Model Selection in High-Dimensional Linear Regression using Boosting with Multiple Testing

Model Selection in High-Dimensional Linear Regression using Boosting with Multiple Testing

\noindentJEL classification: C12, C13, C22, C52, C55.

\noindentKeywords: High-dimensional regression; model selection; variable selection; boosting; multiple testing; family-wise error control; proxy signals; sparsity; oracle properties; latent factor structures.

Introduction

The problem of model selection in high-dimensional regression settings is well-known and challenging. As large datasets have become commonplace, the issue of how to balance parsimony and fit is increasingly delicate. A widely used approach is penalized regression, which estimates the coefficient vector $\boldsymbol{\beta}$ in a regression of $y_{t}$ on $\boldsymbol{x}_{nt}=\left( x_{1t},x_{2t},...,x_{nt}\right) ^{\prime}$, by solving $\boldsymbol{\widehat{\beta}}=\operatorname*{argmin}_{\beta}[\sum_{t=1}^{T} (y_{t}-\boldsymbol{x}_{nt}^{\prime}\boldsymbol{\beta})^{2} +P_{\boldsymbol{\lambda}}\left( \boldsymbol{\beta}\right) ]$. Here $\boldsymbol{x}_{nt}$ denotes the candidate set of $n$ regressors, $P_{\boldsymbol{\lambda}}\left( \boldsymbol{\beta}\right)$ is a penalty function, and $\boldsymbol{\lambda}$ is a vector of tuning parameters, selected by the user. Penalised regressions have been mainly analysed in the statistical literature, starting with the seminal work of tib1996 and followed by many notable contributions such as those of fan2001, ant2001, efr2004, zou2005, can2007, lv2009, bic2009, zha2010, fan2013, and fan2013b, among others. A prominent alternative class of model selection methods is provided by “greedy” algorithms such as boosting (see, for example, fri2001, buh2006 or kap2010). These procedures construct the model in a stepwise fashion by adding or adjusting regressors sequentially in a direction that improves a chosen in-sample fit criterion.

The aforementioned penalized-regression and greedy methods can be highly effective when the number of candidate regressors is very large. Nonetheless, a key challenge is that they do not guarantee parsimonious specifications; in particular, they may fail to reliably exclude pure noise covariates; namely variables with no role in the data generating process that can still deliver spurious improvements in the penalised objective or in-sample fit. For instance, in the case of penalized regressions, the tuning parameter is often chosen by cross validation with primary emphasis on prediction error (see, e.g., LengEtal2006 and Ch. 7 in HastieEtal2009). This typically leads to relatively mild penalisation and to models that still contain regressors with no role in the data generating process. On the other hand, for greedy procedures, performance hinges on the choice of a stopping rule, and commonly used rules based on information criteria such as AIC and BIC tend to favour overparameterised specifications (see, for example, ChenChen2008). In both cases, these approaches do not, in general, provide a sharp filter that consistently removes variables that are truly irrelevant but exhibit spurious explanatory power in finite samples.

One strand of research that seeks to address the problem of filtering out pure noise covariates has focused on sure screening methods, as developed by fan2008, hua2008, fansamworth and fan2010, among others. These procedures are used as a preliminary device and conduct an initial screening of covariates that are selected via pairwise correlations or univariate regressions. The screening step is designed to ensure that all regressors with nonzero coefficients in the data generating process survive with probability tending to one.

Recent work by chu2018 takes a more direct approach to this filtering issue. Their novel One Covariate, Multiple Testing (OCMT) procedure applies multiple tests to assess the statistical significance of each individual regressor and derives sharp bounds for the associated test statistics under the null of no effect. At each stage, all regressors whose test statistics exceed the corresponding threshold values are retained. This yields a decision rule that acts as a sharp filter for pure noise covariates while retaining those with genuine explanatory power.

Although sure screening and OCMT may substantially improve the treatment of pure noise covariates, they do not address a second, conceptually distinct, difficulty, namely the presence of “proxy signal” covariates. These are variables with zero coefficients in the data generating process that nonetheless appear to have explanatory power in univariate regressions or pairwise correlations, solely because they are correlated with covariates that belong to the underlying model. The selection of such proxy-signal covariates is not a failure of the screening or testing stage. Instead, it reflects the design of these procedures: they are built to retain anything that looks predictive, including proxies.

From an applied perspective, the presence of proxy-signal covariates in the selected model can obscure the interpretation of coefficients and induce multicollinearity, inflating standard errors and unstable estimates. As a result, empirical conclusions can become sensitive to small changes in the model or sample, undermining reliable inference. From a policy perspective, this issue also has non-trivial consequences, since interventions calibrated to proxy variables rather than to the underlying drivers may leave key transmission channels unaffected and render policy measures both hard to interpret and potentially ineffective.

Proxy-signal covariates are pervasive in empirical work. For example, macroeconomic studies of how economic uncertainty affects output growth, employment, or asset returns often use several related uncertainty indices (e.g., newspaper-based, forecast-based, or market-based measures, and different horizons such as one-month-ahead, quarterly, or annual). In monetary economics, analyses of how liquidity affects inflation, credit growth, or investment commonly use multiple monetary aggregates, such as $M0$, $M1$, $M2$, and $M3$, which represent progressively broader money concepts. In both cases, the regressor set contains groups of variables aimed at capturing the same underlying concept. More broadly, many macroeconomic and financial series are partially driven by a small number of latent common factors, so that a large collection of observables is correlated with the same underlying shocks. In such environments, allowing all available proxies to enter the model blurs the distinction between genuine drivers and variables that merely co-move with common shocks, undermining the ability to single out regressors with genuine independent explanatory content.

Taken together, these issues motivate our focus, in what follows, on a sparse data generating process in which the underlying model consists of covariates with nonzero coefficients, that is, the set of true signals. Although this sparsity assumption can be restrictive in applications where many regressors have small but nonzero effects, it remains the dominant modelling paradigm in a variety of disciplines, including economics and finance, where the objective typically extends beyond prediction to the estimation, inference, and interpretation of individual coefficients. Therefore, our aim is to design a selection procedure that mitigates the influence of proxy signal covariates and recovers the true signal set as accurately and parsimoniously as possible.

To this end, the present paper proposes a new model selection procedure, Boosting with Multiple Testing (BMT), which combines stepwise forward variable addition (boosting) with a family-wise multiple testing stopping rule. The researcher observes a large pool of candidate regressors but does not know which are true signals, which are proxies, and which are pure noise. The key distinguishing feature of BMT is that, at each stage, only a single regressor is admitted, even when many candidates appear individually significant. The model is therefore constructed iteratively: rather than retaining all regressors that are found significant in marginal tests, as in OCMT, the procedure updates the specification by selecting the most significant regressor conditional on those already included. At each subsequent stage, the remaining covariates are re-tested conditional on the expanded model, and again at most one additional regressor is admitted. For stopping, we follow OCMT and employ a multiple testing based rule, rather than model selection criteria, so that the resulting procedure acts as a sharp filter for pure noise variables.

A key implication of BMT is that proxy regressors need not remain in the selected model once the relevant signals have entered. Variables whose apparent explanatory power arises solely through correlation with true signals may not survive at later stages and be excluded. This contrasts with OCMT, where such proxy regressors are retained whenever they pass the initial marginal testing step. Although the idea of combining sequential model selection with multiple testing has been alluded to in earlier unpublished work such as fit2015, to the best of our knowledge this is the first paper to provide formal theoretical results and a fully operational algorithm that rigorously justify the approach.

On the theoretical side, the paper establishes three main results. First, under assumptions that closely parallel and generalise those in chu2018, the proposed BMT procedure enjoys oracle-type properties with respect to an approximating model that contains all true signals and at most a finite number of proxy signals. In particular, the probability that any pure noise regressor is selected converges to zero, while the probability that every true signal is retained converges to one.

Second, the sequential updating in BMT makes the multiple-testing filter less greedy than in OCMT. As a consequence, with positive probability, BMT stops before all proxy signal regressors are included. In the special case in which the true signal set contains a single regressor, BMT selects that regressor and no proxy or noise variable with probability approaching one at an exponential rate. It does so without imposing beta-min conditions or other restrictions. This property does not hold for OCMT or penalised regression methods under the same assumptions. More generally, we derive a tractable signal-to-proxy dominance condition under which this no-proxy selection property extends to the case of multiple true signals. Under this condition, BMT asymptotically recovers the full true model, whereas OCMT continues to retain all proxy variables that pass the marginal testing step.\footnote{Sharifvaghefi2025 develops an OCMT-based procedure tailored to settings where correlations among all covariates arise from a few latent common factors. The framework examined there is conceptually related but focuses on this specific dependence structure, whereas the present study investigates more general collinearity patterns within a sequential testing framework.}

Third, it is shown that the post-selection least squares estimator based on the BMT-selected model is asymptotically equivalent to the infeasible oracle estimator that would be obtained if the true model were known in advance.

Further, we provide a theoretical comparison between BMT and penalised regression methods, focusing in particular on the Lasso. We show that the global design conditions required for Lasso consistency, such as the irrepresentable condition, are substantially more restrictive than the local, stage-wise dominance condition underpinning BMT. In a stylised proxy-signal design, this leads to a sharp region of the correlation space in which BMT recovers the true model with probability approaching one, while the Lasso is asymptotically inconsistent. As a result, BMT remains reliable in empirically relevant settings with tightly correlated regressors, where penalised regression methods typically fail to recover the true model.

