State-Dependent Local Projections

• State-dependent local projections are models that capture dynamic shock responses by allowing impulse responses to vary across different regimes.
• They integrate techniques from PDE analysis and time-series econometrics, employing Bayesian, SUR, and high-dimensional methods for robust inference.
• Applications span macroeconomics, finance, biology, and physics, offering improved precision in analyzing regime-specific effects and smooth transitions.

State-dependent local projections are a class of statistical and mathematical models designed to capture the dynamic responses of variables to exogenous shocks, allowing those responses to vary as a function of the system's state or regime. This concept spans both continuous-time infinite-dimensional systems (such as parabolic PDEs with state-dependent delays) and discrete-time time-series models (such as horizon-by-horizon impulse response estimation in econometrics), with particular methodological challenges in proving local existence, statistical efficiency, and valid inference under state dependence.

1. Mathematical Characterization of State-Dependence

In the context of infinite-dimensional systems, state-dependence arises when delay terms or system coefficients are explicit functions of the evolving state variable. For example, the delay functional $F(t,\varphi)$ in a non-autonomous parabolic PDE is constructed via a Stieltjes integral:

$$F(t, \varphi) = \int_{-r}^{0} p(t, \varphi(\theta)) \, dg(\theta; t, \varphi)$$

where $g$ may have discrete jump components governed by state-dependent thresholds $n_k(t, \varphi)$ and weights $h_k(t, \varphi)$ (Rezounenko, 2011).

For finite-dimensional time-series and econometric applications, state-dependence is modeled by allowing coefficients in projection regressions to vary according to either observed states $s_t$ or threshold indicators. A typical state-dependent local projection regression takes the form:

$$y_{t+h} = \sum_l [\beta_{l}^{(1)} z_{t-l} 1\{ \text{state}_{t-l} < \text{threshold} \} + \beta_{l}^{(2)} z_{t-l} 1\{ \text{state}_{t-l} \ge \text{threshold} \}] + \varepsilon_t$$

or in more general nonlinear forms involving polynomial and interaction terms (Tanaka, 2018, Inoue et al., 2023, Cha, 12 Feb 2024).

2. Existence, Uniqueness, and Invariance in PDEs with State-Dependent Delays

The foundational existence and uniqueness theory for PDEs with state-dependent delays faces fundamental obstacles due to the non-Lipschitz nature of discrete state-dependent delay terms. Specifically, uniqueness cannot be established using standard contraction arguments; instead, refined conditions are imposed:

• (A4): Local Lipschitz condition on the continuous component of the delay measure.
• (A5): "Ignoring property" — the discrete delay component $g_d$ is constructed so that, over certain intervals $[\,-r,\,-n_{\text{ign}}(t)]$, differences in history functions $\varphi$ are ignored if they coincide, permitting conditional continuity of solutions.

Summing up, mild solutions exist locally under continuity assumptions; uniqueness is obtained via auxiliary properties of the delay term that ensure "modulus-of-continuity"-type estimates even in non-Lipschitz settings. Furthermore, the classical invariance principle is extended by imposing the subtangential condition:

$$\lim_{h \to 0^+} \frac{1}{h} d\left[ e^{-A h} \varphi(0) + \int_t^{t+h} e^{-A(t+h-s)} B(t, \varphi) ds ; D(t+h) \right]=0$$

guaranteeing that solutions starting in a closed set remain in that set (for example, positive cone in population models) (Rezounenko, 2011).

3. Statistical Models: Bayesian and SUR Approaches to State-Dependent LPs

Local projections (LPs) estimate impulse responses directly for each horizon, typically via OLS or IV regressions. In the presence of state dependence, coefficients are permitted to vary non-parametrically or by regime. Recent advances include:

Bayesian LPs with Roughness Penalty Priors: Parameters are modeled as sequences coupled over horizons and, when extended, over states. The prior takes an adaptive form:

$$p(\theta_j | \tau_j, \Lambda_j) \propto \exp\left\{ - \frac{\tau_j}{2} \theta_j^\top D^\top \Lambda_j D \theta_j \right\}$$

with $D$ representing differences (random walk priors), $\tau$ global smoothing, and $\Lambda$ local adaptivity; all are inferred jointly in MCMC routines. Such models readily accommodate state-dependent projections by estimating distinct coefficient sequences for each regime or state (Tanaka, 2018).

General SUR and GP Priors: LP equations are jointly modeled as seemingly unrelated regressions, with forecast errors correlated across horizons. Impulse response vectors receive Gaussian Process priors:

$\mu_\beta \sim \mathcal{GP}(0, K_\beta)$

where $K_\beta(i,j)$ is a covariance kernel (e.g., squared exponential). Extensions allow kernel parameters (length-scale $\xi$ and variance decay $\vartheta$) to vary with state, and missing data across horizons are imputed from the joint Gaussian distribution (Huber et al., 22 Oct 2024).

