{"id":"1c8f7f3b-a263-49fa-ba97-9f69ae531161","arxiv_id":"2411.16192","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper proposes a matrix factor model with an added regression term for known covariates and derives two-step estimation rates matching the baseline matrix factor model.","lead":"The paper adds observable market factors, such as the Fama-French factors, to matrix factor models used for large stock return panels. It proposes a two-step estimator and shows convergence rates for the regression coefficients and the latent factor loadings.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central rate claim rests on an untested exogeneity assumption (Cov(vec(X_t), vec(F_t))=0 and Cov(vec(X_t), vec(E_t))=0), with no proof of Theorem 2; the real-data Fama-French application is exactly where that assumption is doubtful.","rationale":"I read the paper in good faith. The model is a natural extension of Wang et al. (2019), and the two-step estimator is a sensible construction. Under the stated exogeneity assumptions, the claimed loading rates are plausible and the simulation results are broadly consistent with the theory for strong factors. However, the central claim that adding known factors does not degrade the Wang et al. rates is only as secure as the exogeneity condition. The identification of A by least squares, and hence the construction of \\hat W_t used for the factor loading analysis, explicitly requires Cov(vec(X_t), vec(F_t)) = 0 and Cov(vec(X_t), vec(E_t)) = 0. These are not merely technical conveniences: if violated, the residual process \\hat W_t contains omitted latent-factor variation, and the eigen-analysis in Section 3 estimates a contaminated matrix. The manuscript does not discuss this failure mode. It also does not provide proofs for Theorems 1-5, so the control of the (A-\\hat A)X_t term in \\hat W_t is unverified even under exogeneity. The reader's CONDITIONAL verdict already requires proofs and additional validation; my concern specifies one concrete failure mode (endogeneity of known factors) and one concrete test that would show whether the assumption is actually load-bearing. Since this is consistent with the reader's weakest-assumption identification, I do not see a reason to change the verdict.","tokens_in":13724,"tokens_out":18536,"duration_ms":171941,"concrete_test":"Simulate model (1) with p=q=50, k=r=3, \\delta_1=\\delta_2=0, T=500, and 200 replications. Generate X_t as in Section 5, and generate vec(F_t) = \\rho \\Gamma vec(X_t) + noise for \\rho in {0, 0.3, 0.6, 0.9} with a fixed \\Gamma. Estimate \\hat A by OLS, form \\hat W_t, estimate \\hat Q_i, and record D(\\hat Q_i, Q_i). If D grows with \\rho and no longer scales as p^{\\delta_1} q^{\\delta_2} T^{-1/2}, the exogeneity assumption is load-bearing. A complementary analytical check: bound the contribution of (A-\\hat A)X_t to \\hat M_1 in (13) under exogeneity and verify it is o_p(p^{\\delta_1} q^{\\delta_2} T^{-1/2}).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 estimates A by OLS because Cov(vec(X_t), vec(W_t))=0 is asserted to follow from Cov(vec(X_t), vec(F_t))=0 and Cov(vec(X_t), vec(E_t))=0. If either fails, (1/T)\\sum_t W_t X_t' does not vanish, so \\hat A is biased and \\hat W_t = W_t + (A-\\hat A)X_t contains part of the latent factor term. The subsequent \\hat\\Omega_{w,ij}(h) in (12) and \\hat M_1 in (13) then mix known-factor and latent-factor variation, so Theorem 2's \\|\\hat Q_i - Q_i\\|_2 = O_p(p^{\\delta_1} q^{\\delta_2} T^{-1/2}) need not hold, and the central claim that adding AX_t does not degrade Wang et al. rates loses its support. The paper offers no proof of Theorem 2, and the real-data analysis uses Fama-French factors (constructed from market returns) as X_t for individual stock returns, a setting where correlation with latent factors is a real risk. This is the load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a matrix-variate time series factor model with both known covariates and latent factors, Y_t = A X_t + R F_t C' + E_t. The estimation is done in two steps: ordinary least squares for the coefficient matrix A, followed by the Wang et al. (2019) eigenanalysis method applied to the residuals to estimate the latent loading spaces and the number of factors. The authors claim convergence rates for A, the loadings, and the signal part that coincide with those of Wang et al. (2019), and they support the method with simulations and a real-data application using Fama-French factors as known covariates for daily stock returns.","tokens_in":1,"tokens_out":6609,"duration_ms":116871,"significance":"If the theoretical claims are correct, the paper offers a natural and useful extension of high-dimensional matrix factor models by incorporating observable covariates, and the preserved convergence rates for the latent part would be an interesting result. The paper also provides numerical experiments and a real-data demonstration. However, the central theoretical results are stated without proofs, the key exogeneity assumption is not formally listed or tested, and there are internal inconsistencies in the conditions. These issues currently prevent verification of the paper's main claims, so the significance cannot be assessed from the manuscript as it