Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Testing linearity of spatial interaction functions \`a la Ramsey

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A single score statistic, computed from the linear spatial model alone, tests whether the spatial interaction function is truly linear.

desk verdict A genuinely new Ramsey-style test for linearity of the spatial lag, but the null distribution is inherited from Gupta (2018) rather than proved, and the simulations sit outside the stated p^3/n = o(1) condition. read the letter →

arxiv 2412.14778 v3 pith:4RLDX7W5 submitted 2024-12-19 econ.EM

Abhimanyu Gupta , Jungyoon Lee , Francesca Rossi This is my paper · ORCID
classification econ.EM
keywords SpatialautoregressionSeriesexpansionSpecificationtestingNonparametricinteractionfunctionRamseyRESETLagrangemultipliertestHeteroskedasticityrobustness
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

The paper proposes a test of the null hypothesis that the spatial interaction function $f$ in a spatial autoregressive model $y_i = \lambda w_i' y + f(w_i' y) + x_i' \beta + \varepsilon_i$ is identically zero, so that the familiar linear SAR specification is correct. The test expands $f$ in a sieve of $p$ basis functions and builds a Lagrange-multiplier statistic from the gradient of the 2SLS objective function evaluated under the null, which means only the linear SAR model must be estimated. The statistic is centred and rescaled as $T = (n \hat{d}'\hat{H}^{-1}\hat{d} - p)/\sqrt{2p}$, and the paper claims it is asymptotically standard normal under the null when the sieve dimension satisfies $p^3/n \to 0$, consistent against fixed nonlinear alternatives, and able to detect local alternatives of order $p^{1/4}/\sqrt{n}$. Monte Carlo experiments show good size and power across several network designs, and an application to Finnish municipal property tax data corroborates the earlier conclusion of no tax competition when municipality fixed effects are included, while rejecting linearity in the same model without fixed effects. The practical payoff is a specification test for the most common empirical model of social and spatial interaction that avoids estimating any nonparametric function.

What carries the argument

The central object is the truncated series (sieve) expansion $f(z) = \sum_{j=1}^{p} \alpha_j \psi_j(z) + r(z)$ of the unknown spatial interaction function, which turns the hypothesis $H_0: f = 0$ into the linear restriction $\alpha_1 = \cdots = \alpha_p = 0$ on $p$ sieve coefficients. The test statistic is the centred and scaled score (Lagrange multiplier) statistic $T = (n \hat{d}'\hat{H}^{-1}\hat{d} - p)/\sqrt{2p}$, where $\hat{d}$ is the gradient of the 2SLS objective evaluated at the null estimates and $\hat{H}$ is its robust covariance; the centring by $p$ and scaling by $\sqrt{2p}$ mirrors the standardization of a $\chi^2_p$ variable. The asymptotic argument approximates $T$ by a quadratic form in the innovations with population-anchored weights (Theorem A1 in the appendix) and then invokes a central limit theorem for quadratic forms with a growing number of instruments to obtain the standard normal null distribution.

What would settle it

Simulate the null model with spatially dependent heteroskedastic errors under the paper's Assumptions 1-6, choose the sieve dimension as $p \approx n^{1/4}$ so that $p^3/n \to 0$, and check whether the empirical rejection rate of $T$ at the 5% normal critical value converges to 5% over many Monte Carlo replications; a clear size distortion would show that the borrowed CLT does not govern this statistic.

Watch

Extended reading notes

Core claim

Under $H_0: f \equiv 0$ in the spatial autoregressive model with an unknown link function, the statistic $T = (n \hat{d}'\hat{H}^{-1}\hat{d} - p)/\sqrt{2p}$ is shown to be asymptotically standard normal, provided that the sieve approximation error decays at rate $\nu > 5/2$ and that the sieve dimension grows no faster than $p^3/n \to 0$ (Theorem 2). Here $\hat{d}$ is the gradient of the IV objective under the null and $\hat{H}$ is its heteroskedasticity-robust covariance estimator. The paper proves further that the test is consistent against any fixed nonlinear alternative (Theorem 3) and that it detects local alternatives of the form $\alpha_j = p^{1/4} n^{-1/2} \delta_j$ with a non-central normal limit (Theorem 4). The test requires estimation only under the linear null, and the paper recommends comparing the statistic against either standard normal critical values or the standardized $\chi^2_p$ distribution for moderate $p$. The empirical section finds that the test fails to reject linearity for Finnish property tax rates with municipality and year fixed effects, matching the earlier finding of no tax competition, but strongly rejects linearity for the level specification without fixed effects, supporting the interpretation that earlier evidence of tax competition came from a misspecified model.

