ramsey_reset#

cultivars.diagnostics.nonlinearity.ramsey_reset(endog, *, order, powers=3)[source]#

Ramsey’s RESET on an autoregression: do powers of the fitted value explain the residual?

The AR(\(p\)) is fitted, its fitted values \(\hat y_t\) raised to the powers \(2, \ldots, k\), and those powers added to the regression,

\[y_t = \phi^\top w_t + \sum_{j = 2}^{k} \gamma_j\, \hat y_t^{\,j} + e_t,\]

with the joint exclusion \(\gamma_2 = \cdots = \gamma_k = 0\) tested by \(F(k - 1,\; n - p - k)\) with \(n\) the rows of the autoregression. The test names no alternative: any smooth misspecification of the conditional mean, a missing regime, a nonlinear lag, an omitted regressor correlated with the fitted value, raises the powers of the fitted value, so it is the general-purpose check to run when the regime tests disagree, and a rejection here without one from tsay() or terasvirta() points away from a regime model.

Parameters:

  • endog (ArrayLike) – The series, (T,).

  • order (int) – Autoregressive order \(p\) of the linear null, at least 1.

  • powers (int) – Highest power \(k\) of the fitted value added, at least 2; 2 tests the square alone.

Return type:

LinearityTest

Returns:

The LinearityTest named "RESET", with alternative="smooth misspecification", delay=None, and no companions.

Raises:

  • SpecificationError – If the order is not positive, powers is below 2, or the series is too short.

  • NumericalError – If a value is not finite or the augmented regression is degenerate, which a series whose fitted values are nearly constant produces.

Note

High powers of the fitted value are nearly collinear with each other, so powers above 4 buys little and costs conditioning; the default of 3 is Ramsey’s.

See also

References

Ramsey, J. B. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society B, 31(2), 350-371.

Thursby, J. G., & Schmidt, P. (1977). Some properties of tests for specification error in a linear regression model. Journal of the American Statistical Association, 72(359), 635-641.

Example

A linear AR(1) keeps the null; the two-regime series of the module examples rejects it:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> y = np.zeros(300)
>>> for t in range(1, 300):
...     y[t] = 0.5 * y[t - 1] + rng.standard_normal()
>>> verdict = ramsey_reset(y, order=1)
>>> verdict.reject(), verdict.df, verdict.delay
(False, (2, 295), None)
>>> z = np.zeros(300)
>>> for t in range(1, 300):
...     z[t] = (0.8 if z[t - 1] <= 0.0 else -0.5) * z[t - 1] + rng.standard_normal()
>>> ramsey_reset(z, order=1).reject()
True
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