S., Boivin, J., and Eliasz, P

    Bernanke, B. S., Boivin, J., and Eliasz, P. (2005), Measuring the effects of monetary policy: a factor-augmented vector autoregressive (FAVAR) approach, The Quarterly Journal of Economics, 120, 387--422

    work page 2005

Show all 55 references
  1. [9]

    J., Lucas, A., and Schaumburg, J

    Blasques, F., Koopman, S. J., Lucas, A., and Schaumburg, J. (2016), Spillover dynamics for systemic risk measurement using spatial financial time series models, Journal of Econometrics, 195, 211--223

    2016

  2. [10]

    and Chen, Z

    Chen, J. and Chen, Z. (2008), Extended Bayesian information criteria for model selection with large model spaces, Biometrika, 95, 759--771

    2008

  3. [11]

    (2025), Estimating time-varying networks for high-dimensional time series, Journal of Econometrics, 105941

    Chen, J., Li, D., Li, Y.-N., and Linton, O. (2025), Estimating time-varying networks for high-dimensional time series, Journal of Econometrics, 105941

    2025

  4. [12]

    and Qu, A

    Cho, H. and Qu, A. (2013), Model selection for correlated data with diverging number of parameters, Statistica Sinica, 901--927

    2013

  5. [13]

    De Paula, A., Rasul, I., and Souza, P. C. (2025), Identifying network ties from panel data: Theory and an application to tax competition, Review of Economic Studies, 92, 2691--2729

    2025

  6. [14]

    (2001), Infrastructure development and economic growth: an explanation for regional disparities in China? Journal of Comparative Economics, 29, 95--117

    D \'e murger, S. (2001), Infrastructure development and economic growth: an explanation for regional disparities in China? Journal of Comparative Economics, 29, 95--117

    2001

  7. [15]

    Donoho, D. L. and Johnstone, I. M. (1994), Ideal spatial adaptation by wavelet shrinkage, Biometrika, 81, 425--455

    1994

  8. [16]

    (2004), Least angle regression, The Annals of Statistics, 407--451

    Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), Least angle regression, The Annals of Statistics, 407--451

    2004

  9. [17]

    (2008), High dimensional covariance matrix estimation using a factor model, Journal of Econometrics, 147, 186--197

    Fan, J., Fan, Y., and Lv, J. (2008), High dimensional covariance matrix estimation using a factor model, Journal of Econometrics, 147, 186--197

    2008

  10. [18]

    (2022), Estimating number of factors by adjusted eigenvalues thresholding, Journal of the American Statistical Association, 117, 852--861

    Fan, J., Guo, J., and Zheng, S. (2022), Estimating number of factors by adjusted eigenvalues thresholding, Journal of the American Statistical Association, 117, 852--861

    2022

  11. [19]

    Fan, J., Guo, S., and Hao, N. (2012), Variance estimation using refitted cross-validation in ultrahigh dimensional regression, Journal of the Royal Statistical Society Series B: Statistical Methodology, 74, 37--65

    2012

  12. [20]

    and Li, R

    Fan, J. and Li, R. (2001), Variable selection via nonconcave penalized likelihood and its oracle properties, Journal of the American Statistical Association, 96, 1348--1360

    2001

  13. [21]

    (2020), Statistical foundations of data science, Chapman and Hall/CRC

    Fan, J., Li, R., Zhang, C.-H., and Zou, H. (2020), Statistical foundations of data science, Chapman and Hall/CRC

    2020

  14. [22]

    and Liao, Y

    Fan, J. and Liao, Y. (2022), Learning latent factors from diversified projections and its applications to over-estimated and weak factors, Journal of the American Statistical Association, 117, 909--924

    2022

  15. [23]

    (2013), Large covariance estimation by thresholding principal orthogonal complements, Journal of the Royal Statistical Society Series B: Statistical Methodology, 75, 603--680

    Fan, J., Liao, Y., and Mincheva, M. (2013), Large covariance estimation by thresholding principal orthogonal complements, Journal of the Royal Statistical Society Series B: Statistical Methodology, 75, 603--680

