marginal_effects: Marginal Effects of the Variance Determinants on Inefficiency

Description

For a model whose inefficiency scale depends on covariates (the _Z models), the marginal effects of those covariates on the mean and variance of the inefficiency term, $\partial E[u]/\partial z_k$ and $\partial \mathrm{Var}[u]/\partial z_k$.

Usage

marginal_effects(object, average = FALSE, component = c("u", "h"))

Value

With average = FALSE, a data frame with one row per observation: sigma_u, E_u, Var_u, then one dE_u.d<z> and one dVar_u.d<z> column per non-constant variance determinant. Attributes "average" (the average marginal effects), "link" and "family" record the convention used.

With average = TRUE, just the named vector of average marginal effects.

Arguments

object: an object of class "sfareg" from sfm ("NHN_Z", "NE_Z") or psfm ("TRE_Z", "GTRE_Z"), fitted with a variance-determinant formula such as y ~ x1 + x2 | z.
component: which inefficiency component to differentiate. "u" (the default) is the only choice for every model except psfm(model_name = "GTRE_Z"), where it is the transient component; "h" selects that model's persistent component, parameterized by the third formula segment y ~ x | z | zp. The output columns are named after the component, so a table cannot be misread once separated from the call that produced it.
average: if TRUE, return only the named vector of average marginal effects. Default FALSE, which returns the full per-observation table with that vector attached as the "average" attribute.

Details

A $\delta$ coefficient is not interpretable on its own: it sits in a log link for a scale parameter, so its units are neither those of $u$ nor those of $z$, and its magnitude is not comparable across models whose link differs. The marginal effect is on the scale of $u$, and is what applied work reports.

Writing $s = \sigma_u(z)$, every derivative reduces to $\partial s/\partial z_k$ times a constant that cancels back into the moment itself. For the half-normal and the exponential alike:

$$\textrm{SD link:} \quad \partial E[u]/\partial z_k = \delta_k E[u], \qquad \partial \mathrm{Var}[u]/\partial z_k = 2\delta_k \mathrm{Var}[u]$$ $$\textrm{variance link:} \quad \partial E[u]/\partial z_k = (\delta_k/2) E[u], \qquad \partial \mathrm{Var}[u]/\partial z_k = \delta_k \mathrm{Var}[u]$$

The factor of two matters. sfm()'s "NHN_Z" and "NE_Z" place the linear predictor on the standard deviation, $\sigma_u = \exp(z'\delta)$, while psfm()'s "TRE_Z" and "GTRE_Z" place it on the variance, $\sigma_u = \sqrt{\exp(z'\delta)}$. Reading a $\delta$ from one family with the other's convention in mind is wrong by exactly this factor. Reporting the marginal effect instead of the coefficient removes the trap, since the effect is on the scale of $u$ either way.

The link is read from the fit rather than assumed, so a sfm model fitted with z_link = "var" and a psfm model are handled the same way and their effects are directly comparable.

psfm(model_name = "GTRE_Z") separates persistent inefficiency $h_i$ from transient $u_{it}$ and gives each its own determinant block, so both are available: component = "u" for the transient effects and component = "h" for the persistent ones. Each block is located by NAME rather than by position, because the $\sigma_h$ coefficients are the trailing ones and a positional rule would report them under a $\sigma_u$ label.

Effects are per observation, because $\sigma_u$ varies with $z$. Constant columns of the $z$ design are dropped: the derivative with respect to an intercept is not a marginal effect.

Standard errors are deliberately not reported. The effect is a nonlinear function of both $\delta$ and $z$, so its sampling distribution requires the delta method through the full covariance matrix or a bootstrap; a number printed without one would be read as inference.

Examples

Run this code

dat <- data_gen_cs(N = 300, rand = 1, sig_u = 1, sig_v = 0.3,
                   cons = 0.5, beta1 = 0.5, beta2 = 0.5, a = 5, mu = 0.5)
fit <- sfm(y_pcs_z ~ x1 + x2 | z, data = dat, model_name = "NHN_Z")

me <- marginal_effects(fit)
head(me)

## The conventional summary.
marginal_effects(fit, average = TRUE)