stands.","major_comments":[{"comment":"Theorems 1-5 are stated without proofs, and no supplementary file or appendix is provided. This is a fundamental gap for a theory paper: the central claim that the convergence rates coincide with Wang et al. (2019) cannot be verified. The authors should provide complete proofs (or a detailed proof sketch with all steps) for these results.","section":"Section 4"},{"comment":"The exogeneity assumption Cov(vec(X_t), vec(F_t)) = 0 and Cov(vec(X_t), vec(E_t)) = 0 is used to justify the least squares estimator of A, but it is not listed among Conditions 1-7 in Section 4 and is not tested in the simulations or real data. If this assumption fails, \\hat A is biased, the residual \\hat W_t contains latent factor variation, and the rate in Theorem 2 is unsupported. In the real data application, using Fama-French factors as X_t for individual stock returns makes correlation with latent factors plausible. Please state this assumption formally and provide diagnostic evidence or a sensitivity analysis.","section":"Section 3"},{"comment":"Condition 4 states that E(X_t X_t') = P_t with (1/T)Σ P_t → P and λ_min(P) ≍ q. Since X_t is m × q with m fixed, X_t X_t' is m × m, so its eigenvalues cannot grow as q → ∞. This internal inconsistency also appears in Theorem 1, whose rate O_p(p^{1/2}T^{-1/2}) does not depend on q. Please clarify the intended normalization of the covariate matrix.","section":"Section 4, Condition 4"},{"comment":"For δ1 = δ2 = 0.5, the relative frequency of correctly estimating the number of factors is 0.0 in nearly all configurations, even at T = 2pq. The theoretical condition p^{δ1} q^{δ2} T^{-1/2} = o(1) is not satisfied in these settings, which the paper does not mention. The authors should either present simulation settings where the theoretical condition holds and the method works, or explicitly discuss this limitation.","section":"Section 5, Table 3"},{"comment":"Condition 3 states that the rank of Σ_f(h) is k* = max(k, r), but Σ_f(h) is the covariance matrix of vec(F_t), which has dimension kr × kr. The condition is thus unclear as written, and the subsequent conditions on individual row/column covariances are not connected to identifiability of the loading spaces. Please rewrite this condition precisely.","section":"Section 4, Condition 3"}],"minor_comments":[{"comment":"There are several grammatical errors, e.g., 'This article considers to model' should be 'This article considers modeling', and 'provides a extensive framework' should be 'provides an extensive framework'.","section":"Abstract and Introduction"},{"comment":"The notation Ω_{zq,ij}(h) is used before Z_t has been properly defined, and the argument of Cov in (7) is difficult to parse. Please define Z_t and clarify the covariance arguments.","section":"Section 3, equations (7)-(8)"},{"comment":"The text says the estimated dimensions are k = 1 and r = 1 but then states 'we use k = 2 and r = 1 here for accessible illustration.' This is confusing and should be explained.","section":"Section 6"},{"comment":"Table 9 is formatted ambiguously: the columns for R2_K, R2_U, R2_T, and the matrix factor model R2 are not clearly separated, and for the (2,1) row the values appear inconsistent (R2_U exceeds R2_T). Please fix the table layout.","section":"Section 6, Table 9"},{"comment":"Some references appear unrelated to the topic (e.g., Psychogios et al. 2012, Sahay 2005, Tian et al. 2015); these should be removed or replaced with relevant literature.","section":"References"},{"comment":"The out-of-sample R2 notation is not fully defined: it is unclear how the test set is constructed, whether parameters are re-estimated on the training set, and how the fitted values are computed. Please clarify.","section":"Section 6, equation (18)"}],"recommendation":"major_revision","confidential_remarks":"The paper has a potentially useful extension and a reasonable two-step estimation idea, but the absence of proofs for all main theorems is a serious problem for a statistics journal. The unresolved exogeneity issue and the inconsistency in Condition 4 further weaken the current submission. I believe the paper could be acceptable after a major revision that adds complete proofs, corrects the conditions, and addresses the simulation limitations, but in its present form it is not publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a clean extension of Wang et al. (2019) to Y_t = A X_t + R F_t C' + E_t, adding an observable covariate term to the matrix factor model. Estimating A by OLS and then running the Wang procedure on residuals is the natural thing to do, and the simulation shows it works when the covariates are generated independently of the latent factors and errors. The claimed loading rates matching Wang et al. is plausible, and the real-data comparison with the plain matrix factor model is a nice touch: the known-factors term explains a meaningful share of out-of-sample variation.