I want to compute the following conditional expectation

$E_{t}[\phi(A_{t+1}, \eta_{t+1})| A_t]$

where $\log A_{t}=\rho \log A_{t-1} + e_{t}$ and $e_{t}$ is IID $N~(0,\sigma_e)$ and $\eta_{t}$ is another normally distributed variable with $(0,\sigma_{\eta})$. The function $\phi(.)$ is known.

I am unsure if someone wants to compute this through numerical methods how should use the quadrature techniques or any suitable alternatives.


Let's first rewrite this. From your formulas, you have that $$ A_{t+1} = A_t^\rho \exp(e_t) $$ where $A_t$ is just the previous value and, consequently, just a fixed parameter. So what you need to compute is then $$ E_t[\phi(A_{t+1},\eta_{t+1})|A_t] = \frac 1C \int_{-\infty}^\infty \int_{-\infty}^\infty \phi(A_t^\rho e^a,b) e^{-a^2/(2\sigma_e^2)} e^{-b^2/(2\sigma_\eta^2)} \; da \; db. $$ How this is best computed numerically very much depends on the form of $\varphi$.

  • $\begingroup$ Thank you for your answer. Can you please elaborate, just to be in the same page, how C should be defined, and whether in your answer by a and b you mean the different realizations for $e_t$ and $\eta_t$ respectively over which the integration occurs. Second, suppose that the function $\phi(.)$ is some nonlinear function of both arguments if this helps, would then multidimensional quadrature (tensor-product rules) be a reasonable thing to do? $\endgroup$ – user17880 Nov 30 '14 at 14:07
  • $\begingroup$ $C$ is the product of the normalization constants of the two Gaussians. And yes, $a$ and $b$ are the realizations of $e_t$ and $\eta_t$ (I wanted to use different symbols so as not to confuse it with the base of the exponentiation). As for the form of $\phi$: Every possible form of $\phi$ is a nonlinear function of two arguments, with the (not very interesting) exception of the very few functions that happen to be linear in the arguments. The question is whether $\phi$ is oscillatory, is smooth or nonsmooth, decays quickly, allows for an analytic representation, etc. $\endgroup$ – Wolfgang Bangerth Dec 1 '14 at 3:17
  • $\begingroup$ Thanks, again. Yes, $\phi(.)$ you can assume it a smooth function. Are you aware of any available Matlab codes that could implement this multidimensional integration, once I specify my function $\phi$ ? I would appreciate any suggestions here. $\endgroup$ – user17880 Dec 1 '14 at 12:06
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    $\begingroup$ But the class of smooth functions is still very very large. Does it oscillate? Does it decay? Do you know the exact form so that it could be transformed into an integral that's simpler to handle (e.g., if $\phi$ was polynomial). Integration becomes much more efficient if you can use knowledge of the integrand, rather than assuming it is a black box. $\endgroup$ – Wolfgang Bangerth Dec 1 '14 at 13:13

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