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Assume that we have the following equation and the following assumption. The scope is to solve for some particular variables expressed later.

Update

$$E_{t}\left[ b(A_{t+1})^{1-\gamma} *R_{t+1}^{-\gamma}*[r_k(A_{t+1},a)-r_h(A_{t+1})+\eta)] \right]=0$$

where $R=(1-\theta_1-\theta_2)*r^f + \theta_1*r_k(A,a)+\theta_2 *[r_h(A)+\eta]$

Also consider that the number $b(A)$ is pinned down recursively by the following equation:

$$b_t = G[E_t(b_{t+1}*R_{t+1}^{1-\gamma})]$$, where $G(\cdot)$ is a known function

Main Assumptions

  1. $A_t$ is a random variable that follows a Markov process, and we know its value at time $t$, but not at time $t+1$.

  2. $a$ is an $IID$ random variable lognormally distributed with some mean $\mu_a$ and variance $\sigma^2_a$.

  3. $\eta$ is another $IID$ random variable normally distributed with some mean $\mu_{\eta}$ and variance $\sigma^2_\eta$.

  4. The RVs are independent of each other. That is, $a$ is independent of $\eta$, while these two can be correlated with $A$, thought are still independently drawn from $A$.

  5. $b(A)$ is a number that depends on $A$ and I do not know their functional relationship.

  6. $r^f$ is a known number.

  7. by $r_k(A,a) $ and respectively $r_h(A)$ I mean a functional relationship (which I know) between these two that depend on $A, a$ and $A$ respectively.

To sum up, the expectations operator is about the unknowns in the next period $A,a$ and $\eta$. To fix the ideas. Let's suppose a simple case, where $a,\eta$ take two values and $A$ follows a two-state Markov process and also take two values. Let's also suppose that at time $t$ we are in one of these states. In that case, the Expectations operator is defined over the next possible state for $A$, and the possible values for the $IID$ RVs.

THE COMPUTATIONAL TASK: Solve for $\theta_1$ and $\theta_2$

Apparently, the solution for each $\theta$ will depend on the possible values of $A$, so I expect to be a vector.

Any guidance on how to solve this equation will be appreciated.

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  • $\begingroup$ That sounds like a very challenging problem; I'm not sure you have much hope if you can't even evaluate the random function (since you do not know $b$). But I don't think $\theta$ will be a vector (unless you set it up as such). $\endgroup$ – Christian Clason Oct 21 '14 at 12:11
  • $\begingroup$ You might want to search for "parameter estimation" or "calibration" for Markov processes. $\endgroup$ – Christian Clason Oct 21 '14 at 12:13
  • $\begingroup$ @ChristianClason thanks for the response. Let's say that I can calibrate the markov process this is somehow straight forward to me. Can you still believe solve it in the way you described in my other post ? But is not totally clear how I will define the Gaussian-Hermite Quadrature. $\endgroup$ – user17880 Oct 21 '14 at 14:27
  • $\begingroup$ I was thinking to solve the problem, for some given value of $b$, then I can employ some other equations from the model to pin down $b$ for the given values found for $\theta's$ $\endgroup$ – user17880 Oct 21 '14 at 14:30
  • $\begingroup$ In principle, that might work. Your expectation is now a two-dimensional integral $E(f(a,\nu)) = \int\int (t,s)d\phi_a(t)d\phi_\nu(s)$, which you can approximate with tensor-product quadrature $\sum_i\sum_j w_i^a w_j^\nu f(t_i^a, t_j^\nu)$, where $w_i^a$ and $t_i^a$ are the Gauss-Hermite weights and nodes for a lognormal distribution and $w_j^\nu$ and $t_j^\nu$ are those for a normal distribution (as in my answer to your other question). $\endgroup$ – Christian Clason Oct 21 '14 at 16:09

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