\n\nThe soft spots are real. All five theorems in Section 4 are stated without proofs, and no supplementary file is provided. As a referee I could not verify Theorem 2, which is the main rate result. The more serious issue is identification of A. The paper assumes Cov(vec(X_t), vec(F_t)) = 0 and Cov(vec(X_t), vec(E_t)) = 0. That is exactly the kind of exogeneity assumption that matters and is never checked. In the simulations it holds by construction because X_t is generated as an independent VAR. In the empirical application, X_t is the Fama-French factors and Y_t is individual stock returns. Those factors are built from market returns, so they are almost surely correlated with whatever latent factors are left in the residual. If the assumption fails, the least squares estimator of A is biased, the residual contains part of the latent structure, and the claimed rate for the loadings has no support. The paper needs to face this directly, either by proving Theorem 2 under the stated conditions (which would still leave the applicability question) or by discussing what happens under endogeneity and perhaps testing it with the data.\n\nThere is a minor issue too: Table 3 shows that the ratio-based choice of the number of factors fails badly when factors are weak (relative frequencies of 0.0 in several designs). The authors note it but do not explain how practitioners should interpret the estimated dimension in those cases.\n\nBottom line: the model is sensible and the paper is clearly written, but as it stands the theoretical core is unverified and the key assumption is questionable in exactly the application it showcases. It deserves a serious referee, but the referee should demand proofs (or a supplement) and a thorough treatment of the exogeneity issue. I would not cite it in its current form.","headline":"A straightforward extension of matrix factor models to include known covariates, with plausible rates but no proofs and an untested exogeneity assumption that is likely violated in the real-data application.","tokens_in":14487,"tokens_out":2614,"would_cite":false,"duration_ms":24691,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62H25","62M10","62J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding observable covariates to a matrix factor model need not slow the estimation of latent factors.","keywords":["matrix time series","matrix factor model","partially known factors","latent factors","least squares estimation","high-dimensional time series","convergence rates","two-step estimation"],"falsifier":"Simulate from $Y_t = A X_t + R F_t C' + E_t$ with $X_t$ generated partly from $F_t$ (say $X_t = c F_t + \\text{noise}$ with $c>0$), which violates the uncorrelatedness assumption; if the two-step estimator's loading error $\\|\\hat Q_i - Q_i\\|_2$ stops shrinking at the claimed rate as $c$ grows, the assumption is doing the work.","tokens_in":13517,"feed_emoji":"📈","tokens_out":6439,"duration_ms":56246,"temperature":0.7,"pith_summary":"This paper proposes a matrix time series model that combines a regression term for observable covariates with a latent matrix factor model: $Y_t = A X_t + R F_t C' + E_t$. The central aim is to show that adding the known-factor term does not come at a statistical cost: after estimating $A$ by least squares and applying the standard matrix-factor estimation procedure to the residuals, the loading matrices converge at exactly the same rates as in the model without covariates. This matters because observable pricing factors are natural explanatory variables in finance, and a model that uses both known and latent factors is more interpretable without sacrificing accuracy for the latent structure. The paper also gives convergence rates for the coefficient matrix, the estimated signal, and eigenvalue-based dimension estimators, alongside simulations and a stock-return application.","feed_headline":"Known factors don't slow latent factor estimation","feed_subtitle":"A two-step matrix time series estimator matches the rates of the pure latent-factor model.","key_machinery":"The workhorse is the lagged auto-covariance identity for the residual process. With $W_t = R F_t C' + E_t = Q_1 Z_t Q_2' + E_t$ and white-noise errors, the cross-covariance of columns of $W_t$ satisfies $\\Omega_{w,ij}(h) = Q_1 \\Omega_{zq,ij}(h) Q_1'$ for $h \\ge 1$, so the statistic $M_1 = \\sum_{h,i,j} \\Omega_{w,ij}(h) \\Omega_{w,ij}(h)'$ equals $Q_1$ times a sum of factor autocovariances times $Q_1'$. Consequently the eigenspace of $M_1$ coincides with the column space of the row loading matrix $Q_1$, and the analogous construction on transposed residuals gives $Q_2$. The QR decomposition first writes the loadings as semi-orthogonal matrices times nonsingular matrices, making the column spaces identifiable despite the rotational indeterminacy of $R F_t C'$.","core_discovery":"The central claim is that the two-step estimator, least squares for the coefficient matrix $A$ followed by eigen-analysis of a lagged cross-covariance statistic built from the residuals, recovers the latent factor loading spaces with error $O_p(p^{\\delta_1} q^{\\delta_2} T^{-1/2})$, identical to the rate proven for the matrix factor model with no known-factor term. The same rate holds for both the row and column loading spaces, and the eigenvalue ratios used to choose the number of latent factors remain valid. In other words, the regression term $A X_t$ is absorbed in the first step, and the residual process $W_t = R F_t C' + E_t$ behaves like the pure matrix factor model, so the presence of observable covariates does not degrade estimation