Tractable Latent State Filtering for Non-Linear DSGE Models Using a Second-Order Approximation and Pruning

Abstract

This paper develops a novel approach for estimating latent state variables of Dynamic Stochastic General Equilibrium (DSGE) models that are solved using a second-order accurate approximation. I apply the Kalman filter to a state-space representation of the second-order solution based on the ‘pruning’ scheme of Kim et al. (J Econ Dyn Control 32:3397–3414, 2008). By contrast to particle filters, no stochastic simulations are needed for the deterministic filter here; the present method is thus much faster; in terms of estimation accuracy for latent states it is competitive with the standard particle filter. Use of the pruning scheme distinguishes the filter here from the deterministic Quadratic Kalman filter presented by Ivashchenko (Comput Econ, 43:71–82, 2014). The filter here performs well even in models with big shocks and high curvature.

Appendices

Appendix: Computing Moments of the State Vector (for KalmanQ Filter Formula)

The unconditional mean and variance of the state vector $Z_{t+1}$ of the augmented state equation (5) are given by: $E(Z_{t+1} )=(I-G_1 )^{-1}G_0$ and $V(Z_{t+1} )=G_1 V(Z_{t+1} )G_1^ {\prime }+V(u_{t+1} )$, respectively. Stationarity of $Z_{t+1}$ (which holds under the assumption that all eigenvalues of $F_1$ are strictly inside the unit circle) implies $E(Z_{t+1} )=E(Z_t ),\,V(Z_{t+1} )=V(Z_t )$. Once $V(u_{t+1} )$ has been determined, $V(Z_{t+1} )$ can efficiently be computed using a doubling algorithm. Note that $\omega _t^{(1)} =\sum _{i=0}^\infty {(F_1)^{i}F_2 \varepsilon _{t-i} }$ and recall that

$$\begin{aligned} u_{t+1} \equiv G_2 \varepsilon _{t+1} +G_{12} \omega _t^{(1)} \!\otimes \! \varepsilon _{t+1} +G_{22} [P(\varepsilon _{t+1} )-E(P(\varepsilon _{t+1} ))]. \end{aligned}$$

(A.1)

$E(\omega _t^{(1)} )=0$, $E(\omega _t^{(1)} \otimes \varepsilon _{t+1} )=0$, $E((\omega _t^{(1)} \otimes \varepsilon _{t+1} )\varepsilon _{t+1}^{\prime })=0$, $E((\omega _t^{(1)} \otimes \varepsilon _{t+1} )P(\varepsilon _{t+1})^{\prime })=0$ hold as $\varepsilon _{t+1}$ has mean zero and is serially independent. Hence, the covariances between the first and second right-hand side (rhs) terms in (A.1), and between the second and third rhs terms are zero. $\varepsilon _t \sim N(0,\Sigma _\varepsilon )$ implies that the unconditional mean of all third order products of elements of $\varepsilon _{t+1}$ is zero (Isserlis’ theorem): $E\varepsilon _{t+1}^i \varepsilon _{t+1}^j \varepsilon _t^k =0$ for all $i,\,j,\,k=1,\ldots ,\hbox {m}$, where $\varepsilon _{t+1}^h$ is the $h$-th element of $\varepsilon _{t+1}$. Thus the covariance between the first and third rhs terms in (A.1) too is zero. Note that $V(\omega _t^{(1)} \!\otimes \! \varepsilon _{t+1} )=V(\omega _t^{(1)} )\!\otimes \! \Sigma _\varepsilon ,$ with $V(\omega _t^{(1)} )=F_1 V(\omega _t^{(1)} )F_1^{\prime }+F_2 \Sigma _\varepsilon F_2^{\prime }$. Thus,

$$\begin{aligned} V(u_{t+1} )=G_2 \Sigma _\varepsilon G_2^{\prime }+G_{12} (V(\omega _t^{(1)} )\!\otimes \! \Sigma _\varepsilon )G_{12}^ {\prime }+G_{22} V(P(\varepsilon _{t+1} ))G_{22}^{\prime }. \end{aligned}$$

