Stochastic Responses

6. Stochastic Responses#

Authors: Jaroslav Borovicka, Lars Peter Hansen, and Thomas J. Sargent

Date: October 2025 \(\newcommand{\eqdef}{\stackrel{\text{def}}{=}}\)

6.1. Introduction#

Impulse-response methods have been used by economists since [Frisch, 1933] and in other disciplines. For nonlinear stochastic models, impulse responses are themselves stochastic. In contrast to their linear counterparts, they do not simply scale in the size of the impulse. Alternative approaches have been suggested in economics including [Gallant et al., 1993], [Koop et al., 1996], and [Gourieroux and Jasiak, 2005]. This chapter provides stochastic responses in both discrete and continuous time for marginal changes in state variables and shocks. Much, but not all, of the vast literature on vector autoregressions views these responses as ends in and of themselves. For instance, many applied researchers refer to “structural VAR’s”. She applied researchers embrace “causal language” in which identified shocks taken as exogenous inputs into a dynamical system are presumed to “cause” movements in the vector time series of interest. Such measurements, while interesting, can have rather indirect connections to counterfactuals or hypothetical interventions related to perspective policy changes that are at the heart of structural models. Builders of dynamic stochastic equilibrium models use the construct of structural’ in the sense of [Marschak, 1953], [Hurwicz, 1966], and [Lucas, 1976]. They allow for investigating how a dynamical system changes when one portion of it is altered. When [Hurwicz, 1966], for instance, provided formal definition of a structural model, he chose not to use the term “cause.” For us, these stochastic responses reveal model implications as is often done in empirical macroeconomics, but they are also central inputs into marginal valuations that we will use for a variety of purposes.

The responses that we characterize in this section are local in nature. They measure how a full vector time series responds to small change in a state or shock at some initial time period. They are convenient because they have a simplified structure. In contrast to global calculations for nonlinear models, the local responses scale linearly in the magnitude of the initial perturbation. They can be suggestive of the impacts of large changes, but they are not intended as a substitute for global investigations. In subsequent chapters, we provide extensive discussions of two types of such applications and extensions that complement the empirical macroeconomists quantitative characterizations. The first type provides generates what we call shock elasticities that help us characterize the building blocks for exposures to uncertainty and prices of those exposures. The second type provides asset-pricing type representations for endogenous variables including various forms of capital and opens the door dynamic versions of marginal policy assessments as is common in public finance and environmental economics.

6.2. Discrete time#

We first consider a discrete-time specification.

6.2.1. Markov dynamics#

We start with Markov process

(6.1)#\[\begin{split}X_{t+1} = \psi(X_t, W_{t+1}), \\ Y_{t+1} - Y_t = \kappa(X_t, W_{t+1}),\end{split}\]

where there are \(n\) components of \(X,\) \(Y\) is scalar, and \(W\) is \(k\) dimensional. The \(Y\) process allows for growth along a common stochastic path.

6.2.2. Discrete-time variational dynamics#

We construct what is called first variational processes. We denote by \(\Lambda\) the marginal responses of the \(X\) process, and we denote by \(\Delta\) the marginal responses of the \(Y\) process. Both are stochastic processes. We use the initial conditions \(\Lambda_0\) and \(\Delta_0,\) to delineate what responses are of interest. For instance if \(\Lambda_0\) is set to a coordinate vector, we study the responses to a marginal change to the corresponding date zero state variable, \(X_0\). For these and other initializations, \(({\Lambda_t}', \Delta_t)'\) is the date \(t\) state vector stochastic response to the chosen perturbation at the initial date. These variational processes are the stochastic impulse responses to small changes in the underlying state variables.

