Short Articles

The Field of Study of Econometrics 

Our research focuses on the development and correct use of econometric identification, estimation and inference techniques for statistical and economic problems using state-of-the-art methods. At the Christis G. Katsouris Institute of Econometrics and Data Science, we produce impactful research through contributions in both econometric theory and applications. Our passion for addressing the most important and relevant current issues in theoretical and applied econometrics allow us to expand beyond boundaries on how we estimate and forecast economic relations using time series, panel and big data.

© Christis G. Katsouris Institute of Econometrics and Data Science

Let's Get Real:

Business Cycle Dynamics, Dynamic Causal Effects and Some Properties of Estimators and Test Statistics based on Asymptotically Efficient Theory in Structural Vector Autoregressive Models

© Christis G. Katsouris Institute of Econometrics and Data Science

1. Introduction

The identification and estimation of dynamic causal effects, motivated a large body of research in the time series econometrics and macroeconometrics literature (e.g., see Stock & Watson (2018, EJ)). Econometric theory and methods were developed permitting to conduct inference about Granger causality in multivariate time series, to optimally estimate cointegrating regressions, to identify and estimate structural shocks via SVAR models, as well as to evaluate the impact of shocks to the macroeconomy using VAR-based and LP-based impulse response estimators.   

Statistical inference procedures have focused on methods that are robust to HAC standard errors in the presence of serially correlated residuals as in Montiel Olea & Plagborg‐Møller (2021, Ecta) and Braun & Brüggemann (2023, JBES), with impulse responses constructed a la Sims. Moreover, uniform asymptotically valid inference in the presence of persistence is proposed by Inoue & Kilian (2020, JoE) and Montiel Olea & Plagborg‐Møller (2021, Ecta), that allow for controlled uniform sized tests for both stationary and non-stationary data regardless of the length of the horizon. In addition, Plagborg‐Møller & Wolf (2021, Ecta) show that LP-based and VAR-based impulse responses are asymptotically equivalent, with comparable finite-sample properties that depend on the variance-bias trade-off. Recently, Montiel Olea et al. (2026, nber/w32495) establish the double robustness property of local projections which permits inference about dynamic causal effects even in nonlinear environments. Lastly, Lewis & Mertens (2026, cem/wp01-26) develop weak instrument bias inference in impulse response estimators, which allows to examine instrument strength in empirical macroeconomics.  

Despite the growing literature on single-equation methods for impulse response inference which allows to study the dynamics of responses across horizons and across model variables, joint and simultaneous inference based on Wald-type statistics remains a powerful methodology, particularly suited when examining the joint impact of a vector of shocks in possibly nonstationary and unstable environments with information frictions and rigidities. In particular, Dufour & Wang (2024, arXiv:2409.10820) propose a novel two-stage estimation and inference procedure for generalized impulse responses which encompasses all coefficients in a multi-horizon linear projection model, while allowing the projection horizon to scale with the sample size. However, to develop a unified framework that facilitates simultaneous estimation and inference, semiparametric efficiency properties have to be derived. Specifically, Xu (2025, SSRN 5709504) develops uniform asymptotic theory for LP regression when the true lag order of the model is unknown and potentially infinite. This author shows that LPs can achieve semiparametric efficiency at a given horizon provided that the controlled lag order diverges. Therefore, establishing the semiparametric validity of simultaneous inference procedures for both stationary and nonstationary data worth further study. 

2. Econometric Framework and Methodology 

We contribute to the literature of estimation and inference in SVAR models with a focus on dynamic causal effects for both stationary and non-stationary time series data. In particular, recently Montiel Olea, Plagborg-Møller, Qian & Wolf (2026, nber/w32495) develop an econometric framework for a local-to-SVAR which permits to develop estimation and inference robust to the presence of functional form misspecification. In our framework we consider the local-to-unity SVAR model, and develop persistent robust inference procedures. Relevant frameworks are presented in Cheng, Han & Inoue (2022, ET) who propose IV estimation of SVAR models robust to possible nonstationary regressors using GMM-type estimators, and in Chevillon, Mavroeidis & Zhan (2020, ET) who develop persistent robust inference in SVAR models identified long-run restrictions based on IVX-type estimators. Both methodologies imply limiting distributions of test statistics that are nuisance parameter free. Moreover, Dou & Müller (2021, Ecta) introduce a generalization of the local-to-unity model of time series persistence by allowing for p autoregressive roots and p-1 moving average roots to unity. Thus, we focus on developing a unified framework for structural analysis of multivariate time series, possibly highly persistent (SVAR with nonstationary regressors). 

2.1. Asymptotically Equivalent Estimators

Key points to discuss:

VAR-based vis-à-vis LP-based IR Estimators and their CIs: 

• Width of Confidence interval of VAR-based IR Estimators

• Width of Confidence interval of LP-based IR Estimators

An intuitive way to statistically evaluate whether these two estimators are asymptotically equivalence (e.g., based on finite-sample comparisons), is to check that the coverage probability of the confidence intervals around the true value of the estimators under study is numerically equivalent. For example, in the presence of local misspecification (e.g., as in Montiel Olea et al. (2024, nber/w32495)), we can construct consistent specification test, if and only if, the procedure can detect any local departures from correct functional form with probability approaching one almost surely as the sample size increases. In other words, the presence of functional form misspecification can be detected when the coverage probability of the CIs of these two estimators is equivalent, based on any (robust) model selection approach which induces a sufficiently large lag length, such that there is no violation of the asymptotic equivalence property.         

Keywords: impulse response function, local projection, long horizon, uniform inference, Structural Vector Autoregression, non-Gaussianity, local-to-unity, localizing coefficient of persistence. 

Dr. Christis Katsouris, Ph.D. University of Southampton

© Christis G. Katsouris Institute of Econometrics and Data Science

Econometrics Literature:

• Lewis, D., and Mertens, K. (2026). "Weak Instrument Bias in Impulse Response Estimators". Cemmap Working Paper (CWP01/26). Available at cemmap/wp01-26.

• Montiel Olea, J. L., Plagborg-Møller, M., Qian, E., and Wolf, C. K. (2026). "Double Robustness of Local Projections and Some Unpleasant Arithmetic". NBER Working Paper (No. w32495). Available at nber/w32495. 

• Bruns, M., Lütkepohl, H., and McNeil, J. (2025). "Avoiding Unintentionally Correlated Shocks in Proxy Vector Autoregressive Analysis". Journal of Business & Economic Statistics, 1-13. 

• Cocci, M. D., and Plagborg-Møller, M. (2025). "Standard Errors for Calibrated Parameters". Review of Economic Studies, 92(5), 2952-2978. 

• Forneron, J. J., and Qu, Z. (2025). "Fitting Dynamically Misspecified Models: An Optimal Transportation Approach". Preprint arXiv:2412.20204. 

• Holberg, C., and Ditlevsen, S. (2025). "Uniform Inference for Cointegrated Vector Autoregressive Processes". Journal of Econometrics, 247, 105944. 

• Lewis, D. J., and Mertens, K. (2025). "A Robust Test for Weak Instruments for 2SLS with Multiple Endogenous Regressors". Review of Economic Studies, rdaf103. 

• Xu, K. L. (2025). "Local Projection based Inference Under General Conditions". Available at SSRN 5709504. 

• Casini, A., and Perron, P. (2024). "Prewhitened Long-Run Variance Estimation Robust to Nonstationarity". Journal of Econometrics, 242(1), 105794. 

• Dufour, J. M., and Wang, E. (2024). "Simple Robust Two-Stage Estimation and Inference for Generalized Impulse Responses and Multi-Horizon Causality". Preprint arXiv:2409.10820. 

• Forneron, J. J. (2024). "Detecting identification Failure in Moment Condition Models". Journal of Econometrics, 238(1), 105552. 

• Funovits, B. (2024). "Identifiability and Estimation of Possibly Non-Invertible SVARMA Models: The Normalised Canonical WHF Parametrisation". Journal of Econometrics, 241(2), 105766. 

• Gafarov, B., Karamysheva, M., Polbin, A., and Skrobotov, A. (2024). "Wild Inference for Wild SVARs with Application to Heteroscedasticity-based IV". Preprint arXiv:2407.03265. 

• Gorgi, P., Koopman, S. J., and Schaumburg, J. (2024). "Vector Autoregressions with Dynamic Factor Coefficients and Conditionally Heteroskedastic Errors". Journal of Econometrics, 244(2), 105750. 

• Herbst, E. P., and Johannsen, B. K. (2024). "Bias in Local Projections". Journal of Econometrics, 240(1), 105655. 

• Hoesch, L., Lee, A., and Mesters, G. (2024). "Locally Robust Inference for Non‐Gaussian SVAR Models". Quantitative Economics, 15(2), 523-570. 

• Braun, R., and Brüggemann, R. (2023). "Identification of SVAR Models by Combining Sign Restrictions with External Instruments". Journal of Business & Economic Statistics, 41(4), 1077-1089. 

• Casini, A. (2023). "Theory of Evolutionary Spectra for Heteroskedasticity and Autocorrelation Robust Inference in Possibly Misspecified and Nonstationary Models". Journal of Econometrics, 235(2), 372-392. 

• Jentsch, C., and Lunsford, K. G. (2022). "Asymptotically Valid Bootstrap Inference for Proxy SVARs". Journal of Business & Economic Statistics, 40(4), 1876-1891. 

• Dou, L., and Müller, U. K. (2021). "Generalized Local‐to‐Unity Models". Econometrica, 89(4), 1825-1854.

• Montiel Olea, J. L., and Plagborg‐Møller, M. (2021). "Local Projection Inference is Simpler and More Robust than you Think". Econometrica, 89(4), 1789-1823. 

• Plagborg‐Møller, M., and Wolf, C. K. (2021). "Local Projections and VARs Estimate the Same Impulse Responses". Econometrica, 89(2), 955-980. 

Macroeconomics and Monetary Economics Literature:

• McLeay, M., and Tenreyro, S. (2026). "Dollar Dominance and the Transmission of Monetary Policy". Quarterly Journal of Economics, 141(1), 605-666. 

• Matthes, C., and Schwartzman, F. (2025). "The Consumption Origins of Business Cycles: Lessons from Sectoral Dynamics". American Economic Journal: Macroeconomics, 17(4), 82-123. 

• Chang, M., Chen, X., and Schorfheide, F. (2024). "Heterogeneity and Aggregate Fluctuations". Journal of Political Economy, 132(12), 4021-4067. 

