Impulse Response Analysis of Structural Nonlinear Time Series Models

Impulse Response Analysis of Structural Nonlinear Time Series Models

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\noindentKeywords: macroeconometrics, semiparametric, sieve estimation, physical dependence

\noindentJEL: C14, C22, C54, E52, F40

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Introduction

Linearity is a foundational assumption in structural time series modeling. For example, large classes of macroeconomic models in modern New Keynesian theory can be linearized, justifying the use of the linear time series toolbox from a theoretical point of view. The seminal work of simsMacroeconomicsReality1980 on vector autoregressive (VAR) models brought the study of dynamic economic relationships into focus within the macro-econometric literature, for which the estimation and analysis of impulse response functions (IRFs) is key hamilton1994TimeSeriesAnalysis,lutkepohlNewIntroductionMultiple2005,kilianStructuralVectorAutoregressive2017. The local projection (LP) approach of jordaEstimationInferenceImpulse2005 has also gained popularity as a flexible and easy-to-implement alternative.

Linear econometrics models are, however, limited in the kind of effects that they can describe. In nonlinear DGPs, linear VARs as well as standard LP methods can only reconstruct the best linear impulse responses approximation plagborg-mollerLocalProjectionsVARs2021. And even though asymmetries in monetary policy and non-proportional shock effects are now commonly studied, most works still rely on parametric specifications. For example, tenreyro2016pushing study both sign and size effects of monetary policy (MP) shocks using censoring and cubic transformations, respectively. caggianoEconomicPolicyUncertainty2017,pellegrinoUncertaintyMonetaryPolicy2021 and caggianoUncertaintyShocksGreat2021 use multiplicative interacted VAR models to estimate the effects of uncertainty and MP shocks. From a macro-finance perspective, forniNonlinearTransmissionFinancial2023,forniAsymmetricEffectsNews2023 study the economic effects of financial shocks following a quadratic VMA specification debortoliAsymmetricEffectsMonetary2020. gambettiBadNewsGood2022 study news shocks asymmetries by imposing that news changes enter an autoregressive model through a threshold map. Parametric nonlinear specifications are also common prescriptions in time-varying models auerbachMeasuringOutputResponses2012,caggianoEstimatingFiscalMultipliers2015 and state-dependent models ramey2018government,goncalvesStatedependentLocalProjections2024.

In this paper, we aim to design a semiparametric, structural nonlinear time series modeling and estimation framework with explicit theoretical properties. Our primary contribution is the extension and combination of the block-recursive structural framework of goncalvesImpulseResponseAnalysis2021 with the uniform sieve estimation theory of chenOptimalUniformConvergence2015 within a general physical dependence setup wuNonlinearSystemTheory2005. Under classical nonparametric assumptions, we show that a two-step semiparametric series estimation procedure can consistently recover the structural model in a uniform sense. In order to be able to relax the assumptions on the linearity and fixed parametric model specifications, we restrict our study to the case of compactly supported, weakly dependent data. We emphasize that, even in this constrained setting, to the best of our knowledge, our work is the first to offer a formal combination of these approaches. We provide explicit guarantees for semiparametrically-estimated nonlinear IRFs: Nonlinear impulse response function estimates are asymptotically consistent and, thanks to an iterative algorithm, can also be straightforwardly computed.

To illustrate the validity of our proposed methodology, we first evaluate its performance with several simulations. With realistic sample sizes, the efficiency costs of our semiparametric procedure are small compared to correctly specified parametric responses. A second set of simulations provides a simple setup where the nonlinear parametric model is mildly misspecified. Still, the large-sample bias is considerable, while for semiparametric estimates it is negligible. We then evaluate how the IRFs computed using the new method compare with those from two empirical exercises studied in the literature. In a small, quarterly model of the U.S. macroeconomy based on tenreyro2016pushing, we find that point estimates of linear and parametric nonlinear IRFs may underestimate in intensity the GDP responses by up to $13\%$ and $16\%$, respectively, after a large exogenous monetary policy shock. Moreover, sieve responses achieve maximum impact a year before their linear counterparts. Then, we evaluate the effects of interest rate uncertainty on US output, prices, and unemployment following istrefiSubjectiveInterestRate2018. In this exercise, the impact on industrial production of a one-deviation increase in uncertainty is $54\%$ stronger according to semiparametric IRFs than the comparable linear specification. These findings suggest that structural responses based on linear specifications can significantly underestimate the effects of shocks.

\paragraph{Literature Review.} Let us mention some key references directly related to our discussion. On the one hand, jordaEstimationInferenceImpulse2005 already proposed a “{flexible local projection}” approach based on the Volterra expansion. The flexible LP proposal is effectively equivalent to adding polynomial terms to a linear regression, meaning it is a semiparametric method, and it should be analyzed as such. Yet, the Volterra expansion is not formally justified, nor is its truncation, which is key in studying its properties SirotkoSibirskaya2020volterraBootstrap,Movahedifar2023closedloopVolterra. On the other hand, smoothed LP methods, see e.g. plagborg-moller2016essays,barnichonImpulseResponseEstimation2019, can address exclusively concerns of regularity in the shape of estimated impulse responses, but not any potential underlying nonlinearities in the DGP. Recently, goncalvesStatedependentLocalProjections2024 outlined a general, nonparametric LP estimation procedure for nonlinear IRFs, which was later studied in goncalvesNonparametricLocalProjections2024 under high-level conditions on the functional form of the IRF itself. gourieroux2023nonlinear also devise a framework for nonparametric kernel estimation and inference of IRFs via local projections, although they mostly work in the one-dimensional, single lag case. Finally, following the Generalized IRF (GIRF) approach Koop1996,potterNonlinearImpulseResponse2000,gourierouxNonlinearInnovationsImpulse2005,terasvirtaModellingNonlinearEconomic2010, kanazawaRadialBasisFunctions2020 proposed to use radial basis function neural networks to estimate nonlinear reduced-form GIRFs for the U.S. economy. While GIRFs can be essentially characterized as impulse responses with more sophisticated conditioning sets, they are lacking in that they do not inherently address the problem of {structural} identification kilianStructuralVectorAutoregressive2017.

