MFVAR#

class cultivars.multivariate.reduced_form.mixed_frequency.MFVAR(coefficients, sigma_u, *, kinds, period, intercept=None, weights=None, names=None)[source]

Bases: object

A latent high-frequency VAR observed through temporal aggregation.

The model is \(z_t = c + A_1 z_{t-1} + \dots + A_p z_{t-p} + u_t\) at the high frequency; what is observed is \(z\) itself for some variables and a weighted sub-period sum for others, on a calendar expressed as nan in the sample rather than as a time-varying observation matrix. The state stacks \(\max(p, m)\) lags of \(z_t\), deep enough that a reading spanning \(m\) sub-periods is a fixed linear function of the current state, so the observation matrix is constant and the filter’s missing-value masking does the rest.

This class does not estimate. The omission is a finding, not an unfinished feature. Simulating a three-variable monthly VAR with one variable observed only as a quarterly average and estimating by EM:

T

|A err|

Sigma11

|S err|

control

0.051

1.100

0.060

1500

0.129

2.629

1.469

6000

0.183

3.104

1.944

against a truth of Sigma11 = 1.160. The control row is the identical estimator with nothing aggregated, so the machinery is sound. The error in the aggregated rows grows with the sample: this is convergence to a wrong limit, not sampling noise, and the likelihood confirms it – the wrong answer scores about a hundred log-points above the truth. The mechanism is that a quarterly reading constrains a three-month average, and a high monthly innovation variance with weaker persistence is nearly observationally equivalent to a low one with stronger persistence.

Priors do not rescue it. A Minnesota prior shrinks A and cannot reach the unidentified direction, which lives in Sigma. An inverse-Wishart prior on Sigma anchored at an aggregation-implied scale fails for a sharper reason: the correct anchor requires the high-frequency persistence, which is precisely what is not identified. Anchoring even at the true value needs a prior worth half the sample to recover the cell.

The route to an estimated mixed-frequency VAR is Gibbs sampling over (A, Sigma) jointly with a proper prior – integrating rather than maximizing, as Schorfheide and Song do – which belongs to cultivars.bayes once a sampling backend exists. Until then, supply parameters and filter: from_result() takes them from a VAR fitted on whatever high-frequency sample exists, smooth() infers the latent path, nowcast() reads off its last period.

Variables:

  • _coefficients – The validated (order, k, k) autoregressive stack.

  • _sigma_u – The validated, symmetrized (k, k) innovation covariance.

  • _intercept – The (k,) latent intercept, or None.

  • _kinds – One frequency role per variable.

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

  • _weights – Per-variable explicit sub-period weights, or None.

  • _names – Variable labels.

See also

  • MFVARResult – the record smooth() returns.

  • MIDASVAR – the low-frequency model, which estimates.

  • VAR – the estimator whose parameters this class borrows, and the right model when every variable is observed at the high frequency.

  • state_space – the Kalman filter and simulation smoother the inference runs on.

References

Mariano, R. S., & Murasawa, Y. (2003). A new coincident index of business cycles based on monthly and quarterly series. Journal of Applied Econometrics, 18(4), 427-443.

Schorfheide, F., & Song, D. (2015). Real-time forecasting with a mixed-frequency VAR. Journal of Business & Economic Statistics, 33(3), 366-380.

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

Example

>>> import numpy as np
>>> A = np.array([[[0.5, 0.1], [0.0, 0.4]]])
>>> S = np.eye(2)
>>> y = np.full((24, 2), np.nan)
>>> rng = np.random.default_rng(0)
>>> y[:, 0] = rng.standard_normal(24)
>>> y[2::3, 1] = rng.standard_normal(8)
>>> res = MFVAR(A, S, kinds=["high", "flow"], period=3).smooth(y)
>>> res.endog.shape
(24, 2)

Private attributes#

_coefficients

_intercept

_kinds

_names

_period

_sigma_u

_weights

Private methods#

_state_space()

Assemble the observable form.