• Badinger, H., and Schiman, S. (2023). "Measuring Monetary Policy in the Euro Area using SVARs with Residual Restrictions". American Economic Journal: Macroeconomics, 15(2), 279-305. 

• McNeil, J. (2023). "Monetary Policy and the Term Structure of Inflation Expectations with Information Frictions". Journal of Economic Dynamics and Control, 146, 104588. 

• Harding, M., and Wouters, R. (2022). "Risk and State-Dependent Financial Frictions". BoC Working Paper (No. 2022-37). Available at boc/wp2022-37. 

• Ludvigson, S. C., Ma, S., and Ng, S. (2021). "Uncertainty and Business Cycles: Exogenous Impulse or Endogenous Response?". American Economic Journal: Macroeconomics, 13(4), 369-410. 

• Beaudry, P., Galizia, D., and Portier, F. (2020). "Putting the Cycle Back into Business Cycle Analysis". American Economic Review, 110(1), 1-47. 

• Bloom, et al. (2018). "Really Uncertain Business Cycles". Econometrica, 86(3), 1031-1065. 

• Müller, U. K., and Watson, M. W. (2016). "Measuring Uncertainty about Long-Run Predictions". Review of Economic Studies, 83(4), 1711-1740. 

• Sarte, P. D., Schwartzman, F., and Lubik, T. A. (2015). "What Inventory Behavior Tells Us About How Business Cycles Have Changed". Journal of Monetary Economics, 76, 264-283. 

• Ilut, C. L., and Schneider, M. (2014). "Ambiguous Business Cycles". American Economic Review, 104(8), 2368-2399. 

• Alessandria, G., Kaboski, J., and Midrigan, V. (2013). "Trade Wedges, Inventories, and International Business Cycles". Journal of Monetary Economics, 60(1), 1-20. 

• Beaudry, P., Collard, F., and Portier, F. (2011). "Gold Rush Fever in Business Cycles". Journal of Monetary Economics, 58(2), 84-97. 

• Gabaix, X. (2011). "The Granular Origins of Aggregate Fluctuations". Econometrica, 79(3), 733-772. 

• Perron, P.,  and Wada, T. (2009). "Let's Take A Break: Trends and Cycles in US Real GDP". Journal of Monetary Economics, 56(6), 749-765. 

• Hamilton, J. D. (2005). "What's Real About the Business Cycle?". NBER Working Paper (No. w11161). Available at nber/w11161. 

• Forni, M.,  and Reichlin, L. (1998). "Let's Get Real: A Factor Analytical Approach to Disaggregated Business Cycle Dynamics". Review of Economic Studies, 65(3), 453-473. 

• King, R. G., Plosser, C. I., and Rebelo, S. T. (1988). "Production, Growth and Business Cycles: I. The Basic Neoclassical Model". Journal of Monetary Economics, 21(2-3), 195-232. 

• King, R. G., Plosser, C. I., and Rebelo, S. T. (1988). "Production, Growth and Business Cycles: II. New Directions". Journal of Monetary Economics, 21(2-3), 309-341. 

• Shapiro, M. D., and Watson, M. W. (1988). "Sources of Business Cycle Fluctuations". NBER Macroeconomics Annual, 3, 111-148. 

• Long Jr, J. B.,  and Plosser, C. I. (1983). "Real Business Cycles". Journal of Political Economy, 91(1), 39-69. 

Labour and Public Economics Literature:

• Teramoto, K. (2026). "Unequal Wage Cyclicality: Evidence, Theory, and Implications for Labor Market Volatility". Journal of Political Economy Macroeconomics (just-accepted). 

• Cerra, V., Fatás, A., and Saxena, S. C. (2023). "Hysteresis and Business Cycles". Journal of Economic Literature, 61(1), 181-225. 

Should We Go One Step Further?

Methods for Estimation and Inference in Econometrics

© Christis G. Katsouris Institute of Econometrics and Data Science

1. Introduction

In many econometric applications, a researcher must select an instrument vector from a candidate set of instruments. According to Hall & Peixe (2003, Econometric Reviews), when the objective is to perform inference about the unknown parameters using conventional asymptotic theory, then it is desirable for the chosen instrument vector to satisfy certain regularity conditions (e.g., orthogonality, identification restrictions, efficiency and non-redundancy). Moreover, for valid inference with time series models, in the presence of irrelevant instruments, econometric methods are adjusted. In particular, Phillips, P.C.B. (2014, JoE) propose optimal estimation of cointegrated systems with irrelevant instruments. In addition, Kheifets & Phillips, P.C.B. (2025, cowles/wp2885) propose optimal estimation in multicointegrated systems, while Phillips, P.C.B. & Kheifets (2021, cowles/wp2652) develop an estimation approach in semiparametric triangular systems under multicointegration which is robust against integration order of unknown form. This ensures nuisance parameter free inference. Lastly, Kheifets & Phillips, P.C.B. (2023, JoE) examine the fiscal sustainability of the US government. The authors test the hypothesis that government revenue and expenditure are strongly cointegrated with a known cointegration coefficient, where multicointegration naturally arises if bounds are imposed on deviations of debt from revenue.

Using non-causal instruments in time series regressions requires caution, since the instrumentation is constructed using future innovations, which can distort the finite-sample properties of estimators and test statistics. In particular, Lof (2014, OBES) propose GMM estimation in the presence of non-causal instruments and rational expectations. In fact, Lanne & Saikkonen (2011, OBES) have shown that if any of the time series used as instrument is noncausal (i.e., depends on its future values), then the GMM estimator is inconsistent. Moreover, Lanne & Luoto (2013, JEDC) propose an estimation method of the NKPC using a univariate noncausal autoregressive model for the inflation rate, while Lanne & Saikkonen (2013, ET) propose estimation and inference in noncausal VAR models for non-Gaussian time series. Recently, Gourieroux & Jasiak (2025, arXiv:2205.09922) propose innovation filtering procedures for valid identification and inference in mixed Causal-Noncausal VAR models. 

Furthermore, the VAR and Local Projections equivalence for impulse responses, as shown by Plagborg‐Møller & Wolf (2021, Ecta), is a useful property worth further study; both in terms of model specifications for Causal VAR processes (e.g., in the presence of unit roots and multiple instruments), and of model specifications for (mixed) Causal-Noncausal VAR processes. Toward a unified framework, Holberg & Ditlevsen (2025, JoE) propose uniform inference procedures for cointegrated VAR processes, via a properly normalized covariance matrix and simulation-based critical values (e.g., via the least favourable distribution approach of Elliott, Müller & Watson (2015, Ecta)). A numerical analysis of test optimality for nonstandard econometric problems can be found in Ketz, McCloskey & Scherer (2025, arXiv:2512.19843). These authors examine the properties of power envelopes for test statistics when the true parameter vector is near or at the boundary of the parameter space using a novel computational procedure. 

2. Econometric Identification of Structural Shocks

A large body of macroeconomics literature has focused on identification strategies of structural parameters for linear SVAR models. In particular, frequentist approaches include identification via proxies (e.g., see Montiel Olea, Stock & Watson (2021, JoE)), via heteroscedasticity (e.g., see Lanne & Lütkepohl (2008, JMCB) and Lütkepohl & Netšunajev (2017, JEDC)), and via Non-Gaussianity (e.g., see Lanne, Meitz & Saikkonen (2017, JoE)), while Bayesian approaches include set-identified SVARs (e.g., see Gafarov, Meier & Montiel Olea (2018, JoE)), and robust Bayesian inference of structural parameters (e.g., see Giacomini, Kitagawa & Read (2022, JoE)). In addition, identification strategies for nonlinear SVARs have been developed. For example, Virolainen (2025, arXiv:2404.19707) propose statistically identification via Non-Gaussianity in structural smooth transition VAR models.

Specifically, estimating interpretable dynamic causal effects, in nonlinear environments, is a more challenging task. Recently, Kolesár & Plagborg-Møller (2025, JBES) propose expressing the causal estimands as continuous linear functionals of the shocks of interest. This implies that, under standard identification assumptions, linear methods estimate a weighted average of a shock's true marginal effects, even when the effects are arbitrarily non-linear. The particular property applies to linear regression settings, as in Kunievsky (2025, arXiv:2512.13645) who obtains a bias correction for the estimator of weighted average derivatives of the outcome conditional expectation function with respect to the variable of interest. Moreover, Dearing (2025, SSRN 5124490) derives a Frisch-Waugh-Lovell theorem for GMM estimators.     

2.1. Point-Identified Macroeconometric Models 

In this section, we focus on the point identification and estimation methods commonly used in macroeconometrics, for both Gaussian and Non-Gaussian time series. Moreover, the parametric and nonparametric identification of structural parameters affects the asymptotic properties of impulse response function estimators when considering a region of the parameter space which induces weak identification. The IRF allows us to assess how a unit shock in the error term at time h is expected to propagate to the observed variables over time. Under regularity conditions the SVAR model admits a VMA process of an infinity length, which has a unique representation for a future sequence of observations. Our main research objective is to establish the asymptotic behaviour of both structural parameter estimators and impulse responses estimators, thereby facilitating inference. In particular, Keweloh, Hetzenecker & Seepe (2023, JIMF) propose a novel block-recursive SVAR-GMM estimator, which combines conventional identification strategies based on restrictions with the statistical identification approach, which exploits non-Gaussianity and independence in the data. The proposed method is used to estimate a Proxy-SVAR, which allows to derive a proxy for information shocks. However, establishing the asymptotic equivalence between a block-recursive identified SVAR estimator and an IV-based estimator for structural parameters and impulse responses, remains an open problem worth further study. We focus on the more challenging setting where we exploit the presence of non-Gaussianity and persistence in the data. 

2.2. Set-Identified Macroeconometric Models 

The Bayesian econometrics literature focuses on developing Bayesian methods for estimation of structural parameters and hypothesis testing in VAR, SVAR and DSGE models, using computationally efficient algorithms. In particular, Giacomini & Kitagawa (2021, Ecta) propose robust Bayesian inference for set-identified models, while Giacomini, Kitagawa & Read (2022, JoE) develop methods for robust Bayesian inference in Gaussian SVAR models where the parameters are set-identified using external instruments. Moreover, Bayesian estimation approaches for local projections are proposed by Ferreira, Miranda-Agrippino & Ricco (2025, RES) and Huber, Matthes & Pfarrhofer (2025, arXiv:2410.17105). The recent Bayesian econometrics literature has also focused on developing joint Bayesian inference approaches for impulse responses in VAR models as in Inoue & Kilian (2022, JoE) and Arias, Rubio‐Ramírez & Waggoner (2025, Ecta) who extend standard Bayesian inference in set-identified SVAR models. The authors show that joint inference procedures provide necessary and sufficient conditions for using a uniform prior over the set of orthogonal matrices. This property allows to conduct inference based on a uniform joint prior distribution over the identified set for the vector of impulse responses. Therefore, Bayesian inference is conducted based on a uniform joint prior distribution for the vector of impulse responses. Lastly, Wu & Koop (2025, JBES) propose fast and order-invariant Bayesian inference in VARs using the eigen-decomposition of the error covariance matrix.  

