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Utility Functions

BayesianFactorZoo.construct_weight_matrix — Function
construct_weight_matrix(R::Matrix{Float64}, f::Matrix{Float64}, 
                     type::String="OLS", kappa::Float64=1e6)

Construct weighting matrix for GMM estimation of linear SDF models.

Arguments

  • R: Matrix of test assets with dimension $t \times N$
  • f: Matrix of factors with dimension $t \times k$
  • type: "OLS" or "GLS", default="OLS"
  • kappa: Large constant for factor moment conditions, default=1e6

Details

Constructs a $(N+k) \times (N+k)$ block diagonal weighting matrix W:

\[W = \begin{bmatrix} W_R & 0_{N\times k} \\ 0_{k\times N} & \kappa I_k \end{bmatrix}\]

where:

OLS (type="OLS"):

$W_R = I_N$

GLS (type="GLS"):

$W_R = \Sigma_R^{-1}$

The structure reflects GMM moment conditions:

\[E[g_t(\lambda_c,\lambda_f,\mu_f)] = E[(R_t - \lambda_c1_N - R_t(f_t - \mu_f)'\lambda_f), (f_t - \mu_f)] = [0_N, 0_k]\]

Returns

Returns a Matrix{Float64} of size $(N+k) \times (N+k)$ containing the weighting matrix W with structure:

  • Upper-left block: Identity (OLS) or inverse return covariance (GLS)
  • Lower-right block: $\kappa I_k$
  • Off-diagonal blocks: Zero matrices

Note: The returned matrix matches the dimension requirements of SDF_gmm function

Notes

  • Input matrices R and f must have the same number of rows (time periods)
  • The GLS version requires a well-conditioned return covariance matrix
  • κ should be large enough to ensure accurate factor mean estimation
  • Output matches dimensions required by SDF_gmm function
  • Block structure is optimal under conditional homoskedasticity

References

Bryzgalova S, Huang J, Julliard C (2023). "Bayesian solutions for the factor zoo: We just ran two quadrillion models." Journal of Finance, 78(1), 487–557.

Hansen, Lars Peter (1982). "Large Sample Properties of Generalized Method of Moments Estimators." Econometrica, 50(4), 1029-1054.

Examples

# Construct OLS weighting matrix
W_ols = construct_weight_matrix(R, f, "OLS")

# Construct GLS weighting matrix
W_gls = construct_weight_matrix(R, f, "GLS")

# Use custom kappa value
W_custom = construct_weight_matrix(R, f, "OLS", 1e8)

# Use in GMM estimation
results_ols = SDF_gmm(R, f, W_ols)
results_gls = SDF_gmm(R, f, W_gls)

See Also

  • SDF_gmm: Main function using these weighting matrices
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BayesianFactorZoo.psi_to_priorSR — Function
psi_to_priorSR(R::Matrix{Float64}, f::Matrix{Float64}; 
              psi0::Union{Nothing,Float64}=nothing,
              priorSR::Union{Nothing,Float64}=nothing,
              aw::Float64=1.0, bw::Float64=1.0)

Map between prior tightness parameter $\psi$ and prior Sharpe ratio.

Arguments

  • R: Matrix of test assets with dimension $t \times N$
  • f: Matrix of factors with dimension $t \times k$
  • psi0: Prior tightness parameter to convert to SR
  • priorSR: Target SR to convert to psi0
  • aw,bw: Beta prior parameters

Returns

Returns a Float64 value either:

  • The implied prior Sharpe ratio if psi0 provided
  • The required psi0 value if priorSR provided

Note: Returns error message string if neither or both arguments are provided

Notes

  • Exactly one of psi0 or priorSR must be provided
  • Input matrices R and f must have the same number of rows (time periods)
  • The mapping helps choose priors based on economic intuition about achievable Sharpe ratios
  • Default aw=bw=1 implies 50% prior probability of factor inclusion
  • The relationship is monotonic: higher ψ implies higher prior Sharpe ratio
  • Useful for calibrating priors in continuoussssdf and continuoussssdf_v2

References

Bryzgalova S, Huang J, Julliard C (2023). "Bayesian solutions for the factor zoo: We just ran two quadrillion models." Journal of Finance, 78(1), 487–557.

Examples

# Load test data
# Convert psi0 to implied prior Sharpe ratio
implied_sr = psi_to_priorSR(R, f; psi0=5.0)
println("Psi0 = 5.0 implies prior SR = $implied_sr")

# Find psi0 needed for target Sharpe ratio
required_psi = psi_to_priorSR(R, f; priorSR=0.5)
println("For prior SR = 0.5, need psi0 = $required_psi")

# Use custom Beta prior parameters
psi_sparse = psi_to_priorSR(R, f; 
                          priorSR=0.3,
                          aw=1.0, 
                          bw=9.0)  # Prior favoring sparsity

# Helper functions also available:
using BayesianFactorZoo: calculate_prior_SR, find_psi_for_target_SR

# Get prior SR for a given psi
sr = calculate_prior_SR(5.0, R, f)

# Get psi for a target SR
psi = find_psi_for_target_SR(0.5, R, f)

See Also

  • continuous_ss_sdf: Main function using this prior calibration
  • continuous_ss_sdf_v2: Version for tradable factors using this calibration
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