We have a system of differential equations:

\begin{equation} \begin{cases} \dv{I}{t} = -0.73 U(t) + 0.0438 + 0.4 \dv{M}{t} \\ \dv{U}{t} = 0.4I-0.012 \\ \dv{G}{t} = \dv{M}{t} - I(t) \end{cases} \end{equation}

where, \(M\) is a sinusoidal function which we can control.

We hope for this system to be as stable as possible.

First, let’s try to get a general solution of the system. The linearized(ish) solution takes the shape of:

\begin{equation} \dv t \mqty(I \\ U \\ G) = \mqty(0 & -x_1 & 0 \\ x_4 & 0 & 0 \\ -1 & 0 & 0 ) \mqty(I \\ U \\ G)+ \dv{M}{t}\mqty(x_3 \\ 0 \\ 1) + \mqty(x_2 \\ x_5 \\ 0) \end{equation}

A reading: (Krugman 2009)

Reflection

The discussion here of the conflict between “saltwater” and “freshwater” (Keynesian and Neoclassical) economists is very interesting when evaluated from the perspective of our recent impending recession.

One particular statement that resonated with me in the essay was the fact that a crisis simply “pushed the freshwater economists into further absurdity.” It is interesting to see that, once a theory has been well-established and insulated in a community, it becomes much more difficult to parcel out as something that could be wrong.

Because cursed languages work best together. Repo is here.

Some Motivation

We wanted an Emacs OS, but that required implementing libc syscalls and also just getting glib to compile, which is obviously not happening. Instead we are going for the second-best thing, which is to be able to interactively write an OS in Emacs.

And thus this project essentially involves enabling the process of incremental development of an operating system kernel. That is, beyond the bare basics and the important, atomic things, everything else should be able to be written and interpreted as they are written (e.g., instead of being compiled into the kernel before).

Jax stuff news!

“Feel the API”

jit -> sharding hints -> explicit sharding -> shard_map collectives -> pallas -> FFI

Refs

There are refs now! Dynamics slicing! Plumbing out metrics is hard! Batch-norms! How do we express things in Jax.

“I want to keep this thing in memory at the same place.”

For stuff like gradient accumulation / sparse updating this is very good.

history

• we can’t express mutability
• and we think the change

fine.

There’s refs now!

CAPM, a Review

Note that we will be using the Sharpe-Linter version of CAPM:

\begin{equation} E[R_{i}-R_{f}] = \beta_{im} E[(R_{m}-R_{f})] \end{equation}

\begin{equation} \beta_{im} := \frac{Cov[(R_{i}-R_{f}),(R_{m}-R_{f})]}{Var[R_{m}-R_{f}]} \end{equation}

Recall that we declare \(R_{f}\) (the risk-free rate) to be non-stochastic.

Let us begin. We will create a generic function to analyze some given stock.

Data Import

We will first import our utilities

import pandas as pd
import numpy as np

Let’s load the data from our market (NYSE) as well as our 10 year t-bill data.