Comparing Solution Methods for DSGE Models with Labor Market Search

Abstract

I compare the performance of solution methods in solving a standard real business cycle model with labor market search frictions. Under the conventional calibration, the model is solved by the projection method using the Chebyshev polynomials as its basis, and the perturbation methods up to third order in both levels and logs. Evaluated by two accuracy tests, the projection approximation achieves the highest degree of accuracy, closely followed by the third order perturbation in levels. Although different in accuracy, all the approximated solutions produce simulated moments similar in value.

Appendix

Starting with the capital grid, for any element of the set, $k_t^i\in [k_{min}, k_{max}]$ with i being a positive integer for indexing purpose, the linear transformation

$$\begin{aligned} \upvarphi (k_t^i) = \frac{2(k_t^i-k_{min})}{k_{max}-k_{min}} - 1, \;\;\; i= 1,2,... \end{aligned}$$

(50)

ensures that $\upvarphi (k_t^i)$ is bounded to the set $[-1, 1]$. I choose $n_k$ elements from the set, collected in the vector $k_t = \begin{bmatrix} k_t^1&k_t^2&\ldots&k_t^{n_k} \end{bmatrix}'$, such that after applying the linear transformation (50) to $k_t$, the elements of the resulting vector $\upvarphi (k_t)=\begin{bmatrix} \upvarphi (k_t^1)&\upvarphi (k_t^2)&\ldots&\upvarphi (k_t^{n_k}) \end{bmatrix}'$ are the $n_k$ roots of the following $n_k$th Chebyshev polynomial basis

$$\begin{aligned} T\left( \upvarphi (k_t)\right) = \begin{bmatrix} T_0&T_1\left( \upvarphi (k_t)\right)&T_2\left( \upvarphi (k_t)\right)&\ldots&T_{n_k}\left( \upvarphi (k_t)\right) \end{bmatrix} \end{aligned}$$

(51)

where $T_i(\cdot ) \equiv \cos (i\arccos (\cdot ))$ is the ith Chebyshev polynomial with $T_0=1$ and $T\left( \upvarphi (k_t)\right)$ is of dimension $(n_k+1)\times (n_k+1)$.

Analogous to my choice of elements from the capital set, I choose $n_n$ and $n_z$ elements from these two sets, $n_t = \begin{bmatrix} n_t^1&n_t^2&\ldots&n_t^{n_n} \end{bmatrix}'$ and $z = \begin{bmatrix} z_t^1&z_t^2&\ldots&z_t^{n_z} \end{bmatrix}'$, that after being transformed by $\upvarphi (\cdot )$, are the $n_n$ and $n_z$ roots of the following $n_n$th and $n_z$th Chebyshev polynomial basis respectively

$$\begin{aligned}&T\left( \upvarphi (n_t)\right) = \begin{bmatrix} T_0&T_1\left( \upvarphi (n_t)\right)&T_2\left( \upvarphi (n_t)\right)&\ldots&T_{n_n}\left( \upvarphi (n_t)\right) \end{bmatrix} \end{aligned}$$

(52)

$$\begin{aligned}&T\left( \upvarphi (z_t)\right) = \begin{bmatrix} T_0&T_1\left( \upvarphi (z_t)\right)&T_2\left( \upvarphi (z_t)\right)&\ldots&T_{n_z}\left( \upvarphi (z_t)\right) \end{bmatrix} \end{aligned}$$

(53)

where $T\left( \upvarphi (n_t)\right)$ and $T\left( \upvarphi (z_t)\right)$ are of dimension $(n_n+1)\times (n_n+1)$ and $(n_z+1)\times (n_z+1)$ respectively.

As in Judd (1992), Aruoba et al. (2006) and Caldara et al. (2012), the multidimensional basis of the approximated policy function is the Kronecker product of the above three one-dimensional basis

$$\begin{aligned} X(k_t,n_t,z_t) = T(\upvarphi (k_t)) \otimes T(\upvarphi (n_t)) \otimes T(\upvarphi (z_t)) \end{aligned}$$

(54)

with dimension $(n_g\times n_g)$ where $n_g = (n_k+1)\times (n_n+1)\times (n_z+1)$ is the number of all triplets of the collocation points along three dimensions, i.e., the number of grid points in the three-dimensional state space $[k_{min}, k_{max}]\times [n_{min}, n_{max}]\times [z_{min}, z_{max}]$. With this multidimensional basis, the approximated policy function of consumption and vacancy writes

$$\begin{aligned}&{\hat{c}}_t = X(k_t,n_t,z_t){\Theta }_c = P_c(k_t,n_t,z_t;{\Theta }_c) \end{aligned}$$

(55)

$$\begin{aligned}&{\hat{v}}_t = X(k_t,n_t,z_t){\Theta }_v = P_v(k_t,n_t,z_t;{\Theta }_v) \end{aligned}$$

(56)

where ${\hat{}}$ indicates these are approximated policy functions, and ${\Theta }_c$ and $\Theta _v$ are two vectors of coefficients to be determined. Both ${\hat{c}}_t$ and ${\hat{v}}_t$ are of dimension $(n_g\times 1)$.

