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poisson_glm module

Estimates the semilinear Choo and Siow homoskedastic (2006) model using Poisson GLM.

choo_siow_poisson_glm(muhat, phi_bases, tol=1e-12, max_iter=10000, verbose=1)

Estimates the semilinear Choo and Siow homoskedastic (2006) model using Poisson GLM.

Parameters:

Name Type Description Default
muhat Matching

the observed Matching

required
phi_bases np.ndarray

an (X, Y, K) array of bases

required
tol Optional[float]

tolerance level for linear_model.PoissonRegressor.fit

1e-12
max_iter Optional[int]

maximum number of iterations for linear_model.PoissonRegressor.fit

10000
verbose Optional[int]

defines how much output we want (0 = least)

1

Returns:

Type Description
PoissonGLMResults

a PoissonGLMResults instance

Example
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n_households = 1e6
X, Y, K = 4, 3, 6
# we setup a quadratic set of basis functions
phi_bases = np.zeros((X, Y, K))
phi_bases[:, :, 0] = 1
for x in range(X):
    phi_bases[x, :, 1] = x
    phi_bases[x, :, 3] = x * x
    for y in range(Y):
        phi_bases[x, y, 4] = x * y
for y in range(Y):
    phi_bases[:, y, 2] = y
    phi_bases[:, y, 5] = y * y

lambda_true = np.random.randn(K)
phi_bases = np.random.randn(X, Y, K)
Phi = phi_bases @ lambda_true

# we simulate a Choo and Siow sample from a population
#  with equal numbers of men and women of each type
n = np.ones(X)
m = np.ones(Y)
choo_siow_instance = ChooSiowPrimitives(Phi, n, m)
mus_sim = choo_siow_instance.simulate(n_households)
muxy_sim, mux0_sim, mu0y_sim, n_sim, m_sim = mus_sim.unpack()

results = choo_siow_poisson_glm(mus_sim, phi_bases)

# compare true and estimated parameters
results.print_results(
    lambda_true,
    u_true=-np.log(mux0_sim / n_sim),
    v_true=-np.log(mu0y_sim / m_sim)
)
Source code in cupid_matching/poisson_glm.py
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def choo_siow_poisson_glm(
    muhat: Matching,
    phi_bases: np.ndarray,
    tol: Optional[float] = 1e-12,
    max_iter: Optional[int] = 10000,
    verbose: Optional[int] = 1,
) -> PoissonGLMResults:
    """Estimates the semilinear Choo and Siow homoskedastic (2006) model
        using Poisson GLM.

    Args:
        muhat: the observed Matching
        phi_bases: an (X, Y, K) array of bases
        tol: tolerance level for `linear_model.PoissonRegressor.fit`
        max_iter: maximum number of iterations
            for `linear_model.PoissonRegressor.fit`
        verbose: defines how much output we want (0 = least)

    Returns:
        a `PoissonGLMResults` instance

    Example:
        ```py
        n_households = 1e6
        X, Y, K = 4, 3, 6
        # we setup a quadratic set of basis functions
        phi_bases = np.zeros((X, Y, K))
        phi_bases[:, :, 0] = 1
        for x in range(X):
            phi_bases[x, :, 1] = x
            phi_bases[x, :, 3] = x * x
            for y in range(Y):
                phi_bases[x, y, 4] = x * y
        for y in range(Y):
            phi_bases[:, y, 2] = y
            phi_bases[:, y, 5] = y * y

        lambda_true = np.random.randn(K)
        phi_bases = np.random.randn(X, Y, K)
        Phi = phi_bases @ lambda_true

        # we simulate a Choo and Siow sample from a population
        #  with equal numbers of men and women of each type
        n = np.ones(X)
        m = np.ones(Y)
        choo_siow_instance = ChooSiowPrimitives(Phi, n, m)
        mus_sim = choo_siow_instance.simulate(n_households)
        muxy_sim, mux0_sim, mu0y_sim, n_sim, m_sim = mus_sim.unpack()

        results = choo_siow_poisson_glm(mus_sim, phi_bases)

        # compare true and estimated parameters
        results.print_results(
            lambda_true,
            u_true=-np.log(mux0_sim / n_sim),
            v_true=-np.log(mu0y_sim / m_sim)
        )
        ```

    """
    try_sparse = False
    X, Y, K = phi_bases.shape
    XY = X * Y
    n_rows = XY + X + Y
    n_cols = X + Y + K

    # the vector of weights for the Poisson regression
    w = np.concatenate((2 * np.ones(XY), np.ones(X + Y)))
    # reshape the bases
    phi_mat = _make_XY_K_mat(phi_bases)

    if try_sparse:
        w_mat = spr.csr_matrix(
            np.concatenate((2 * np.ones((XY, n_cols)), np.ones((X + Y, n_cols))))
        )