2.3. Partial and Weak Identification 

Detecting for presence of weak instrument bias in impulse response estimators has been recently examined by Lewis & Mertens (2026, cemmap/wp01-26). Potential mis-identification in structural macroeconomic models affect the accuracy and reliability of estimated point estimators, which are used for evaluating the impact and transmission mechanism of policy shocks. For example, Adolfson et al. (2019, EER) examine the identification and misspecification problems in standard closed and open-economy empirical NK DSGE models used in monetary policy analysis. Moreover, Wolf (2020, AEJ: Macro) examines whether SVAR mis-identification of monetary policy shocks has an impact when evaluating the transmission mechanism to real economic activity. 

The partial identification in structural econometrics should not be confused with the partially identified SVAR setting. In particular, Lütkepohl, Shang, Uzeda & Woźniak (2025, JoE) propose partial identification of SVARs with non-centred stochastic volatility. Moreover, Bacchiocchi et al. (2024, arXiv:2403.06879) propose an identification strategy in SVAR models by exploiting breaks in the variances of the structural shocks. For this class of models, point-identification relies on an eigen-decomposition involving the covariance matrices of reduced-form errors based on the condition that eigenvalues are distinct, while fails in the presence of multiplicity of eigenvalues. In addition, Hoesch, Lee & Mesters (2024, QE) propose locally robust semiparametric estimation and inference in SVARs, which permits to conduct hypothesis testing on partially identified parameters and to construct confidence sets. This semiparametric approach implies the presence of both finite and infinite-dimensional components, which are estimated using semiparametric score functions. Constructing a robust inference approach in the presence of weak identification in Non-Gaussian SVARs, is indeed an interesting direction for future research. Lastly, developing asymptotically efficient estimators for LP-based and VAR-based estimators in partial and weak identification settings using semiparametric methods worth further study.  

3. Econometric Estimation and Inference Methods

In this section, we discuss aspects of asymptotic theory for estimation and inference procedures across the econometric frameworks we consider here. 

3.1. Univariate Time Series Models

We begin our discussion by reviewing recent advances in the time series econometrics literature with respect to hypothesis testing in univariate settings. In particular, Astill, Harvey, Leybourne & Taylor (2025, essex/wp42258) propose a regression-based test for asset price bubbles using an unobserved components time series model, whose components correspond to the fundamental and bubble parts of the general solution. Based on the locally best invariant testing principle, the authors derive the asymptotic theory of a statistic for testing the null hypothesis that no bubble component is present, against the alternative that a bubble episode occurs in a given subsample of the data (see also Scheinkman & Xiong (2003, JPE)). Moreover, Bykhovskaya & Duffy (2025, arXiv:2512.12110) propose econometric estimation and inference in dynamic tobit model (nonlinear time series model) with a unit root. The authors develop novel econometric theory that establishes weak convergence for the nonlinear functionals of interest to the corresponding nonlinear OU process equipped with the J1 topology (see also Bykhovskaya & Duffy (2024, JoE)).          

Furthermore, the sequential estimation of autoregressive parameters in general VAR models has been examined by Basu & Mukhopadhyay (1999, IJoS Series A) and Lee & Sriram (1999, SPA). These frameworks focus on establishing that the sequential OLS estimator is asymptotically risk efficient and the stopping rule is shown to be asymptotically efficient. However, equivalent results in the case of nearly unstable autoregressions (e.g., nonstandard processes) as well in the case of sequential confidence interval estimation or even in the case of a two-stage procedure are limited. In particular, estimation and inference in simultaneous equation models using single-equation arguments, is a research area which has been extensively studied in the econometrics literature. Recently, De Vos & Everaert (2026, JoE) develop a GLS estimation approach of local projection regressions. These authors propose GLS transformations of local projections, which allows to mitigate the robustness-efficiency trade-off.   

3.2. Multivariate Time Series Models

Structural identification and estimation with multivariate time series requires the use of a valid identification strategy to recover structural shocks, and parametric or semiparametric methods for estimating structural parameters and impulse responses. Bayesian methods typically rely on algorithmic procedures (e.g., see Inoue & Kilian (2013; 2022, JoE)), while iterative and sequential estimators are used in online settings (e.g., online DSGE model estimation with real-time data). In fact, the recent pandemic posed significant challenges when estimating and forecasting in VAR models during the post-pandemic period. In particular, an outlier-robust estimation approach requires that the reduced-form error is decomposed into a regular component and an outlier component, in which case VAR coefficients are estimated using machine learning tools. Moreover, Cui (2025, JTSA) propose a smoothing spline semi-parametric Non-Gaussian SVAR, while Keweloh, Klein & Prüser (2025, arXiv:2302.13066) propose an estimation approach for Bayesian non-Gaussian SVARs with potentially endogenous proxy variables. In addition, estimating dynamic causal estimands in possibly unstable environments is an issue addressed by several authors (e.g., see Inoue, Rossi & Wang (2024, JoE)). Efficient estimation of the parameter path in unstable time series environments is studied by Müller & Petalas (2010, RES). 

Furthermore, Greenaway-McGrevy (2013, ET) develops an econometric framework for multistep prediction of panel VAR processes, while Greenaway-McGrevy (2019, ET) proposes an asymptotically efficient forecast selection procedure for panel VARs. In the context of VARs, Ludwig (2024, SSRN 4882149) shows that local projections are VAR predictions of increasing order; which allows to estimate long-lag VARs. Thus, developing bias-corrected estimators in the presence of truncation bias, so that the correction will asymptotically eliminate the bias, can yield valid confidence intervals for impulse responses. Lastly, the availability of real-time data motivated the implementation of online estimation schemes. For example, Cai et al., (2021, Econometrics Journal) develop a Bayesian framework for online estimation of DSGE models.

3.3. Inference in Macroeconometric Models 

Inference in macroeconometric models include Wald-type statistics for testing linear restrictions on matrix coefficients, GMM-type test for testing overidentifying restrictions, Wald-type tests for testing the null of parameter constancy against the alternative of structural breaks in VAR/SVAR models, as well as F-type statistics for testing for the absence of weak instrument bias in impulse response estimators of SVAR models (e.g., see Lewis & Mertens (2026, cemmap/wp01-26)). In particular, we are interested in the asymptotic properties of Wald-type statistics when testing the null hypothesis of no weak instrumentation in impulse response estimators. For example, Windmeijer (2025, JoE) propose a robust F-statistic as a test for weak instruments in linear IV regressions with a single endogenous regressor. Econometric inference in SVAR models robust to the presence of weak identification using Wald-type statistics worth further study. For example, Lewis & Mertens (2025, RES) develop a robust eigenvalue ratio test for weak instruments based on the 2SLS estimator in local projection regression models with multiple endogenous instruments. Our proposed Wald-type test will contribute to the available toolkit of inference procedures for SVARs.      

4. Inequality, Heterogeneity and Aggregate Fluctuations

We study the literature that concentrates on inequality, heterogeneity and aggregate fluctuations, with a focus on economic theory, as well as the identification and estimation methods. To begin with, Chu & Peretto (2023, EER) examine the income inequality dynamics in an economy featuring an endogenous transition from stagnation to growth. These authors find that endogenous labour supply introduces a channel through which inequality contributes to shaping the transition path of the economy and that households sort into a class that supplies zero labour and a class that supplies labour. In particular, Eeckhout & Lindenlaub (2019, AEJ: Macro) argue that even in the absence of exogenous shocks labour markets can create cyclical outcomes. The authors propose a theory in which the search behaviour of the employed has profound aggregate implications for the unemployed, due to a strategic complementarity between active on-the-job search and vacancy posting by firms, which leads to multiple equilibria. Moreover, Angeletos & La’o (2020, JPE) study optimal monetary policy in a business-cycle setting in which information frictions create nominal and real rigidity; verifying that learning against the wind helps maximizes production efficiency. Recently, Majeed, Hambur & Breunig (2025, JoM) examine whether monetary policy affects innovation activity and productivity in a small open economy, while Bertolotti (2026, RES) provides novel evidence on the rich dynamics induced by policy shocks aimed at stimulating R&D and innovation in the long run. Understanding the links between inequality, heterogeneity, and aggregate fluctuations over changing demographics, is another important dimension. For example, Liu, Mian & Sufi (2022, Ecta) develop a theory which shows that in low interest rate environment, leads to rising market concentration and falling productivity growth. Using a rich semi-endogenous growth model of firm dynamics, Peters & Walsh (2021, nber/w29424) show analytically that a decline in population growth reduces creative destruction, increases average firm size and concentration, raises market power and misallocation, and lowers aggregate growth in the long-run (i.e., connecting the decline in population growth and changes in firm dynamics). Furthermore, Bhandari et al. (2024, nber/w32948) study the nature of entrepreneurship using a novel panel dataset which is found to have a positive role in enhancing economic stabilization. These authors find stable entry and exit rates and self-employment with no significant decline in the share of entrepreneurs over the sample period. Lastly, Fiess, Fugazza & Maloney (2010, JDE) study the impact of self-employment on macro fluctuations, while Bilal & Lhuillier (2025, nber/w29348) develop a tractable quantitative macro model to study the equity-efficiency trade-off between large wage declines that outsourced workers experience, and the increase of aggregate productivity due to domestic outsourcing.  

Second, a growing body of research in the macroeconomics and the macroeconometrics literature has focused on developing structural identification and estimation methods for combining macro-level with micro-level data. Relevant frameworks are developed by Chang, Chen & Schorfheide (2024, JPE) and Andersen et al. (2023, JoF). In particular, Chang, Chen & Schorfheide (2024, JPE) propose identification and estimation for state-space models in functional space which allows to examine the impact of structural shocks on aggregate macro variables. The state-transition equation takes the form of a functional VAR which stacks macro aggregates and a cross-section density. The measurement equation captures the error in estimating log densities via repeated cross-sectional samples, while the log densities and their transition kernels are approximated by sieves. This macroeconometric environment leads to a finite-dimensional VAR model for macro aggregates with sieve coefficients. Lastly, Ettmeier, Kim & Schorfheide (2024) develop a cross-sectional VAR model which combines aggregate variables with unit-specific outcomes, thereby allowing to study the dynamic effects of aggregate shocks across the cross section. Therefore, the asymptotic properties of estimation and inference procedures worth further study. 