\paragraph The remainder of this paper is organized as follows. Section (ref) provides the general framework for the structural model. Section (ref) describes the two-step semiparametric estimation strategy, and Section (ref) discusses nonlinear impulse response function computation, validity, and consistency. In Section (ref) we give a brief overview of simulation results, while Section (ref) contains the empirical analyses. Finally, Section (ref) concludes. All proofs and additional content can be found in the Appendix. Concerning notation: scalar and vector random variables are denoted in capital or Greek letters, e.g. $Y_t$ or $\epsilon_t$, while realizations are shown in lowercase Latin letters, e.g. $y_t$. For a process $\{Y_t\}_{t\in \mathbb{Z}}$, we write $Y_{t:s} = (Y_t, Y_{t+1}, \ldots, Y_{s-1}, Y_s)$, as well as $Y_{*:t} = (\ldots, Y_{t-2}, Y_{t-1}, Y_t)$ for the left-infinite history and $Y_{t:*} = (Y_t, Y_{t+1}, Y_{t+2}, \ldots)$ for its right-infinite history. The same notation is also used for random variable realizations. For a matrix $A \in \mathbb{R}^{d \times d}$ where $d \geq 1$, $\lVert A \rVert$ is the spectral norm, $\lVert A \lVert_\infty$ is the supremum norm and $\lVert A \rVert_r$ for $0 < r < \infty$ is the $r$-operator norm. For a random vector or matrix, we will use $\lVert \,\cdot\, \rVert_{L^r}$ to denote the associated $L^r$ norm.

Model Framework

In this section, we introduce the general nonlinear time series model, which is a generalization of the one developed in goncalvesImpulseResponseAnalysis2021. In terms of structural shocks identification, the idea is straightforward: A scalar series, $X_t$, is chosen to be the structural variable identifying shocks, and it explicitly determines the dynamic effects on the remaining data, vector $Y_t$. This will enable the derivation of economically meaningful (structural) impulse responses due to an exogenous shock impacting $X_t$.

General Model

This paper focuses on the family of nonlinear autoregressive models of the form

where $X_t \in \mathcal{X} \subseteq \mathbb{R}$ and $Y_t \in \mathcal{Y} \subseteq \mathbb{R}^{d_Y}$ are scalar and $d_Y$-dimensional time series, respectively, $u_t = (u_{1t}, u_{2t}')' \in \mathcal{U} \subseteq \mathbb{R}^d$ are innovations, $d = 1 + d_Y$, $G_2 : \mathbb{R}^{1 + pd} \to \mathbb{R}$ is a generic nonlinear map, and $A_{12}(L)$ and $A_{11}(L)$ are lag polynomials lutkepohlNewIntroductionMultiple2005. We let $Z_t := (X_t, Y_t')' \in \mathbb{R}^{d}$ be the full data vector. Let us provide some examples for the model classes nested by (ref).

Structural Framework

Model (ref) involves only reduced-form innovations $u_{1t}$ and $u_{2t}$, and additional assumptions are necessary to provide a structural interpretation. Many such assumptions have been devised in the macroeconomic literature, but few can be directly applied to nonlinear models kilianStructuralVectorAutoregressive2017. We follow the block-recursive identification strategy outlined in goncalvesImpulseResponseAnalysis2021 and originally due to kilian2011responses.

From (ref) we derive

where, without loss of generality, we have assumed (as in Example (ref)) that we can separate the linear and nonlinear ($\mathpalette\arc@arc{G}_2$) components from $G_2$. In general, it can be the case that $\mu_2 = 0$, $A_{22}(L) = 0$ or $A_{21}(L) = 0$ if e.g. $G_2$ is strictly nonlinear. In vector form:

We can now formalize the structural specification of our model.

Note that inverting $B_0^{-1}$ gives

and, multiplying both sides of (ref) by $B_0$, we find

where $b = (b_1, b_2')' \in \mathbb{R}^d$ and $\mathpalette\arc@arc{F}(Z_{t:t-p}) = (0, \mathpalette\arc@arc{F}_2(Z_{t:t-p}))'$ for $\mathpalette\arc@arc{F}_2 : \mathbb{R}^{1+pd_Y} \to \mathbb{R}^{d_Y}$, $F_2 = B_{0,22} \mathpalette\arc@arc{G}_2$. In practice, to estimate the model's coefficients, we will leverage (ref). This latter form was termed the pseudo-reduced form by goncalvesImpulseResponseAnalysis2021.

Observe that $\mathpalette\arc@arc{G}_2(Y_{t-1:t-p}, X_{t:t-p})$ is correlated with $u_{2t}$ through $B_0^{21} \epsilon_{1t}$. As $X_t$ depends linearly on $\epsilon_{1t}$, if $B_0^{21} \not= 0$ and $\mathpalette\arc@arc{G}_2(Y_{t-1:t-p}, X_{t:t-p})$ is not independent of $X_t$, there is an endogeneity problem. goncalvesImpulseResponseAnalysis2021 address the issue by proposing a two-step estimation procedure wherein one proxies for $\epsilon_{1t}$ with residual $\widehat{\epsilon}_{1t}$. As we prove in Section (ref) below, this approach also generally allows for consistent semiparametric estimation.

Structural Nonlinear Impulse Responses

Starting from pseudo-reduced equations (ref), we begin by assuming that the linear autoregressive component is stable.

