Parameterized expectationsΒΆ

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 \(\psi^{n+1}(s_t) = h(s_t, \varphi^{n+1}(s_t))\)