Load-bearing premise

The asymptotic null distribution of $T$ is the load-bearing premise, and it is imported from a central limit theorem proved in earlier work in a different (non-spatial) setting, with the paper asserting rather than verifying that the required quadratic-form CLT holds under its spatial, heteroskedastic, growing-instruments assumptions.

Editorial extensions

If this is right

  • Estimating only the null linear SAR model, practitioners can now test for hidden nonlinearity in the spatial lag, avoiding the computational cost of nonparametric estimation of the link function.
  • The test is consistent: any fixed nonlinearity in the spatial interaction function will be detected asymptotically.
  • Local alternatives of order $p^{1/4}/\sqrt{n}$ are detectable, a dampened rate typical of series-based specification tests and noticeably slower than the parametric $1/\sqrt{n}$ rate.
  • Using standardized $\chi^2_p$ critical values gives reliable size for small $p$ and moderate samples, complementing the asymptotic normal approximation.

Reading between the lines

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

  • Inference: If the null approximation holds uniformly in the sieve dimension, the same statistic could be applied to test linearity of the link function in other network- and peer-effect models where $f$ acts on a network-weighted average of outcomes.
  • Inference: The Monte Carlo design $p = \lfloor n^{1/3} \rfloor$ makes $p^3/n = O(1)$, outside the theorem's condition; rerunning the size experiments with $p \approx n^{1/4}$ would test whether the claimed normal approximation holds where the theory says it should.
  • Inference: The paper's empirical contrast — linearity not rejected with municipality fixed effects but rejected without them — suggests the test can serve as a diagnostic for omitted cross-sectional confounders, not only for functional form.
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

2 major / 5 minor

Summary. The paper proposes a test for the null hypothesis that the spatial interaction function in a spatial autoregressive model is linear. The test is constructed from a sieve approximation to the nonlinear component and follows the Lagrange Multiplier principle, so that only the linear SAR model is estimated. The test statistic T in (3.9) is a standardized quadratic form in the 2SLS gradient. The paper claims that T is asymptotically standard normal under the null (Theorem 2), that the test is consistent (Theorem 3), and that it detects local alternatives of order p^{1/4}/sqrt(n) (Theorem 4). Monte Carlo evidence and an empirical application to Finnish municipal tax competition are also provided.

Significance. If the asymptotic results are correct, the paper offers a computationally simple and economically motivated specification test for a practically important question, complementing the existing literature on testing nonlinearity in spatial lags. The paper is clearly written, includes several structural models that justify the null hypothesis, and presents a credible empirical illustration. The main weakness is that the central null distribution is not proved in the paper: Theorem 2 is delegated to a 'trivial modification' of a result in the first author's earlier paper, and the Monte Carlo design violates the condition p^3/n=o(1). These are load-bearing issues that prevent acceptance in the current form.