The recent pandemic exacerbated income inequalities. According to Kartashova & Zhou (2021, SSRN 3967802), the economic downturn during the pandemic showed that policy responses had heterogeneous effects over the income and wealth distributions, due to unequal impact across population subgroups. In particular, Lippi & Perri (2023, JME) using a micro-founded growth model find that changes in households' earnings dynamics that are consistent with the micro data imply unequal growth across the earnings distribution (i.e., violation of mean preserving property). Specifically, Bertheau et al. (2023, AER) study the unequal consequences of job loss across countries. Moreover, the provision of unemployment insurance benefits during the pandemic had an unprecedented fiscal policy response. In particular, Kong & Prinz (2020, JPE) examine which shutdown policies affected unemployment, while Finamor & Scott (2021, EL) test whether generosity on provision of unemployment insurance contributed to differential  employment outcomes. In fact, Adams-Prassl et al. (2022, LE) argue that both WFH policies and the distribution of the share of tasks that can be done from home have an impact on labour market outcomes (e.g., see also Bloom et al. (2015, QJE)). Using a large-scale survey of firms, Kudlyak et al. (2025, SSRN 5142316) study how and why firms adjust their labour (i.e., lay off workers instead of cutting wages) in response to adverse economic shocks. Their findings show that layoffs are more prevalent that pay cuts, but pay cuts are not rare in the firms that experience a reduction in revenue. Using a statistical identification approach, Miescu & Rossi (2021, EER) show that pandemic-induced shocks had distributional effects on economic and financial conditions. For example, Gulyas, Meier & Ryzhenkov (2024, EER) analyze the effects of monetary policy across firms and workers and show that low-paid workers who were originally employed by low-paying firms are prone to falling down the firm wage ladder. Lastly, Bald et al. (2022, JPE) study the causal impact of removing children from abusive/neglected homes, while Deshpande & Mueller-Smith (2022, QJE) show that the costs to taxpayers of enforcement and incarceration from supplemental security income (SSI) removal are so high that they nearly eliminate the savings to taxpayers from reduced SSI benefits.   

4.1. Research in Macroeconometrics

A formal macro framework that focuses on the relationship between inequality dynamics and business cycle fluctuations is proposed by Bayer, Born & Luetticke (2024, AER). These authors use a heterogeneous agent New Keynesian model with incomplete markets and portfolio choice and propose a Bayesian approach for estimating structural parameter in state space. A considerable body of research in the macroeconomics literature focuses on frameworks that use quantitative macro models and econometric estimation approaches that combine both micro-level and macro-level data. In particular, Gagliardone et al. (2025, AER) study the micro evidence and macro implications of the NKPC using quarterly micro data on prices, costs, and output from the Belgian manufacturing sector. Moreover, Liu & Plagborg‐Møller (2023, QE) propose a full-information estimation approach for heterogeneous agent models using macro and micro data (e.g., see also Bilal (2023, nber/w31103) and Reiter (2023, econstor/wp274596)), while Poledna, Miess, Hommes & Rabitsch (2023, EER) develop an economic forecasting framework using an agent-based model. The macro impact of the unequal pandemic recession is examined by Cirelli & Gertler (2025, AEJ: Macro), who use both firm and industry data with a three sector NK model to obtain quantitative estimates on economic winners and losers. Toward a unified framework, Winberry et al. (2025, nber/w34611) show that the techniques used in HANK models for the transmission of monetary policy where heterogeneous households face idiosyncratic income risk and borrowing constraints, apply to investment using a model where heterogeneous firms face idiosyncratic productivity risk and financial frictions. Thus, examining the role of marginal propensities to invest for heterogenous firms using quasi-experimental evidence worth further study.       

Specifically, Villalvazo (2025, JME) studies the relationship between inequality and asset prices via the cross-section dimension of Fisher's debt-inflation mechanism that triggers sudden stop crises. The author proposes a small open economy, asset-pricing model with heterogeneous-agents and aggregate risk to measure the effects of inequality during crises. The quantitative macro model is solved via a recursive form representation and parameter calibration. Using data from a quarterly household rotating panel survey with a representative sample of households, the author implements an over-identified GMM to obtain estimates for structural parameters. Moreover, McCrary & Janssens (2025, SSRN 5282668) develop a computationally efficient full-information estimation approach for nonlinear DSGE models. The novelty of the method is that in contrast to the conventional approach of separating the model solution and the filtering procedures into two potentially computationally intensive steps, it integrates the solution into the filtering procedure. As a result, the computational method constructs a sequence of local linear solutions at the best forecast of the current state, yielding a time-varying linear state-space system for likelihood evaluation via the Kalman filter. Lastly, Gonzalez, et al. (2025, arXiv:2508.16817) examine which nonlinear state space models can be efficiently parallelized in order to reduce computational complexity and time. 

We focus on methodological approaches for estimation and forecasting in macroeconometric  models. Forecasts of the variables relevant for the policy problem, and their impulse responses to anticipated policy shocks, constitute sufficient information to construct valid couterfactuals. In particular, Hebden & Winkler (2026, JEDC) propose an efficient procedure to solve for policy counterfactuals in sequence space. These authors compute counterfactuals of the U.S. economy after the pandemic shock under several monetary policy regimes. Moreover, Beraja (2023, JPE) propose a semi-structural methodology for constructing policy counterfactuals in SVAR models, based on policy rule changes over time. Lastly, Forneron & Qu (2025, arXiv:2412.20204) develop specification testing for DSGE-VARs by fitting dynamically misspecified state-space models using an optimal transportation approach.   

4.2. Research in Microeconometrics

To begin with, a large body of applied microeconometrics literature examines the impact of inequalities on economic and/or health outcomes. In particular, Melly & Pons (2025, arXiv:2502.18242) examine the impact of the food stamp program on birth weights. The authors propose a minimum distance estimation approach for quantile panel data models where unit effects may be correlated with covariates. The empirical findings of these authors show that the intervention increased birth weights predominantly at the lower end of the distribution. Moreover, Bick, Blandin & Rogerson (2025,  nber/w32997) study the relationship between lifetime hours worked and lifetime earnings based on a calibrated structural model, while Balke, Bonhomme & Lamadon (2025, wp) develop an econometric framework for variational inference in nonlinear state-space models which allows to evaluate the variational posterior choice problem. The authors apply the proposed estimation approach to a subsample of the PSID dataset, and find the presence of nonlinearities in the conditional volatility and persistence of earnings. In fact, replicating these findings for workers in developing countries who face substantial constraints to job search, worth further study. From the economic theory perspective, VanVuren (2025, wp) develops a search-and-matching model that incorporates key features of developing economics including a large self-employment sector, savings-constrained households, and capital-constraint firms. From the computational econometric perspective, these frameworks typically rely on sparse grid methods for solving medium-dimension economic models (e.g., see discussion in VanVuren (2024, wp)). 

Second, a large body of literature spanning both labour economics (e.g., see Léné (2011, LE)) as well as international trade and migration (e.g., see discussion in Lebow (2024, JDE)), focuses on measuring the impact of occupational up- and downgrading over the business cycles on economic outcomes. Contrary to expectations, the rate of downgrading of workers (e.g., such as immigrants) is found to be greater in the boom, while upgrading is more likely to exhibit procyclicality. In particular, Osikominu (2013, RES) investigates the dynamic effects of alternative training schemes. Although from a fiscal perspective, only the low-cost short-term training schemes are cost efficient in the short run; the author finds evidence showing that long-term training programs imply entering employment at a faster rate, than without training, and substantially more stable employment spells and higher earnings. These findings have implications to the design of optimal labour program treatment effects and guaranteed universal basic income schemes (e.g., see Daruich & Fernández (2024, AER)). Moreover, Yagan (2019, JPE) studies the long-term impact of employment hysteresis from the Great Recession, while Mueller & Spinnewijn (2025, JPE) examine the nature of long-term unemployment using rich administrative data from Sweden. Cohen, Johnston & Lindner (2025, AEJ: AE) examine the depreciation of skills among unemployed German workers using a panel of skill measures linked to administrative data. The authors find no decline in a wide range of cognitive and noncognitive skills while workers remain unemployed. These empirical findings imply that skill depreciation in human capital is unlikely to be a major explanation for observed duration dependence in reemployment outcomes. Lastly, Seong & Seo (2024, ET) propose a functional IV approach to estimate the impact of immigration on native labour market outcomes in the US.

From the econometric theory perspective, we discuss recent advances on panel data methods for microeconometrics. In particular, Callaway & Karami (2023, JoE) examine the finite-sample and large sample properties of treatment effect estimators in panel data models with interactive fixed effects and a small number of time periods. Moreover, Botosaru, Giacomini & Weidner (2024, SSRN 4815865) propose estimation and inference for the effects of a policy in the absence of an untreated or control group. The authors use the proposed method to replicate the findings of some previous empirical studies (i.e., the impact of divorce laws on average suicide mortality rate), but without an untreated or control group. However, in applied settings, researchers are interested in measuring the time-varying impact of treatment effects due to unobserved heterogeneity. Specifically, Botosaru & Liu (2025, arXiv:2509.13698) develop identification and estimation of heterogeneous treatment effects (HTEs) in event studies. These authors propose a semiparametric dynamic (short) panel data model with correlated random coefficients to simultaneously capture outcome persistence and treatment effect heterogeneity. In addition, Botosaru & Liu (2026, arXiv:2601.05493) develop a dynamic panel data event study regression framework which permits to decompose direct treatment effects from indirect effects using endogenous covariate adjustment. Under sequential exogeneity and homogeneous effects, point identification can be established for the common parameters that govern the outcome and treatment effect dynamics, the distribution of HTEs, and the covariate feedback process. The method allows to demystify the determinants of the treatment effect dynamics (such as the firm productivity responds to the minimum wage; e.g., see discussion in Rao & Risch (2026, QJE)). Understanding the dynamic impact of economic phenomena requires novel methods that capture time-varying information in data.     