This standard stability assumption enables us to write impulse responses in a manner that can yield useful simplifications for additively separable models.\footnote{Stability of the linear VAR component is neither necessary nor sufficient for ensuring stability and stationarity of the entire nonlinear process, cf. Assumption (ref) in Section (ref) below.} Then, letting $\Psi(L) = (I_{d} - A(L) L)^{-1}$, one can write

where $\eta := \Psi(1) (\mu_1, \mu_2')'$, $\Theta(L) := \Psi(L) B_0^{-1}$ and $\Gamma(Z_{t:*}) := \Psi(L)(0, \mathpalette\arc@arc{G}_2(Y_{t-1:t-p}, X_{t:t-p})')'$. We emphasize that the nonlinear term $\Gamma(Z_{t:*})$ generally depends on the entire past of the process $Z_t$, as $\Psi(L)$ is an infinite-order MA polynomial. To define impulse responses, we partition the polynomial $\Theta(L)$ according to $\Theta(L) := [\Theta_{\cdot 1}(L) \:\vert\: \Theta_{\cdot 2}(L)]$ , where $\Theta_{\cdot 1}(L)$ represents the first column of matrices in $\Theta(L)$, and $\Theta_{\cdot 2}(L)$ the remaining $d_Y$ columns.

Given impulse $\delta \in \mathbb{R}$ at time $t$, define the shocked innovation process as $\epsilon_{1 s}(\delta) = \epsilon_s$ for $s \not= t$ and $\epsilon_{1 t}(\delta) = \epsilon_{1 t} + \delta$, as well as the shocked structural variable as $Z_s(\delta) = Z_s$ for $s < t$ and $Z_s(\delta) = X_s(\epsilon_{s:t+1}, \epsilon_t + \delta, \epsilon_{t-1:*})$ for $s \geq t$. Further, for a given horizon $h \geq 0$, let

be the time-$t$ baseline and shocked series, respectively. Then,

is the unconditional impulse response at horizon $h$ due to shock $\delta$. The difference between series is $ Z_{t+h}(\delta) - Z_{t+h} = \Theta_{h,\cdot 1} \delta + \Gamma(Z_{t:*}(\delta)) - \Gamma(Z_{t:*}) $, hence

Closed-form computation of nonlinear IRFs is highly non-trivial. Even in the separable case (ref), while one can linearly separate expectations in the impulse response formula, terms $\mathbb{E}\left[ \Gamma_j(Z_{t+j}(\delta)) - \Gamma_j(Z_{t+j}) \right]$ for $0 \leq j \leq h$ cannot be meaningfully simplified further. Moreover, these expectations involve nonlinear functions of lags of $Z_t$ and are impractical to derive explicitly. To avoid working with $\Theta(L)$ and $\Gamma(L)$, we now present an iterative algorithm which allows one to easily and efficiently compute nonlinear IRFs.\footnote{The algorithm we propose is a natural counterpart to the one in Proposition 3.1 of goncalvesImpulseResponseAnalysis2021, wherein they suggest to estimate the MA form coefficients recursively. Our approach instead relies on directly iterating forward the model's equations, which is more computationally straightforward.}

Proposition (ref) follows directly from the definition of the unconditional impulse response (ref) combined with a direct forward iteration of (ref), sidestepping the explicit MA($\infty$) formulation in (ref). This approach dispenses from the need to simulate innovations $\{\epsilon_{t+j}\}_{j=1}^{h-1}$, as the joint distribution of $\{X_{t+h-1}, X_{t+j-1}, \ldots, X_{t}\}$ contains all relevant path information.

When the model is estimated from data, for residuals $\widehat{\epsilon}_{1t}$ and $\widehat{\xi}_{2t}$ it trivially holds

In practice, this means that one can numerically construct the shocked sequence as

for $j = 1, \ldots, h$ where $\widehat{X}_t(\delta) = X_t + \delta$, $\widehat{X}_{t-s} = X_{t-s}$ for all $s \geq 1$, and similarly for $\widehat{Y}_t(\delta)$.

Estimation

To discuss estimation, we will rewrite the equations in (ref) with some minor reordering as

where $\xi_{2t} = B_0^{22} \epsilon_{2t}$, $\Pi_1 := ( \eta_1, A_{1,11}, \cdots, A_{p,11}, A_{1,12}', \cdots, A_{p,12}' )' \in \mathbb{R}^{1 + p d}$,

$W_{1t} := ( 1, X_{t-1}, \ldots, X_{t-p}, Y_{t-1}', \ldots, Y_{t-p}' )' \in \mathbb{R}^{1 + p d}$, and $W_{2t} := (1, X_t, \allowbreak X_{t-1}, \ldots, X_{t-p}, \allowbreak Y_{t-1}', \ldots, \allowbreak Y_{t-p}', \allowbreak \epsilon_{1t} )' \in \mathbb{R}^{2 + p d}$. With a slight abuse of notation, similar that of Example (ref), we have written the functional terms in $\Pi_2$ as a “vector product”, $G_2 \cdot (X_{t:t-p}', Y_{t-1:t-p}')' \equiv G_2(X_{t:t-p}, Y_{t-1:t-p})$, where $G_2$ is a vector of functions, one for each component of $Y_t$.

Whenever $\Pi_1 \not= 0$, $W_{2t}$ is an infeasible vector of regressors due to term ${\epsilon}_{1t}$. To estimate $\Pi_2$, one can use $\widehat{W}_{2t} = (1, X_t, \allowbreak X_{t-1}, \ldots, X_{t-p}, \allowbreak Y_{t-1}', \ldots, Y_{t-p}', \allowbreak \widehat{\epsilon}_{1t})'$ instead, which contains generated regressors in the form of residual $\widehat{\epsilon}_{1t}$. A valid two-step estimation procedure goncalvesImpulseResponseAnalysis2021 is: (I) Regress $X_t$ on $W_{1t}$ to get estimate $\widehat{\Pi}_1$ and residuals $\widehat{\epsilon}_{1t} = X_t - \widehat{\Pi}_1' W_{1t}$; (II) Semiparametrically regress $Y_t$ on $\widehat{W}_{2t}$ to get estimate $\widehat{\Pi}_2$.

There are many ways to implement step (II), given that the literature on non- and semiparametric regression is mature. We rely on the sieve framework of chenOptimalUniformConvergence2015 as the workhorse to derive the main theoretical results. The sieve framework is quite rich, encompassing e.g. neural networks chenImprovedRatesAsymptotic1999,shenAsymptoticPropertiesNeural2023.

