Origins and Nature of Macroeconomic Instability in Vector Autoregressions

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From Nonlinear DSGE to TVP-VAR: A Derivation

Consider a nonlinear DSGE model linearized to second order around a deterministic path. The resulting system for the endogenous variables \( y_t \) and state vector \( x_t \) can be expressed as:

equation[equation omitted — 135 chars of source]

where:

itemize• \( x_t \in \mathbb{R}^n \) is the state vector, including current and forward-looking variables. • \( e_t \sim \mathcal{N}(0, \Sigma_e) \) is a vector of structural shocks. • \( \eta_t = y_t^2 - E_t[y_{t+1}^2] \) is a second-order expectations error. • \( P_t = E_t[y_{t+1} y_{t+1}'] \) contains forward-looking second moments. • All coefficient matrices (e.g., \( \Gamma_{0,t}(\theta), \Gamma_{1,t}(\theta) \)) are time-varying due to the nonlinear solution's dependence on state trajectories.

Assume there exists a time-varying policy function for the observables:

equation[equation omitted — 29 chars of source]

where \( C_t \) is derived from the solution of the model and inherits time variation from the policy rule and underlying steady state.

Substituting into (ref) gives:

equation[equation omitted — 160 chars of source]

This is nonlinear due to the quadratic terms and expectations.

We seek a reduced-form for \( y_t \). Using the recursive structure of \( x_t \), we can attempt to iterate forward and isolate \( y_t \). Write: \[ x_t = \Gamma_{0,t}^{-1} \left( \Gamma_{1,t} x_{t-1} + \Psi e_t + \Pi \eta_t + P_t \right) \] \[ y_t = C_t x_t = C_t \Gamma_{0,t}^{-1} \Gamma_{1,t} x_{t-1} + C_t \Gamma_{0,t}^{-1} \Psi e_t + C_t \Gamma_{0,t}^{-1} \Pi \eta_t + C_t \Gamma_{0,t}^{-1} P_t \]

Define:

align*[align* omitted — 176 chars of source]

Then:

equation[equation omitted — 74 chars of source]

If we substitute recursively \( x_{t-1}, x_{t-2}, \ldots \) and similarly write higher lags of shocks, we obtain:

equation[equation omitted — 87 chars of source]

where \( A_{j,t} \), \( B_{k,t} \) are time-varying reduced-form coefficients and \( u_t \) contains residual higher-order terms (e.g., approximations of \( \eta_t \), higher-order expectations, etc.).

\paragraph{Conclusion.} The time variation in \( A_{j,t} \) and \( B_{k,t} \) arises naturally from the nonlinear solution, where each coefficient depends on the deterministic path or state-contingent policy functions. The resulting structure is a Time-Varying Parameter VAR with MA terms (TVP-VAR-MA), providing a reduced-form representation of a second-order DSGE solution.