5. Time-Varying Effects

5.1. Parametric and Nonparametric Time-Varying Models 

5.1.1. Linear Models with Time-Varying Parameters

To begin with, Gorgi, Koopman & Schaumburg (2024, JoE) propose a novel methodology for analyzing VAR models with time-varying coefficient matrices and conditionally heteroscedastic disturbances. These authors consider an unobserved factor process to specify the time-variation in the VAR coefficients and a GARCH specification for modelling volatility in the disturbance vector. Moreover, Chan & Eisenstat (2017, JAE) propose efficient estimation of Bayesian VARMA with time-varying coefficients. Specifically, TVP-VAR models with stochastic volatility are routinely used to capture the time-varying effects of the impact of policies (e.g., see Gao, Peng & Yan (2024, JoE)), while TVP-SVAR-SV models allow to capture the time-varying effects of structural shocks on aggregate macro variables (e.g., see Boufateh & Saadaoui (2021, EE)). However, a drawback of time-varying parameter VARs with stochastic volatility specifications is that the time-varying coefficients are modelled as random walks. To improve the predictive ability of these model specifications when forecasting inflation, Korobilis (2021, JBES) propose a methodology that can be used to estimate time-varying parameter regressions with arbitrarily large number of exogenous predictors. Recently, Amir-Ahmadi, Mlikota & Stevanović (2025, arXiv:2512.20152) propose a TVP-VAR-SV model where the time-varying coefficients have a factor structure, which is shown to improve the predictive ability of forecasts. In fact, the empirical findings of Korobilis (2021, JBES) verify that once the random walk dynamics are removed, the role of time-varying parameters in forecasting becomes less important and the most significant feature is the information included in exogenous predictors. 

All the above settings require necessary and sufficient conditions (e.g., invertibility and ergodic stationarity) for the stability of solutions in the presence of time-variation. In constant coefficient settings, Funovits (2024, JoE) propose a identifiability and estimation approach of possibly non-invertible SVARMA models. In time dependent coefficient models, Yan, Gao & Peng (2025, ET) develop asymptotic theory for time-varying VMA infinite-order models. In fact, modelling multivariate time series with time-varying dynamics has seen growing attention in the macroeconometrics literature, due to the dynamic effects of macroeconomic conditions which cannot be captured adequately by the conventional time series regression models with time-invariant coefficients. In particular, Arias, Rubio-Ramirez, Shin & Waggoner (2026, RES) propose a Bayesian approach for estimation and inference in time-varying SVAR models identified via sign restrictions.    

5.1.2 Nonlinearity and Temporal Dependence

To begin with, Chen, Hansen & Carrasco (2010, JoE) study how nonlinearity induces temporal dependence in continuous-time Markov models using spectral density representations. Second, Ashley & Verbrugge (2008, ER) propose a new class of nonlinear time series models in which one of the coefficients is frequency dependent. Third, a considerable body of research in the time series econometrics literature focuses on the properties of estimators and test statistics arising from models which are specified with time-dependent coefficients (e.g., see Andrews & Li (2025, QE) and references therein; as well as Zhang & Wu (2015, AoS) and references therein). Modelling nonlinear dynamics with time-varying linear models via computationally efficient algorithms, in the context of SVAR models worth further study. In particular, Braun, Kapetanios & Marcellino (2025, RES) examine the estimation and inference of time-varying impulse response functions in SVARs identified with external instruments. Using kernel estimators for nonparametric variation, the authors develop an estimation approach that accommodates possible weak identification in the impulse responses. 

5.2. Parametric and Nonparametric Time-Invariant Models 

• Bernstein-type inequalities: Nonparametric estimation under near-epoch dependence (e.g., see Yuan & Spindler (2025, JoE) and Gao, Peng, Wu & Yan (2024, JoE)). These non-asymptotic results are useful when deriving error bounds on the growth rate of network ties.  For example, Krebs (2018, CiS) establish a Bernstein-type inequality for exponentially growing graphs. In particular, Higgins & Martellosio (2023, JoE) propose a penalised quasi-maximum likelihood estimator which is robust to possible network misspecification by shrinking the coefficients of irrelevant weights matrices to exactly zero. Moreover, Chernozhukov, et al. (2025, JBES) propose a high-dimensional GMM estimator for dimension reduction and debiased to correct for shrinkage bias and inference on large scale spatial panel network models.  

• Score-driven vis-à-vis observation-driven time series models: To begin with, Meitz & Saikkonen (2021, JoE) develop a novel econometric inference procedure for observation-dependent regime switching in mixture autoregressive models. Second, recent literature on estimation methods for observation-driven models include Krabbe (2025, arXiv:2412.19555) who derive the asymptotic properties of the MLE for Markov-switching observation-driven models and Blasques et al. (2018, EJS) who establish feasible invertibility conditions and the asymptotic properties of MLE for observation-driven models. Third, recent literature on estimation methods for score-driven models include van Heel et al. (2025, arXiv:2502.05021) who derive stability conditions and performance guarantees for misspecified multivariate score-driven filters. Moreover, Buccheri, Corsi & Dzuverovic (2024, arXiv:2412.01367) propose conditions for scalar invariance which enhances parameter identifiability in score-driven factor models. In addition, Francq & Zakoian (2023, ET) establish the local asymptotic normality property for general conditionally heteroscedastic and score-driven time series models. Lastly, Amengual et al. (2025, JoE) propose score-type tests for normal mixtures (testing normality against discrete normal mixtures) with an application to employment regression equations.  

6. Minimax Estimation and Asymptotics in Econometrics

In this section, we review recent literature on the use of minimax asymptotics for regular econometric models as well as for regression models with weakly dependent data. In particular, Meitz & Shapiro (2025, arXiv:2504.11269) consider asymptotics of the optimal value and the optimal solutions of parametric minimax estimation problems, which facilitate the development of statistical inference methods. Moreover, Dalle & De Castro (2025, EJoS) propose minimax estimation of partially-observed vector autoregressions, while Tan, Guo & Zhu (2025, JoE) propose hypothesis testing for the parametric forms of the mean and variance functions in regression models under diverging-dimension settings. To mitigate the curse of dimensionality, weighted residual empirical process-based tests are introduced, both with and without martingale transformations (see also Chan & Zhang (2013, Bernoulli) and Goldenshluger & Zeevi (2001, AoS)). Lastly, Krampe, Paparoditis & Trenkler (2023, JoE) develop structural inference in sparse high-dimensional VARs via bootstrap.  

From the econometric theory perspective, the minimax asymptotics developed by Meitz & Shapiro (2025, arXiv:2504.11269) allow to establish the asymptotic properties of minimax linear estimators as well as to construct bias-aware confidence intervals for structural parameters. In particular, Kong (2025, arXiv:2510.16661) study a weighting estimator via a minimax procedure which entails solving the convex optimization problem that trades-off worst-case conditional bias against variance (see also Kunievsky (2025, arXiv:2512.13645)). These techniques are useful when considering the joint distribution of income, consumption and income expectations. For example, Bonhomme & Denis (2023, wp) propose a regression-based approach to estimate how individuals' expectations influence their responses to counterfactual change using a three-step estimation method that relies on iid data. Extending these approaches to temporally dependent data worth further study.  

In this article, we focus on parametric and semiparametric identification strategies as well as on algorithmic procedures for estimating structural parameters, under a broad array of econometric settings, thus we shall also mention some further applications. In particular, Khan, Lan, Tamer & Yao (2024, JoE) propose estimating high-dimensional monotone index models using iterative convex optimization. Moreover, Yao (2025, arXiv:2511.21948) and Zeleneev & Zhang (2025, arXiv:2511.15427) propose low-rank estimation of nonlinear panel data models using the ADMM algorithm. In addition, Zheng (2025, JASA) propose an interpretable and efficient infinite-order VAR model for high-dimensional time series, while Ando & Hoshino (2025, arXiv:2502.13431) propose a novel functional VAR framework for analyzing network interactions of functional outcomes in panel data settings, with an application to the demand for a bike-sharing service. Lastly, Chen, et al. (2024, arXiv:2401.14535) propose CaRiNG; which is a statistical framework for learning temporal causal representation under non-invertible generation process.  

7. How Persistent is Persistent Enough?

Recall that inference in LP-based VAR regressions using the lag-augmentation approach is examined by Montiel Olea & Plagborg‐Møller (2021, Ecta). The lag-augmentation approach was proposed as a suitable method for robust inference in VAR models against the unknown integration order of time series data. For example, Kurozumi & Yamamoto (2000, ER) propose a modified lag augmented VAR estimation method which is robust to the misspecification of the lag length, while Lütkepohl & Saikkonen (1999, Economic Letters) propose a lag augmentation test for the cointegration rank of a VAR process. Recently, Xu (2023, SSRN 4372388) develops uniform asymptotic theory for local projections under general conditions. Moreover, finite-sample size distortions due to the presence of high persistence in time series data can be examined using the local-to-unity asymptotic framework [1]. Comparing the performance of VAR-based vis-à-vis LP-based impulse responses in finite samples while permitting for the presence of high persistence in time series data, worth further study. Estimation and inference in settings where the local-to-unity parametrization is used to capture the persistence properties of time series require econometric theory tools from nonstationary time series econometrics. Recently there is also growing interest in implementing weak instrument bias F-tests for impulse response estimators [2]. These frameworks rely on the local-to-zero asymptotic framework to establish exact sample distribution theory [3]. Extending these tests to the case of high persistent data based on the local-to-unity asymptotic framework, is a considerably more challenging task. For example, Yang & Xu (2015, SSRN 2847956) develop estimation and inference under weak identification and persistence in dynamic linear regression models with an application to coefficients in the forecast-based monetary policy reaction function.

We focus on uniform inference for econometric models commonly used in the macroeconometrics literature such as VARs and SVARs. Specifically, we consider estimation and inference for nonstationary time series processes (e.g., nearly unstable processes). An econometric framework on uniform inference for cointegrated VAR processes is proposed by Holberg & Ditlevsen (2025, JoE), who employ the notion of uniform convergence to derive asymptotic theory. An instrumental variable approach for conducting uniform inference in univariate predictive regressions and local projection regressions is proposed by MP (2025). However, for the macroeconometric settings we examine, we restrict the parameter space of nonstationarities in the vicinity of unity. We do not consider more technically challenging cases such as explosive roots (e.g., see Wang (2025, ET) and references therein [4]); which are beyond the scope of our current research projects. Examples of theoretical-driven frameworks with applications in macroeconometrics include the study of Duffy, Mavroeidis & Wycherley (2025, JoE) who develop estimation and inference for censored and kinked structural VAR models. The proposed structural econometric model specification allows to capture threshold-type nonlinearities, which arise due to occasionally binding constraints (e.g., such as the zero lower bound constraint on short-term nominal interest rates). Examples of empirical-driven frameworks with applications in macroeconometrics include the study of Antolin-Diaz & Surico (2025, AER) who examine the long-run effects of government spending in the presence of cointegration as well as Guerini et al. (2020, MD) who investigate the causal effects of public and private debts on US output dynamics using statistically identified Cointegrated SVAR models.    