Semiparametric Series Estimation

The semiparametric regression step we require is more readily analyzed by working on each component of $Y_t$. For $i \in \{1, \ldots, d_Y\}$, consider

Let then $\pi_{2,i} := [ \mu_{2,i}, \: G_{2,i}, \: B^{21}_{0,i} ]'$. The regression equation for $\pi_{2,i}$ is thus $Y_{i} = \pi_{2,i}' W_{2} + \xi_{2i}$, where $Y_{i} = (Y_{1,i}, \ldots, Y_{n,i})'$ and $\xi_{2i} = (\xi_{2t,1}, \ldots, \xi_{2t,n})'$. The estimation target is the conditional expectation $\pi_{2,i}(w) = \mathbb{E}[ Y_{t,i} \:\vert\: W_{2t} = w ]$ under the assumption $\mathbb{E}[ \xi_{2t,i} \:\vert\: W_{2t} ] = 0$.

Assume that $G_{2,i} \in \Lambda$, where $\Lambda$ is a sufficiently regular function class to be specified in the following. Given a collection $b_{1\kappa}, \ldots, b_{\kappa\kappa}$ of $\kappa \geq 1$ basis functions belonging to sieve $\mathcal{B}_\kappa$, define $b^\kappa(\cdot) := \left( b_{1\kappa}(\cdot), \ldots, b_{\kappa\kappa}(\cdot) \right)'$ and $ B_\kappa := \big( b^\kappa(Y_{0:1-p}, X_{1:1-p}), \ldots, b^\kappa(Y_{n-1:n-p}, X_{n:n-p}) \big)' $. For univariate functions, one can directly apply spline, wavelet and Fourier sieves; in the multivariate case, tensor-product sieves are straightforward generalizations chenOptimalUniformConvergence2015. To construct the final semiparametric sieve for $\pi_{2,i}$, indicated by $\mathcal{B}_\pi$, let $b_{\pi,1K}, \ldots, b_{\pi,KK}$ be the sieve basis in $\mathbb{R} \times \mathcal{B}_\kappa \times \mathbb{R}$ for $\kappa \geq 1$ and $K = 2 + \kappa$ given by $b_{\pi,1 K}(W_{2t}) = 1$, $b_{\pi,\ell K}(W_{2t}) = b_{\ell \kappa}(Y_{t-1:t-p}, X_{t:t-p})$, for $2 \leq \ell \leq \kappa+1$, and $b_{\pi,K K}(W_{2t}) = \epsilon_{1t}$. Note that $K$, the overall size of the sieve, grows linearly in $\kappa$, which itself controls the effective dimension of the nonparametric component of the sieve, $b_{\pi,2 K}, \ldots, b_{\pi,(\kappa+1) K}$. Introducing $b^K_\pi(w) := ( b_{\pi,1K}(w), \ldots, b_{\pi,KK}(w) )'$ and $B_\pi := ( b^K_\pi(W_{21}), \ldots, b^K_\pi(W_{2n}) )'$, the generally infeasible least squares series estimator $\widehat{\pi}_{2,i}^*(w)$ is given by $\widehat{\pi}^*_{2,i}(w) = b^K_\pi(w)' ({B}_\pi' {B}_\pi)^{-1} {B}_K' Y_i$. Similarly, the feasible series regression matrix $\widehat{B}_\pi := ( b^K_\pi(\widehat{W}_{21}), \ldots, b^K_\pi(\widehat{W}_{2n}) )'$ yields the feasible least squares series estimator, $\widehat{\pi}_{2,i}(w) = b^K_\pi(w)' (\widehat{B}_\pi' \widehat{B}_\pi)^{-1} \widehat{B}_K' Y_i$.

To further streamline notation, wherever it does not lead to confusion, we will let $\pi_2$ be a generic coefficient vector belonging to $\{\pi_{2,i}\}_{i=1}^p$, as well as define $\widehat{\pi}_{2}$, $Y$ and $u_2$ associated to the same regression equation.

Distributional and Sieve Assumptions

To derive asymptotic consistency results, we begin by stating conditions on the basic probability structure of the model.

Assumption (ref) follows both goncalvesImpulseResponseAnalysis2021 and chenOptimalUniformConvergence2015. Note that, as $W_{2t}$ depends only on $X_{t:t-p}$, $Y_{t-1:t-p}$ and $\epsilon_{1t}$, the entries of $\xi_{2t}$ in (ref) are independent of $W_{2t}$, so that $\mathbb{E}[ u_{2it} \:\vert\: W_{2t} ] = 0$.

Assumption (ref) implies that $X_t$, $Y_t$, as well as $\epsilon_t$ are bounded random variables. In (semi-)nonparametric estimation, imposing that $X_t$ is bounded almost surely is a standard assumption. Since lags of $Y_t$ and innovations $\epsilon_t$ contribute linearly to all components of $Z_t$, it follows that they too must be bounded. In practice, Assumption (ref) is not particularly restrictive, as many credibly stationary economic series often have reasonable implicit (e.g., inflation) or explicit bounds (e.g., employment rate). The analysis of impulse responses on compact domains is, however, non-trivial. In Section (ref) below, we provide an IRF shock relaxation framework that can accommodate this setting.

Without loss of generality, let $\mathcal{Y} = [0,1]^{d_Y}$ and $\mathcal{X} = [0,1]$.

Assumptions (ref) and (ref) are classical in the nonparametric regression literature. Let then $\mathcal{W}_2 \subset \mathbb{R}^d$ be the domain of $W_{2t}$. By assumption, $\mathcal{W}_2$ is compact and convex and is given by the direct product $ \mathcal{W}_2 = \{1\} \times \mathcal{Y}^{\, p} \times \mathcal{X}^{1+p} \times \mathcal{E}_1 $, where $\mathcal{E}_1$ is the domain of structural innovations $\epsilon_{1t}$ i.e. $\mathcal{E} \equiv \mathcal{E}_1 \times \mathcal{E}_2$.

