Parameterized expectations¶

We consider an fgh model, that is a model with the form:

$$s_t = g\left(s_{t-1}, x_{t-1}, \epsilon_t \right)$$

$$0 = f\left(s_{t}, x_{t}, E_t[h(s_{t+1}, x_{t+1})] \right)$$

where $$g$$ is the state transition function, $$f$$ is the arbitrage equation, and $$h$$ is the expectations function (more accurately, it is the function over which expectations are taken).

The parameterized expectations algorithm consists in approximating the expectations function, $$h$$, and solving for the associated optimal controls, $$x_t = x(s_t)$$.

At step $$n$$, with a current guess of the control, $$x(s_t) = \varphi^n(s_t)$$, and expectations function, $$h(s_t,x_t) = \psi^n(s_t)$$ :
• Compute the conditional expectation $$z_t = E_t[\varphi^n(s_t)]$$

• Given the expectation, update the optimal control by solving $$0 = \left( f\left(s_{t}, \varphi^{n+1}(s), z_t \right) \right)$$

• Given the optimal control, update the expectations function $$\psi^{n+1}(s_t) = h(s_t, \varphi^{n+1}(s_t))$$