In effect, we deduce the evolution of the variational processes by “differentiating” the state dynamics in a generalized sense that accommodates the underling stochastic structure. To obtain a recursive representation for \((\Lambda, \Delta),\) we differentiate (6.1) and apply the chain rule:

(6.2)#\[\begin{split}\Lambda_{t+1} = \frac {\partial \psi}{\partial x'} (X_t, W_{t+1}) \Lambda_t \\ \Delta_{t+1} - \Delta_t = \frac {\partial \kappa}{\partial x}(X_t, W_{t+1}) \cdot \Lambda_t .\end{split}\]

In this calculation, \(\Lambda_{t+1}\) and \(\Delta_{t+1}\) are stochastic as they inherit the stochastic dependence of \(X_{t+1}\) and \(Y_{t+1}.\) By differentiating the process at a given initial calendar date, we are allowing for date \(t\) variables to change as a function of date \(t\) information. Not surprisingly, the variational process dynamics depend explicitly on the original state dynamics.

Example 6.1

The evolution of the variational processes is nonstochastic if \( \frac {\partial \psi}{\partial x'}\) and \( \frac {\partial \kappa}{\partial x}\) are constant as is true when \(\psi\) and \(\kappa\) are affine in \(x\). Otherwise, variational processes are stochastic.

Example 6.2

Consider the following quadratic specification:

\[\begin{align*} X_{t+1}^i & = {\sf a}_i \cdot X_t+ {\frac 1 2} {X_t}'{\mathbb A}_i X_t + {X_t}'{\mathbb B}_i W_{t+1} + {\sf b}_i \cdot W_{t+1}, \hspace{.3cm} i=1,...,n\cr Y_{t+1} - Y_t & = {\sf d} \cdot X_t + {\frac 1 2} {X_t}'{\mathbb D} X_t + {X_t}' {\mathbb F}W_{t+1} + {\sf f} \cdot W_{t+1} . \end{align*}\]

where \({\mathbb A}_i\) and \({\mathbb D}\) are normalized to be symmetric. A simple calculation shows:

\[\begin{align*} \Lambda_{t+1}^i & = {\sf a}_i \cdot \Lambda_t + {\Lambda_t}'{\mathbb A}_i X_t + {\Lambda_t}'{\mathbb B}_i W_{t+1} , \hspace{.3cm} i=1,...,n\cr \Delta_{t+1} - \Delta_t & = {\sf d} \cdot \Lambda_t + {\Lambda_t}'{\mathbb D} X_t + {\Lambda_t}' {\mathbb F}W_{t+1} . \end{align*}\]

This illustrates a specific stochastic structure for the variational processes. As we have seen in Chapter 9:Exploring Recursive Utility, quadratic specifications of this type emerge as second-order approximations to nonlinear stochastic models of state dynamics.

6.3. Continuous-time dynamics#

We now consider the continuous-time counterpart for Brownian motion shocks.

6.3.1. Markov diffusion dynamics#

As a part of a more general derivation, we begin with state dynamics modeled as a Markov diffusion:

\[\begin{split}\begin{align*} dX_t & = \mu(X_t) dt + \sigma(X_t) dW_t \\ dY_t & = \nu(X_t) dt + \varsigma(X_t) dW_t. \end{align*}\end{split}\]

where \(W\) is now a \(k\)-dimensional standard Brownian motion. We denote the filtration (family of specifications of conditioning information events) \({\mathfrak F} \eqdef \left\{ {\mathfrak F}_t : t\ge 0\right\}\) constructed from the Brownian motion and any pertinent date zero information.

6.3.2. Variational processes#

Following [Borovička et al., 2014], we construct marginal impulse response functions using what are called variational processes. We build the dynamics for what is called the first variational, \(\Lambda\) by following the construction in [Fournie et al., 1999]. The first variational process tells the marginal impact on future \(X\) of a marginal change in one of the initial states analogous to the \(\Lambda\) process that we constructed in discrete time. Thus this process has the same number of components as \(X\). By initializing the process at one of the alternative coordinate vectors, we again isolate an initial state of interest.[1].

