learn_beta_convergence

source

learn_beta_convergence(
    n_units=60,
    n_years=21,
    rho=0.9,
    gamma=0.6,
    corr=0.7,
    noise=0.05,
    seed=0,
)

Show unconditional vs conditional β-convergence on a known-parameter panel.

Simulates an AR(1) panel in logs x_{t+1} = a + rho*x_t + gamma*z_i + e where z_i is a fixed steady-state determinant correlated (corr) with each unit’s initial level. The AR(1) persistence pins the truth exactly: over a horizon T = n_years - 1 the slope of growth on the initial level is beta = (rho**T - 1) / T and the structural speed of convergence is lambda = -ln(rho) (half-life ln 2 / lambda). Because z_i is omitted, the unconditional regression is biased — units look like they barely converge (or even diverge); conditioning on z_i recovers the true convergence slope. This is the classic distinction between absolute and conditional convergence, demonstrated with :func:expdpy.analyze_beta_convergence.

Parameters

Name	Type	Description	Default
n_units	int	Panel dimensions (units and annual periods). The horizon is `T = n_years - 1`.	`60`
n_years	int	Panel dimensions (units and annual periods). The horizon is `T = n_years - 1`.	`60`
rho	float	AR(1) persistence in `(0, 1)`; it sets the true speed `-ln(rho)` (closer to 1 means slower convergence).	`0.9`
gamma	float	Loading of the steady-state determinant `z` (drives the omitted-variable bias of the unconditional estimate).	`0.6`
corr	float	Correlation between `z` and the initial level (also drives the bias).	`0.7`
noise	float	Idiosyncratic shock standard deviation.	`0.05`
seed	int	Random seed.	`0`

Returns

Name	Type	Description
	SandboxResult	`df` (unconditional vs conditional vs true slope), `fig`, `summary` and `topic`.

Examples

This sandbox simulates its own AR(1) panel, so the call needs no DataFrame:

import expdpy as ex

res = ex.learn_beta_convergence()
res.fig