Assumption (ref) provides mild regularity conditions on the families of sieves that can be used for the series estimator. More generally, letting $\mathcal{W}_2$ be compact and rectangular makes Assumptions (ref)(i)-(ii) hold for commonly used basis functions chenOptimalUniformConvergence2015. The approximation properties of these sieves are well understood chenChapter76Large2007.\footnote{See also chenPenalizedSieve2013,belloniNewAsymptoticTheory2015 for additional discussion and examples of sieve families.} In particular, Assumption (ref)(i) holds with $\omega_1 = 0$ since the domain is fixed over the sample size. What is also needed is that the nonparametric components of the sieve given by $b_{\pi,1K}, \ldots, b_{\pi,KK}$ are able to approximate $G_{2,i}$ with an error that decays sufficiently fast with $K$. Lastly, Assumption (ref)(iii) is a mild assumption on the conditioning of the semiparametric sieve.

We define $\widetilde{b}^K_\pi(w) := \mathbb{E}[\, {b}^K_\pi(W_{2t}) {b}^K_\pi(W_{2t})' \,]^{-1/2}\, {b}^K_\pi(w)$ and $\widetilde{B}_\pi := \big( \widetilde{b}^K_\pi(W_{21}), \allowbreak \ldots, \allowbreak \widetilde{b}^K_\pi(W_{2n}) \big)'$ to be the orthonormalized vector of basis functions and the orthonormalized regression matrix, respectively. To derive uniform converges rates under dependence, we require that the Gram matrix of orthonormalized sieve converges to the identity matrix.

chenOptimalUniformConvergence2015 introduced Assumption (ref) as a key ingredient for their proofs, while also showing that it holds whenever $\{W_{2t}\}_{t\in\mathbb{Z}}$ is either an exponential or algebraic $\beta$-mixing process. Unfortunately, mixing conditions are difficult to verify or test with respect to model specification, as they rely on bounding the worst-case “independence gap” between probability events (see Appendix (ref)). We extend their approach to the case of geometrically decaying physical dependence, a metric proposed by wuNonlinearSystemTheory2005. This is a setting where many estimation and inference results have been derived, see for example wuKernelEstimationTime2010,wuAsymptoticTheoryStationary2011a,chenSelfnormalizedCramertypeModerate2016 and references within.

Assumption (ref) is subsumed by Assumption (ref). Using a physical dependence measure, we argue that it is also possible to swap mixing conditions with more explicit, primitive conditions derived exclusively in terms of model specification (ref). In particular, for specific semiparametric model specifications, it is possible to verify Assumption (ref) directly by leveraging stability/contractivity theory of dynamic systems. We refer the reader to Appendix (ref) for an additional, in-depth discussion of dependence and physical conditions.

Uniform Convergence and Consistency

We can now state our main result, which shows that the two-step estimation procedure for (ref) provides consistent estimates.

Sup-norm bounds for $\lVert \widehat{\Pi}^*_2 - \Pi_2 \rVert_\infty$ may be obtained from Lemma 2.3 and Lemma 2.4 in chenOptimalUniformConvergence2015. Assuming $s \geq 1$ and $d = 1$, such as in the setting of the additively separable model in Example (ref) and in our empirical applications, it is possible to show that, if the optimal nonparametric rate for $K$ is used and the additive sieve inherits the conditioning of the underlying sieve bases, then $\widehat{\Pi}_2$ is sup-norm consistent.

The requirement $\lambda_{K,n} \lesssim 1$ for additively separable sieves is mild given the known properties of B-spline and wavelet sieves, although nontrivial. Since one cannot exploit sparsity in the case of non-locally supported bases, as is the case with linearly separable sieves, we assume $\lambda_{K,n}$ is upper bounded by a constant to streamline the analysis of the empirical sieve projection operator (see also the discussion in huangLocalAsymptoticsPolynomial2003a, Section 7).

Impulse Response Analysis

After discussing the estimation of the structural model's coefficients, we can now address the derivation of nonlinear impulse responses. To ensure compatibility with bounded support assumptions, we introduce an extension of the classical IRF definition, termed relaxed impulse response function, which differs only in the form of the shock applied to the model. We then show that nonlinear relaxed IRFs can be consistently estimated, and uniformly so for shocks picked within a compact range.

Relaxed Shocks

Under Assumptions (ref) and (ref), the standard construction of impulse responses following Section (ref) is, unfortunately, improper. This is immediately seen by noticing that, at impact, $X_t(\delta) = X_{t} + \delta$, meaning that $\mathbb{P}( X_t(\delta) \not\in \mathcal{X} ) > 0$ since there is a translation of size $\delta$ in the support of $X_t$. To address this problem, we introduce an extension to the standard additive shock that is used to define impulse responses.

We begin by defining mean-shift shocks, that is, shocks such that the distribution of time $t$ innovations is shifted to have mean $\delta$, while retaining compact support almost surely.

With a mean-shift shock, at impact it holds $X_{t}(\delta) = X_{t} + (\epsilon_{1t}(\delta) - \epsilon_{1t})$. In the standard setting, where $\mathbb{E}[\epsilon_t] = 0$ and $\mathcal{E}_1 \equiv \mathbb{R}$, $\epsilon_{1t}(\delta) = \epsilon_{1t} + \delta$ is clearly valid. More generally, however, imposing $\mathbb{E}[\epsilon_{1t}(\delta)] = \delta$ requires the distribution of $\epsilon_{1t}$ to be known. If instead one is willing to assume only that $\mathbb{E}[\epsilon_{1t}(\delta)] \approx \delta$, it is possible to sidestep this need by introducing a shock relaxation function.

By setting $\epsilon_{1t}(\delta) = \epsilon_{1t} + \delta \rho(\epsilon_{1t})$ for a $\rho$ compatible with $\delta$, it follows that $ X_{t}(\delta) = X_{t} + \delta \rho(\epsilon_{1t})$ and $\lvert \mathbb{E}[\epsilon_{1t}(\delta)] \rvert = \lvert \delta \mathbb{E}[\rho(\epsilon_{1t})] \rvert \leq \lvert \delta \rvert$ since $\mathbb{E}[\rho(\epsilon_{1t})] \in [0, 1]$ by definition of $\rho$. If $\rho$ is a bump function, a relaxed shock is a structural shock that has been mitigated proportionally to the density of innovations at the edges of $\mathcal{E}_1$ and the squareness of $\rho$.