        # construct the Z matrix
        ones_X = spr.csr_matrix(np.ones((X, 1)))
        ones_Y = spr.csr_matrix(np.ones((Y, 1)))
        zeros_XK = spr.csr_matrix(np.zeros((X, K)))
        zeros_YK = spr.csr_matrix(np.zeros((Y, K)))
        zeros_XY = spr.csr_matrix(np.zeros((X, Y)))
        zeros_YX = spr.csr_matrix(np.zeros((Y, X)))
        id_X = spr.csr_matrix(np.eye(X))
        id_Y = spr.csr_matrix(np.eye(Y))
        Z_unweighted = spr.vstack(
            [
                spr.hstack(
                    [
                        -spr.kron(id_X, ones_Y),
                        -spr.kron(ones_X, id_Y),
                        phi_mat,
                    ]
                ),
                spr.hstack([-id_X, zeros_XY, zeros_XK]),
                spr.hstack([zeros_YX, -id_Y, zeros_YK]),
            ]
        )
        Z = Z_unweighted / w_mat
    else:
        ones_X = np.ones((X, 1))
        ones_Y = np.ones((Y, 1))
        zeros_XK = np.zeros((X, K))
        zeros_YK = np.zeros((Y, K))
        zeros_XY = np.zeros((X, Y))
        zeros_YX = np.zeros((Y, X))
        id_X = np.eye(X)
        id_Y = np.eye(Y)
        Z_unweighted = np.vstack(
            [
                np.hstack([-np.kron(id_X, ones_Y), -np.kron(ones_X, id_Y), phi_mat]),
                np.hstack([-id_X, zeros_XY, zeros_XK]),
                np.hstack([zeros_YX, -id_Y, zeros_YK]),
            ]
        )
        Z = Z_unweighted / w.reshape((-1, 1))

    _, _, _, n, m = muhat.unpack()
    var_muhat, var_munm = _variance_muhat(muhat)
    (
        muxyhat_norm,
        var_muhat_norm,
        var_munm_norm,
        n_households,
        n_individuals,
    ) = _prepare_data(muhat, var_muhat, var_munm)

    clf = linear_model.PoissonRegressor(
        fit_intercept=False,
        tol=tol,
        verbose=verbose,
        alpha=0,
        max_iter=max_iter,
    )
    clf.fit(Z, muxyhat_norm, sample_weight=w)
    gamma_est = clf.coef_

    # we compute the variance-covariance of the estimator
    nr, nc = Z.shape
    exp_Zg = np.exp(Z @ gamma_est).reshape(n_rows)
    A_hat = np.zeros((nc, nc))
    B_hat = np.zeros((nc, nc))
    for i in range(nr):
        Zi = Z[i, :]
        wi = w[i]
        A_hat += wi * exp_Zg[i] * np.outer(Zi, Zi)
        for j in range(nr):
            Zj = Z[j, :]
            B_hat += wi * w[j] * var_muhat_norm[i, j] * np.outer(Zi, Zj)

    A_inv = spla.inv(A_hat)
    varcov_gamma = A_inv @ B_hat @ A_inv
    stderrs_gamma = np.sqrt(np.diag(varcov_gamma))

    beta_est = gamma_est[-K:]
    varcov_beta = varcov_gamma[-K:, -K:]
    beta_std = stderrs_gamma[-K:]
    Phi_est = phi_bases @ beta_est

    # we correct for the effect of the normalization
    n_norm = n / n_individuals
    m_norm = m / n_individuals
    u_est = gamma_est[:X] + np.log(n_norm)
    v_est = gamma_est[X:-K] + np.log(m_norm)

    # since u = a + log(n_norm) we also need to adjust the estimated variance
    z_unweighted_T = Z_unweighted.T
    u_std = np.zeros(X)
    ix = XY
    for x in range(X):
        n_norm_x = n_norm[x]
        A_inv_x = A_inv[x, :]
        var_log_nx = var_munm_norm[ix, ix] / n_norm_x / n_norm_x
        slice_x = slice(x * Y, (x + 1) * Y)
        covar_term = var_muhat_norm[:, ix] + np.sum(var_muhat_norm[:, slice_x], 1)
        cov_a_lognx = (A_inv_x @ z_unweighted_T @ covar_term) / n_norm_x
        ux_var = varcov_gamma[x, x] + var_log_nx + 2.0 * cov_a_lognx
        u_std[x] = sqrt(ux_var)
        ix += 1

    v_std = stderrs_gamma[X:-K]
    iy, jy = X, XY + X
    for y in range(Y):
        m_norm_y = m_norm[y]
        A_inv_y = A_inv[iy, :]
        var_log_my = var_munm_norm[jy, jy] / m_norm_y / m_norm_y
        slice_y = slice(y, XY, Y)
        covar_term = var_muhat_norm[:, jy] + np.sum(var_muhat_norm[:, slice_y], 1)
        cov_b_logmy = (A_inv_y @ z_unweighted_T @ covar_term) / m_norm_y
        vy_var = varcov_gamma[iy, iy] + var_log_my + 2.0 * cov_b_logmy
        v_std[y] = sqrt(vy_var)
        iy += 1
        jy += 1

    results = PoissonGLMResults(
        X=X,
        Y=Y,
        K=K,
        number_households=n_households,
        number_individuals=n_individuals,
        estimated_gamma=gamma_est,
        estimated_Phi=Phi_est,
        estimated_beta=beta_est,
        estimated_u=u_est,
        estimated_v=v_est,
        varcov_gamma=varcov_gamma,
        varcov_beta=varcov_beta,
        stderrs_gamma=stderrs_gamma,
        stderrs_beta=beta_std,
        stderrs_u=u_std,
        stderrs_v=v_std,
    )

    return results