bvarHyperopt#

Purpose#

Optimize Minnesota prior hyperparameters by maximizing the log marginal likelihood.

Format#

ho = bvarHyperopt(y)#
ho = bvarHyperopt(y, ctl)
ho = bvarHyperopt(y, ctl, xreg=X)
Parameters:

  • y (TxM matrix or dataframe) – endogenous variables.

  • ctl (struct) –

    Optional input, a bvarControl structure with initial hyperparameter values. The optimization mode is determined by which lambda values are nonzero:

    • lambda6 = 0, lambda7 = 0: optimize lambda1 only

    • lambda6 > 0: optimize lambda1 + lambda6 (SOC)

    • lambda7 > 0: optimize lambda1 + lambda7 (SUR)

    • Both > 0: optimize lambda1 + lambda6 + lambda7

  • xreg (TxK matrix) – Optional keyword, exogenous regressors.

  • quiet (scalar) – Optional keyword, set to 1 to suppress output. Default = 0.

Returns:

ho (struct) –

An instance of a hyperoptResult structure containing:

ho.lambda1

Scalar, optimized overall tightness.

ho.lambda3

Scalar, optimized lag decay (if included in optimization).

ho.lambda6

Scalar, optimized SOC tightness (if included).

ho.lambda7

Scalar, optimized SUR tightness (if included).

ho.log_ml

Scalar, maximized log marginal likelihood.

ho.converged

Scalar, 1 if optimizer converged.

ho.n_evals

Scalar, number of function evaluations.

ho.ctl

bvarControl struct, pre-populated with all optimal values. Ready to pass directly to bvarFit().

Examples#

One-Line Optimal BVAR#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));

// Optimize and estimate in two lines
ho = bvarHyperopt(data);
result = bvarFit(data, ho.ctl);

print "Optimal lambda1:" ho.lambda1;
print "Log ML:" ho.log_ml;

Optimize with SOC and SUR#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));

ctl = bvarControlCreate();
ctl.p = 4;
ctl.lambda6 = 1;      // Enable SOC (initial value)
ctl.lambda7 = 1;      // Enable SUR (initial value)

ho = bvarHyperopt(data, ctl);
result = bvarFit(data, ho.ctl);

print "Optimal lambda1:" ho.lambda1;
print "Optimal lambda6:" ho.lambda6;
print "Optimal lambda7:" ho.lambda7;

Remarks#

Implements the Giannone, Lenza & Primiceri (2015) approach to hyperparameter selection. The marginal likelihood is maximized using L-BFGS-B constrained optimization, treating the Minnesota hyperparameters as continuous variables with positivity constraints.

The returned ho.ctl structure is a complete bvarControl with all fields populated — the optimal lambda values plus all other settings carried over from the input. Pass it directly to bvarFit() without further modification.

Model#

The marginal likelihood of the data under the conjugate Minnesota prior is:

\[p(Y | \lambda) = \pi^{-\frac{T m}{2}} \frac{|\bar\Phi|^{m/2}}{|\Omega|^{m/2}} \frac{|\bar{S}|^{-\bar\alpha/2}}{|S_0|^{-\alpha_0/2}} \prod_{i=1}^{m} \frac{\Gamma((\bar\alpha + 1 - i)/2)}{\Gamma((\alpha_0 + 1 - i)/2)}\]

where \(\lambda = (\lambda_1, \lambda_6, \lambda_7)\) are the hyperparameters being optimized, and the posterior parameters \(\bar\Phi, \bar{S}, \bar\alpha\) depend on \(\lambda\) through the prior.

The optimum \(\lambda^* = \arg\max_\lambda \log p(Y | \lambda)\) is the empirical Bayes or “type II maximum likelihood” estimate. Algorithm ———

  1. Evaluate \(\log p(Y | \lambda)\) analytically (closed form for conjugate NIW).

  2. Maximize using L-BFGS-B with positivity constraints on all \(\lambda\).

  3. Starting values: the input ctl lambda values (defaults: lambda1=0.2).

The optimization is fast because each function evaluation is \(O(K^2 m)\) (no MCMC). Typical wall-clock time is 0.01-0.05 seconds. Troubleshooting —————

Optimizer returns lambda1 at the upper bound: The data wants a very loose prior (lambda1 → ∞ approaches OLS). This suggests the sample is large enough that the prior doesn’t help. Consider using OLS (varFit()) or reducing the search bounds.

lambda6 or lambda7 optimized to near zero: The data does not support sum-of-coefficients or single-unit-root priors. This is informative — the prior is not needed for this dataset. Verification ————

GLP hyperparameter optimization verified against R BVAR::bvar() with hyper = "auto" on multiple datasets. Optimal lambda values and maximized log marginal likelihoods agree within optimization tolerance.

See crossval/23_glp_crossval.R. References ———-

  • Giannone, D., M. Lenza, and G. E. Primiceri (2015). “Prior selection for vector autoregressions.” Review of Economics and Statistics, 97(2), 436-451.

Library#

timeseries

Source#

bvar.src

See also

Functions bvarFit(), bvarControlCreate()

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