    2013

  16. [24]

    and Lv, J

    Fan, J. and Lv, J. (2011), Nonconcave penalized likelihood with NP-dimensionality, IEEE Transactions on Information Theory, 57, 5467--5484

    2011

  17. [25]

    (2021), A shrinkage principle for heavy-tailed data: High-dimensional robust low-rank matrix recovery, Annals of statistics, 49, 1239

    Fan, J., Wang, W., and Zhu, Z. (2021), A shrinkage principle for heavy-tailed data: High-dimensional robust low-rank matrix recovery, Annals of statistics, 49, 1239

    2021

  18. [26]

    R., and Venables, A

    Fujita, M., Krugman, P. R., and Venables, A. (2001), The spatial economy: Cities, regions, and international trade, MIT press

    2001

  19. [27]

    and Tracy, J

    Gyourko, J. and Tracy, J. (1991), The structure of local public finance and the quality of life, Journal of Political Economy, 99, 774--806

    1991

  20. [28]

    D., Raftery, A

    Hoff, P. D., Raftery, A. E., and Handcock, M. S. (2002), Latent space approaches to social network analysis, Journal of the American Statistical Association, 97, 1090--1098

    2002

  21. [29]

    Holland, P. W. and Leinhardt, S. (1981), An exponential family of probability distributions for directed graphs, Journal of the American Statistical Association, 76, 33--50

    1981

  22. [30]

    H., and Wang, H

    Huang, D., Lan, W., Zhang, H. H., and Wang, H. (2019), Least squares estimation of spatial autoregressive models for large-scale social networks, Electronic Journal of Statistics, 13, 1135--1165

    2019

  23. [31]

    (2021), Feature screening for network autoregression model, Statistica Sinica, 31, 1239

    Huang, D., Zhu, X., Li, R., and Wang, H. (2021), Feature screening for network autoregression model, Statistica Sinica, 31, 1239

    2021

  24. [32]

    Kelejian, H. H. and Prucha, I. R. (1998), A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances, The Journal of Real Estate Finance and Economics, 17, 99--121

    1998

  25. [33]

    (1987), The primary, secondary, tertiary and quaternary sectors of the economy, Review of Income and Wealth, 33, 359--385

    Kenessey, Z. (1987), The primary, secondary, tertiary and quaternary sectors of the economy, Review of Income and Wealth, 33, 359--385

    1987

  26. [34]

    and Yao, Q

    Lam, C. and Yao, Q. (2012), Factor modeling for high-dimensional time series: inference for the number of factors, The Annals of Statistics, 694--726

    2012

  27. [35]

    (2004), Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models, Econometrica, 72, 1899--1925

    Lee, L.-F. (2004), Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models, Econometrica, 72, 1899--1925

    2004

  28. [36]

    --- (2007), GMM and 2SLS estimation of mixed regressive, spatial autoregressive models, Journal of Econometrics, 137, 489--514

    2007

  29. [37]

    (2010), Specification and estimation of social interaction models with network structures, The Econometrics Journal, 13, 145--176

    Lee, L.-F., Liu, X., and Lin, X. (2010), Specification and estimation of social interaction models with network structures, The Econometrics Journal, 13, 145--176

    2010

  30. [38]

    and Yu, J

    Lee, L.-f. and Yu, J. (2010), Estimation of spatial autoregressive panel data models with fixed effects, Journal of Econometrics, 154, 165--185

    2010

  31. [39]

    (2018), Interpolation theory, vol

    Lunardi, A. (2018), Interpolation theory, vol. 16, Springer

    2018

  32. [40]

    (2009), Contour projected dimension reduction, The Annals of Statistics, 3743--3778

    Luo, R., Wang, H., and Tsai, C.-L. (2009), Contour projected dimension reduction, The Annals of Statistics, 3743--3778

    2009

  33. [41]

    and Snijders, T

    Nowicki, K. and Snijders, T. A. B. (2001), Estimation and prediction for stochastic blockstructures, Journal of the American Statistical Association, 96, 1077--1087

    2001

  34. [42]

    (1993), Linear model selection by cross-validation, Journal of the American statistical Association, 88, 486--494