From a methodological perspective, the paper contributes by exploiting sharp probability inequalities for strongly mixing processes developed in den2022. These results are strong enough to carry the martingale-difference arguments in chu2018 over to heterogeneous strong mixing settings. This, in turn, permits dynamic specifications with autoregressive behaviour that are ruled out under the original OCMT conditions and are typically excluded in the literature on sure screening and penalised regression. The same inequalities also accommodate polynomial (rather than subexponential) probability tails, at the cost of restricting the growth rate of the candidate regressors. This is better suited to applications with heavier-tailed data because it relaxes the standard subexponential-tail assumption, which effectively requires that all moments of regressors and errors exist.

In addition to the theoretical results outlined above, we provide extensive simulation evidence in support of the proposed approach. The Monte Carlo design spans 180 configurations, by varying sample size, sparsity, and the strength and source of multicollinearity induced by latent common and local factors. When the strength of the common factor is high, virtually all regressors in the candidate set behave as proxy signals, so these designs are particularly demanding for procedures that aim to recover the true signals. We compare BMT with OCMT, Lasso and Adaptive Lasso, and evaluate performance using the Matthews Correlation Coefficient (MCC) Matthews1975, and standard selection metrics, including the true positive rate (TPR, share of true signals selected), true discovery rate (TDR, share of selected regressors that are true signals), false positive rate (FPR, share of noise or proxy variables selected) and false discovery rate (FDR, share of selected regressors that are not true signals). MCC can be interpreted as a correlation between the true and selected inclusion indicators, aggregating all four classification outcomes (true positives, true negatives, false positives and false negatives) into a single balanced index that remains informative in sparse, highly unbalanced settings. Across all designs, BMT delivers MCC values close to one, with high TPR and TDR, low FPR and FDR, and model sizes that closely track the true sparsity level. By contrast, OCMT, Lasso and Adaptive Lasso tend to overselect or exhibit much more volatile performance, especially under strong multicollinearity.

Finally, we provide two empirical illustrations based on high-dimensional information from the FRED-QD database of McCrackenNg2021. In the first, aggregate annual emissions for the United States economy are modelled as a function of penalties imposed on emitting facilities together with a broad set of macro-financial indicators. In the second, we study a high-dimensional predictive Phillips curve in which changes in inflation are modeled using the same data environment. Across both applications, BMT achieves the strongest in-sample fit, selects sparse and interpretable specifications, and delivers lower out-of-sample forecast errors than OCMT and Lasso-based alternatives. These results highlight the ability of BMT to balance parsimony, interpretability, and predictive performance in high-dimensional settings.

The remainder of the paper is organised as follows. Section (ref) outlines the BMT algorithm, and Section (ref) develops its asymptotic properties. Section (ref) presents the Monte Carlo design and finite-sample results, while Section (ref) reports the empirical illustrations. Section (ref) concludes. Supplements provide additional theoretical results and proofs, as well as the full set of Monte Carlo outcomes for all designs.

Notations: Generic positive finite constants are denoted by $C_{i}$ for $i=0,1,2,...$ . They can take different values at different instances. If $\left\{ f_{n}\right\} _{n=1}^{\infty}$ is any real sequence and $\left\{g_{n}\right\} _{n=1}^{\infty}$ is a sequence of positive real numbers, then $f_{n}=O(g_{n})$, if there exists a positive finite constant $C_{0}$ such that $\left\vert f_{n}\right\vert /g_{n}\leq C_{0}$ for all $n$. $f_{n}=o(g_{n})$ if $f_{n}/g_{n}\rightarrow0$ as $n\rightarrow\infty$. If $\left\{f_{n}\right\} _{n=1}^{\infty}$ and $\left\{ g_{n}\right\} _{n=1}^{\infty}$ are both positive sequences of real numbers, then $f_{n}=\ominus\left(g_{n}\right) $ if there exists $N_{0}\geq1$ and positive finite constants $C_{0}$ and $C_{1}$, such that $\inf_{n\geq N_{0}}\left( f_{n}/g_{n}\right) \geq C_{0},$ and $\sup_{n\geq N_{0}}\left( f_{n}/g_{n}\right) \leq C_{1}$. $\rightarrow_{p}$ denotes convergence in probability as $n,T\rightarrow\infty$.

Setup

We consider the following data generating process (DGP):

where $\boldsymbol{z}_{t}$ is a known vector of $\zeta$ pre-selected variables, $x_{1t},x_{2t},...,x_{kt}$ are the $k$ unknown true signal variables, $0<\left\vert \beta_{i}\right\vert \leq C<\infty$, for $i=1,2,...,k$, and $u_{t}$ is an error term. It is assumed that $\boldsymbol{z}_{t}$ and $x_{it}$, $i=1,2,...,k,$ are contemporaneously uncorrelated with $u_{t}$ at time $t$. $\boldsymbol{z}_{t}$ may include deterministic components such as a constant, a linear trend, and seasonal dummy variables, as well as stochastic variables, such as lagged values of $y_{t}$ or observed common factors, that are typically chosen on a priori theoretical grounds.

Further suppose that the $k$ signals are contained in the candidate set $\mathcal{S} _{n}=\left\{ x_{it},i=1,2,...,n\right\}$, with $n$ being potentially larger than $T$.\footnote{It is assumed that the set of candidate regressors contains all signal variables. However, the BMT procedure remains applicable even when some signals are missing from the candidate set. In this case, the true model, or any model that nests it, cannot be recovered. Nevertheless, BMT is still able to eliminate regressors that are uncorrelated with the signals.} In addition to the $k$ signals, the candidate set is comprised of noise or distant variables, which have zero correlation with the signals once the effects of $\boldsymbol{z}_{t}$ are filtered out, and a remaining group of variables which, conditional on $\boldsymbol{z}_{t}$, are correlated with the signals. We refer to the latter as pseudo-signals or proxy variables, since they may be mistakenly viewed as signals. Denote the sets of true signals, pseudo-signals, and noise/distant variables by $\mathcal{S}_{s}$, $\mathcal{S}_{p}$, and $\mathcal{S}_{d}$, respectively.

We now present the full details of the Boosting with Multiple Testing (BMT) algorithm, which is a multi-stage procedure. In the first stage, we run the $n$ regressions of $y_{t}$ on $x_{it}$, for $i = 1,\dots,n$. Let $t_{i}$ denote the $t$-ratio from the regression involving $x_{it}$, and write this as $t_{i,(1)}$, where the subscript $(1)$ indexes the stage. We select the variable with the largest absolute $t$-ratio, that is, the $x_{it}$ for which $|t_{i,(1)}|$ is maximised over $i$. Denote the index of this selected variable by $\mathcal{S}_{(1)}^{c}$, and let $\boldsymbol{x}_{(1)}^{o}$ be its $T \times 1$ observation vector. Define $\boldsymbol{X}_{(1)}=\left(\boldsymbol{Z}, \boldsymbol{x}_{(1)}^{o}\right)=\left(\boldsymbol{x} _{(1),1},\dots,\boldsymbol{x}_{(1),T}\right)^{\prime}$, a $T \times (\zeta+1)$ matrix, where $\boldsymbol{Z} = \left(\boldsymbol{z}_{1},\dots,\boldsymbol{z}_{T}\right)'$, and set $\mathcal{S}_{(1)} = \mathcal{S}_{(1)}^{c}$ and $\mathfrak{A}_{(2)} = \{1,\dots,n\} \setminus \mathcal{S}_{(1)}$. In the second stage, we run regressions of $y_{t}$ on $\boldsymbol{x}_{(1),t}$ and $x_{it}$, for $i \in \mathfrak{A}_{(2)}$, and obtain the corresponding $t$-ratios for $x_{it}$, denoted by $t_{i,(2)}$. We construct selection indicators, given by

where $c_{p}(n,\delta)$ is the critical value function, defined by

$\Phi^{-1}\left( .\right) $ is the inverse of standard normal distribution function, $f\left( n,\delta\right) =cn^{\delta}$ for some positive constants $\delta$ and $c$, and $p$ ($0<p<1$) is the nominal size of the individual tests to be set by the investigator. We refer to $\delta$ as the critical value exponent. Among regressors that pass the multiple-testing filter, only the single regressor with the largest absolute t-statistic is selected, and all others are deferred to subsequent stages. We denote the index of the selected unit by $\mathcal{S}_{\left( 2\right) }^{c}$ and the $T\times1$ observation vector of the selected variable by $\boldsymbol{x}_{(2)}^{o}$. We let $\boldsymbol{X}_{(2)}=\left(\boldsymbol{X}_{(1)},\boldsymbol{x}_{(2)}^{o} \right)=\left(\boldsymbol{x}_{(2),1} ,...,\boldsymbol{x}_{(2),T}\right)^{\prime}$, a $T \times \left(\zeta+2 \right)$ matrix, and $\mathcal{S}_{\left( 2\right) }=\mathcal{S}_{\left( 1\right) }\cup\mathcal{S}_{\left( 2\right) }^{c}$, $\mathfrak{A}_{\left( 3\right) }=\left\{ 1,...,n\right\} \setminus\mathcal{S}_{\left( 2\right) }$. We proceed to subsequent stages $j = 3,4,\ldots$ in the same manner, defining $\boldsymbol{X}_{(j)} = \left(\boldsymbol{X}_{(j-1)}, \boldsymbol{x}_{(j)}^{o}\right)$, $\mathcal{S}_{(j)} = \mathcal{S}_{(j-1)} \cup \mathcal{S}_{(j)}^{c}$ and $\mathfrak{A}_{(j+1)} = \{1,\dots,n\} \setminus \mathcal{S}_{(j)}$. At each stage $j$, the value of $n$ in $c_p(n,\delta)$ is set equal to the cardinality of $\mathfrak{A}_{(j)}$.