major comments (2)
  1. [Section 4.1, Theorem 2 (proof in Appendix A)] The proof of Theorem 2 does not establish the asymptotic normality of the quadratic form. After Theorem 1, the proof says: 'By Remark 3 of Gupta (2018), the claim follows by a trivial modification of the arguments in the proof of Theorem 3.3 therein to allow for heteroskedastic innovations.' Neither Remark 3 nor Theorem 3.3 of Gupta (2018) is stated, and no verification is provided that the conditions of that result hold for the present SAR model with spatial dependence, a growing number of instruments (m ~ p), and heteroskedastic innovations. The required CLT for the studentized quadratic form nd'H^{-1}d, with H defined in (4.6), needs explicit conditions on the weight matrix, the instrument set, and the innovation distribution; these are not given. Since Theorem 2 provides the null distribution used in practice and is also invoked in the proofs of Theorems 3 and 4, this is a load-bearing gap.
  2. [Section 5 (Monte Carlo)] The Monte Carlo design sets p = floor(n^{1/3}). For the reported sample sizes n = 100, 200, 400, 700, 1000, 2000, this gives p = 4, 5, 7, 8, 10, 12 and p^3/n approximately 0.64, 0.63, 0.86, 0.73, 1.00, and 0.86, so p^3/n is O(1) and not o(1). Theorems 1 and 2 require p^3/n = o(1). Hence the simulations are outside the stated theoretical conditions, and the reported size results based on the normal approximation do not validate Theorem 2. The authors should either re-run the simulations with p = o(n^{1/3}) (for example p = floor(n^{1/4}) or p = floor(n^{0.3})), or extend the theory to allow p = n^{1/3}.
minor comments (5)
  1. [Section 4, Assumption 5] The notation 'n/p(ν+1/2) = o(1)' is ambiguous; it should be written as n / p^{ν+1/2} = o(1).
  2. [Appendix A, proof of Theorem 1] In the text after (A.17), the first two terms are said to be Op(p^2/n^{3/2}); based on the bounds \|\hat d - d\| = Op(p^{3/2}/n) and \|d\| = Op(\sqrt{p}/n), these terms are actually Op(p^2/n^2). The conclusion is unaffected under p^3/n = o(1), but the stated rate is a typo.
  3. [Section 5, Tables 1 and 2] The note says 'For lattice, n = 100, 210, 400, 702, 992, 1980', but the table rows list n = 100, 200, 400, 700, 1000, 2000; the lattice sample sizes should be clearly displayed in the table itself to avoid confusion.
  4. [Section 2, Example 1] In equation (2.2), the parentheses are likely misplaced: the expression inside the large parentheses should be \tilde{k}_i + \sum_{j\ne i} \tau_j / c(d_{ij}), not \tilde{k}_i + \sum_{j\ne i} \tau_j / c(d_{ij}) as currently appears to place \tilde{k}_i inside the sum.
  5. [Section 6, Tables 5 and 6] The table notes define significance stars, but no stars appear in the tables; either add the stars or remove the note.

Circularity Check

1 steps flagged · score 5.0 of 10

Theorem 2's null distribution is imported from Gupta (2018) without stated conditions or proof; the central asymptotic result therefore rests on a first-author self-citation.

  1. self citation load bearing [Appendix A, Proof of Theorem 2]
    "By Theorem 1, it suffices to show that nd′H−1d−p√2p d→N(0,1). Under the null hypothesis, the above CLT is for a SAR model with an increasing number of instruments. By Remark 3 of Gupta (2018), the claim follows by a trivial modification of the arguments in the proof of Theorem 3.3 therein to allow for heteroskedastic innovations (see e.g. Korolev (2019) and the justification of claim (D.3) in Robinson (2008))."

    The central distributional claim of the paper—the asymptotic standard normality of T under H0—is not proved here. The proof stops at an appeal to Remark 3 of Gupta (2018), a first-author self-citation, with only a 'trivial modification' asserted for heteroskedasticity. The paper does not state Gupta's Remark 3 or Theorem 3.3, nor verify that their conditions cover this spatial SAR, growing-instrument, heteroskedastic setting; Theorem 1 only replaces estimates by population quantities and does not deliver the required quadratic-form CLT. Thus the load-bearing step reduces to an unverified prior result by the same author, rather than a derivation contained in this paper.

full rationale

The paper is not circular in the fitting or definitional sense: the test statistic is not calibrated to the data, and the Monte Carlo and empirical exercises use external benchmarks. However, the asymptotic null distribution of T, which is the paper's pivotal result, is imported from the first author's earlier work, Gupta (2018), with no statement or verification of the borrowed CLT's assumptions. This is a load-bearing self-citation, because Theorem 2 is what justifies normal critical values and the entire testing procedure. The paper does contain substantial independent material—Theorem 1's approximation result, the consistency proof, local power analysis, and simulation evidence—so the central claim is not entirely reducible to the self-citation. Yet the distributional step itself is unsupported within the paper. Separately, the Monte Carlo design sets p = floor(n^{1/3}), making p^3/n = O(1) rather than o(1), so the simulations do not satisfy Theorem 2's stated conditions; this is a correctness concern rather than a circularity concern. Overall score 5 reflects a central result resting on an unverified first-author self-citation while other parts of the paper retain independent content.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The test is an adaptation of existing sieve and LM methods, so no new physical entities are introduced. The burden sits in the rate and regularity conditions plus the borrowed CLT.

