theil_u2 – Theil’s U2 inequality coefficient – ratio of forecast MSE to no-change MSE.#

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Operational op under axis point_metrics, sub-layer L5_A_metric_specification, layer l5. Standalone callable: mf.functions.theil_u2.

Function signature#

mf.functions.theil_u2(
    y_true: np.ndarray | pd.Series,
    y_pred: np.ndarray | pd.Series,
    y_prev: np.ndarray | pd.Series,
) -> float

Parameters#

name

type

default

constraint

description

y_true

`np.ndarray

pd.Series`

—

—

y_pred

`np.ndarray

pd.Series`

—

—

y_prev

`np.ndarray

pd.Series`

—

—

Returns#

float — scalar result.

Behavior#

Point-forecast metric theil_u2. U₂ = √(Σ (ŷ_t - y_t)² / Σ (y_{t-1} - y_t)²). U₂ < 1 means the forecast beats the random-walk benchmark. Standard sanity-check ratio in macro forecasting – if U₂ ≥ 1 the model is no better than ‘tomorrow looks like today’.

When to use

Sanity-checking against the random-walk benchmark; macro-forecasting tradition.

When NOT to use

When a custom benchmark (not random walk) is preferred – use relative_mse instead.

In recipe context#

Set params.point_metrics = "theil_u2" in the relevant layer to activate this op within a recipe:

# Layer L5 recipe fragment
params:
  point_metrics: theil_u2

References#

  • macroforecast design Part 3, L5: ‘evaluation = (metric × benchmark × aggregation × decomposition × ranking).’

  • Theil (1966) ‘Applied Economic Forecasting’, North-Holland (Chapter 2: Inequality coefficients).