learn_first_differences

source

learn_first_differences(
    n_units=150,
    n_periods=2,
    beta=2.0,
    unit_effect_corr=0.8,
    noise_sd=0.5,
    seed=0,
)

Show that first differencing removes the unit effect — matching the within estimator.

Simulates a balanced panel y_it = beta * x_it + alpha_i + e_it where the regressor x is correlated with each unit’s fixed effect alpha_i (so pooled OLS is biased). Differencing (Δy on Δx) cancels alpha_i; on a two-period panel the first- differences estimate equals the within (demeaning) estimate, and both recover beta.

Parameters

Name	Type	Description	Default
n_units	int	Panel dimensions. With `n_periods=2` first differences and the within estimator coincide exactly; for more periods they differ slightly in finite samples.	`150`
n_periods	int	Panel dimensions. With `n_periods=2` first differences and the within estimator coincide exactly; for more periods they differ slightly in finite samples.	`150`
beta	float	True within-unit slope.	`2.0`
unit_effect_corr	float	Correlation between `x` and the unit effect (drives the pooled bias).	`0.8`
noise_sd	float	Idiosyncratic noise standard deviation.	`0.5`
seed	int	Random seed.	`0`

Returns

Name	Type	Description
	SandboxResult	`df` (pooled vs first differences vs within vs true slope), `fig`, `summary` and `topic`.

Examples

This sandbox simulates its own two-period panel, so the call needs no DataFrame:

import expdpy as ex

res = ex.learn_first_differences()
res.fig