MIDASVARResult#

class cultivars.multivariate.reduced_form.mixed_frequency.MIDASVARResult(*, endog, names, order, trend, sigma_u, sigma_ml, resid, fittedvalues, design, llf, nobs, n_params, posterior=None, prior_label='none', coefficients, deterministic, exog_high, exog, exog_names, midas_coefficients, theta, period, midas_lags)[source]

Bases: _VectorResult, _SummaryMixin, _ComparisonMixin, _VectorInferenceMixin, _VectorPropagationMixin

A fitted MIDAS vector autoregression.

The low-frequency system

\[y_t = D d_t + \sum_{i=1}^{p} A_i\, y_{t-i} + B\, x_t(\theta) + u_t, \qquad x_{t}(\theta)_q = \sum_{j=1}^{L} w_j(\theta_q)\, x^{(q)}_{t, j},\]

where \(x^{(q)}_{t,j}\) is the \(j\)-th most recent sub-period reading of high-frequency series \(q\) at the end of low-frequency period \(t\), and the weights are the normalized exponential Almon curve \(w_j \propto \exp(\theta_1 j + \theta_2 j^2)\) summing to one, so the scale of each regressor lives in its column of \(B\) and the two entries of \(\theta_q\) carry only the shape of the window. Estimation profiles \(\theta\) – least squares conditional on it, a derivative-free search over it – and this record holds the estimate at the maximizer.

The reduced-form surface is inherited unchanged, and everything it reports is conditional in exactly one place: the coefficient standard errors, and every test built from them, take the estimated lag polynomial as known. joint_stderr() is the honest alternative and says what it costs.

forecast() is conditional in the same sense a VARX forecast is: the model holds no process for the high-frequency block, so it asks for the future sub-period path and refuses without it.

Note

The propagation surface – companion, stability, impulse responses, variance and historical decompositions – reads the endogenous block only. The compressed regressors are conditioned on, not shocked, so they enter no moving-average representation, and a response here is the response of the low-frequency system to its own innovations holding the high-frequency path fixed. exog is the compressed block at the estimated polynomial, so it is a function of the data and of theta, not a supplied regressor; the inherited design treats it as data, which is the conditioning the Note above describes.

Variables:

  • endog – The low-frequency sample.

  • exog_high – The high-frequency panel as supplied, chronological.

  • exog – The compressed regressor block over the effective sample, one column per high-frequency series at the estimated polynomial.

  • names – Endogenous labels, in Cholesky order.

  • exog_names – High-frequency series labels.

  • order – Endogenous autoregressive order.

  • period – Sub-periods per low-frequency period.

  • midas_lags – Sub-periods each weight curve spans.

  • trend – Deterministic specification.

  • coefficients – (p, k, k) stack of A_1, ..., A_p.

  • midas_coefficients – (k, m) slopes on the compressed regressors.

  • theta – (m, 2) estimated lag-polynomial parameters, one row per high-frequency series.

  • deterministic – Deterministic coefficients, one row per term.

  • sigma_u – Residual covariance with the degrees-of-freedom correction.

  • sigma_ml – Residual covariance divided by the effective sample.

  • resid – Residuals over the effective sample.

  • fittedvalues – One-step conditional means over the effective sample.

  • design – The regressor matrix as estimated, compressed block last.

  • llf – Gaussian log-likelihood.

  • nobs – Effective sample size.

  • n_params – Free parameters – coefficients, covariance, and the two polynomial parameters per high-frequency series.

See also

  • MIDASVAR – the model whose fit() returns this record.

  • MFVARResult – the high-frequency alternative, inferred rather than estimated.

  • VARXResult – the same conditional surface with an unrestricted exogenous block, the model this collapses to at period == 1.

References

Ghysels, E., Santa-Clara, P., & Valkanov, R. (2004). The MIDAS touch: Mixed data sampling regression models. CIRANO Working Paper 2004s-20.

Ghysels, E., Sinko, A., & Valkanov, R. (2007). MIDAS regressions: Further results and new directions. Econometric Reviews, 26(1), 53-90.

Ghysels, E. (2016). Macroeconomics and the reality of mixed frequency data. Journal of Econometrics, 193(2), 294-314.

Example

A quarterly pair in which the first variable loads on the last two months of a monthly series with weights 0.7 and 0.3. The profiled polynomial recovers the decaying curve and a unit slope, the second equation’s slope is nothing, the joint standard errors widen the conditional ones where the polynomial matters, and the forecast needs the three future months:

>>> import numpy as np
>>> rng = np.random.default_rng(0)
>>> x = rng.standard_normal((360, 1))
>>> window = 0.7 * x[2::3, 0] + 0.3 * x[1::3, 0]
>>> noise = 0.1 * rng.standard_normal(120)
>>> y = np.column_stack([window + noise, rng.standard_normal(120)])
>>> res = MIDASVAR(y, x, order=1, period=3).fit()
>>> res.midas_weights.ravel().round(2), res.midas_coefficients.round(1).ravel() + 0.0
(array([0.69, 0.3 , 0.01]), array([1., 0.]))
>>> res.nobs, res.n_params, res.k_exog, res.midas_lags
(119, 13.0, 1, 3)
>>> joint = res.joint_stderr()
>>> bool(joint["y1: x1"] > res.bse["y1: x1"]), sorted(k for k in joint if "theta" in k)
(True, ['theta1[x1]', 'theta2[x1]'])
>>> res.forecast(1, exog_high_future=np.ones((3, 1))).shape
(1, 2)

Private methods#

_comparison_label()

Short specification label for a ranking table.

_summary_table()

Build the structured summary.

_trailing_blocks()

The compressed-regressor coefficients, in design order.

_trailing_labels()

Compressed-regressor column names.
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