[WIP] Class representing discretized transitions

[Mv77]

I'm making public versions of a few methods that I have used to work for simulations with discretized/matrix transitions.

The idea is to have a simple class that holds all the ingredients needed to, say, find the steady state distribution of an infinite horizon or life-cycle model, or to iterate a distribution forward a single time period.

Ideally, people will use this and make it more efficient and find useful sparse representations and who knows what else.

• Tests for new functionality/models or Tests to reproduce the bug-fix in code.
• Updated documentation of features that add new functionality.
• Update CHANGELOG.md with major/minor changes.

[Mv77]

This is a work in progress, do not merge yet. I need to, for example, add tests.

[mnwhite]

I finally had a chance to look at this. Main comments are related to both prongs of find_steady_state_dstn.

For the infinite horizon version, you don't need to loop on _iterate_dstn_forward_ih until convergence. You can just combine your single living_tmat with a "newborn transition matrix" that has newborn_dstn in every row, with weights surv_prob and 1-surv_prob. That's a "master" transition matrix. Then transpose it and use sp.linalg.eig to get k=1 eigenvectors, and normalize it. The largest eigenvector of (the transpose of) a non-singular, non-degenerate Markov matrix is always 1, and the corresponding (normalized) eigenvector is the steady state distribution. For extra speed, cast the transition matrix to sp.sparse.csr_matrix and use sp.sparse.linalg.eigs instead. Note that this method is used in HARK.ConsumptionSaving.ConsNewKeynesianModel.NewKeynesianConsumerType.find_steady_state(), minus the sparse arrays.

For the lifecycle model, your steady state age distribution is wrong. You can't just take the cumulative survival probabilities and normalize them, because that ignores death and replacement. Instead, make a square array A of "age transitions", with two non-zero elements in each row: A_{i,i+1} = LivPrb_i and A_{i,0} = 1-LivPrb_i. Just like above, take the transpose of A, then find the eigenvector associated with its largest eigenvalue and normalize it. That's the steady state age distribution. Your survival-conditional state distributions are correct. See L168-202 of estimation.py in the DistributionOfWealthMPC REMARK for where this was done in cstwMPC, but note that when I wrote the function there, I didn't know that the largest eigenvalue is always 1 and/or eig wasn't necessarily sorted back then.

EDIT: Ignore that last paragraph, it's wrong.

[mnwhite]

Addendum to the above: The lifecycle method assumes that you want a constant population, with newborns exactly replacing decedents. If you instead want a constant growth rate of the population, the math is slightly different (but you can probably work it out).

[mnwhite]

Hit pause on my comment about the steady state age distribution. Normalized cumulative survival probabilities and the first normalized eigenvector might be equivalent. I'm on a train right now and can't check.

[mnwhite]

Yeah nevermind, the methods are equivalent.

[Copilot]

Pull Request Overview

This PR introduces the DiscreteTransitions class for simulating population transitions in discretized state spaces for both life-cycle and infinite-horizon models. Key changes include the addition of a new dataclass with methods to iterate distributions forward, calculate conditional age distributions, and compute steady-state distributions.

[Copilot]

In infinite-horizon mode, the properties 'self.living_tmat' and 'self.surv_prob' are referenced, but the class defines 'living_tmats' and 'surv_probs'. Please update the variable names to ensure consistency.

Suggested change

	dstn_init[0], self.living_tmat[0], self.surv_prob[0], self.newborn_dstn
	dstn_init[0], self.living_tmats[0], self.surv_probs[0], self.newborn_dstn

[Copilot]

Correct the typo 'constatn' to 'constant'.

Suggested change

	- Newborns draw their state from a distribution that is constatn over time.
	- Newborns draw their state from a distribution that is constant over time.