Model Uncertainty, State Uncertainty, and State-Space Models

Abstract

State-space models have been increasingly used to study macroeconomic and financial problems. A state-space representation consists of two equations, a measurement equation which links the observed variables to unobserved state variables and a transition equation describing the dynamics of the state variables. In this chapter, we show that a classic linear-quadratic macroeconomic framework which incorporates two new assumptions can be analytically solved and explicitly mapped to a state-space representation. The two assumptions we consider are the model uncertainty due to concerns for model misspecification (robustness) and the state uncertainty due to limited information constraints (rational inattention). We show that the state-space representation of the observable and unobservable can be used to quantify the key parameters on the degree of model uncertainty. We provide examples on how this framework can be used to study a range of interesting questions in macroeconomics and international economics.

Appendix

1.1 Solving the Current Account Model Explicitly Under Model Uncertainty

To solve the Bellman equation (4.7), we conjecture that

$$\displaystyle{v\left (s_{t}\right ) = -As_{t}^{2} - Bs_{ t} - C,}$$

where A, B, and C are undetermined coefficients. Substituting this guessed value function into the Bellman equation gives

$$\displaystyle\begin{array}{rcl} -As_{t}^{2} - Bs_{ t} - C =\max _{c_{t}}\min _{\nu _{t}}\left \{-\frac{1} {2}{\left (\overline{c} - c_{t}\right )}^{2} + \boldsymbol{\eta }E_{ t}\left [\vartheta \nu _{t}^{2} - As_{ t+1}^{2} - Bs_{ t+1} - C\right ]\right \}.& &{}\end{array}$$

(4.48)

We can do the min and max operations in any order, so we choose to do the minimization first.  The first-order condition for ν _t is

$$\displaystyle{2\vartheta \nu _{t} - 2AE_{t}\left [\omega _{\zeta }\nu _{t} + Rs_{t} - c_{t}\right ]\omega _{\zeta } - B\omega _{\zeta } = 0,}$$

which means that

$$\displaystyle{ \nu _{t} = \frac{B + 2A\left (Rs_{t} - c_{t}\right )} {2\left (\vartheta -A\omega _{\zeta }^{2}\right )} \omega _{\zeta }. }$$

(4.49)

Substituting (4.49) back into (4.48) gives

$$\displaystyle\begin{array}{rcl} -As_{t}^{2} - Bs_{ t} - C& =& \max _{c_{t}}\left \{-\frac{1} {2}{\left (\overline{c} - c_{t}\right )}^{2} + \boldsymbol{\eta }E_{ t}\left [\vartheta {\left [\frac{B + 2A\left (Rs_{t} - c_{t}\right )} {2\left (\vartheta -A\omega _{\zeta }^{2}\right )} \omega _{\zeta }\right ]}^{2}\right.\right. {}\\ & & \qquad \quad \quad -\left.\left.As_{t+1}^{2} - Bs_{ t+1} - C\right ]\right \}, {}\\ \end{array}$$

where

$$\displaystyle{ s_{t+1} = Rs_{t} - c_{t} +\zeta _{t+1} +\omega _{\zeta }\nu _{t}. }$$

The first-order condition for c _t is

$$\displaystyle{\left (\overline{c} - c_{t}\right ) - 2\boldsymbol{\eta }\vartheta \frac{A\omega _{\zeta }} {\vartheta -A\omega _{\zeta }^{2}}\nu _{t} + 2\boldsymbol{\eta }A\left (1 + \frac{A\omega _{\zeta }^{2}} {\vartheta -A\omega _{\zeta }^{2}}\right )\left (Rs_{t} - c_{t} +\omega _{\zeta }\nu _{t}\right ) + \boldsymbol{\eta }B\left (1 + \frac{A\omega _{\zeta }^{2}} {\vartheta -A\omega _{\zeta }^{2}}\right ) = 0.}$$

Using the solution for ν _t the solution for consumption is

$$\displaystyle{ c_{t} = \frac{2A\boldsymbol{\eta }R} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A}s_{t} + \frac{\overline{c}\left (1 - A\omega _{\zeta }^{2}/\vartheta \right ) + \boldsymbol{\eta }B} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A}. }$$

(4.50)

Substituting the above expressions into the Bellman equation gives

$$\displaystyle\begin{array}{rcl} & & -As_{t}^{2} - Bs_{ t} - C {}\\ & =& -\frac{1} {2}{\left ( \frac{2A\boldsymbol{\eta }R} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A}s_{t} + \frac{-2\boldsymbol{\eta }A\overline{c} + \boldsymbol{\eta }B} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A}\right )}^{2} {}\\ & +& \frac{\boldsymbol{\eta }\vartheta \omega _{\zeta }^{2}} {{\left (2\left (\vartheta -A\omega _{\zeta }^{2}\right )\right )}^{2}}{\left (\frac{2AR\left (1 - A\omega _{\zeta }^{2}/\vartheta \right )} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A} s_{t} + B -\frac{2\overline{c}\left (1 - A\omega _{\zeta }^{2}/\vartheta \right )A + 2\boldsymbol{\eta }AB} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A} \right )}^{2} {}\\ & -& \boldsymbol{\eta }A\left ({\left ( \frac{R} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A}s_{t} -\frac{-B\omega _{\zeta }^{2}/\vartheta + 2c + 2B\boldsymbol{\eta }} {2\left (1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A\right )}\right )}^{2} +\omega _{ \zeta }^{2}\right ) {}\\ & -& \boldsymbol{\eta }B\left ( \frac{R} {1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A}s_{t} -\frac{-B\omega _{\zeta }^{2}/\vartheta + 2c + 2B\boldsymbol{\eta }} {2\left (1 - A\omega _{\zeta }^{2}/\vartheta + 2\boldsymbol{\eta }A\right )}\right ) -\boldsymbol{\eta }C. {}\\ \end{array}$$

Given η R = 1, collecting and matching terms, the constant coefficients turn out to be

$$\displaystyle\begin{array}{rcl} A& =& \frac{R\left (R - 1\right )} {2 - R\omega _{\zeta }^{2}/\vartheta },{}\end{array}$$

(4.51)

$$\displaystyle\begin{array}{rcl} B& =& - \frac{R\overline{c}} {1 - R\omega _{\zeta }^{2}/\left (2\vartheta \right )},{}\end{array}$$

(4.52)

$$\displaystyle\begin{array}{rcl} C& =& \frac{R} {2\left (1 - R\omega _{\zeta }^{2}/2\vartheta \right )}\omega _{\zeta }^{2} + \frac{R} {2\left (1 - R\omega _{\zeta }^{2}/2\vartheta \right )\left (R - 1\right )}{\overline{c}}^{2}.{}\end{array}$$

(4.53)

Substituting (4.51) and (4.52) into (4.50) yields the consumption function. Substituting (4.53) into the current account identity and using the expression for s _t yields the expression for the current account.