(31 December 2025)

Dr. Christis Katsouris, Ph.D. University of Southampton

© Christis G. Katsouris Institute of Econometrics and Data Science

Appendix A. Econometric Theory and Methods

Remark 1. 

Recall that the local-to-unity asymptotic framework has been previously used in the context of SVAR models. For example, Chevillon, Mavroeidis & Zhan (2020, ET) propose a robust inference approach for a bivariate SVAR model with nonstationary regressors identified via long-run restrictions. Specifically, the high persistent regressors are assumed to be generated from a local-to-unity process which can lead into nonstandard inference when test statistics are constructed based on the OLS estimator (e.g., see Cavanagh, Elliott & Stock (1995, ET), Phillips, P.C.B. (2023, ET), Brien, Jansson & Nielsen (2024, ET) and references therein). However, the authors establish a robust inference approach against the unknown persistence using the IVX estimator. The authors show that the asymptotic distribution of the test statistic does not involve nuisance parameters (e.g., persistence or endogeneity parameter), which ensures nuisance parameter free inference. In addition, the IVX estimator allows to conduct conventional inference which is robust even when error terms are conditionally heteroscedastic, and can be easily implemented in multivariate settings as well (e.g., see discussion in Magdalinos (2022, ET)). Moreover, robust inference methods for predictive regressions that address the key challenges posed by endogenously persistent regressors are also proposed by Ibragimov, Kim & Skrobotov (2025, arXiv:2511.09249), who extend their earlier work on this topic to the case of predictive regressions with multiple regressors (e.g., see KMS (2015, RES)).    

Remark 2. 

Recall that in the context of linear IV regression with weak instruments, Andrews & Marmer (2008, JoE) propose an exactly distribution-free inference approach. Several authors have also derived finite-sample distribution theory in linear IV models under weak identification. More recently, Lewis & Mertens (2026, cemmap/wp01-26) propose an F-type statistic for testing for weak instrument bias in impulse response estimators of SVAR models based on the local-to-zero asymptotic framework. 

• An exactly distribution-free inference approach for testing weak instrument bias in SVARs doesn't currently exist in the literature (e.g., see Andrews & Marmer (2008, JoE)). 

• When considering the performance of competing estimators/ estimation methods, researchers focus on relative efficiency. In particular, Gallant & Tauchen (1999, JoE) compare the asymptotic relative efficiency of efficient method of moments when implemented with a semi-nonparametric auxiliary model to that of conventional method of moments when implemented with polynomial moment functions. Moreover, Prokhorov (2009, Economic Letters) compares the relative efficiency of QMLE and GMM estimators of covariance structure models.  

• Extending the comparisons of VAR-based vis-à-vis LP-based impulse response estimators for the case of persistent data, worth further study. In particular, both Lewis & Mertens (2026, cemmap/wp01-26), who examine the weak identification bias problem for impulse response estimators of covariance stationary time series, and Montiel Olea, Plagborg-Møller, Qian & Wolf (2026, nber/w32495), mention that they do not consider high persistent time series; another potentially significant source of finite sample bias. According to Montiel Olea, Plagborg-Møller, Qian & Wolf (2026, nber/w32495) their asymptotics abstract from any biases that arise from persistent macro data. The problem of finite-sample bias in local projections is discussed by Herbst & Johannsen (2024, JoE) and Kilian & Kim (2009, SSRN 1405063)).  

• Moreover, a large body of research in the time series econometrics literature consider estimation and inference of impulse responses that correspond to persistent processes (e.g., see Phillips, P.C.B. (1998, JoE), Wright (2000, JBES), Gospodinov (2004, EJ), Pesavento & Rossi (2007, JEDC)). The uniform validity of impulse response inference in autoregressions is examined by Inoue & Kilian (2020, JoE), Montiel Olea & Plagborg‐Møller (2021, Ecta) and MP (2025a); where the first two frameworks correspond to the lag augmentation approach, while the third framework corresponds to the IVX approach. Specifically, the IVX instrumentation approach proposed by MP (2025a) covers uniform inference for general autoregressive processes and thus more general persistence classes than the IVX approach of KMS (2015, RES). Moreover, the IVX instrument of KMS (2015, RES) controls the degree of persistence by constructing a mildly integrated process as an instrument (when regressors belong to the vicinity of unity), while the IVX approach in KMS (2015, RES) controls the degree of persistence using as an instrument a 'shifting combination' of a mildly integrated process with a mildly explosive process, depending on where the root is located (e.g., vicinity of unity vs explosive region). Lastly, Dufour & Wang (2024, arXiv:2409.10820) propose an IV-type method using as instrument the lagged innovation terms.

Remark 3. 

Recall that Phillips, P.C.B. & Gao (2017, JoE) develop exact finite sample and asymptotic distributions for structural equation tests based on partially restricted reduced form estimates. The authors establish exact and asymptotic sample distribution theory for the case of strong identification, and the case where the instruments are all totally irrelevant. Recall that Choi & Phillips, P.C.B. (1992, JoE) study the effect of partial identification on the finite sample and asymptotic distributions of estimators and Wald statistics. The technique of isolating identifiable parts of the coefficient vectors via a rotation of the coordinate system developed by Phillips, P.C.B (1989, ET), who studies partially identified econometric models.  

• Lewis & Mertens (2026, cemmap/wp01-26) approximate the finite-sample distribution of impulse response function estimators that are just-identified with a weak instrument using the local-to-zero asymptotic framework. These authors obtain analytical expressions of the bias of the F-statistic for testing the null hypothesis of weak identification by expressing the distribution of the ratio w.r.t the Cauchy distribution. 

• Forchini & Hillier (2005, cemmap/wp04-05) examined the links between ill-conditioned problems, Fisher information and weak instruments in structural equation models.  

Remark 4. 

Recall that Wang, Q. (2025, ET) provides a general asymptotic theory framework for mildly explosive autoregression in which it is shown that Cauchy limit theory remains invariant across a broad range of error processes, including general linear processes with martingale difference innovations. The notion of uniform asymptotic theory in this framework, should not be confused with the uniform inference approach in Holberg & Ditlevsen (2025, JoE) who employ the notion of uniform convergence for nonstationary processes. Some studies focus on constructing uniform asymptotically valid test statistics and confidence intervals (uniform sized) using the local-to-unity asymptotics. Developing uniform asymptotic valid inference procedures via the IVX estimator in the context of local projection regressions, such as when constructing LP-based impulse response estimates or VAR-based impulse response estimates, worth further study. Lastly, it is worth mentioning that these asymptotic frameworks rely on different techniques and asymptotic approximations as the sample size increases. Large-sample theory via local-to-unity triangular arrays is considerably more involved (e.g., see MP (2025b) and Phillips, P.C.B. & Magdalinos (2007, JoE)). This strand of literature has proposed tools for establishing the asymptotic distribution of estimators and test statistics for near to unity processes. In particular, Aue & Horváth (2007, ET) develop limit theory for mildly explosive autoregression with stable errors, while Magdalinos (2012, JoE) develop limit theory for mildly explosive autoregression under both weakly and strongly dependent innovation errors. For example, the undergraduate paper of Shin (2024) derives OLS asymptotics along drifting parameter sequences for AR(2) processes.  

Econometric theory tools that are commonly used for establishing the asymptotic behaviour of estimators and test statistics in nearly unstable autoregressive models include:

(i).   Lindeberg-Levy Central Limit Theorem, 

(ii).  Functional Central Limit Theorems for Partial Sum Processes,

(iii). Exact finite sample distribution theory for the ratio of sufficient statistics in autoregressions where the autoregressive coefficient is parameterized as a local-to-unity process (via moderate deviations from unity; e.g., see Phillips & Magdalinos (2007, JoE)).         

Appendix B. Inference on Structural Parameters

Structural Econometric Models vis-à-vis SVAR Models 

Weak Instruments: For example, Wang & Zivot (1998, Ecta) consider the problem of conducting asymptotically valid inference on structural parameters in IV regressions with weak instruments. These authors using the local-to-zero asymptotics of Staiger & Stock (1997, Ecta), derive the asymptotic distributions of LR- and LM-type statistics for testing simple hypotheses on structural parameters based on the MLE and GMM estimation methods. The authors show that, in contrast to the nonstandard limiting distribution of Wald statistics, the limiting distributions of LM and LR statistics are bounded by a chi-square distribution with degrees of freedom given by the number of instruments. These authors construct asymptotically valid confidence sets for structural parameters by inverting these statistics. In contrast to Staiger & Stock (1997, Ecta), the methods of Wang & Zivot (1998, Ecta) do not require estimating nuisance parameters since the test statistics are asymptotically boundedly pivotal. Generally, competing estimators are typically evaluated by their bias and risk properties, such as their mean bias and mean-squared error, or their median bias (e.g., see Müller & Wang (2019, JoE)). For example, Andrews, D. W. (1993, Ecta) propose exactly median-unbiased estimation and exact confidence intervals for impulse responses and cumulative impulse responses in autoregressive models, while Fair, R. C. (1996, JAE) propose simulation-based estimation of median unbiased estimates in macroeconometric models. A discussion on bimodality and weak instrumentation in structural equation estimation can be found in Phillips, P.C.B. (2006, ET). Currently, estimation and inference procedures that address the weak instrumentation bias in impulse responses are scarce. For example, Yu, Liao & Phillips, P.C.B. (2024, ET) propose new control function approaches in threshold regression with endogeneity. An econometric framework for estimation and inference in nonlinear SVAR models, robust to weak instrumentation bias of dynamic causal effects worth further study. These considerations are particularly important in applied macroeconomics research work. The issue of using instrumental variables that are possibly invalid, weak or both, such as in growth regressions is referred to as 'Blunt instruments' by Bazzi & Clemens (2013, AEJ: Macro); who discuss how to avoid common pitfalls. Lastly, Martínez-Iriarte, Sun & Wang (2020, JoE) develop asymptotic F-tests for weak identification in the presence of temporal dependence.           