I solve for the unknown coefficients ${\Theta }_c$ and $\Theta _v$ from the two Euler equations (12) and (15) using Den Haan and Marcet’s (1990) functional iteration: at each grid point i

• 1. use j-th iteration of the coefficients, ${\Theta }_c^j$ and ${\Theta }_v^j$, to compute

  $$\begin{aligned} n_{t+1}^i&= (1-\mu )n_t^i + m_0 P_v\left( k_t^i,n_t^i,z_t^i;{\Theta }_v^j\right) ^{1-\upeta }(1-n_t^i)^{\upeta },\;\; i = 1,2,...,n_g \end{aligned}$$

  (57)
  $$\begin{aligned} k_{t+1}^i&= (1-{\updelta })k_t^i + e^{z_t^i} (k_t^i)^{{\upalpha }} (n_t^i)^{1-{\upalpha }} - P_c\left( k_t^i,n_t^i,z_t^i;{\uptheta }_c^j\right) \\&\quad - {\upkappa }_vP_c(k_t^i,n_t^i,z_t^i;{\Theta }_v^j) - m_0{\upkappa }_u(1-n_t^i)\nonumber \end{aligned}$$

  (58)
  $$\begin{aligned} c_{t+1}^i&= P_c\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+{\upvarepsilon }_t ;{\uptheta }_c^j\right) \end{aligned}$$

  (59)
  $$\begin{aligned} v_{t+1}^i&= P_v\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+{\upvarepsilon }_t ;{\uptheta }_v^j\right) \end{aligned}$$

  (60)

• 2. given (57)–(60) and approximating the conditional expectation with the Gauss–Hermite quadrature, the Euler equation for consumption (12) writes

  $$\begin{aligned} \left( {\hat{c}}_t^i\right) ^{-1}&= {\upbeta }\sum _{r=1}^{m} \biggl [P_c\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r ;{\Theta }_c^j\right) ^{-1}\\&\times \left( 1-{\updelta }+{\upalpha }\exp \left( {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r\right) (k_{t+1}^i)^{{\upalpha }-1} (n_{t+1}^i)^{1-{\upalpha }}\right) \frac{\omega _r}{\sqrt{\pi }}\nonumber \biggl ] \end{aligned}$$

  (61)

  where $\zeta _r$ and $\omega _r$ are Gauss–Hermite quadrature points and weights. From the foregoing solve for ${\hat{c}}_t^i$. Analogously, the Euler equation for employment (15) writes

  $$\begin{aligned} \left( {\hat{v}}_t^i\right) ^{\upeta }&= \frac{(1-\upeta )m_0}{{\upkappa }_v} (1-n_t^i)^{\upeta }\left( {\hat{c}}_t^i\right) {\upbeta }\\&\quad \times \sum _{r=1}^{m} \biggl [ P_c\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r ;{\Theta }_c^j\right) ^{-1}\nonumber \\&\quad \times \biggl (-\left( n_{t+1}^i\right) ^{-1/{\upgamma }}P_c\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r ;{\Theta }_c^j\right) \nonumber \\&\quad + (1-{\upalpha })\exp \left( {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r\right) \left( k_{t+1}^i\right) ^{{\upalpha }}\left( n_{t+1}^i\right) ^{-{\upalpha }} +{\upkappa }_u\nonumber \\&\quad + \frac{{\upkappa }_v(1-\mu )P_v\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r ;{\Theta }_v^j\right) ^{\upeta }}{(1-\upeta )m_0\left( 1-n_{t+1}^i\right) ^{\upeta }}\nonumber \\&\quad -\frac{\upeta {\upkappa }_v}{1-\upeta }\frac{P_v\left( k_{t+1}^i, n_{t+1}^i, {\uprho }z_t^i+\sqrt{2}{\upsigma }\zeta _r ;{\Theta }_v^j\right) }{1-n_{t+1}^i} \biggl )\frac{\omega _r}{\sqrt{\pi }} \biggl ]\nonumber \end{aligned}$$

  (62)

  from the foregoing solve for ${\hat{v}}_t^i$.

• 3. repeat step 1–2 for all $n_g$ grid points, get an estimation of the new coefficients with the following regression

  $$\begin{aligned} {\hat{\Theta }}^{j+1} = \begin{bmatrix} {\uptheta }_c^{j+1}&{\Theta }_v^{j+1} \end{bmatrix} = \left[ X(k_t,n_t,z_t)'X(k_t,n_t,z_t)\right] ^{-1}X(k_t,n_t,z_t)' \begin{bmatrix} {\hat{c}}_t&{\hat{v}}_t \end{bmatrix} \end{aligned}$$

  (63)

  where $X(k_t,n_t,z_t)$ is the multidimensional basis defined by (54). Then obtain the $(j+1)$-th iteration of the coefficients with the following updating rule

  $$\begin{aligned} \Theta ^{j+1} = \Lambda {\hat{\Theta }}^{j+1} + (1-\Lambda )\Theta ^j \end{aligned}$$

  (64)

  where $\Lambda \in (0,1]$ is a parameter for stabilizing the iteration.

• 4. repeat step 1–3 till $\Vert \Theta ^{j+1} - \Theta ^j \Vert$ is smaller than a desired level of tolerance.

The choice of parameters for the iteration is summarized in Table 7.

Table 7 Parameters of the iteration