free parameters (1)
  • p, number of sieve terms = floor(n^{1/3}) in the Monte Carlo design (p = 4, 5, 7, 8, 10, 12); practitioner choice elsewhere
    The test statistic and all asymptotic rates depend on the diverging sieve length p, but the simulation choice p = floor(n^{1/3}) gives p^3/n of order 1, not o(1) as Assumption 5 and Theorems 1-4 require.
assumptions (6)
  • domain assumption Assumption 1: independent errors with variances in [k,K] and finite fourth moments, allowing heteroskedasticity.
    Invoked throughout to bound the covariance matrices and the quadratic form; Section 4.
  • domain assumption Assumptions 3-4: W has zero diagonal, bounded row and column sums, entries O(1/h), and sup_lambda ||(I-lambda W)^{-1}||_infty bounded.
    Controls spatial dependence and ensures existence and uniqueness of the SAR reduced form; standard but restrictive for dense or near-singular W.
  • domain assumption Assumption 5: rank and eigenvalue conditions on M and L, plus sieve approximation error sup_z r(z) = O_p(p^{-nu}) with nu > 5/2 and p^3/n = o(1).
    The rate p^3/n = o(1) is the load-bearing rate used in Theorems 1-4; the Monte Carlo design with p = floor(n^{1/3}) does not satisfy it.
  • domain assumption Assumption 6: dependence and fourth-moment condition Delta_{Z'Z} + Delta_{Z'U} = O(n), ensuring a weak law for sample second moments.
    Needed for Lemma 1 and all O_p(p/n^{1/2}) bounds on sample moments.
  • domain assumption Assumption 7: uniform boundedness and invertibility of Omega_U and its derivatives under alternatives.
    Used only for consistency and local power proofs, not for the null distribution.
  • standard math Quadratic-form CLT for growing-dimensional projections from Gupta (2018), Remark 3, with a heteroskedastic modification asserted but not proved here.
    This is the main unproved ingredient for Theorem 2; it is a first-author self-citation and no derivation is given in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Testing linearity of spatial interaction functions \`a la Ramsey." pith.science (2026). https://pith.science/paper/4RLDX7W5

@misc{pith2026241214778,
  author       = {Pith},
  title        = {Pith review of: Testing linearity of spatial interaction functions \`a la Ramsey},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RLDX7W5}},
  note         = {Machine review of arXiv:2412.14778}
}
read the original abstract

We propose a computationally straightforward test for the linearity of a spatial interaction function. Such functions arise commonly, either as practitioner imposed specifications or due to optimizing behaviour by agents. Our conditional heteroskedasticity robust test is nonparametric, but based on the Lagrange Multiplier principle and reminiscent of the Ramsey RESET approach. This entails estimation only under the null hypothesis, which yields an easy to estimate linear spatial autoregressive model. Monte Carlo simulations show excellent size control and power. An empirical study with Finnish data illustrates the test's practical usefulness, shedding light on debates on the presence of tax competition among neighbouring municipalities.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Nonparametric Test for Cross-Unit Spillovers

    econ.EM 2026-07 conditional novelty 6.0 of 10

    A new nonparametric test detects whether people are influenced by their peers' traits or outcomes, catching nonlinear spillovers that linear regressions miss.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · cited by 1 Pith paper

  1. [1]

    Ozdaglar, and A

    Acemoglu, D., A. Ozdaglar, and A. Tahbaz-Salehi (2016). Networks, shocks, and systemic risk. In Y. Bramoull \'e , A. Galeotti, and B. Rogers (Eds.), The Oxford Handbook of the Economics of Networks , pp.\ 569--608. Oxford University Press

    work page 2016

  2. [2]

    Allen, F. and D. Gale (2000). Financial contagion. Journal of Political Economy\/ 108 , 1--33

    work page 2000

  3. [3]