In summary, at stage $j$ we condition on $\boldsymbol{z}_{t}$ and all regressors selected up to stage $j-1$, and regress $y_{t}$ on each remaining variable in the candidate set in turn. Let $\widehat{J}$ denote the last (random) stage at which a regressor is admitted. The final set of selected regressors, including the vector of pre-selected regressors, is denoted as $\boldsymbol{X}_{(\widehat{J})}$ and is of dimension $T \times (\zeta + \widehat{J})$.

It is important to emphasize that the aforedescribed stages of BMT differ fundamentally from those of OCMT. BMT is a stepwise selection procedure in which each stage adds exactly one variable to the model.\footnote{The procedure can be generalised to select a prespecified, possibly data-dependent, number of variables per stage; this extension is examined in our Monte Carlo study.} By contrast, OCMT is much more aggressive, selecting all signals and pseudo-signals that pass the multiple-testing threshold in a single stage. Consequently, additional stages beyond the first are required for OCMT only in the presence of “hidden” signals.\footnote{A hidden signal is a pathological case in which a regressor belongs to the DGP for $y_t$ (its true coefficient is non-zero), but owing to its correlation with other true signals, its net effect on $y_t$ is exactly offset by a linear combination of the effects of those true signals; see chu2018 for more details. In what follows, this case is deliberately excluded: the parameter-equality restrictions required to generate hidden signals appear highly unlikely in practice, whereas treating them explicitly would impose substantial additional notational and technical burden. Conceptually, however, extending the analysis to cover hidden signals is straightforward. Importantly, because BMT proceeds via sequential conditional testing, a hidden signal would be revealed at later stages once the true signals that offset its marginal effect are controlled for, so that such cases are naturally accommodated by the multi-stage structure of the procedure.}

A key implication of the sequential conditioning employed by BMT is that proxy regressors need not survive the selection procedure once the relevant signals have entered the model. By construction, the inclusion of a true signal alters the conditional testing environment faced by the remaining candidates, so that regressors whose marginal significance is driven solely by correlation with that signal may no longer exhibit explanatory power once conditioning is imposed. To illustrate this mechanism, consider $y_t=\beta x_{1t}+u_t$ and a proxy regressor $x_{2t}=\rho x_{1t}+v_t$, with $v_t$ uncorrelated with $(x_{1t},u_t)$. In the marginal regression of $y_t$ on $x_{2t}$, $x_{2t}$ has non-zero explanatory power whenever $\rho\neq0$. By contrast, once $x_{1t}$ is included in the model, the population partial coefficient on $x_{2t}$ is zero, so $x_{2t}$ need not survive subsequent testing stages. Thus, proxies that appear significant marginally can be filtered out once the relevant signal is controlled for. This mechanism underlies the early-stopping property of BMT formalised in Remark 2 below.

Theoretical Results

Assumptions

We employ the following set of assumptions.

The above assumptions closely resemble those used for OCMT in chu2018. In particular, they (i) impose a well-behaved sample second moment matrix for the non-noise regressors (Assumption (ref)); (ii) rule out excessively heavy tails for the regressors in the candidate set and the error term via sub exponential tail bounds (Assumption (ref)); (iii) control the covariance structure of the regressors selected by requiring that the eigenvalues of $E(\mathbf{S}'\mathbf{S}/T)$ are uniformly bounded away from zero and, apart from a finite number, remain bounded as $l_{T}$ grows (Assumption (ref)); (iv) impose a sparse and uniformly bounded projection of any candidate regressor on the set of already selected variables and pre-selected controls (Assumption (ref)); and (v) restrict the number and minimal strength of signals so that their associated $t$-statistics diverge and non-negligible effects can still be detected consistently as $T$ grows (Assumption (ref)). The main departures are in the treatment of dependence and tail behaviour. Both regressors and errors are allowed to be heterogeneous strongly mixing processes rather than martingale difference sequences (Assumptions (ref)-\textbf{(ref)}), which permits dynamic specifications with autoregressive behaviour that are ruled out under the original OCMT conditions. Moreover, instead of the nonsharp exponential inequalities discussed in Appendix C of chu2018, we rely on the sharp probability inequalities of den2022, which are strong enough to carry over OCMT type results to the mixing case and also allow for polynomial rather than exponential tails in (ref)-(ref). Allowing for polynomial tails relaxes the usual subexponential tail requirement, which in particular implies the existence of all moments and can be restrictive in applications with heavier tailed data, at the cost of restricting the growth rate of the number of regressors.\footnote{We do not fully develop these polynomial-tail extensions for simplicity, but they can be shown straightforwardly by using the fat-tail part of Lemma (ref).}

Next, we present the theoretical properties of BMT. In Subsection (ref) we show that, despite being far less aggressive than OCMT in admitting proxy signals, BMT still enjoys the key OCMT properties established in chu2018 in a more general framework. In particular, working under mixing rather than martingale-difference assumptions, we establish oracle-type properties with respect to an approximating model that contains the true signals together with a small number of additional regressors. Subsection (ref) in turn derives new results on recovery of the true model and on no proxy selection, identifying settings and conditions under which BMT dominates OCMT in the sense that the probability of selecting pseudo signals vanishes at an exponential rate. It further shows that the post-selection least squares estimator based on the BMT-selected model is asymptotically equivalent to the infeasible oracle estimator that knows the true model. Subsection (ref) contrasts the conditions required for consistency of BMT with those imposed in the Lasso literature, showing that BMT remains reliable under strong correlation between true signals and inactive regressors, where the Lasso may fail to recover the true model.

Results relative to the approximating model

We begin by deriving standard rates. Let $\widehat{\mathcal{L}}_{i}$ denote the final selection indicator for regressor $i$, equal to one if the variable is selected and zero otherwise, and let $\mathcal{A}_{0}$ denote the event that the approximating model is recovered:

namely the event that all $k$ true signals are selected and no distant/noise variables are included. In other words, the procedure recovers the entire set $\mathcal{S}_{s}$ without any false positives from $\mathcal{S}_{d}$, although it may still select some pseudo/proxy signals in $\mathcal{S}_{p}$.

Further, define the numbers of true positives and false positives (relative to the approximating model), by

and let $|\mathcal{S}_{s}|=k$ and $|\mathcal{S}_{d}|=n-k-k^{\ast}$. We then define the True Positive Rate, as well as False Positive Rate and False Discovery Rate relative to $\mathcal{A}_{0}$, as

Selections from $\mathcal{S}_{p}$ do not enter $FPR_{n,T}^{(0)}$ and $FDR_{n,T}^{(0)}$, so that, unlike $TPR_{n,T}$, these measures evaluate performance with respect to the approximating model $\mathcal{A}_{0}$.\footnote{One can instead define $FPR_{n,T}^{(0)}$ relative to the true model, so that false positives include both proxy signals and pure noise. In that case, the contribution of a finite number of selected proxy signals is asymptotically negligible relative to $|\mathcal{S}_{p}\cup\mathcal{S}_{d}|$, which grows with $n$. By contrast, this argument does not straightforwardly extend to $FDR_{n,T}^{(0)}$, since both the numerator and denominator involve the finite numbers of selected signals and proxies, and the ratio need not converge to zero unless the number of selected proxy signals is asymptotically negligible. For clarity we retain the approximating model perspective here, and strengthen the analysis for the true model in Section (ref).}

Theorem (ref) shows that, under the stated conditions, BMT behaves in a standard “oracle-like’’ fashion with respect to the approximating model $\mathcal{A}_{0}$. Part (a) establishes that the probability that the procedure continues beyond the number of stages required to add all $k$ true signals and all $k^{\ast}$ pseudo signals becomes negligible. In this sense, overfitting in terms of selecting additional noise variables is rare. Part (b) states that the approximating model itself is selected with probability tending to one; that is, with high probability all $k$ true signals are included and no noise variables from $\mathcal{S}_{d}$ are selected, even though some pseudo signals in $\mathcal{S}_{p}$ may still be admitted. Part (c) translates these results into familiar performance measures $TPR_{n,T}$, $FPR_{n,T}^{(0)}$ and $FDR_{n,T}^{(0)}$, defined relative to $\mathcal{A}_{0}$: the true positive rate converges in probability to one, while the false positive rate and false discovery rate converge in probability to zero (up to the vanishing contribution $k^{\ast}/(n-k)$ of pseudo signals), provided that $\delta$ exceeds the threshold $\max\left\{1,2-\kappa_{1}/3\right\}$.

