Asymptotic Variances for Predictions and Marginal Effects
Two results of interest after estimating the model are: the predictions F(x’в) and the marginal effects dF/dx = f (Xв) в• For example, given the characteristics of an individual x, we can predict his or her probability of purchasing a car. Also, given a change in x, say income, one can estimate the marginal effect this will have on the probability of purchasing a car. The latter effect is constant for the linear probability model and is given by the regression coefficient of income, whereas for the probit and logit models this marginal effect will vary with the Xj’s, see (13.7) and (13.8). These marginal effects can be computed with Stata using the dprobit command. The default is to compute them at the sample mean x. There is also the additional problem of computing variances for these predictions and marginal effects. Both F(xв) and f (x в)в are nonlinear functions of the в’s. To compute standard errors, we can use the following linear approximation which states that whenever в = F(в) then the asy. var(e) = (dF/двуУ(l3)(dF/d/3). For the predictions, let z = X/3 and denote by F = F(X/3) and f = f (x’P), then
dF/dp = (dF/dz)(dz/dP) = fx and asy. var(F) = f2x’ У (ft) x.
For the marginal effects, let р = f/3, then
asy. var(p) = (д’р/др )У(Р)(др/др )’ (13.30)
where др/др’ = fIk + P(df/dz)(dz/d3′) = fIk + (df/dz)(Px’).
For the probit model, дf/дz = дф/дz = —zф• So, др/дв = ф[Ік — zвX and
asy. var(3) = ф [Ik — ХррХ]У(P)[Ik — x’PPx’]’ (13.31)
For the logit model, f = Л(1 — Л), so
дp/дz = (1 — 2Л )(дЛ ^z) = (1 — 2Л )(f) = (1 — 2Л )Л (1 — Л)
др/др = Л (1 — Л )[Ik + (1 — 2Л )3x’]
and (13.30) becomes
asy. var(3) = [Л (1 — Л )]2 [Ik + (1 — 2Л)px’]V 0)[h + (1 — 2Л)px’ ]’ (13.32)