    Shao, J. (1993), Linear model selection by cross-validation, Journal of the American statistical Association, 88, 486--494

    1993

  35. [43]

    (2012), Semiparametric GMM estimation of spatial autoregressive models, Journal of Econometrics, 167, 543--560

    Su, L. (2012), Semiparametric GMM estimation of spatial autoregressive models, Journal of Econometrics, 167, 543--560

    2012

  36. [44]

    (1996), Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society Series B: Statistical Methodology, 58, 267--288

    Tibshirani, R. (1996), Regression shrinkage and selection via the lasso, Journal of the Royal Statistical Society Series B: Statistical Methodology, 58, 267--288

    1996

  37. [45]

    Wainwright, M. J. (2019), High-dimensional statistics: a non-asymptotic viewpoint, vol. 48, Cambridge University Press

    2019

  38. [46]

    (2012), Factor profiled sure independence screening, Biometrika, 99, 15--28

    Wang, H. (2012), Factor profiled sure independence screening, Biometrika, 99, 15--28

    2012

  39. [47]

    (2009), Shrinkage tuning parameter selection with a diverging number of parameters, Journal of the Royal Statistical Society Series B: Statistical Methodology, 71, 671--683

    Wang, H., Li, B., and Leng, C. (2009), Shrinkage tuning parameter selection with a diverging number of parameters, Journal of the Royal Statistical Society Series B: Statistical Methodology, 71, 671--683

    2009

  40. [48]

    (2007), Tuning parameter selectors for the smoothly clipped absolute deviation method, Biometrika, 94, 553--568

    Wang, H., Li, R., and Tsai, C.-L. (2007), Tuning parameter selectors for the smoothly clipped absolute deviation method, Biometrika, 94, 553--568

    2007

  41. [49]

    and Lee, L.-f

    Yang, K. and Lee, L.-f. (2017), Identification and QML estimation of multivariate and simultaneous equations spatial autoregressive models, Journal of Econometrics, 196, 196--214

    2017

  42. [50]

    (2022), Spatial heterogeneity of the economic growth pattern and influencing factors in formerly destitute areas of China, Journal of Geographical Sciences, 32, 829--852

    Yin, J., Li, S., Zhou, L., Jiang, L., and Ma, W. (2022), Spatial heterogeneity of the economic growth pattern and influencing factors in formerly destitute areas of China, Journal of Geographical Sciences, 32, 829--852

    2022

  43. [51]

    (2016), Strategic interaction in political competition: Evidence from spatial effects across Chinese cities, Regional Science and Urban Economics, 57, 23--37

    Yu, J., Zhou, L.-A., and Zhu, G. (2016), Strategic interaction in political competition: Evidence from spatial effects across Chinese cities, Regional Science and Urban Economics, 57, 23--37

    2016

  44. [52]

    (2022), Joint latent space models for network data with high-dimensional node variables, Biometrika, 109, 707--720

    Zhang, X., Xu, G., and Zhu, J. (2022), Joint latent space models for network data with high-dimensional node variables, Biometrika, 109, 707--720

    2022

  45. [53]

    (2024), Variable selection and subgroup analysis for high-dimensional censored data, Statistical Theory and Related Fields, 8, 211--231

    Zhang, Y., Wang, J., and Zhang, W. (2024), Variable selection and subgroup analysis for high-dimensional censored data, Statistical Theory and Related Fields, 8, 211--231

    2024

  46. [54]

    (2023), Spatial and temporal evolution characteristics and spillover effects of China's regional carbon emissions, Journal of Environmental Management, 325, 116423

    Zhou, K., Yang, J., Yang, T., and Ding, T. (2023), Spatial and temporal evolution characteristics and spillover effects of China's regional carbon emissions, Journal of Environmental Management, 325, 116423

    2023

  47. [55]

    (2020), Multivariate spatial autoregressive model for large scale social networks, Journal of Econometrics, 215, 591--606

    Zhu, X., Huang, D., Pan, R., and Wang, H. (2020), Multivariate spatial autoregressive model for large scale social networks, Journal of Econometrics, 215, 591--606

    2020

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.