Theorem (ref) complements Theorem (ref) by showing that the selection properties of BMT translate into standard rates for both fit and parameter estimation in the final regression. The quantity $F_{\widetilde{u}}$ is the average squared residual from the selected model, so the benchmark is the innovation variance $\sigma^{2}$ that would be obtained by an oracle that knew exactly which regressors have nonzero coefficients. Equation (ref) states that $F_{\widetilde{u}}$ converges in probability to $\sigma^{2}$ at rate $n^{-\kappa_{1}/2}$, which corresponds to the usual $T^{-1/2}$ rate since $T$ behaves like $n^{\kappa_{1}}$. Thus, asymptotically the BMT selected model attains the same residual variance as the oracle model. The second relation (ref) concerns $F_{\boldsymbol{\widetilde{\beta}}}$, the Euclidean norm of the error in the full $n$ dimensional coefficient vector, and shows that this norm converges to zero at rate $n^{-\kappa_{1}}$ (that is, at the standard $T^{-1}$ rate in squared norm). In particular, the coefficients on the true signals are estimated consistently, and the inclusion of a finite number of pseudo signals does not prevent the overall estimation error from vanishing. Together with Theorem (ref), this establishes that BMT is consistent for the approximating model and achieves standard parametric rates for both residual variance and coefficient estimation.

Recovery of the true model and no proxy selection

The results in this subsection move beyond recovery of the approximating model and address exact model selection. Theorems 3 and 4 establish conditions under which BMT avoids proxy selection and recovers the true signal set, while Theorem 5 shows that the post-BMT least squares estimator is asymptotically equivalent to the infeasible oracle estimator that knows the true model. Recall that the approximating model $\mathcal{A}_{0}$ requires that all $k$ true signals are selected and that no noise variables from $\mathcal{S}_{d}$ are included, while still allowing for the possibility that some pseudo signals in $\mathcal{S}_{p}$ are also selected. By contrast, the true model requires that all true signals are selected and that no other regressors, neither noise variables nor proxy signals, enter the final specification. Formally, we define

Thus $\mathcal{A}_{1}$ is the event that the procedure selects all $k$ true signals in $\mathcal{S}_{s}$ and excludes all variables in $\mathcal{S}_{d}$ and $\mathcal{S}_{p}$.

Further, define the numbers of true and false positives, relative to the true model $\mathcal{A}_{1}$, by

so that $|\mathcal{S}_{s}|=k$ and $|\mathcal{S}_{p}\cup\mathcal{S}_{d}|=n-k$. The True Positive Rate, $TPR_{n,T}$, is defined as before by $TPR_{n,T}=TP_{n,T}/k$. We now define the False Positive Rate, False Discovery Rate and True Discovery Rate relative to $\mathcal{A}_{1}$ as

We first show that in the simple case of a single true signal, under the same set of assumptions (Assumptions (ref)-(ref)) as in Subsection (ref), BMT recovers this true model with probability approaching one and does not select any proxy variables.

In other words, when there is a single true signal, BMT selects that signal and excludes all proxy and noise variables with probability tending to one, which substantially strengthens the selection properties relative to OCMT, without any additional assumptions.

Finally, we extend the above results by showing that the no-proxy property in Theorem (ref) can carry over to cases with multiple true signals. To this end, consider the local configuration of a given true signal and its proxies. Write

where $\boldsymbol{y}=(y_{1},\ldots,y_{T})^{\prime}$ is the $T\times 1$ vector of observations on the dependent variable, $\boldsymbol{x}_{1}$ is the true signal under consideration, $\boldsymbol{X}_{2}$ collects the remaining true signals and $\boldsymbol{X}_{3}$ collects the proxy regressors, so that

and

The condition in Eq. (ref) can be interpreted as a signal-to-proxy dominance requirement. The left-hand side measures the contribution of the true signal $x_{1}$ to the fit of the model, after accounting for its correlation with both the other true signals and a given proxy $x_{i}$. The right-hand side provides an upper bound on the contribution that can be attributed to the proxy through its indirect association with the outcome via the other true signals. When Eq. (ref) holds, the true signal delivers a strictly larger marginal gain than any proxy at the relevant stages of the BMT procedure. As a result, BMT selects true signals ahead of proxies and, asymptotically, recovers the true model $\mathcal{A}_{1}$. By contrast, OCMT would continue to retain all proxy variables that pass the marginal testing step. Note that when $\boldsymbol{\beta}_{2}=\boldsymbol{0}$, corresponding to the presence of a single true signal, the cross-signal terms vanish and the right-hand side of (ref) equals zero. In this case, the condition is automatically satisfied under nondegenerate designs.

The signal-to-proxy dominance condition in Eq. (ref) is particularly relevant in empirical settings where regressors exhibit strong common dependence. In many macroeconomic and financial datasets, large collections of variables are driven by a small number of latent factors, so that regressors are highly correlated through common shocks. Such dependence patterns are fully accommodated by condition (ref) on the covariance structure in Theorem (ref).

To further clarify the content of condition (ref), it is useful to consider two canonical and empirically relevant dependence structures.

Section (ref) provides a stylised example in which a proxy is correlated with multiple true signals, illustrating how the signal-to-proxy dominance condition operates under factor-driven dependence and highlighting the contrast with the global design conditions required for Lasso consistency. The implications of the factor-driven dependence case are further explored in our Monte Carlo analysis. We show that, even in the presence of unobserved common factors, BMT selects the true regressors, and only the true regressors, with high probability.

Theorem (ref) formalises the oracle property of post-selection OLS based on BMT. Theorems (ref) and (ref) establish that, under the stated assumptions, BMT recovers the true model $\mathcal{A}_{1}$ with probability tending to one and excludes all proxy and noise regressors. Although the true signal set is unknown in finite samples, asymptotically the selected model coincides with the data generating process with probability approaching one. Conditional on correct model selection, the post-selection estimator reduces to the usual OLS estimator applied to the true model, so standard arguments yield consistency and asymptotic normality for the $k$ coefficients in the true signal set, with asymptotic covariance matrix $\sigma^{2}\mathbf{\Sigma}_{\mathcal{S}_{s}}^{-1}$. Since the probability of incorrect model selection vanishes asymptotically, the post-selection estimator attains the same first-order asymptotic distribution as the infeasible oracle estimator that knows the true signal set in advance. This yields a pointwise oracle result for each data generating process satisfying Assumptions (ref)-(ref) and (ref)-(ref).\footnote{This contrasts with the generic post-model selection setting analysed by LeebPoetscher2008, where post-selection estimators (including post-Lasso OLS) can exhibit highly non-normal behaviour and no uniformly valid normal approximation exists. Our result is pointwise rather than uniform, but under our assumptions the BMT-selected model coincides with the true model with probability tending to one, so standard regression asymptotics apply on the true signal set.}

For indices $i\notin\mathcal{S}_{s}$, the corresponding components $\widetilde\beta_i$ converge in probability to zero. These coefficients are retained in $\boldsymbol{\widetilde\beta}_n$ purely for notational convenience; asymptotically, the associated regressors are excluded from the model and no non-degenerate limiting distribution is defined for them.

Assumption (ref) imposes a lower bound on the magnitude of nonzero coefficients on the $\sqrt{T}$ scale, thereby ruling out sequences of local alternatives under which model selection remains asymptotically unstable. Moreover, the parsimonious nature of BMT, together with the design conditions in Assumptions (ref) and (ref), ensures that the number of selected regressors remains finite and that the associated second-moment matrices are well behaved, so that the usual OLS variance estimator (and its robust analogue) is consistent for the asymptotic variance of the post-selection estimator on the true signal set.

Comparison with design conditions for Lasso

It is useful to compare the dominance condition in Eq. (ref) with the design assumptions typically imposed for variable-selection consistency of penalised regression methods such as the Lasso. In that literature, two types of high-level conditions are standard. First, one assumes a design (or correlation) condition, such as the irrepresentable or mutual-incoherence condition; see, for example, ZhaoYu2006,MeinshausenBuhlmann2006. Writing the population covariance matrix of the regressors in block form as

where $\mathbf{\Sigma}_{11}$ corresponds to the true signals and $\mathbf{\Sigma}_{22}$ to the “inactive regressors” (i.e., regressors with zero coefficients in the DGP), a typical version of this condition requires

for some $\eta>0$, with $\boldsymbol{\beta}_{1}$ denoting the coefficient vector on the true signals. This is a global restriction: it must hold uniformly over the inactive regressors and over the sign pattern of the nonzero coefficients, and in effect rules out very strong correlations between active and inactive variables. Second, one usually imposes a “beta-min” condition, namely that the nonzero coefficients are not too small relative to the penalty level and the noise, for instance

so that active coefficients remain detectable after shrinkage; see, for example, Wainwright2009.