It is important to emphasize that shock relaxation is a generalization of standard shock designs. Indeed, when $\mathcal{X} = \mathbb{R}$ and $\mathcal{E}_1 = \mathbb{R}$, $\rho = 1$ is a relaxation function compatible with all $\delta \in \mathbb{R}$. Nonetheless, we may also wonder how much information on the nonlinear term $G_2$ we can recover at the “boundary” of a finite sample. If $X_t$ is unbounded but well-concentrated, even under strong smoothness conditions and strictly positive density, little can be learned about the local structure of regression functions in regions of low density.\footnote{In our regression setting, for example, Theorem 1 in kohlerOptimalGlobalRates2009 on $L_2$ kernel regression error, assuming $\mathbb{E}[\lvert X_t \rvert^{\beta}] \leq M < \infty$ for some constant $\beta > 2s$, would require the bandwidth to grow over $\mathcal{X}$ faster than $\lvert X_t \rvert$. This question is also linked to issues in kernel density estimation over sets with boundary, see e.g. karunamuni2005on,malec2014nonparametric,berry2017density and references therein.}

Relaxed Impulse Response Consistency

We will now study relaxed impulse responses in the setting of additively separable models. Additive separability is a common assumption in applied work, as we shall impose it in the empirical analyses of Section (ref) and (ref). Further, collecting nonlinear terms over lags significantly streamlines notation and analysis, and aligns with the setup of Corollary (ref). It would be straightforward, if tedious, to extend our derivations below to the more general setting of Theorem (ref).

Given $\delta \in \mathbb{R}$ and compatible shock relaxation function $\rho$, let $\widetilde{\delta}_t := \delta \rho({\epsilon}_{1t})$. Starting from a path $X_{t+j:t}$ and (ref), the relaxed shock path is

The relaxed-shock impulse response is thus given by

For $1 \leq \ell \leq d$, we define ${V}_{j,\ell}(\delta)$ to be the sample analog of the horizon $j$ nonlinear effect on the $\ell$th variable,

where ${\Gamma}_{j,\ell}$ is the $\ell$th component of functional vector ${\Gamma}_{j}$. As $\epsilon_{1t}$ is not universally observable, we introduce its residual counterpart, $\widehat{\widetilde{\delta}}_t = \delta \rho(\widehat{\epsilon}_{1t})$. The associated plug-in sample estimates are $\widehat{V}_{j,\ell}(\delta) = ({n-j})^{-1} \sum_{t=1}^{n-j} \widehat{v}_{j,\ell}\big( X_{t+j:t}; \widehat{\widetilde{\delta}}_t \big)$, $\widehat{v}_{j,\ell}(X_{t+j:t}; \widehat{\widetilde{\delta}}_t) = \widehat{\Gamma}_{j,\ell} \widehat{\gamma}_{j}(X_{t+j:t}; \widehat{\widetilde{\delta}}_t) - \widehat{\Gamma}_{j,\ell} X_{t+j}$, and

Our next theorem proves the consistency of the relaxed impulse responses estimator based on semiparametric series estimates. We leverage the sup-norm bounds of Theorem (ref) to derive a result that is uniform in $\delta$ over a compact interval $[-\mathcal{D}, \mathcal{D}]$, $\mathcal{D} > 0$. This allows us to make valid comparisons between IRFs due to shocks of different sizes.

Simulations

To analyze the performance of the two-step semiparametric estimation strategy discussed above, we begin by considering the two simulation setups employed by goncalvesImpulseResponseAnalysis2021. We compare the bias and MSE of the estimated relaxed shocked impulse response functions for different methods. The population responses we consider in this section are also constructed using the same shock relaxation scheme; therefore, both relaxed IRF estimators are correctly specified. Appendix (ref) includes a robustness analysis wherein non-relaxed nonlinear population IRFs are targeted. We also provide simulations under a misspecified design, which highlight how, in larger samples, the nonparametric sieve estimator consistently recovers impulse responses, whereas a least-squares estimator constructed with a pre-specified nonlinear transform may not.\footnote{Population impulse responses are estimated with $10^5$ replications, while MSE and bias of both semiparametric and parametric IRFs are computed with $10^4$ Monte Carlo replications. In all setups, a cubic B-spline sieve is used.}

\paragraph*{Benchmarks.}

Like in goncalvesImpulseResponseAnalysis2021, we consider two simulation setups: A bivariate design with identified shocks (DGPs 1-3) and a three-variable design with partial block-recursive identification (DGPs 4-6). In both, we set a sample size of $n = 240$, which is realistic for most macroeconomic data settings: this is approximately equivalent to 20 years of monthly data or 60 years of quarterly data goncalvesImpulseResponseAnalysis2021. We discuss here only the bivariate simulation design with shock $\delta = +1$, and refer the reader to Appendix (ref) for the block-recursive setup.

We set either $X_t = \epsilon_{1t}$ (DGP 1), $X_t = 0.5 X_{t-1} + \epsilon_{1t}$ (DGP 2) or $X_t = 0.5 X_{t-1} + 0.2 Y_{t-1} + \epsilon_{1t}$ (DGP 3), and

Innovations $\epsilon_{1t}$ and $\epsilon_{2t}$ are drawn as independent, truncated standard Gaussian variables over $[-3, 3]$. The shock relaxation function is $\rho(z) = \mathbb{I}\{|z| \leq 3 \}\exp\left( 1 + ( \lvert {z}/{3} \rvert^4 - 1 )^{-1} \right)$, cf. Remark (ref). In Figure (ref) we show MSE and bias curves for the IRF on $Y_t$ in DGP 2, where $X_t$ is an exogenous AR(1) process. One can see that the sieve IRF leads only to a minor increase in mean squared error at short horizons compared to directly estimating the parameter of the true specification. This marginal increase in MSE is consistent across DGPs 1 through 3.