of the latent factors.","pith_inferences":["Beyond the paper, the orthogonality assumption suggests a natural stress test: if $X_t$ is allowed to correlate with $F_t$ or $E_t$, the residual $W_t$ is no longer a valid matrix factor model, and one would expect the claimed rates to fail; extending the model to endogenous covariates would require instrumental-variable or control-function ideas.","The same two-step structure could be adapted to the two-sided regression form $A X_t B'$ with observable matrix covariates, at the cost of a bilinear rather than linear first step; the paper notes this form is more natural but does not pursue it.","The rate-matching result hints that latent-factor estimation is first-order unaffected by the regression term, so any efficiency loss from estimating $A$ is asymptotically negligible; a finite-sample comparison of standard errors could quantify this.","Because the real-data column loading on the U.S. is near zero after including known factors, one testable implication is that for markets with strong observable pricing factors, latent column structure may be superfluous; this could be checked by fitting the model with and without the known-factor term on other multi-market panels."],"forward_implications":["Known factors can be added to a matrix factor model without slowing down latent-factor estimation: the loading error rate is the same whether or not the term $A X_t$ is present.","The least-squares estimator of the coefficient matrix is consistent when $p/T \\to 0$, with rate $\\|\\hat A - A\\|_F = O_p(p^{1/2} T^{-1/2})$.","The ratio-based eigenvalue estimator for the number of latent factors remains theoretically valid, though weak factors make correct dimension selection harder in finite samples, as the simulations show.","The signal part $A X_t + R F_t C'$ is consistently estimable as $p$ and $q$ grow, so the fitted model can be used for forecasting and out-of-sample evaluation.","In the stock-return application, the known-factor term accounts for a substantial share of explained variation, and the latent column factor has near-zero loading on the U.S. market once known factors are included."],"supporting_citations":[{"why":"Establishes the matrix factor model, the estimation procedure, and the convergence rates that the second step of this paper inherits and extends.","marker":"Wang et al. (2019)"},{"why":"Shows that the column spaces of the factor loadings are uniquely defined despite rotational indeterminacy, which justifies estimating $M(Q_1)$ and $M(Q_2)$.","marker":"Lam et al. (2011)"},{"why":"Provides the ratio-based estimator for the number of latent factors used in this paper.","marker":"Lam and Yao (2012)"},{"why":"Supplies the vector special case of regression with latent factors that the model generalizes, and the distance measure used for comparing loading spaces.","marker":"Chang et al. (2015)"}],"fun_headline_variants":["Known factors don't degrade latent factor recovery","Matrix time series: covariate term no speed bump","Latent factors estimated at same rate despite covariates","Regression term absorbed, latent rates preserved","Two-step matrix estimator matches no-covariate benchmark"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire argument depends on the observable covariates $X_t$ being uncorrelated with the latent factors $F_t$ and with the noise $E_t$, so that ordinary least squares isolates $A$ and leaves residuals that are exactly a matrix factor model plus white noise.","fun_headline_variants_meta":{"raw":{"variants":["Known factors don't degrade latent factor recovery","Matrix time series: covariate term no speed bump","Latent factors estimated at same rate despite covariates","Regression term absorbed, latent rates preserved","Two-step matrix estimator matches no-covariate benchmark"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000185,"raw_usage":{"total_tokens":1240,"prompt_tokens":780,"completion_tokens":460,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":396,"completion_tokens_details":{"reasoning_tokens":390}},"tokens_in":396,"tokens_out":460,"duration_ms":4898,"temperature":1.0,"reasoning_tokens":390,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:25:42.510739+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate from $Y_t = A X_t + R F_t C' + E_t$ with $X_t$ generated partly from $F_t$ (say $X_t = c F_t + \\text{noise}$ with $c>0$), which violates the uncorrelatedness assumption; if the two-step estimator's loading error $\\|\\hat Q_i - Q_i\\|_2$ stops shrinking at the claimed rate as $c$ grows, the assumption is doing the work.","supporting_citations":[{"cited_title":", author Yao, Q","cited_arxiv_id":null,"evidence_quote":"Shows that the column spaces of the factor loadings are uniquely defined despite rotational indeterminacy, which justifies estimating $M(Q_1)$ and $M(Q_2)$."},{"cited_title":", author Yao, Q","cited_arxiv_id":null,"evidence_quote":"Provides the ratio-based estimator for the number of latent factors used in this paper."},{"cited_title":", author Guo, B","cited_arxiv_id":null,"evidence_quote":"Supplies the vector special case of regression with latent factors that the model generalizes, and the distance measure used for comparing loading spaces."}],"review_version":1}