By contrast, the dominance condition in Eq. (ref) that underpins Theorem (ref) is local and tailored to the proxy-signal setting. It neither imposes a global incoherence bound between $\mathcal{S}_{s}$ and $\mathcal{S}_{p}\cup\mathcal{S}_{d}$ nor relies on a Lasso-type beta-min condition tied to shrinkage. Accordingly, it does not require correlations between true signals and proxy regressors to be uniformly small. Instead, (ref) is formulated as a pairwise, stage-wise comparison between a given true signal $x_{1}$ and a given proxy $x_{i}$, conditional on the remaining true signals $\mathbf{X}_{2}$: the left-hand side measures the contribution of $x_{1}$ to model fit after partialling out its correlation with $\mathbf{X}_{2}$ and $x_{i}$, while the right-hand side provides an upper bound on the contribution that can be attributed to the proxy through $\mathbf{X}_{2}$. This restriction is local in coefficient space, as it depends on the realised values of $(\beta_{1},\boldsymbol{\beta}_{2})$ rather than on worst-case sign patterns, and local in variable space, as it compares each true signal with each proxy separately rather than imposing a uniform bound over all inactive regressors. In this sense, Eq. (ref) is less stringent than classical Lasso design conditions and remains compatible with empirically realistic settings featuring tightly correlated clusters of regressors, while still guaranteeing the no-proxy selection property of BMT and asymptotic recovery of the true model $\mathcal{A}_{1}$.

A stylised comparison in a three-regressor design

To illustrate further how the global design restrictions required by the Lasso differ from the local, stage-wise dominance condition underlying BMT, we consider the following stylised three-regressor design. The example isolates a simple but empirically relevant “proxy problem”, in which an inactive regressor is correlated with multiple true signals and can therefore mimic their combined effect.

Let the data be generated according to

where $x_{1t}$ and $x_{2t}$ are the true signals and let $x_{3t}$ be an inactive proxy variable with zero coefficient. The coefficients satisfy $\beta_1 = 1$ and $\beta_2 = \alpha$, with $0<\alpha<1$, so that $x_{1t}$ is the dominant signal while $x_{2t}$ is comparatively weak. The proxy $x_{3t}$ does not enter the DGP directly but is correlated with both true signals, such that \[ \mathrm{Corr}(x_{1t},x_{3t})=\mathrm{Corr}(x_{2t},x_{3t})=\rho>0, \] while the true signals themselves are orthogonal, $\mathrm{Corr}(x_{1t},x_{2t})=0$.

This configuration captures a common identification problem in macroeconomic and financial applications, where multiple observables load on the same underlying drivers (e.g., StockWatson2002). The proxy $x_{3t}$ can be interpreted as a variable that aggregates information from both structural signals and may therefore appear spuriously important in marginal regressions.

We now characterise conditions under which Lasso and BMT recover the true model $\mathcal{S}=\{1,2\}$ as the sample size grows. Let $\widehat{\mathcal{S}}_{\text{Lasso}}$ and $\widehat{\mathcal{S}}_{\text{BMT}}$ denote the sets selected by the two procedures.

The wedge $\mathcal{W}$ highlights a fundamental difference between the two procedures. Lasso fails when the aggregate correlation between the proxy and the active set becomes too large: since $x_{3t}$ is correlated with both signals, its marginal association with $y_{t}$ scales with $2\rho$, and once $2\rho>1$ the proxy dominates individual true regressors. By contrast, BMT relies on a stage-wise ordering principle. At the initial stage, the proxy can only displace the dominant signal if its marginal correlation exceeds the contribution of the strongest regressor, which occurs when $\rho(1+\alpha)>1$.

Importantly, the condition $\rho<1/(1+\alpha)$ governs only the avoidance of proxy selection at early stages. The subsequent detectability of the weak signal $x_{2t}$ depends separately on the magnitude of $\alpha$ and standard power considerations, rather than on the correlation structure itself. For example, when $\alpha=0.2$, Lasso fails for any $\rho>0.5$, whereas BMT correctly recovers the structural model for correlations up to $\rho\approx0.83$. Appendix (ref) generalises this result to multiple true signals.

Simulation Study

This section evaluates the finite sample performance of BMT using a range of diagnostic measures: the Matthews Correlation Coefficient (MCC) Matthews1975,ChiccoJurman2020, the F1 score, the True Discovery Rate (TDR), the False Discovery Rate (FDR), the True Positive Rate (TPR) and the False Positive Rate (FPR).\footnote{We have also experimented with a variant of BMT that allows for selecting more than one regressor per stage. While this extension can improve finite-sample performance in very small samples, we found no systematic gains once the time dimension exceeds $T=100$. These results are available upon request. Given the additional tuning choices and algorithmic complexity involved, we therefore focus on the BMT procedure as described earlier in the paper.}

TDR, often referred to as “precision”, measures the proportion of selected variables that are true signals, while TPR, or “recall”, captures the proportion of true signals that are correctly identified. FDR quantifies the share of selected variables that are in fact non-signals, and FPR reflects the share of non-signals that are erroneously selected. The F1 score, defined as the harmonic mean of TDR and TPR, provides a single summary of the precision-recall trade off and is particularly informative when the objective is to detect as many true signals as possible while keeping false discoveries under control. The Matthews Correlation Coefficient (MCC) is a comprehensive measure of binary classification performance, combining true positives, true negatives, false positives and false negatives into a single balanced index that remains informative in sparse, highly unbalanced settings, such as ours, where true signals are few relative to the total number of candidate variables. It ranges from $-1$ (complete disagreement) to $+1$ (perfect classification), with $0$ corresponding to performance no better than random guessing. Among the above metrics, MCC is arguably the most comprehensive, as it jointly reflects both Type I and Type II errors and provides a robust overall assessment of variable selection performance. By contrast, component measures such as TPR or TDR considered in isolation can give a partial, and in imbalanced designs even misleading, picture. For example, a method may achieve a high TPR simply by selecting many variables, at the cost of an elevated FDR. For this reason, while we report TPR, TDR, FPR and FDR as useful diagnostics, we rely primarily on MCC to summarise performance in high-dimensional settings with sparse and correlated signals. Further details on the computation of these metrics are provided in the Appendix.

In addition to these classification based criteria, we report the model size, defined as the average number of variables selected by each method, which serves as an estimate of the true number of signals. We also consider the root mean squared error (RMSE) of the estimated coefficients for the selected variables, as a measure of estimation accuracy, and the root mean squared forecast error (RMSFE), which captures out-of-sample predictive accuracy.

As a benchmark, we also assess the performance of three additional methods: OCMT (Chudik et al., 2018), Lasso, and Adaptive Lasso.\footnote{The latter two methods are implemented using MATLAB's built-in functionality for regularized variable selection, specifically the lasso function.}

Design

The true DGP is given by

where $u_{t}\sim i.i.d.N(0,1)$ $\forall t$. The autoregressive parameter $\alpha$ alternates between $0.4$ and $0.8$.

We consider the same number of pseudo-signals, $k$, denoted $x_{k+1,t}, x_{k+2,t} \dots, x_{2k,t}$. Both sets of variables are generated as

Noise variables are generated as

The terms $\varepsilon_{it}$ represent idiosyncratic variations across variables, while $g_{t}$ and $f_{t}$ capture “local” and common factors, respectively. The local factor $g_{t}$ affects only the true signals and pseudo signals, whereas the common factor $f_{t}$ influences all variables.

$g_{t}$, $f_{t}$ and $\varepsilon_{it}$ are drawn as AR(1) processes with mean zero and variance one. In particular, we specify

where all errors, $\upsilon_{\varepsilon,it}$, $\upsilon_{f,t}$ and $\upsilon_{g,t}$, are $i.i.d.N(0,1)$.

The inclusion of local and common factors serves two main purposes. First, it introduces multicollinearity among regressors, as is typical in real-world high-dimensional datasets where correlated signals and pseudo-signals arise due to shared latent structures. Second, it captures the common variation often observed in big data settings, where many covariates are driven by latent common factors and shared underlying shocks. Examples of such drivers include macroeconomic trends, technological innovations, regulatory changes, demographic shifts, geopolitical developments, and sector-wide shocks such as fluctuations in commodity prices, global supply chain disruptions, or monetary policy shifts. These factors can lead to strong co-movement among otherwise distinct variables, thereby shaping the correlation structure of the candidate set of regressors. Finally, modeling $g_t$, $f_t$, and $\varepsilon_{it}$ as stationary AR(1) processes induces serial correlation over time and persistence typical of economic time series, so the design features both cross-sectional and temporal dependence.

The conditional (partial) goodness-of-fit coefficient of the model, conditional on $y_{t-1}$, is defined as

where $D = \left( B + \gamma b_{s}^{2} \right) \left( 1 + \gamma\right)$, $B = \|\boldsymbol{\beta}\|^{2}$, $b_{s} = \boldsymbol{\beta}^{\prime} \boldsymbol{\iota}$ with $\boldsymbol{\iota}=\left( 1,1,\dots,1\right) ^{\prime}$, and $\gamma= v_{f}^{2} + v_{g}^{2}$. Intuitively, $\gamma$ summarises how much of the regressors' variance is explained by common factors rather than idiosyncratic noise: $\nu_{f}^{2}$ captures the strength of a global common factor, whereas $\nu_{g}^{2}$ captures the strength of the local factor. Solving for $\varsigma$ yields

It follows that $\varsigma$ adjusts endogenously in response to changes in $\boldsymbol{\beta}$, and $\gamma$, so as to maintain a constant signal-to-noise ratio.