    Allers, M. and J. P. Elhorst (2005). Tax mimicking and yardstick competition among local governments in the N etherlands. International Tax and Public Finance\/ 12 , 493--513

    work page 2005

  4. [4]

    Arraiz, I., D. M. Drukker, H. H. Kelejian, and I. R. Prucha (2010). A spatial Cliff-Ord -type model with heteroskedastic innovations: S mall and large sample results . Journal of Regional Science\/ 50 , 592--614

    work page 2010

  5. [5]

    Calv \'o -Armengol, and Y

    Ballester, C., A. Calv \'o -Armengol, and Y. Zenou (2006). Who's who in networks. Wanted: The key player . Econometrica\/ 74 , 1403--1417

    work page 2006

  6. [6]

    Blume, L. E., W. A. Brock, S. N. Durlauf, and Y. M. Ioannides (2011). Identification of Social Interactions . In J. Benhabib, A. Bisin, and M. O. Jackson (Eds.), Handbook of Social Economics , Volume 1, pp.\ 853--964. North-Holland

    work page 2011

  7. [7]

    Bramoull \'e , Y. and R. Kranton (2007). Public goods in networks. Journal of Economic Theory\/ 135 , 478--494

    work page 2007

  8. [8]

    Kranton, and M

    Bramoull \'e , Y., R. Kranton, and M. D'Amours (2014). Strategic interaction and networks. American Economic Review\/ 104 , 898--930

    work page 2014

Show all 45 references
  1. [9]

    Patacchini, and Y

    Calv \'o -Armengol, A., E. Patacchini, and Y. Zenou (2009). Peer effects and social networks in education. The Review of Economic Studies\/ 76 , 1239--1267

    2009

  2. [10]

    Song, and J

    Chen, S., X. Song, and J. Yu (2025). Oracally efficient estimation and specification testing of partially linear additive spatial autoregressive models. Forthcoming: Econometric Reviews\/ , 1--24

    2025

  3. [11]

    Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models, In: Handbook of Econometrics , Volume 6B, Chapter 76, pp.\ 5549--5632. North Holland

    2007

  4. [12]

    Cliff, A. D. and J. K. Ord (1973). Spatial Autocorrelation . London: Pion

    1973

  5. [13]

    Cliff, A. D. and K. J. Ord (1968). The problem of spatial autocorrelation. Joint Discussion Paper, University of Bristol: Department of Economics, Department of Geography, Series A\/ 15 , i--88

    1968

  6. [14]

    de Jong, R. M. and H. J. Bierens (1994). On the limit behavior of a C hi-square type test if the number of conditional moments tested approaches infinity. Econometric Theory\/ 10 , 70--90

    1994

  7. [15]

    de Paula, \'A . (2017). Econometrics of Network Models . In B. Honor \'e , A. Pakes, M. Piazzesi, and L. Samuelson (Eds.), Advances in Economics and Econometrics: Eleventh World Congress , Volume 1 of Econometric Society Monographs , pp.\ 268–--323. Cambridge University Press

    2017

  8. [16]

    Devereux, M. P., B. Lockwood, and M. Redoano (2007). Horizontal and vertical indirect tax competition: T heory and some evidence from the USA . Journal of Public Economics\/ 91 , 451--479

    2007

  9. [17]

    Devereux, M. P., B. Lockwood, and M. Redoano (2008). Do countries compete over corporate tax rates? Journal of Public Economics\/ 92 , 1210--1235

    2008

  10. [18]

    Donald, S. G., G. W. Imbens, and W. K. Newey (2003). Empirical likelihood estimation and consistent tests with conditional moment restrictions . Journal of Econometrics\/ 117 , 55--93

    2003

  11. [19]

    Gupta, A. (2018). Nonparametric specification testing via the trinity of tests . Journal of Econometrics\/ 203 , 169--185

    2018

  12. [20]

    Gupta, A. and X. Qu (2024). Consistent specification testing under spatial dependence. Econometric Theory\/ 40 , 278--319

    2024

  13. [21]

    Gupta, A. and M. H. Seo (2023). Robust inference on infinite and growing dimensional time-series regression. Econometrica\/ 91 , 1333--1361