The drift for the \(i^{th}\) component of \(\Lambda\) is

\[\lambda' {\frac {\partial \mu_i} {\partial x} }(x) \]

and the coefficient on the Brownian increment is

\[\lambda' \frac {\partial \sigma_i }{\partial x}(x)\]

for \(\lambda\) a hypothetical realization of \(\Lambda_t\) and \(x\) a hypothetical realization of \(X_t,\) where \('\) denotes vector or matrix transposition. The implied evolution of the process \(\Lambda^i\) is[2]

\[d\Lambda_{t}^i = \left(\Lambda_t\right)'\frac{\partial \mu_i}{\partial x}(X_t) dt + \left({\Lambda_t}\right)'\frac{\partial \sigma_i}{\partial x}(X_t) dW_t.\]

With the appropriate stacking, the drift for the composite process \((X,\Lambda)\) is:

(6.3)#\[\begin{split}\mu^a(x,\lambda) \overset{\text{def}}{=} \begin{bmatrix} \mu(x) \\ \lambda'{\frac {\partial \mu_i} {\partial x} }(x) \\ ... \\ \lambda'{\frac {\partial \mu_n} {\partial x} }(x) \end{bmatrix},\end{split}\]

and the composite matrix coefficient on \(dW_t\) is given by

(6.4)#\[\begin{split}\sigma^a(x,\lambda) \overset{\text{def}}{=} \begin{bmatrix} \sigma(x) \\ \lambda'\frac {\partial \sigma_1 }{\partial x}(x)\\ ... \\ \lambda' \frac {\partial \sigma_n }{\partial x}(x) \end{bmatrix}.\end{split}\]

Let \(\Delta\) be the scalar variational process associated with \(Y.\) Then

\[d \Delta_t = \Lambda_t \cdot \frac {\partial \nu}{\partial x} (X_t)dt + {\Lambda_t}' \frac {\partial \varsigma}{\partial x'} dW_t \]

Analogous to the discrete-time outcome, the variational dynamics depend explicitly on the original diffusion dynamics. As in discrete time, by initializing the vector \(\Lambda_0\) at a coordinate vector, the resulting processes give marginal responses to a corresponding state vector.

Example 6.3

Consider the case of linear dynamics:

\[\begin{matrix} \mu(x) = {\mathbb A}x & \sigma(x) = {\mathbb B} \cr \nu(x) = {\mathbb D} x & \varsigma(x) = {\mathbb F} \end{matrix} .\]

Then

\[\begin{align*} \mu^a(x, \lambda) & = \begin{bmatrix} {\mathbb A}x \cr {\mathbb A} \lambda \end{bmatrix} \cr & \cr \sigma^a(X, \lambda) & = \begin{bmatrix} {\mathbb B} \cr 0 \end{bmatrix}. \end{align*} \]

Thus

\[\Lambda_t = \exp \left( {\mathbb A} t \right) \Lambda_0,\]

and

\[\begin{align*} \Delta_t & = {\mathbb D} \int_0^t \Lambda_u du + \Delta_0 \cr & = {\mathbb D} \left[\int_0^t \exp\left( {\mathbb A}u \right) du \right]\Lambda_0 + \Delta_0 \cr & = - {\mathbb D} {\mathbb A}^{-1} \left[ {\mathbb I} - \exp\left( {\mathbb A} t \right) \right]\Lambda_0 + \Delta_0 . \end{align*}\]

Given the underlying linearity, the global responses follow directly from local responses with a scale adjustment for the size of the increment.

In the calculations that follow, we will have cause to do a forward shift \({\mathbb S}^\tau \) of these process by which we shift the time units on all of the variables used in the the construction and the initialization period forward \(\tau\) time periods.

6.3.3. Responses to initial shocks#

So far, we have characterized stochastic responses to initial changes in the state variables. From these, we deduce vector of responses to the initial shocks by changing the initial conditions for the variational processes. Let \(\sigma_i\) denotes the \(i^{th}\) column of \(\sigma\) and \(\varsigma_i\) the \(i^{th}\) entry of \(\varsigma\) and initialize:

\[\begin{align*} \Lambda_0^i & = \sigma_i(X_0) \cr \Delta_0^i& = \varsigma_i(X_0) \end{align*} \]