The Variance Inflation Factor (VIF) for any true signal $x_{it}$ is defined as $VIF_{i}=1 / \left( 1-R_{i}^{2}\right) $, where $R_{i}$ is the coefficient of determination from the population regression of $x_{it}$ on all remaining regressors in the candidate set (excluding $x_{it}$). Since

we have $VIF_{i}=VIF=1+k \gamma$ for all $i\leq 2k$, where $\gamma=\nu_{g}^{2}+\nu_{f}^{2}$. Solving for $\gamma$ yields $\gamma=\left( VIF-1\right) /k$. We specify a proportion $\pi$ that determines the share of $\gamma$ attributed to $\nu_{f}^{2}$, such that $\nu_{f}^{2}=\pi\left( VIF-1\right) /k$ and $\nu_{g}^{2}=\left( 1-\pi\right) \left( VIF-1\right) /k$. Solving for $\nu_{f}$ and $\nu_{g}$ yields

This parametrisation allows control over how total multicollinearity, as measured by VIF, is allocated between the local and global factors, $g_{t}$ and $f_{t}$. When $\pi= \frac{3}{4} $, 75% of total multicollinearity is attributable to the global factor $f_{t} $, reflecting pervasive cross-sectional dependence driven by a latent global factor, while the remaining 25% arises from the local factor $g_{t} $. Conversely, when $\pi= \frac{1}{4} $, local dependence dominates, accounting for 75% of total multicollinearity, so that noise variables are weakly correlated with signals.

We set $R^{2} = 0.7$, $\varrho = 0.6$, and $\delta=1$, which corresponds to a relatively mild choice of the critical value exponent.\footnote{This choice allows us to assess the performance of BMT in designs featuring both local and global latent factors, which generate proxy signals through different correlation channels. Larger values of $\delta$ impose more conservative thresholds and further reduce the probability of selecting proxies, but at the cost of admitting fewer variables at each stage. Focusing on a mild value of $\delta$ therefore provides an assessment of BMT’s ability to control proxy selection under weak penalisation.} Without loss of generality, we normalise the signal coefficients by setting $\beta_{i} = 1 $ for all $i = 1, \dots, k $. We explore two levels of model complexity, with $k \in\{1,4 \}$ and assess the role of multicollinearity by varying the Variance Inflation Factor (VIF), specified as $\text{VIF} \in\{1, 2, 4\} $. When $\text{VIF} = 1 $, the regressors are mutually uncorrelated, implying that the strengths of the local and global latent factors are set to zero, $\nu_{g} = \nu_{f} = 0 $. In this case, true signals are uncorrelated with pseudo signals. In contrast, higher values of VIF represent stronger multicollinearity; for instance, when $\text{VIF} = 4 $, the standard errors of the estimated coefficients for the true and pseudo signals are, on average, four times larger than in the case of no multicollinearity.

Finally, we set $n=T-2k$ and we evaluate performance across all combinations of $T \in\{100, 200, 300\}$ and $n$. This setup allows us to assess how performance varies with dimensionality, without confounding effects from changes in signal strength or sparsity. For each value of $k$, the analysis covers a total of 90 configurations.

Results

We focus primarily on three criteria: the Matthews Correlation Coefficient (MCC), the RMSE of the estimated coefficients corresponding to the selected models, and the model size (number of selected variables). Tables (ref)-(ref) report summary statistics for these performance measures across 180 designs, split by sparsity level $k=1$ and $k=4$.

The MCC results in Tables (ref)-(ref) show that BMT dominates the alternatives in terms of variable selection accuracy. When $k=1$, BMT attains a median MCC of $0.973$ with a narrow inter-quartile range (IQR) of $0.018$, and $MCC>0.8$ in all 90 designs (the maximum possible value is 1). It ranks essentially first on average (mean rank $1.022$) and is the best method in $97.78\%$ of the designs (and in all designs once cases with VIF$=1$ are excluded). For $k=4$, performance remains very strong: the median MCC is $0.985$ with an IQR of $0.041$, $MCC>0.8$ in $95.60\%$ of designs, and BMT attains the top rank in $94.44\%$ of cases (and again in all cases once VIF$=1$ designs are excluded).

By contrast, OCMT, Lasso and Adaptive Lasso exhibit substantially weaker and more volatile selection performance. With $k=1$, OCMT has a median MCC of $0.280$ and a very wide IQR of $0.577$, and achieves $MCC>0.8$ in only $20\%$ of designs. Lasso and Adaptive Lasso have median MCC values between $0.457$ and $0.515$, with relatively low dispersion but no instances in which $MCC>0.8$ or the methods rank first. When $k=4$, OCMT improves (median MCC $0.669$) but remains highly variable, while Lasso and Adaptive Lasso MCC values remain around $0.5$-$0.55$ and never achieve MCC above $0.8$. Overall, BMT is the only procedure that consistently delivers MCC values close to one.

Tables (ref)-(ref) report the root mean squared error (RMSE) of the estimated coefficient vector.\footnote{RMSE is defined as $\sqrt{\frac{1}{r} \sum_{j=1}^{r} \left| \widetilde{\boldsymbol{\beta}}^{(j)}{n} - \boldsymbol{\beta}{n} \right|^{2}}$, where $\widetilde{\boldsymbol{\beta}}^{(j)}_{n}$ coincides with the post-selection OLS estimates on the selected coordinates and is zero elsewhere, and $r$ is the number of Monte Carlo replications.} BMT again performs best: for $k=1$ the median RMSE is $0.082$ and BMT has the lowest RMSE in $80\%$ of designs (and in more than $97\%$ of designs once VIF$=1$ is excluded). For $k=4$, the median RMSE is $0.313$, with BMT ranked first in about three quarters of all designs and in more than $90\%$ of designs when VIF$>1$. OCMT occasionally attains very low RMSE but does so at the cost of extreme instability, as reflected in the very large upper tail of its RMSE distribution (with maxima of 725.473 for $k=1$ and 375.073 for $k=4$). Lasso and Adaptive Lasso produce more stable RMSE distributions, but with substantially higher medians than BMT and no cases in which they are the best performing method.

Model size results in Tables (ref)-(ref) highlight the parsimony of BMT. When $k=1$, the median model size for BMT is $1.093$ with a very small IQR and a mean absolute deviation (MAD) from the true size of only $0.091$. OCMT, in contrast, has a median model size above $21$ with extremely large dispersion and a mean absolute deviation exceeding $17$, reflecting a tendency to select many additional regressors beyond the single true signal. Lasso and Adaptive Lasso systematically overselect, with median sizes of about $8$ and $6.6$, respectively. When $k=4$, BMT again tracks the true model size closely, with a median of $4.080$ and a negligible mean absolute deviation of $0.097$. OCMT, Lasso and Adaptive Lasso all select much larger models on average, with median sizes roughly between $9$ and $16$ and substantial overfitting relative to the true sparsity level.

To gain further insight into how performance varies with $(T,n)$ and with the structure of multicollinearity, Figures (ref)-(ref) present a visual summary of MCC-based and relative RMSE-based performance under different multicollinearity regimes defined by the VIF and the parameter $\pi$. As noted earlier, higher VIF values correspond to more severe multicollinearity. The parameter $\pi$ governs the source of this dependence: when $\pi=0.75$, most of the multicollinearity arises from the global factor $f_{t}$, whereas when $\pi=0.25$ it is primarily driven by the local factor $g_{t}$.

The visual evidence mirrors the tabulated results. BMT attains MCC values close to one across all $(T,n)$ configurations and multicollinearity regimes, indicating reliable detection of true signals with high precision and recall even in severely collinear, high-dimensional settings. In contrast, OCMT performs satisfactorily only when multicollinearity is modest and predominantly local. For any VIF greater than one its MCC tends to deteriorate as both $T$ and $n$ increase. For example, in the design with VIF $=4$ and $\pi=0.75$, the MCC for OCMT falls from about $0.32$ when $(T,n)=(100,100)$ to about $0.14$ when $(T,n)=(200,100)$ and to about $0.06$ when $(T,n)=(300,100)$, whereas BMT improves from about $0.82$ to $0.98$ and $0.99$ over the same sequence. This pattern reflects the fact that as $T$ grows, the marginal tests used by OCMT gain power not only for the true signals but also for proxy signals, which are correlated with the outcome through their association with the true signals. Under substantial multicollinearity there are many such proxies, and OCMT includes all of them once they pass their individual $t$ tests. The resulting proliferation of false positives drives MCC down as the sample size and cross sectional dimension increase.

Lasso and Adaptive Lasso display more stable but uniformly moderate performance. Their MCC values are clustered around $0.5$-$0.6$ with relatively little variation across $(T,n)$ or multicollinearity regimes. The penalty in the objective function limits the number of proxies that can enter, so these methods do not suffer the same explosive growth in false positives as OCMT. However, they continue to admit a non-negligible number of proxy signals, leading to systematic overestimation of model size and preventing MCC from approaching one in the designs considered here.

Additional insight is provided by the RMSE ratios of BMT relative to the other methods, plotted as BMT/OCMT (red line), BMT/Lasso (green line) and BMT/Ad.\ Lasso (grey line). Ratios below one indicate that BMT attains a lower RMSE. Across all designs, the green and grey lines are typically in the 0.45-0.60 range, implying that the RMSE for BMT is often about one half of that for Lasso and Adaptive Lasso. This pattern is remarkably stable across $(T,n)$, VIF and $\pi$, and persists even in the most challenging configurations (e.g.\ VIF $=4$, $\pi\in\{0.25,0.75\}$), where BMT/Lasso and BMT/Ad.\ Lasso ratios remain around 0.5 as $T$ and $n$ increase.