SVAR-IV Models: When proxy variables (external instruments) used to identify target structural shocks are 'weak', inference in proxy-SVARs (or SVAR-IVs) is nonstandard and construction of asymptotically valid confidence sets for the impulse responses of interest requires weak-identification robust methods. In fact, Angelini, Cavaliere & Fanelli (2024, JoE) show that frequentist asymptotic inference can be conducted via Minimum Distance estimation using standard asymptotic methods if the proxy-SVAR can be identified via 'strong' instruments for the non-target shocks. Earlier work extends the Anderson-Rubin tests to construct weak-proxy robust methods that facilitates standard asymptotic inference. Using weak-identification robust methods under departures from Gaussianity has seen growing interest. Regarding inference methods, Windmeijer (2025, JoE) propose the robust F-statistic as a test for weak instruments, while Lee et al. (2023, arXiv:2311.15952) propose a robust conditional Wald-type test for over-identified IVs. In Proxy-SVARs, Bruns & Lütkepohl (2025, JEDC) compare the performance of impulse response estimands when constructed via external vis-à-vis internal instruments, while develop a test for strong exogeneity in Proxy-VARs using information from the proxies of non-target shocks. Moreover, Braun & Brüggemann (2023, JBES) propose identification of SVAR models by combining sign restrictions with external instruments, while Carriero, Marcellino & Tornese (2024, JME) propose a blended identification approach in SVARs that combines identification via heteroscedasticity with narrative restrictions and IVs. The identification strategy with external instruments in SVARs is studied by Miranda-Agrippino & Ricco (2023, JME) as well. An instrumentation method for estimation and inference of dynamic causal effects robust to the strength of identification and the unknown persistence in the data, worth further study. For example, Ganics, Inoue & Rossi (2021, JBES) construct confidence intervals for bias and size distortion in IV and LP-IV models, but their approach does not cover both weak identification and persistence.    

Cointegrated SVAR Models: Estimation and inference for SVAR models with nonstationary regressors is examined by Chevillon, Mavroeidis & Zhan (2020, ET). These authors propose robust inference on structural parameters of SVAR models identified via long-run restrictions. Considering alternative identification strategies in similar settings, worth further study (e.g., via Non-Gaussianity as in Lanne, Meitz & Saikkonen (2017, JoE)). Specifically, Cheng, Han & Inoue (2022, ET) propose an instrumental variable estimation approach of SVAR models robust to possible nonstationarity. Generally, several aspects regarding econometric identification, estimation and inference in these settings worth further study. Comparing the statistical properties of impulse response functions obtained from different estimation methods (e.g., LP-based vis-à-vis VAR-based), is another important research area in the macroeconometrics literature. For example, Holberg & Ditlevsen (2025, JoE) propose uniform inference for cointegrated linear VAR processes, while Duffy & Jiao (2025, arXiv:2507.22869) develop inference on common trends in a cointegrated nonlinear SVAR model. The literature on Wald-type statistics for inference in cointegrating regressions is vast and covers a broad array of applications. These statistics have asymptotic chi-square distributions which enable inferences to be made straightforwardly.   

Appendix C. Empirical Applications

We examine the implementation of our proposed estimation and inference methods to four structural VAR studies that cover economically relevant applications with persistent dynamics. Relevant applications include: (i) the impact of monetary policy shocks on the real economy and/or the long-run relationships and short-run dynamics between stock markets and monetary policy, (ii) the impact of fiscal policy shocks on the real economy (e.g., such as government spending on output), (iii) the distributional effects of financial and uncertainty shocks, and (iv) the impact of oil price shocks on economic activity. 

Particular emphasis will be given to the VAR representation of New Keynesian DSGE models with financial frictions as well as on model calibration and estimation techniques such as Bayesian methods and the simulated method of moments. In the case of nonlinear dynamic models we shall implement suitable non-linear solution methods; which can be found in the macroeconomics literature.     

Semiparametric Methods in Time Series and Panel Data Econometrics:

Identification, Estimation, Inference and Applications

© Christis G. Katsouris Institute of Econometrics and Data Science

1. Introduction

Semiparametric methods are widely used in econometrics and macroeconometrics with both time series (e.g., see Gourieroux & Jasiak (2017, JoE)) and panel data (e.g., see Buchinsky, Li & Liao (2022, JoE)). For example, de Paula, Gualdani, Pastorino & Salgado (2025) examine a large class of dynamic matching models of labour market with ex-ante heterogeneous firms and workers, symmetric uncertainty and learning about worker's ability, and firm monopsony power. The authors examine the ability of empirical measures of the assortativeness of matching to detect the degree of sorting. Importantly, these authors show that the typical measures of sorting severely understate the value derived from the interaction between sorting and options, because such measures ignore the expected option value of the acquired human capital and information about workers ability for future sorting in the labour market (e.g., see Graves, Huckfeldt & Swanson (2023, nber/w31770)). Moreover, Doerr, Drechsel & Lee (2024, nber/w33137) propose a novel channel through which rising income inequality affects job creation and macroeconomic outcomes. These authors build a general equilibrium macro model with heterogeneous households and heterogeneous firms, to study the impact of rising inequality on labour market share and aggregate output. Their econometric identification approach relies on households' bank deposits data and exploits variation across US states using an instrument for firm job creation. We are interested in the econometric analysis (methods and theory) of semiparametric panel data models under more general dependence, which are particularly useful when evaluating the impact of trade shocks on the manufacturing industry across different locations under spillover linkages. 

2. Econometric Analysis with Generated Regressors

The generated regressors problem is discussed in the econometric and macroeconometric literature, when estimated shocks are used either as additional covariates in regression models or as dependent variables in two-stage estimation procedures. In particular, De Jonghe & Lewis (2025, cemmap/wp0825) propose a nonparametric identification approach for supply and demand shocks using bipartite data (i.e., firm-bank loan transactions). These estimated network-based supply and demand shocks are used as dependent variables in panel data regression models to identify the effect of the shock on the bank-firm pair of interest. Moreover, Lippi (2023, ET) discusses the issue of using variables with measurement errors in SVARs to construct validation procedures for DSGE models. Specifically, the author develops an alternative validation approach based on high-dimensional dynamic factor models which are robust against measurement errors in endogenous variables. In addition, Levine, et al. (2025, OBES) examine the source of shock contamination and non-fundamentalness in DSGE models. The authors propose a robust validation method for DSGE models using SVARs under imperfect information. Moreover, Angelini, Cavaliere & Fanelli (2022, JAE) propose bootstrap inference and model adequacy testing in state space models, while Cavaliere, Fanelli & Georgiev (2025, arXiv:2509.01351) propose bootstrap diagnostic tests for more general econometric frameworks, which encompasses the case of DSGE models under weak identification. Lastly, Lapenta (2023, arXiv:2212.11112) propose a specification test for semiparametric models with nonparametrically generated regressors. An empirical application tests the specification of a semiparametric model for labour force participation.     

3. Parametric and Semiparametric Estimation of Time Series and Panel Data Models

3.1 Structural VAR Models  

We discuss the parametric and semiparametric identification and estimation approaches for structural VAR models. In particular, Gouriéroux, Monfort & Renne (2017, JoE) propose onsider consistent semiparametric estimators and impulse response functions in SVAR models, while Hoesch, Lee & Mesters (2024, QE) propose locally robust inference for Non-Gaussian SVAR models, motivated by the Non-Gaussian identification scheme proposed by Lanne, Meitz & Saikkonen (2017, JoE). These econometric frameworks consider identification and estimation of Non-Gaussian models under the assumption of covariance-stationary processes. For example, Lütkepohl & Claessen (1997, JoE) present an econometric analysis for cointegrated VARMA processes, while Dufour & Jouini (2014, CSDA) derive the asymptotic distributions for quasi-efficient estimators in echelon VARMA models. Recently, Hardy & Korobilis (2025, arXiv:2512.03763v1) propose a new class of time-varying parameter VARs that incorporates deterministic adjustments driven by observable exogenous variables, which facilitates adaptive estimation. The main novelty of their approach lies in the departure from the conventional TVP model specification which replaces the unobserved state innovation with an observable drift. Using Monte Carlo simulations the authors show that the proposed adaptive parameter estimation approach provides increased parsimony than conventional TVP estimation. In an out-of-sample forecasting exercise using quarterly macroeconomic time-series data, the authors show that the AVP-VAR model consistently improves forecasts, especially during periods of increased connectedness and heightened volatility. Lastly, Huber, Marcellino & Tornese (2024, arXiv:2411.12655) examine the distributional effects of economic uncertainty using a functional vector autoregression approach which combines cross-sectional densities with aggregate macro time-series. Moreover, Huber & Koop (2024, JAE) propose fast and order-invariant inference in Bayesian VAR models with nonparametrically estimated Non-Gaussian shocks. 

3.2 Panel Time-Series Data Models

Semiparametric estimation methods for panel time-series data models are used when the functional form includes a parametric and a nonparametrically estimated component (e.g., generated regressors). We discuss estimation and inference approaches for panel data regression models. To begin with, Korolev (2018, arXiv:1810.07620) propose a consistent heteroscedasticity robust Lagrange Multiplier type specification test for semiparametric panel data models. Furthermore, Liang, Gao & Gong (2022, JBES) propose a semiparametric spatial autoregressive panel data model with fixed effects with time-varying coefficients. Second, semi-nonparametric approaches are more robust to functional-form misspecification and are better able to discover nonlinear economic relations. Compared to fully nonparametric methods, semi-nonparametrics suffer less from the "curse of dimensionality" and allow for more accurate estimation of structural parameters of interest (e.g., see discussion in Chen, X. (2013)). In particular, Yang, Song & Yu (2025, JoE) propose a sieve GMM method for the estimation of spatial autoregressive panel data models with nonparametric endogenous effect. These authors apply their method to estimate the environmental Kuznets curve and the knowledge spillover effects for a cross-section of countries.

Combining semiparametric estimation and machine learning methods permits to construct econometric frameworks that project infinite-dimensional parameter spaces into settings with finite-dimensional parameters. In particular, Argañaraz (2025, arXiv:2512.08423) develops an automatic debiased machine learning approach of structural parameters with general conditional moments. The methodology allows to automatically construct Neyman-Orthogonal moments in dynamic models defined by a finite number of conditional moment restrictions, with possibly different conditioning variables and endogenous regressors. In addition, Argañaraz & Escanciano (2025, arXiv:2507.13788)) propose debiased machine learning methods for modelling unobserved heterogeneity with applications to high-dimensional panels and measurement error models. Furthermore, Semenova et al. (2023, QE) develop estimation and inference methods for conditional average treatment effects characterized by a high-dimensional parameter in both homogeneous cross-sectional and unit-heterogeneous dynamic panel settings. These authors examine the theoretical and computational properties of the cross-fitting method for weakly dependent data. Lastly, Cao & Leung (2025, arXiv:2511.10995) examine the validity of debiased machine learning methods without cross-fitting, and establish neighborhood stability conditions  for weakly dependent data in a general metric space that accommodates spatial and network observations.      