4. Inference and Model Selection Under State-Dependence

Inference for state-dependent local projections (LPs) presents specialized challenges:

• Pointwise and Simultaneous Inference: Standard HAC (e.g., Newey–West) or Driscoll–Kraay estimators are used for pointwise error bands. System GMM and lag augmentation are employed to stabilize standard errors and account for serial correlation across LP horizons. Simultaneous confidence intervals ("significance bands") are generated via sup-$t$ procedures that simulate joint coverage under the estimated covariance structure (Inoue et al., 2023).
• High-Dimensional Covariate Selection Without Sparsity: The Orthogonal Greedy Algorithm plus High-Dimensional AIC (OGA+HDAIC) enables robust selection of controls when coefficient vectors are dense ("approximately sparse"). The procedure minimizes predictive error while debiasing for inference, crucial in state-dependent environments with many interactions or high-dimensional state variables (Cha, 12 Feb 2024).

5. Applications and Empirical Implications

State-dependent LPs enable empirical researchers to uncover and quantify regime-dependent impulse responses in macroeconomics, finance, biology, and PDE-driven physical systems:

• Macroeconomic State Dependence: Conditional impulse responses reveal, for example, that inflation reacts in line with rational expectations in "good" states (low recession risk) but is highly subjective and amplifies pessimism in "bad" states (high uncertainty), supporting theoretical models of beliefs and expectation-driven dynamics (Cha, 12 Feb 2024).
• PDE Models in Biology and Physics: The developed existence and invariance theory allows state-dependent delays (e.g., population thresholds in Lotka–Volterra systems) to be rigorously incorporated, ensuring physically meaningful properties like positivity are preserved throughout evolution (Rezounenko, 2011).
• Robustness to High-Dimensional Controls: Empirical applications demonstrate the superiority of OGA+HDAIC selection in dense settings (e.g., assessing causal effects of democratization), maintaining nominal coverage and precision when standard LP and LASSO-based approaches fail due to high dimensionality or persistence (Cha, 12 Feb 2024).
• Efficient Estimation and Smoothness: Bayesian formulations (with roughness penalty or GP priors) systematically improve impulse response smoothness, finite-sample efficiency, and uncertainty quantification, whether in estimating monetary policy effects or tracing nonlinear state-dependent shocks (Tanaka, 2018, Huber et al., 22 Oct 2024).

6. Limitations and Directions for Future Research

While state-dependent LP frameworks offer greater flexibility and empirical realism, several methodological and computational limitations persist:

• In the PDE setting, unique existence proofs require stringent "ignoring" or regularity conditions not always verifiable in applications, particularly in high-dimensional systems with complex boundary conditions.
• Bayesian and joint system approaches demand greater computational resources (especially when imputing missing data or modeling joint covariance structures) and may require large sample sizes to reliably estimate state-dependent kernel parameters or smooth impulse response surfaces.
• Identifying state-dependence separately from shock effects can induce identification challenges, particularly when underlying instruments or regimes are endogenous or not clearly separable (Huber et al., 22 Oct 2024).
• Panel data and multi-variable extensions pose additional hurdles, as interaction effects and cross-sectional dependence introduce estimation and inference complexities beyond those present in simple LP or IV settings (Inoue et al., 2023).

A plausible implication is that future developments will focus on scalable algorithms for joint modeling, more robust identification via hierarchical or latent factor approaches, and systematic theoretical analysis of existence and invariance properties under weaker regularity assumptions in both continuous-time and econometric frameworks.

7. Summary Table: Core Features of State-Dependent Local Projections in Recent Literature

Paper (arXiv)	State Dependence Manifested In	Key Technical Innovation
(Rezounenko, 2011)	Delay terms in PDEs	Existence/uniqueness with non-Lipschitz delays, invariance principle
(Tanaka, 2018)	Impulse response coefficients	Bayesian LP with roughness penalty priors for smoothness/efficiency
(Inoue et al., 2023)	Functional, nonlinear response	Joint/simultaneous inference: sup-$t$ bands, dynamic evolution
(Cha, 12 Feb 2024)	State-based regime analysis	OGA+HDAIC high-dimensional selection; empirical state-dependent LP
(Huber et al., 22 Oct 2024)	Joint model structure	Bayesian SUR system, GP prior, data imputation, multi-instrument extension

The technical progression across these papers reflects growing sophistication in modeling, inference, and empirical analysis for state-dependent local projections, addressing both theoretical regularity and practical statistical efficiency in multifaceted dynamic environments.