The behaviour of the red line (BMT/OCMT) depends more strongly on the multicollinearity regime. When multicollinearity is modest and largely local (VIF $=2$, $\pi=0.25$), the ratio is close to one for small samples (e.g.\ 0.98-1.01 for $(T,n)=(100,300)$ and $(100,200)$), whereas it declines toward roughly 0.75 as $T$ and $n$ grow, indicating a substantial improvement in RMSE for BMT. When multicollinearity is stronger or more global, the gains become dramatic. For instance, with VIF $=4$ and $\pi=0.25$, the BMT/OCMT ratio falls from about 0.83-0.90 at $(T,n)=(100,100)$-$(100,300)$ to roughly 0.53-0.59 at $(T,n)=(300,200)$-$(300,300)$. Under VIF $=4$ and $\pi=0.75$, the contrast is starker still: the red line drops from around 0.23-0.26 at $(T,n)=(100,100)$-$(100,300)$ to values close to zero at $(T,n)=(300,300)$, reflecting very large RMSE for OCMT in those designs. These patterns are consistent with the selection behaviour discussed above. OCMT follows a relatively aggressive, or “greedy”, rule: at each stage it includes all regressors that are individually significant in marginal $t$-tests. As $T$ increases, marginal tests gain power not only for true signals but also for highly correlated proxy and noise variables, especially under strong common-factor dependence. Since OCMT admits all regressors whose $t$-statistics cross the threshold, proxy selection can expand rapidly, leading to unstable estimates and rising RMSE. BMT instead admits one variable at a time and prioritises the largest conditional gain in fit, which limits proxy accumulation and stabilises estimation. Lasso-type methods avoid explosive RMSE through shrinkage, but the same shrinkage slows the decline in RMSE as $T$ grows.

Detailed numerical results for MCC, RMSE, model size and the remaining evaluation metrics (F1, TDR, FDR, TPR, FPR and RMSFE) are reported in the Appendix. These confirm that the few cases where BMT does not clearly dominate correspond to designs with a short time dimension ($T=100$) and low or negligible multicollinearity (for example, VIF close to 1 or 2 and a weak global factor, $\pi=0.25$). Taken together, the results indicate that BMT delivers consistently strong performance across a wide range of dimensions and dependence structures, combining high selection accuracy (MCC), low estimation error (RMSE) and parsimonious model size even in the presence of substantial multicollinearity.

Empirical Illustrations

It this Section we present two empirical illustrations that highlight the potential of BMT as a model selection procedure.

Corporate Emissions

Climate change and pollution sit at the center of the global policy agenda, not only because they shape long-run environmental outcomes, but because they map directly into quality-of-life metrics such as air and water safety, morbidity risks, productivity, and property values. Regulators, investors, and communities increasingly need credible, data-driven evidence on what drives emissions in order to design effective penalties, target enforcement, and allocate capital responsibly.

In what follows, we analyse annual emissions and model them as a function of (i) penalties imposed on emitting facilities and (ii) indicators of economic activity and broader macro-financial conditions. While overall economic activity is likely correlated with aggregate emissions, it is far less clear which specific drivers, such as sectors, energy intensity, trade exposure, investment cycles, capacity utilization, or other factors, are most influential.

Methodologically, this is a high-dimensional problem. The dependent variable is aggregate annual emissions for the US economy (total releases, in million pounds) observed over 32 years, while the pool of candidate economic drivers runs into the hundreds, so that $n \gg T$. These covariates include pre-selected variables, such as penalties imposed on emitting facilities, as well as a large set of indicators of economic activity and broader macro financial conditions. In this illustration, we use our model selection procedure to isolate a sparse, interpretable set of predictors and to examine whether enforcement penalties relate to aggregate emissions once we control for the relevant components of economic activity.

Our environmental data are drawn from the U.S. Environmental Protection Agency’s (EPA) Toxics Release Inventory (TRI), a public database that records facility-level releases of listed toxic chemicals. Facilities in TRI-covered sectors that exceed reporting thresholds are required to report annual releases by medium (air, water and land). Although the data are self-reported, TRI applies internal quality-assurance and quality-control (QA/QC) checks and provides rich facility-level attributes (e.g.\ location and industry classification). For our purposes, we use TRI to construct the dependent variable. Our outcome of interest is aggregate annual air emissions for the US economy, defined as the sum of reported releases to air across all facilities, measured in pounds (lb), which yields a national time series over the sample period.

Penalty measures for emitting facilities are derived from EPA enforcement data, with the primary measure being Total Penalty Assessed (the dollar amount assessed in all settlements for concluded enforcement actions). This measure decomposes into penalties from Administrative-Formal (AFR) and Judicial (JDC) actions.\footnote{These penalty fields are defined in EPA’s ECHO/ICIS data dictionaries and reflect assessed amounts (not necessarily amounts collected). See the EPA ECHO data dictionary at \url{https://echo.epa.gov/help/reports/dfr-data-dictionary?check_logged_in=1}.} In what follows, we use Total Penalty Assessed and treat it as the aggregate (AFR + JDC).

Indices of economic activity and broader macro-financial conditions are obtained from the FRED-QD (Federal Reserve Economic Data - Quarterly Database) introduced by McCrackenNg2021. This publicly available dataset is updated monthly through the FRED database. In what follows, the 2025-03 vintage is used, which covers data through 2024Q4. The dataset comprises 245 U.S. macroeconomic time series, most of which begin in 1959Q1. The dataset follows the structure of stock_disentangling_2012, but with broader coverage. The variables are classified into 14 thematic blocks: (i) NIPA output and income aggregates, (ii) industrial production and capacity utilization, (iii) labor market indicators, (iv) housing activity, (v) inventories, orders, and sales, (vi) price level and inflation measures, (vii) earnings and productivity, (viii) interest rates and term spreads, (ix) monetary and credit aggregates, (x) household balance sheet ratios, (xi) exchange rates, (xii) survey-based and sentiment indicators, (xiii) stock market variables, and (xiv) non-household (corporate and government) balance sheet items. Full details of these variables are provided in McCrackenNg2021.

In addition to the variables described above, the regression specification includes a lagged dependent variable to capture persistence, a deterministic time trend to account for the secular decline in emissions, and the first principal component from the FRED-QD database, extracted from the full set of economic and macro-financial indicators. This component summarises common variation across these series and helps mitigate collinearity among the regressors.

The model is specified as in Eq. (ref):

where $y_{t}$ denotes the natural logarithm of aggregate U.S.\ air emissions in year $t$. The vector of pre-selected controls $\boldsymbol{z}_{t}=\left(z_{1,t}, z_{2,t}, z_{3,t}, z_{4,t} \right)^{\prime}$ is $4\times 1$ and consists of: the lagged dependent variable $y_{t-1}$, the natural logarithm of lagged Total Penalty, a deterministic time trend $t$, and the first principal component of the FRED-QD macro-financial indicators. The remaining candidate regressors $\{x_{it}\}_{i=1}^{n}$ are the $n=245$ individual FRED-QD indicators, with $n\gg T$, and we denote by $\mathcal{S}_{s}\subset\{1,\dots,n\}$ the (unknown) set of true signals.

Model selection is conducted using four alternative methods: BMT, OCMT, Lasso, and Adaptive Lasso. For the Lasso estimator, two stopping rules are considered: one based on the Bayesian Information Criterion (BIC), and another based on 10-fold cross-validation, where the tuning parameter is selected to minimize the mean squared prediction error.

Table (ref) below presents results for the benchmark specification in Eq. (ref).

Under BMT (column `BMT'), the autoregressive coefficient on lagged emissions, $z_{1}$, is estimated at 0.663 and is highly statistically significant. Conditional on a deterministic trend, this relatively large coefficient points to strong persistence in U.S. air emissions. By contrast, the coefficient on the lagged penalty variable ($z_{2}$) is positive, very small in magnitude and statistically insignificant. This sign is consistent with penalties being largely reactive rather than preventive: higher penalties tend to follow episodes of unusually high emissions, which themselves display persistence. This might suggest that the current system of penalties is neither sufficiently strong nor sufficiently timely to alter emission behavior in a meaningful way.

The coefficient on the trend variable ($z_{3}$) is negative, reflecting that emissions exhibit a gradual downward trajectory over the sample period. This aligns with the broad historical trend in U.S. emissions, reflecting structural changes in the economy and energy production.

Among the additional regressors in the candidate set, only one variable, permitmw (new private housing units authorized by building permits in the Midwest), is selected, with a positive and highly significant coefficient (0.238). This finding is consistent with the well-documented role of the Midwest as a leading emitter within the U.S. If treated as a separate country, the region would be the fifth-largest emitter globally Kvam2025Midwest. The Midwest’s emissions profile is driven by several structural factors: high transportation demand, heavy reliance on coal and natural gas for electricity generation, widespread natural gas use in residential heating and buildings, and emissions-intensive agricultural practices in the Corn Belt. Against this backdrop, new housing activity in the Midwest is naturally associated with higher emissions, making the positive coefficient on permitmw both statistically and economically meaningful.

