Use new Tauchen

[jlperla]

A new Tauchen was added to the interpolation/quadrature lectures in #370

We should now convert over to using it across the lectures. For this, you can copy/paste a minimal version in each lecture that requires tauchen and just make sure the link to that new lecture is there.

function tauchen(N, rho, sigma, mu, m = 3)
    mu_X = mu / (1 - rho)
    sigma_X = sigma / sqrt(1 - rho^2)
    z = range(-m*sigma_X, m*sigma_X, length = N)
    midpoints = (z[1:(end - 1)] .+ step(z)/2)'
    F = cdf.(Normal(), (midpoints .- rho .* z) ./ sigma)
    P = [F[:, 1] diff(F, dims = 2) (1 .- F[:, end])]
    return (;P, mu_X, sigma_X, x = z .+ mu_X)
end

The challenge is that we won't be able to complete this before implementing the simulate.

[jlperla]

Lets start by getting this into the following lectures (just doublecheck that the tauchen in the interpolation lecture looks good and gives the same output as the above).

• in markov_asset.md
  • Put the taucehn code to replace the call to the QuantEcon mikron chain. Unpack so code like

  mc = tauchen(n, 0.96, 0.02)
  K = mc.p .* exp.(mc.state_values)'
  v = (I - beta * K) \ (beta * K * ones(n, 1))

  • becomes something like

  (;P, x) = tauchen(n, 0.96, 0.02)
  K = P .* exp.(x)'
  v = (I - beta * K) \ (beta * K * ones(n, 1))

  • Instead of the mc in the parameters for the AssetPriceModel etc, you should put the P and x in as args. Something like

  function asset_pricing_model(; beta = 0.96, gamma = 2.0, g = exp, n = 25, rho = 0.9, sigma = 0.02)
      (;P, x) = tauchen(n, rho, sigma)
      return (; beta, gamma, g, x, P)
    end

  • Rename AssetPriceModel to asset_price_model to be consistent, and any others you see.
  • And swap out the Markov chain simulate with the one with copy/pasted in the previous changes.
  • Remove the QuantEcon include. If there are other features of the Markov chain we need, let me know and we can discuss.
• in arellano.md do the same things