    2023

  14. [22]

    Hong, Y. and H. White (1995). Consistent specification testing via nonparametric series regression. Econometrica\/ 63 , 1133--1159

    1995

  15. [23]

    Hoshino, T. (2022). Sieve IV estimation of cross-sectional interaction models with nonparametric endogenous effect. Journal of Econometrics\/ 229 , 263--275

    2022

  16. [24]

    Jenish, N. (2016). Spatial semiparametric model with endogenous regressors. Econometric Theory\/ 32 , 714--739

    2016

  17. [25]

    Jenish, N. and I. R. Prucha (2009). Central limit theorems and uniform laws of large numbers for arrays of random fields . Journal of Econometrics\/ 150 , 86--98

    2009

  18. [26]

    Jenish, N. and I. R. Prucha (2012). On spatial processes and asymptotic inference under near-epoch dependence. Journal of Econometrics\/ 170 , 178 -- 190

    2012

  19. [27]

    Kanbur, R. and M. Keen (1993). Jeux sans fronti\`eres: T ax competition and tax coordination when countries differ in size. American Economic Review\/ 83 , 877--892

    1993

  20. [28]

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

    1998

  21. [29]

    Kelejian, H. H. and I. R. Prucha (2010). Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics\/ 157 , 53--67

    2010

  22. [30]

    K \"o nig, M. D., D. Rohner, M. Thoenig, and F. Zilibotti (2017). Networks in conflict: T heory and evidence from the G reat W ar of A frica. Econometrica\/ 85 , 1093--1132

    2017

  23. [31]

    Korolev, I. (2019). A consistent heteroskedasticity robust LM type specification test for semiparametric models. arXiv:1810.07620v3

    2019 arXiv

  24. [32]

    Lee, J., P. C. B. Phillips, and F. Rossi (2024). Heteroskedasticity robust specification testing in spatial autoregression. Econometric Theory, forthcoming\/ , 1–49

    2024

  25. [33]

    Lee, J. and P. M. Robinson (2016). Series estimation under cross-sectional dependence . Journal of Econometrics\/ 190 , 1--17

    2016

  26. [34]

    Lee, L. F. (2002). Consistency and efficiency of least squares estimation for mixed regressive, spatial autoregressive models. Econometric Theory\/ 18 , 252--277

    2002

  27. [35]

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

    2004

  28. [36]

    Lyytik\"ainen, T. (2012). Tax competition among local governments: E vidence from a property tax reform in F inland. Journal of Public Economics\/ 96 , 584--595

    2012

  29. [37]

    Malikov, E. and Y. Sun (2017). Semiparametric estimation and testing of smooth coefficient spatial autoregressive models . Journal of Econometrics\/ 199 , 12--34

    2017

  30. [38]

    Pinkse, J. and M. E. Slade (2010). The future of spatial econometrics. Journal of Regional Science\/ 50 , 103--117

    2010

  31. [39]

    Ramsey, J. B. (1969). Tests for specification errors in classical linear least squares regression analysis. Journal of the Royal Statistical Society, Series B.\/ 31 , 350--371

    1969

  32. [40]

    Redoano, M. (2014). Tax competition among E uropean countries. D oes the EU matter? European Journal of Political Economy\/ 34 , 353--371

    2014

  33. [41]

    Robinson, P. M. (2008). Correlation testing in time series, spatial and cross-sectional data. Journal of Econometrics\/ 147 , 5--16

    2008

  34. [42]

    Su, L. and X. Qu (2017). Specification test for spatial autoregressive models . Journal of Business & Economic Statistics\/ 35 , 572--584

    2017

  35. [43]

    Tincani, M. (2018). Heterogenous peer effects in the classroom. Working Paper

    2018

  36. [44]

    Tullock, G. (1980). Efficient rent seeking. In J. Buchanan, R. Tollison, and G. Tullock (Eds.), Toward a Theory of the Rent-Seeking Society . Texas A & M University Press

    1980

  37. [45]

    Song, and J

    Yang, Z., X. Song, and J. Yu (2024). Model checking in partially linear spatial autoregressive models. Journal of Business & Economic Statistics\/ 42 , 1210--1222

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.