Turning to the OCMT method (column `OCMT'), the results for the pre-selected covariates $z_{1}$-$z_{4}$ are broadly similar to those obtained under BMT. The key difference lies in variable selection: whereas BMT identifies only permitmw, OCMT selects this variable along with nine additional covariates. A striking feature of these is the inclusion of several highly correlated permit-based indicators. Specifically, permit refers to new private housing units authorized for the United States as a whole, while permits and permitw represent the corresponding measures for the South and the West, respectively. Their joint selection highlights the strong collinearity among regional housing permit series, which track construction activity at different levels of geographical aggregation. This outcome is not surprising, since OCMT admits all variables that are deemed statistically significant when considered individually, so that collinear regressors may enter simultaneously even if not all of them are truly relevant.

Similarly, the OCMT method selects a series of housing-related variables: houst (total new private housing units authorized in the U.S.), and its regional counterparts houstmw (Midwest), housts (South), and houstw (West). These variables, by construction, are highly correlated with one another as well as with the permit measures. Their inclusion again illustrates how OCMT, in contrast to BMT, tends to admit clusters of collinear regressors.

Two additional variables selected by OCMT are uslah and prfix. The former denotes average weekly hours of production and nonsupervisory employees in manufacturing (U.S. Bureau of Labor Statistics series), which proxies for labor input intensity in the industrial sector. The latter is the price index for fixed investment, which captures changes in the cost of capital goods. Both variables are not statistically significant in this specification, and their selection alongside the housing series indicates that OCMT tends to overfit by including regressors with overlapping or weak explanatory power.

Turning to the Lasso-type methods (columns `Lasso with BIC', `Lasso with CV', and `Adaptive Lasso'), none of these procedures select any variables from the candidate set beyond the mandatory covariates. As a result, their estimated specifications are identical, producing the same coefficients, standard errors, RMSE and forecast metrics.

Comparing across methods, BMT emerges as the strongest performer, in terms of both in-sample fit and out-of-sample forecast accuracy. First, in terms of in-sample fit, it achieves the lowest RMSE (0.069) relative to OCMT (0.080) and the Lasso-type methods (0.075). Second, in terms of out-of-sample performance, BMT yields markedly lower forecast errors. This ranking is corroborated by the information criterion: BMT also delivers the lowest BIC (-69.745), which is substantially below that of OCMT (-56.158) and smaller than the values for the Lasso specifications (-65.363), indicating that BMT remains preferred even after penalising for model complexity. Second, in terms of out-of-sample performance, BMT yields markedly lower forecast errors. The RMSFE over a two year horizon is only 0.007, far smaller than that of OCMT (0.119) and also well below the Lasso approaches (0.035). A similar advantage holds at the four year horizon, where BMT delivers an RMSFE of 0.037 compared with 0.093 for OCMT and 0.071 for the Lasso variants.

Overall, these results reinforce the robustness and superior performance of BMT in balancing interpretability, parsimony, and predictive power.

High Dimensional Phillips Curve

To further illustrate the practical advantage of the proposed procedure, we next consider the classic problem of specifying a U.S. Phillips Curve.

The stability and specification of the Phillips curve, the structural link between inflation and real activity, remain a subject of intense debate. Since the breakdown of the stable 1960s trade-off, researchers have questioned whether the curve has flattened, disappeared, or simply become harder to identify amidst high-dimensional noise atkeson2001phillips, stock2008phillips. In a data-rich environment, the key challenge is distinguishing between the true structural drivers of inflation (e.g., specific monetary or labor market signals) and the hundreds of correlated macroeconomic indicators that act as noisy proxies blanchard2016phillips, gali2019structural.

We compare the performance of BMT against two benchmarks: the One Covariate at a Time Multiple Testing (OCMT) procedure of chu2018 and the Lasso tib1996. We show that BMT recovers a sparse, theoretically interpretable specification that maximizes out-of-sample forecasting power, whereas Lasso and OCMT suffer from over-selection in the presence of correlated proxies.

We assume that the change in the inflation rate is determined by a high-dimensional set of lagged macroeconomic indicators. Specifically, we estimate the following predictive regression:

where $\Delta \pi_{t+h}$ denotes the change in the quarterly inflation rate $h$ periods ahead. We focus on the one-quarter-ahead horizon ($h=1$). The predictors $x_{j,t}$ are drawn from the FRED-QD quarterly database McCrackenNg2021. The sample spans from 1959:Q1 to 2024:Q4.

This specification is deliberately reduced-form and predictive in nature. Our objective is not structural identification of the Phillips-curve mechanism, but rather to assess whether a sparse subset of lagged macroeconomic indicators contains incremental forecasting information for future inflation changes in a high-dimensional environment. This approach follows the modern forecasting literature (e.g., Atkeson and Ohanian, Stock and Watson), where the emphasis is on predictive content rather than structural interpretation.

The target variable is derived from the Consumer Price Index (CPIAUCSL). To ensure a rigorous evaluation of structural drivers rather than mechanical identities, we exclude all price indices (e.g., PPI, PCE, sectoral CPIs) from the set of predictors, resulting in a final candidate set of $N=201$ macroeconomic indicators. All predictors are standardized prior to selection. We perform a pseudo-out-of-sample forecasting exercise with an 80%/20% split, training on the initial 80% of the sample and evaluating on the remaining 20% (approximately 43 quarters). Performance is evaluated using the Out-of-Sample Coefficient of Determination ($R^2_{OOS}$) and the Bayesian Information Criterion (BIC).

Table (ref) reports the selected model size, information criteria, and forecasting performance.

Consistent with the findings in the emissions application, the results highlight the ability of BMT to isolate the most informative predictors in a noisy environment. BMT selects a remarkably sparse model consisting of only three variables: Real M1, Real M2, and the 3-Month Treasury Bill rate. This specification is consistent with a monetarist interpretation of inflation dynamics friedman1968role, though our focus is predictive rather than structural. By achieving the lowest BIC and the highest out-of-sample $R^2$, BMT demonstrates that parsimony aids forecasting in this context.

In contrast, Lasso selects two of the core predictors (Real M2 and the 3-Month Treasury Bill rate) but augments them with 10 additional predictors, including Real Disposable Income and Industrial Production. This supports our theoretical argument that Lasso, being sensitive to the aggregate correlation of proxies, struggles to distinguish the primary signal from correlated noise. The inclusion of these redundant variables degrades out-of-sample performance ($R^2 = 1.21\%$). OCMT performs better than Lasso but selects an intermediate model (9 variables) including sectoral employment data, achieving an $R^2$ of 1.67%.

Concluding Remarks

High-dimensional regression analysis sits at the centre of modern empirical work in statistics, econometrics and machine learning, underpinning applications that range from macroeconomic forecasting to genomics and text as data. Existing approaches include regularisation methods such as Ridge and Lasso and greedy algorithms such as stepwise procedures, while multiple-testing based procedures such as OCMT offer an alternative route to parsimony. Each of these classes delivers useful properties but also faces limitations, particularly when regressors are numerous, strongly correlated and potentially dynamic.

This paper has proposed Boosting with Multiple Testing (BMT), a hybrid procedure that combines forward stepwise variable addition with a family-wise multiple testing filter. Methodologically, BMT inherits from OCMT the use of sharp multiple testing thresholds to screen out irrelevant regressors with high probability, while departing from OCMT in a crucial respect: at each stage it adds at most one variable rather than all regressors that pass a marginal test. This less aggressive updating rule is specifically designed to limit the propagation of proxy variables and to stabilise the final specification.

Theoretical results show that BMT enjoys strong asymptotic properties under assumptions that allow for heterogeneous strongly mixing processes and, via the probability inequalities of den2022, for non trivial temporal dependence and heavier tailed distributions. Relative to an approximating model that includes all true signals and excludes noise variables, BMT behaves in an oracle like fashion: the approximating model is selected with probability tending to one and the resulting estimator achieves standard parametric rates for both prediction error and coefficient estimation. We then strengthen these results by deriving conditions under which BMT recovers the exact true model and, in particular, avoids selection of proxy signals. In the single signal case this holds without additional restrictions, while in the general case a signal-to-proxy dominance condition, weaker than standard incoherence assumptions used for Lasso, ensures no proxy selection.

Monte Carlo experiments confirm that these asymptotic features translate into favourable finite sample performance. Across a wide range of designs with varying sample sizes, signal strengths and multicollinearity patterns, BMT reliably attains high Matthews correlation coefficients, small coefficient root mean squared errors and model sizes close to the true sparsity level. Compared with OCMT and Lasso type procedures, BMT is especially effective in strongly collinear environments, where OCMT tends to accumulate proxy variables and penalised regressions struggle to balance selection and shrinkage. Overall, the results show that BMT provides a principled alternative to both greedy multiple-testing procedures and penalised regression methods when regressors are numerous, strongly correlated, and potentially dynamic. Its stagewise structure delivers parsimonious and interpretable models without sacrificing asymptotic efficiency.

The present framework also opens up several natural directions for further work. One is the extension of BMT to panel data settings and to models with endogenous regressors, potentially combining the multiple testing step with instrumental variables or control function approaches. At the same time, it would be of interest to explore variants of BMT in nonlinear frameworks, for example in quantile or count regression, where high-dimensional covariate sets and complex dependence structures are increasingly common.