Using these initial conditions, we obtain the continuous time stochastic responses to shock \(i\) at date zero corresponding to entry \(i\) of \(dW_0\), which we denote \(dW_0^i\). Specifically, the impacts of this date zero shock on \(X\) and \(Y\) are, respectively:

\[\begin{split} & {\mathbb E} \left( \Lambda_t^i \mid {\mathfrak F}_0 \right) dW_0^i \cr & {\mathbb E} \left( \Delta_t^i \mid {\mathfrak F}_0 \right) dW_0^i. \end{split}\]

For the special case of linear dynamics given in Example 6.3,

\[\begin{align*} \Lambda_t^i & = \exp\left( {\mathbb A} t \right) {\mathbb B}_i \cr \Delta_t^i & = - {\mathbb D} {\mathbb A}^{-1} \left[ {\mathbb I} - \exp\left( {\mathbb A} t \right) \right]{\mathbb B}_i + {\mathbb F}_i , \end{align*}\]

which are the continuous-time counterparts of the familiar impulse responses.

6.3.4. Moving-average representation#

With Markov diffusions, we also have a state-dependent counterpart to a moving-average representation that is well known from linear time series models. The aim is represent the \(X\) processes in terms of the shock contributions at different dates and then “add up’’ all of the response across time and components of the Brownian increments.

Stack the shock-specific responses into matrices and vectors:

\[\begin{align} \Phi_t & \eqdef\begin{bmatrix} \Lambda^1_t \cdots \Lambda^k_t \end{bmatrix} \cr \Psi_t & \eqdef \begin{bmatrix} \Delta^1_t \cdots \Delta^k_t \end{bmatrix} \end{align}\]

where \(\Phi_t\) is \(n \times k\) and \(\Psi_t\) is \(1\times k\). To capture the impact of future shocks, we will have cause to do a forward shift \({\mathbb S}^\tau \) of these processes. The operator, \({\mathbb S}^\tau,\) we shift the time units on all of the variables used in the the construction and the initialization period forward \(\tau\) time periods. The resulting formula is known as the Haussmann-Clark-Ocone representation and is given by

\[\begin{align*} X_t & = \int_0^t {\mathbb E} \left( {\mathbb S}^u\Phi_{t-u} \mid {\mathfrak F}_u \right) dW_u + {\mathbb E} \left( X_t \mid {\mathfrak F}_0 \right) \cr Y_t & = \int_0^t {\mathbb E}\left( {\mathbb S}^u \Psi_{t-u} \mid {\mathfrak F}_u \right) dW_u + {\mathbb E} \left( Y_t \mid {\mathfrak F}_0 \right). \end{align*} \]

Note that we form conditional expectations of time shifted stochastic responses to form the random coefficients in the moving-average representations as given by \({\mathbb E} \left( {\mathbb S}^u\Phi_{t-u} \mid {\mathfrak F}_u \right)\) and \({\mathbb E}\left( {\mathbb S}^u \Psi_{t-u} \mid {\mathfrak F}_u \right)\). When the responses turn out not to be stochastic, as in the case of the Remark 2 example, the conditional expectations and the shift are inconsequential. In this case, we recover the familiar convolution formulas for moving-average representations.

Remark 6.1

Many empirical researchers estimate directly what macroeconomists call Jorda projections. These are implemented by regressing a forward sequence of a scalar process on current variable and a shock or particular interest. One can interpret the ambition as wanting to infer impulse responses from direct regressions of future variables on the initial ones. One can view the ambition as a way to measure impulse responses. For instance, the aim could be to infer:

\[{\mathbb E} \left( \Phi_t \mid {\mathcal F}_0 \right) \hspace{.2cm} {\rm and} \hspace{.2cm} {\mathbb E} \left( \Psi_t \mid {\mathcal F}_0 \right), \hspace{.2cm} t \ge 0\]

by regressing \(X_t\) and \(Y_t\) on a measured shock of interest and including additional variables to purge some of the variation in the measured shock. Many applied papers will include cross terms that are pre-determined in advance of the shock to accommodate a form of nonlinearity. For this to be coherent, as our analysis makes clear, one has to think through how the nonlinearity compounds within the stochastic system. The shock of interest can alter other variables that in turn influence the variable of interest in future time periods. Our use of variational processes captures this perspective when the ambition is to measure local impacts.