$\varepsilon _t \sim N(0,\Sigma _\varepsilon )$ also implies that the covariance between $\varepsilon _{t+1}^i \varepsilon _{t+1}^j$ and $\varepsilon _{t+1}^r \varepsilon _{t+1}^s$ is

$$\begin{aligned} Cov(\varepsilon _{t+1}^i \varepsilon _{t+1}^j, \varepsilon _{t+1}^r \varepsilon _{t+1}^s ) = \sigma _{i,r} \sigma _{j,s} +\sigma _{i,s} \sigma _{j,r} \hbox {for i,j,r,s}=1,\,\ldots ,\,\hbox {m}, \end{aligned}$$

where $\sigma _{i,r} =E(\varepsilon _{t+1}^i \varepsilon _{t+1}^r )$. (See, e.g., Triantafyllopoulos 2002). This formula allows to compute $V(P(\varepsilon _{t+1} ))$, the covariance matrix of the vector

$$\begin{aligned} P(\varepsilon _{t+1} )&\equiv (\varepsilon _{t+1}^1 \varepsilon _{t+1}^1, \varepsilon _{t+1}^1 \varepsilon _{t+1}^2, \ldots ,\varepsilon _{t+1}^1 \varepsilon _{t+1}^m, \varepsilon _{t+1}^2 \varepsilon _{t+1}^2, \ldots ,\varepsilon _{t+1}^2 \varepsilon _{t+1}^m,\ldots , \\&\varepsilon _{t+1}^{m-1} \varepsilon _{t+1}^{m-1}, \varepsilon _{t+1}^{m-1} \varepsilon _{t+1}^m, \varepsilon _{t+1}^m \varepsilon _{t+1}^m ). \end{aligned}$$

Conditional Variance of State-Space Disturbance

To derive the formula for the conditional variance of $u_{t+1}$ [(10) in main text] these facts are used:

• (i) $E((\omega _t^{(1)} \!\otimes \! \varepsilon _{t+1} )\varepsilon _{t+1}^{\prime }|\Upsilon ^{t})=\omega _{t,t}^{(1)} \!\otimes \! \Sigma _\varepsilon ,$ with $\omega _{t,t}^{(1)} \equiv E(\omega _t^{(1)} |\Upsilon ^{t})$.

• (ii) $E((\omega _t^{(1)} \!\otimes \! \varepsilon _{t+1} )(\omega _t^{(1)} \!\otimes \! \varepsilon _{t+1} )^{\prime }|\Upsilon ^{t})=E((\omega _t^{(1)} \omega _t^{{(1)}{\prime }})\!\otimes \! (\varepsilon _{t+1} \varepsilon _{t+1}^{\prime })|\Upsilon ^{t})=E((\omega _t^{(1)} \omega _t^{{(1)}{\prime }})|\Upsilon ^{t})\!\otimes \! \Sigma _\varepsilon =(V_{t,t}^{{\omega ^{(1)}}} +\omega _{t,t}^{(1)} \omega _{t,t}^{(1)}{\prime })\!\otimes \! \Sigma _\varepsilon$. (Note that $V_{t,t}^{\omega ^{{(1)}}} =E(\omega _t^{(1)} \omega _t^{(1){\prime }}|\Upsilon ^{t})-E(\omega _t^{(1)} |\Upsilon ^{t})E(\omega _t^{(1)} |\Upsilon ^{t})^{\prime }= E(\omega _t^{(1)} \omega _t^{{(1)}{\prime }}|\Upsilon ^{t})-\omega _{t,t}^{(1)} \omega _{t,t}^{(1){\prime }}.)$

• (iii) $E(P(\varepsilon _{t+1} )\varepsilon _{t+1}^{\prime }|\Upsilon ^{t})=0, \quad E(P(\varepsilon _{t+1} )(\omega _t^{(1)} \!\otimes \! \varepsilon _{t+1} )^{\prime }|\Upsilon ^{t})=0$ (due to Isserlis’ theorem). Thus, the conditional covariance between the $1^{\mathrm{st}}$ and $3^{\mathrm{rd}}$ rhs terms in (A.1) and between the $2\mathrm{nd}$ and $3{\mathrm{rd}}$ rhs terms is zero.