These simulations show that there is negligible loss of efficiency in terms of either MSE or bias when implementing the fully flexible semiparametric estimates at realistic sample sizes. We confirm these results when studying DGPs 4-6, where estimation of the structural matrix $B_0$ is included in the regression problem. Detailed results can be found in Appendix (ref).

\paragraph*{Misspecified Model.}

To assess the robustness of the proposed semiparametric approach versus the parametric nonlinear model, we consider a modified process (DGP 7):

where $\varphi(x) := (x - 1)(0.5 + \tanh(x - 1)/2)$. In this design, we assume that the researcher's prior is $\varphi(x) = \max(0, x)$, as in the benchmark simulations. To emphasize the difference in estimated IRFs, in this setup we focus on $|\delta| = 2$ and $n = 2400$; innovations $\epsilon_{1t}$ and $\epsilon_{2t}$ are drawn from a standard Gaussian distribution truncated over $[-5, 5]$, and $\rho(z) = \mathbb{I}\{|z| \leq 5 \}\exp( 1 + ( \lvert {z}/{5} \rvert^{3.9} - 1 )^{-1} )$. As Figure (ref) shows, positive-shock parametric nonlinear IRF estimates are severely biased, while semiparametric sieve IRFs have comparatively negligible error: This yields an up to 4 times reduction of overall MSE at short horizons. Appendix (ref) provides additional simulation results showing that the same improvements hold when $\delta = - 2$. There, we also discuss the setting where $\varphi(x)$ is replaced with map $\widetilde{\varphi}(x) = \varphi(x+1)$, which agrees closely with $\max(0, x)$. In this last setting, we find that parametric nonlinear regression dominates in MSE and bias terms. As one might expect, therefore, parametric modeling is reliable only in cases where a sufficiently good model prior is available.

Empirical Applications

In this section, we showcase the practical utility of the proposed semiparametric sieve estimator by considering two applied exercises. In line with previous work applying nonlinear IRF methods to macroeconomic data, such as e.g. kilian2011responses,goncalvesImpulseResponseAnalysis2021,goncalvesStatedependentLocalProjections2024 and goncalvesNonparametricLocalProjections2024, our discussion is focused on point impulse response estimates. In both cases, we will consider sieve IRFs constructed with shock relaxation: Appendix (ref) shows that our analysis remains valid also when evaluating non-relaxed semiparametric responses.

Monetary Policy Shocks

We first consider a four-variable model identical to the one analyzed by goncalvesImpulseResponseAnalysis2021, and based on tenreyro2016pushing. Let $Z_t = (X_t, \textnormal{FFR}_t, \textnormal{GDP}_t, \textnormal{PCE}_t)'$, where $X_t$ is the series of narrative U.S. monetary policy shocks, $\textnormal{FFR}_t$ is the federal funds rate, $\textnormal{GDP}_t$ is log-real GDP and $\textnormal{PCE}_t$ is PCE inflation.\footnote{In goncalvesImpulseResponseAnalysis2021 p. 122, it is mentioned that CPI inflation is included in the model, but both in the replication package made available by one the authors (\url{https://sites.google.com/site/lkilian2019/research/code}) from which we source the data, and in tenreyro2016pushing, PCE inflation is used instead. Moreover, the authors say that both the FFR and PCE enter the model in first differences, yet, in their code, these variables are kept in levels. We thus consider a model in levels to allow for a proper comparison between estimation methods, although the series are highly persistent.} As a pre-processing step, GDP is transformed to log GDP and then linearly detrended. The data is available quarterly and spans from 1969:Q1 to 2007:Q4. As in tenreyro2016pushing, we use a model with one lag, $p=1$. The narrative shock $X_t$ is considered to be an i.i.d. sequence, i.e. $X_t = \epsilon_{1t}$, therefore we assume no dependence on lagged variables when implementing the pseudo-reduced form (ref). Like in goncalvesImpulseResponseAnalysis2021, we consider positive and negative shocks of size $|\delta| = 1$ and choose $\rho(z) = \mathbb{I}\{ |z| \leq 4 \} \exp( 1 + ( \lvert {z}/{4} \rvert^{6} - 1 )^{-1} )$ to be the shock relaxation function. Figure (ref) in the Online Appendix provides a check for the compatibility of $\rho$ given the sample distribution of $X_t$. Knots for sieve estimation are located at $\{-1, 0, 1\}$. The model is block-recursive, and U.S. monetary policy shocks are identified without the need to impose additional assumptions on the remaining shocks. goncalvesImpulseResponseAnalysis2021, like tenreyro2016pushing, use two nonlinear transformations, $F(x) = \max(0, x)$ and $F(x) = x^3$, to try to gauge how negative versus positive and large versus small shocks, respectively, affect the U.S. macroeconomy. They find that the two maps yield very similar responses, so we focus on comparing the IRFs estimated via sieve regression with the ones obtained by setting $F(x) = \max(0, x)$, as well as linear IRFs.

Figure (ref) plots estimated impulse responses to both positive and negative monetary policy shocks. The impact on the federal funds rate is consistent across all three procedures. The semiparametric nonlinear response for GDP, unlike in the case of linear and parametric nonlinear IRFs, is nearly zero at impact and has a monotonic decrease until around 10 quarters ahead. The change in shape is meaningful, as the procedure of goncalvesImpulseResponseAnalysis2021 still yields a small short-term upward jump in GDP when a monetary tightening shock hits. Moreover, after the positive shock, the sieve GDP responses reaches its lowest value 4 and 2 quarters before the linear and parametric nonlinear responses, while its size is 13% and 16% larger, respectively.\footnote{The strength of this effect changes across different shock sizes, as Figure (ref) in Appendix (ref) proves. As shock sizes get smaller, nonlinear IRFs, both parametric and sieve, show decreasing negative effects.} Finally, the sieve PCE response is positive for a shorter interval, but looks to be more persistent once it turns negative, also 10 months after impact.