4. Evaluating Counterfactuals in Time Series and Panel Data Models

Structural vector autoregressive models are commonly used in empirical macroeconomic research to evaluate the effects of identified structural shocks on macroeconomic variables. A particular application is the evaluation of monetary policy rules based on blocks with the policy shocks vis-à-vis the non-policy shocks. In fact, the so-called Taylor-rule suggests that when implementing quantitative investigations under rational expectations, macroeconomic stabilization policies ought to optimize the unconditional expectations of the policymaker's objective function (e.g., see Damjanovic et al. (2011, SSRN 1804065)); although the timing of monetary policy matters (e.g., see discussion in Olivei & Tenreyro (2007, AER)). Moreover, the identification, estimation and inference procedures developed for SVAR-type models are commonly used to identify fiscal shocks, financial and uncertainty shocks (e.g., see De Santis & van der Veken (2025, OBES)) and trade shocks, among others. In particular, Beraja (2018, wp) discusses the counterfactual equivalence in structural macro models and presents a methodology for constructing counterfactual in SVARs with respect to policy rule changes. Recently, Hebden & Winkler (2025, JEDC) propose an efficient procedure for constructing policy counterfactuals in linear models with occasionally binding constraints in sequence space. Lastly, Caravello, McKay & Wolf (2025, nber/w32988) develop a macroeconometric systems approach which allows to evaluate monetary policy counterfactuals under alternative policy rules, while McKay & Wolf (2023, Ecta) show that a general family of linearized structural macro models with estimable causal effects of contemporaneous and news shocks to the prevailing policy rule provides sufficient conditions for counterfactuals under alternative policy rules. 

Furthermore, counterfactual methods for macroeconometric models are also used to evaluate the impact of alternative policies on macro outcomes such as inflation targeting (e.g., see Svensson (1997, EER)). Constructing inflation-targeting policy counterfactuals, is particularly useful for developing economies under different exchange rate regimes. In fact, Bambe et al. (2024, JEDC) argue that estimates of dynamic causal effects of such policies in developed economies vis-à-vis in emerging market economies, typically reflect local economic conditions, since estimation and inference is conducted based on a conditioning set that captures the degree of reduction in macroeconomic instability. Moreover, Lim & McNelis (2012, PER) evaluate the macroeconomic adjustments with an estimated DSGE, and implement a counterfactual inflation-targeting simulation experiment, which allows to obtain estimates of the welfare gains from switching exchange rate regimes. Thus, macroeconometric settings under the assumption that policy rules are set optimally, allow to reverse engineer policy objectives from observed time series data (e.g., see Kam, Lees & Liu (2009, JMCB)), permitting to construct counterfactuals for different macro conditions (see also Choi & Foerster (2021, RED)). We consider parametric and semiparametric methods for estimating conditional and unconditional counterfactuals, in both stabilization policy settings (e.g., monetary policy) and non-stabilization policy settings, using time series and panel data. For example, Wang (2024, arXiv:2408.09271) propose a novel counterfactual imputation method that combines the dimension reduction capabilities of PCA to handle high-dimensional datasets with the flexibility of factor models, which accommodate a wide range of DGPs.   

From the macroeconomic theory perspective, Nakamura, Riblier & Steinsson (2025, nber/w34200) study how violations of the conventional structural identification assumptions such as indeterminacy, non-fundamentalness and perfect-foresight, can affect the evaluation of the Taylor rule in New Keynesian models. The authors argue that deviations from the Taylor rule, do not constitute evidence of suboptimal monetary policy. Moreover, Moura & de Carvalho, A. (2010, JoM) examine the way monetary policy has been conducted in the seven largest Latin American economics. These authors select the most appropriate functional form through out-of-sample measures of forecasting performance, and find strong empirical evidence support for endogenous monetary policy reacting to macroeconomic variables. Furthermore, Charemza, Francq, Lupu, Makarova & Zakoïan (2025, PloS One) propose a novel forecast evaluation test for the effects of (stabilization) economic policy in forecast errors. The testing approach of these authors does not rely on detailed policy intervention data, therefore constructing forecast evaluation tests with vis-à-vis without intervention data, worth further study. In particular, Qu, Timmermann & Zhu (2024, IJF) develop new methods for testing equal predictive accuracy for panels of forecasts, exploiting information in both the time-series and cross-sectional data dimensions. Alternative approaches for constructing counterfactuals such as the constrained optimal policy projection and as well as the synthetic control method can be also examined. The projection-based constrained policy optimization approach is used in econometrics and statistics. For example, Yang et al. (2020, arXiv:2010.03152) study the problem of learning control policies that optimize a reward function based on statistical guarantees such as fairness and honesty (e.g., see Hartford et al. (2016, arXiv:1612.09596)). Moreover, Zhang & Lu (2024, Mathematics) propose a time series synthetic control method for policy evaluation using heteroscedastic time series data (see also discussion in Kinn (2018, arXiv:1803.00096)).       

Further econometric applications for data-rich environments which address the program evaluation problem can be found in Carvalho, Masini & Medeiros (2018, JoE) who develop counterfactual methods for high-dimensional panel time-series data as well as in Masini & Medeiros (2021, JASA) who develop counterfactual analysis with artificial controls in the presence of both nonstationarity and high-dimensionality. Extending these approaches in the presence of unknown forms of persistence using local-to-unity parametrizations (e.g., see Dou & Müller (2021, Ecta) and references therein), worth further study. Thus, estimation and inference methods that address the spurious regression issue with good finite-sample properties, regardless whether treated and untreated units are cointegrated or not, as well as being robust to unknown forms of persistence, are particularly useful for counterfactual analysis with persistent data. An econometric framework for evaluating counterfactuals with nonstationary data is proposed by Shi, Xi & Xie (2025, arXiv:2505.22388). These authors develop a synthetic business cycle control method, which allows to investigate the causal effects of a 'natural experiment' on the trajectory of macro outcomes during the post-intervention period. Lastly, Chen, Phillips & Shi (2025, SSRN 5195264) develop counterfactual analysis and treatment effect inference, which allows to evaluate the causal effects of bubble mitigation policies.

5. Dynamic Causal Effects in High-Dimensional Settings

In this section we discuss aspects of estimation and inference for impulse responses obtained from SVAR models in a data-rich environment (e.g., see Bernanke & Boivin (2003, JME)). We also motivate the implementation of these methodologies using a macroeconomic example.  

During periods of credit supply shocks, tracing the impact of bank liquidity shocks and investment dynamics, which can affect the timely provision of credit to firms, has a detrimental role in understanding the related macroeconomic effects. For example, Melcangi (2024, AEJ: Macro) examines the extent to which financial frictions affect firm's propensity to hire. In particular, the authors build a heterogeneous-firm model with shocks to firms' idiosyncratic productivity and aggregate credit uncertainty, in which precautionary savings in cash arise endogenously from the interaction between real and financial frictions, and thus affect the transmission mechanism of credit supply shocks onto labour demand. Moreover, Jeenas (2025, JPE forthcoming) studies the role of firms' balance sheet liquidity in the transmission of monetary policy to investment, based on a heterogeneous-firm macro model with financial constraints, debt issuance costs and differential returns on firms' cash and borrowing. Under a counterfactual scenario, model-based estimates indicate that monetary transmission to investment has evolved in its distributional effects across firms and their contributions to the aggregate response. Therefore, understanding the origins and effects of macroeconomic uncertainty in the long run; especially in the presence of nonlinearities, is important for both business cycle analysis and macroeconomic forecasting (e.g., see Carriero, Marcellino & Tornese (2023, EL), Bianchi, Kung & Tirskikh (2023, QE), Bandi & Tamoni (2023, JoE), Bianchi, Ilut & Schneider (2018, RES), Müller & Watson (2016, RES) and Giannone, Reichlin & Sala (2006, JoE)). 

From the macroeconometric perspective, considering finite-sample comparisons of VAR-based and LP-based impulse responses, provides researchers a mechanism to select an estimation approach based on the bias-variance trade-off (e.g., see discussion in Li, Plagborg-Møller & Wolf (2024, JoE)). We discuss relevant issues to model selection and impulse response construction. To begin with, Schmidt & Guilkey (1976, JoE) investigate the small sample problem of truncation lags in the estimation of autoregressive models with autocorrelated errors (see also Pesaran (1973, IER)). Using Monte Carlo simulations the authors show that in small samples dropping higher-order terms does have an effect on hypothesis testing. Second, a large body of time series econometric literature propose data-dependent methods for the selection of the truncation lag (e.g., see Ng & Perron (1995, JASA)). Moreover, Plagborg‐Møller & Wolf (2021, Ecta) show that VAR-based and LP-based impulse responses are asymptotically equivalent. Using model selection procedures that are robust to dynamic misspecification under unknown lag order with possibly unequal lag-lengths, can improve the impulse response estimates. Third, in addition to the optimal truncation lag choice, testing sufficient conditions for infinite step Granger Noncausality in infinite order VAR processes (e.g., see Triacca, Damette & Giovannelli (2020, SSRN 3630340)), allows to validate the finite-sample properties of impulse responses constructed via sequentially estimated VARs vis-à-vis impulse responses constructed via local projections (e.g., see Ludwig (2024, SSRN 4882149)). For example, De Graeve & Westermark (2025, JME) show that longer lag length simultaneously reduces misspecification, which in turn reduces variance. Using data generated by frontier DSGE models, the authors show that long-lag VARs can reduce bias and variance with better mean-squared error; while structural conclusions about the impact of technology and monetary policy shocks on the economy different significantly from shorter lag VARs. Lastly, Adamek, Smeekes & Wilms (2024, EJ) propose local projection inference in high dimensions, while Dettaa & Wang (2024, arXiv:2410.04330) propose inference in high-dimensional linear projections with applications to multi-horizon granger causality and network connectedness. From the econometric theory perspective, developing asymptotic theory for VAR-based impulse responses when the number of impulse responses exceeds the number of VAR parameters, is an important task to facilitate inference (e.g., see Guerron-Quintana, Inoue & Kilian (2017, JoE)). For instance, Krampe, Paparoditis & Trenkler (2023, JoE) develop structural inference procedures in sparse high-dimensional VARs using bootstrap techniques.