Using the “Chandrasekhar Recursions” for Likelihood Evaluation of DSGE Models

Abstract

In likelihood-based estimation of linearized Dynamic Stochastic General Equilibrium (DSGE) models, the evaluation of the Kalman Filter dominates the running time of the entire algorithm. In this paper, we revisit a set of simple recursions known as the “Chandrasekhar Recursions” developed by Morf (Fast Algorithms for Multivariate Systems, Ph.D. thesis, Stanford University, 1974) and Morf et al. (IEEE Trans Autom Control 19:315–323, 1974) for evaluating the likelihood of a Linear Gaussian State Space System. We show that DSGE models are ideally suited for the use of these recursions, which work best when the number of states is much greater than the number of observables. In several examples, we show that there are substantial benefits to using the recursions, with likelihood evaluation up to five times faster. This gain is especially pronounced in light of the trivial implementation costs—no model modification is required. Moreover, the algorithm is complementary with other approaches.

Appendix

1.1 Verification of recursions for $F_t, K_t, \text{ and } K_{g, t}$.

To show that Chandrasekhar recursions work for the special case discussed above, we show how to write the recursions for $F_{t}$ and $K_{t}$ in terms of $\Delta P_{t|t-1}$. For $F_t$:

$$\begin{aligned} F_{t}&= ZP_{t|t-1}Z^\prime + H\nonumber \\&= ZP_{t|t-1}Z^\prime + H + F_{t-1} - F_{t-1}\nonumber \\&= F_{t-1} + ZP_{t|t-1}Z^\prime - ZP_{t-1|t-2}Z^\prime \nonumber \\&= F_{t-1} + Z\Delta P_{t|t-1}Z^\prime . \nonumber \end{aligned}$$

$K_t$ and $K_{g, t}$ are similar. For $K_{g, t}$:

$$\begin{aligned} K_{g, t} = (K_{g, t-1}F_{t-1} + T\Delta (P_{t})Z^\prime )F_{t}^{-1}. \end{aligned}$$

(22)

1.2 Verification of the difference equation for $\Delta P_{t|t-1}$.

Proof of Lemma

From the Kalman Filter, we have that

$$\begin{aligned} P_{t+1|t} = TP_{t|t-1}T^\prime + RQR - K_{g, t}F_{t}K_{g, t}^\prime . \end{aligned}$$

Subtracting $P_{t|t-1}$ from both sides, we have that

$$\begin{aligned} \Delta P_{t+1|t} = T\Delta P_{t|t-1}T^\prime - K_{g, t}F_{t}K_{g, t}^\prime + K_{g, t-1}F_{t-1}K_{g, t-1}. \end{aligned}$$

Using the recursions for $F_t$ shown in the previous lemma, we have

$$\begin{aligned} \Delta P_{t+1|t} = T\Delta P_{t|t-1}T^\prime - K_{g, t}(F_{t-1}+Z\Delta P_{t-1|t}Z^\prime )K_{g, t}^\prime + K_{g, t-1}F_{t-1}K_{g, t-1}. \end{aligned}$$

Grouping like terms and completing the square for $(T-K_{g, t}Z)$, we have

$$\begin{aligned} \Delta P_{t+1|t}&= (T\!-\!K_{}{g, t}Z)\Delta P_{t|t-1}(T\!-\!K_{g, t}Z)^\prime \!+\! K_{g, t}Z\Delta P_{t|t-1}T^\prime \!+\! T\Delta P_{t|t-1} Z^\prime K_{g, t}^\prime \nonumber \\&- 2K_{g, t}\Delta P_{t|t-1}K_{g, t}^\prime - K_{g, t}F_{t-1}K_{g, t}^\prime + K_{g, t-1}F_{t-1}K_{g, t-1}^\prime . \end{aligned}$$

(23)

Note that we can write the final product in the equation, using (22), (19), and tediously expanding terms, as,

$$\begin{aligned} K_{g, t-1}F_{t-1}K_{g, t-1}&= (K_{g, t}F_t - T\Delta P_{t|t-1}Z^\prime )F_t^{-1}(K_{g, t}F_t - T\Delta P_{t|t-1}Z^\prime ) \nonumber \\&= (K_{g, t}(F_{t-1} + Z\Delta P_{t|t-1}Z^\prime )-T\Delta P_{t|t-1}Z^\prime )F_{t}^{-1}(K_{g, t}(F_{t-1} \nonumber \\&+ Z\Delta P_{t|t-1}Z^\prime ) T\Delta P_{t|t-1}Z^\prime ) \nonumber \\&= (K_{g, t}F_{t-1} + (K_{g, t}Z - T)\Delta P_{t|t-1}Z^\prime )F_{t-1}^{-1}(K_{g, t}F_{t-1} \nonumber \\&+ (K_{g, t}Z - T)\Delta P_{t|t-1}Z^\prime ) \nonumber \\&= (T - K_{g, t}Z)\Delta P_{t|t-1}Z^\prime F_{t-1}^{-1}Z\Delta P_{t|t-1}(T - K_{g, t}Z) \nonumber \\&+ K_{g, t}F_{t-1}K_{g, t} \nonumber \\&+ K_{g, t}Z\Delta P_{t|t-1}(K_{g, t}Z - T)^\prime +(K_{g, t}Z - T)\Delta P_{t|t-1}Z^\prime K_{g, t}^\prime \nonumber \\&= (T - K_{g, t}Z)\Delta P_{t|t-1}Z^\prime F_{t-1}^{-1}Z\Delta P_{t|t-1}(T - K_{g, t}Z) \nonumber \\&+ K_{g, t}F_{t-1}K_{g, t} \nonumber \\&- K_{g, t}Z\Delta P_{t|t-1}T^\prime - T\Delta P_{t|t-1}Z^\prime K_{g, t}^\prime + 2K_{g, t}\Delta P_{t|t-1}K_{g, t}^\prime .\nonumber \\ \end{aligned}$$

(24)

Combining (23) and (24), we have verified the lemma.