When the shock is expansionary, one sees that the semiparametric FFR response is marginally mitigated compared to the alternative estimates. An important puzzle is due to the negative impact on GDP: Both types of nonlinear responses show a drop in output in the first 5 quarters. Such a quick change seems unrealistic, as one does not expect inflation to suddenly reverse sign, but, as goncalvesImpulseResponseAnalysis2021 also remark, the overall impact on inflation of both shocks is small when compared to the change in federal funds rate.

Uncertainty Shocks

Traditional central bank policymaking is heavily guided by the principle that a central bank can and should influence expectations. Therefore, controlling the (perceived) level of ambiguity in current and future commitments is key. istrefiSubjectiveInterestRate2018 provide an analysis of the impact of unforeseen changes in the level of subjective interest rate uncertainty on the macroeconomy. For the sake of simplicity, our evaluation will focus only on their 3-month-ahead uncertainty measure for short-term interest rate maturities (3M3M) and the U.S. economy. Like in istrefiSubjectiveInterestRate2018, let $Z_t = (X_t, \textnormal{IP}_t, \textnormal{CPI}_t, \textnormal{PPI}_t, \textnormal{RT}_t, \textnormal{UR}_t)'$ be a vector where $X_t$ is the chosen uncertainty measure, $\textnormal{IP}_t$ is the (log) industrial production index, $\textnormal{CPI}_t$ is the CPI inflation rate, $\textnormal{PPI}_t$ is the producer price inflation rate, $\textnormal{RT}_t$ is (log) retail sales and $\textnormal{UR}_t$ is the unemployment rate. The nonlinear model specification is given by $Z_t = \mu + A_1 Z_{t-1} + A_2 Z_{t-1} + F_1(X_{t-1}) + F_2(X_{t-2}) + D W_t + u_t ,$ where $W_t$ includes a linear time trend and oil price $\textnormal{OIL}_t$.\footnote{Inclusion of linear exogenous variables in the semiparametric theoretical framework in Section (ref) is straightforward as long as one can assume that they are stationary and weakly dependent. The choice of using $p=2$ is identical to that of the original authors, based on BIC.} The data has a monthly frequency and spans the period between May 1993 and July 2015.\footnote{We utilize the original data employed by the authors, who kindly shared it upon request. However, we rescale retail sales ($\textnormal{RT}_t$) so that the level in January 2000 equals 100.} Note here that nonlinear functions $F_1$ and $F_2$ are assumed not to affect $X_t$, which is the structural variable. The linear VAR specification of istrefiSubjectiveInterestRate2018 is recovered by simply assuming $F_1 = F_2 = 0$ prior to estimation. Since they use recursive identification and order the uncertainty measure first, this model too is block-recursive. We consider a positive shock with intensity $\delta = \sigma_{\epsilon,1}$, where $\sigma_{\epsilon,1}$ is the standard deviation of structural innovations. In this empirical exercise, the relaxation function is $\rho(z) = \mathbb{I}\left\{ |z| \leq 1/4 \right\} \exp( 1 + ( \lvert 4 x \rvert^{8} - 1 )^{-1} ) $ and we set $\{0.1, 0.3\}$ to be the cubic spline knots. As 3M3M is a non-negative measure of uncertainty, some care must be taken to make sure that the shocked paths for $X_t$ do not reach negative values. Figure (ref) in Appendix (ref) shows that the relaxation function is compatible, and also that the shocked nonlinear paths of $X_t$ with impulse $\delta$ and $\delta'$ all do not cross below zero.

Figure (ref) presents both the linear and nonlinear structural impulse responses obtained. Importantly, even though istrefiSubjectiveInterestRate2018 estimate a Bayesian VAR model and here we consider a frequentist vector autoregressive benchmark, the shape of the IRFs is retained, cf. the median response in the top row of their Figure 4. When uncertainty increases, industrial production drops, and the size and extent of this decrease are intensified in the nonlinear responses. The sieve IP response reaches a value that is $54\%$ lower than that of the respective linear IRF.\footnote{Figure (ref) in Appendix (ref) confirms that this difference is consistent over a range of shock sizes, too.} A similar behavior holds true for retail sales ($38\%$ lower) and unemployment ($23\%$ higher), proving that this shock is more profoundly contractionary than suggested by the linear VAR model. Further, CPI and PP inflation both display short-term fluctuations, which strengthen the short- and medium-term impact of the shock. CPI and PP nonlinear inflation responses are $76\%$ and $41\%$ stronger than their linear counterpart, respectively. These differences show that linear IRFs might be both under-estimating the short-term intensity and misrepresenting the long-term persistence of inflation reactions. Given the strength of nonlinear IRFs, this discrepancy may also suggest that the 3M3M uncertainty measure partially captures the financial channel, too. Hence, we believe our analysis provides some evidence that the linear VAR used by istrefiSubjectiveInterestRate2018 may miss some key impulse response features.\footnote{See also Figure (ref), which plots the estimated functions of the endogenous variables.}

Conclusion

This paper studies the application of semiparametric series estimation to the problem of structural impulse response analysis for time series. After first discussing partial block-recursive identification, we have shown that, for models with moderate physical dependence, series estimation can be employed and structural IRFs are consistently estimated. Simulations showcase that this approach is both valid in moderate samples and has the added benefit of being robust to misspecification of the nonlinear model components. Finally, two empirical applications showcase the potential insights gained by departing from either linear or parametric nonlinear specifications when estimating structural responses.

A key aspect that we have not touched upon is inference in the form of confidence intervals. The development of inferential theory appears feasible in light of the uniform inference results obtained by e.g. belloniNewAsymptoticTheory2015 in the i.i.d. setting and liUniformNonparametricInference2020 for time series data, and it is an important direction for future research. Studying other sieve spaces, such as neural networks chenImprovedRatesAsymptotic1999,farrellDeepNeuralNetworks2021a or shape-preserving sieves chenChapter76Large2007, would also be highly desirable. Finally, in the spirit of kangINFERENCENONPARAMETRICSERIES2021, deriving new inference results that are uniform in the selection of series terms is important, as, in practice, the sieve should be tuned in a data-driven way.