Marginal Effects and Inference for Fractional Responses

From Fractional Response Models to Marginal Effects

A coefficient in a fractional response model describes how a covariate changes the model's index, not directly how much the conditional mean changes. Because the link function is nonlinear, the same coefficient can imply different changes in the expected fraction at different covariate values. Marginal effects translate fitted models into changes on the outcome scale.

The calculations below use the conditional mean specification \(E(Y_i\mid x_i)=F(x_i'\beta)\), where \(Y_i\) is a fraction between zero and one, \(x_i\) is a vector of regressors including an intercept, and \(F\) is a link function. Fractional logit uses the logistic cumulative distribution function; fractional probit uses the standard normal cumulative distribution function. The outcome need not be restricted to values strictly between zero and one.

Definition: A marginal effect is the change in the model's conditional mean associated with a change in a regressor, evaluated at specified covariate values. For a continuous regressor it is usually a derivative. For a discrete regressor it is usually a difference between predicted conditional means. An average partial effect (APE) is the sample average of individual marginal effects, each evaluated at that observation's covariates.

Write \(\eta_i=x_i'\beta\) for the index. For a continuously varying regressor \(x_{ij}\), suppose the index is linear in that regressor with coefficient \(\beta_j\). The chain rule gives the derivative of the conditional mean. More generally, if the index includes transformations or interactions, differentiate the full index with respect to the regressor rather than using \(\beta_j\) alone.

\frac{\partial E(Y_i\mid x_i)}{\partial x_{ij}} = f(\eta_i)\frac{\partial \eta_i}{\partial x_{ij}}, \qquad \text{and if }\frac{\partial \eta_i}{\partial x_{ij}}=\beta_j,\quad ME_{ij}=f(\eta_i)\beta_j,

Here \(f=F'\) is the link's density. For fractional logit, \(F(\eta)=\Lambda(\eta)=1/(1+\exp(-\eta))\), so \(f(\eta)=\Lambda(\eta)[1-\Lambda(\eta)]\). For fractional probit, \(F(\eta)=\Phi(\eta)\) and \(f(\eta)=\phi(\eta)\), the standard normal density.

Worked Example: Continuous Effects Under Logit and Probit

Suppose a hypothetical fitted fractional logit model has index \(\hat{\eta}=-1+0.8x\), where \(x\) is measured in tens of service hours. At \(x=1\), the index is \(\hat{\eta}=-1+0.8(1)=-0.2\). The fitted conditional mean is \(\Lambda(-0.2)=0.4502\), and the logit density there is \(0.4502(1-0.4502)=0.2475\). Thus the marginal effect is

\widehat{ME}(1)=0.2475(0.8)=0.1980.

At this covariate value, an additional ten service hours is associated with an estimated increase of 0.1980 in the conditional mean fraction, or 19.80 percentage points. This is a local derivative, not a claim that every ten-hour increase has that same effect.

For comparison, suppose a separate hypothetical fractional probit model has index \(\hat{\eta}=-0.5+0.6x\). At \(x=1\), its index is \(0.1\), and \(\phi(0.1)=0.3970\). Its marginal effect is \(0.3970(0.6)=0.2382\), or 23.82 percentage points per ten hours at that value of \(x\). The two effects come from different fitted models; the coefficients themselves are not directly comparable as outcome-scale effects.

Discrete Changes and Average Partial Effects

A derivative is not the natural effect measure for a binary regressor. Instead, hold the other regressors fixed and compare the fitted conditional mean when the binary variable is set to one with the fitted mean when it is set to zero. This finite change is meaningful even though a binary variable does not vary continuously.

\Delta_{ij} = F\!\left(x_i(1)'\beta\right) - F\!\left(x_i(0)'\beta\right),

In this expression, \(x_i(d)\) is the regressor vector for observation \(i\) with the binary variable set to \(d\), and all other covariates held at their observed values. If there are interactions, changing the binary variable also changes the relevant interaction terms. The result is a change in the conditional mean fraction, not a change in the link index.

An effect evaluated at one selected covariate vector can be useful for illustrating a profile. To summarize effects across the estimation sample, calculate each observation's effect and average them. For a continuous regressor this averages derivatives; for a binary regressor it averages discrete changes. In either case the APE is on the scale of the fractional outcome.

Worked Example: Average Discrete Change for a Binary Regressor

Consider a hypothetical fractional logit model with \(\hat{\eta}=-1+0.7d+0.4w\), where \(d\) indicates program participation and \(w\) is a measured covariate. For two sample observations, let \(w=0\) and \(w=1\). Calculate each participation effect by predicting twice for the same observation, first with \(d=1\) and then with \(d=0\).

For the observation with \(w=0\), the two indices are \(-1+0.7(1)+0.4(0)=-0.3\) and \(-1+0.7(0)+0.4(0)=-1\). The discrete change is

\Lambda(-0.3)-\Lambda(-1)\approx 0.1566.

For the observation with \(w=1\), the indices are \(0.1\) and \(-0.6\). The change is \(\Lambda(0.1)-\Lambda(-0.6)\approx 0.1706\). The two-observation APE is therefore

\widehat{APE} = \frac{0.1566+0.1706}{2} =0.1636.

In this hypothetical sample, participation is associated with an average fitted increase of 0.1636 in the conditional mean fraction, or 16.36 percentage points. The average is not the effect at the average value of \(w\); it averages two separately calculated changes.

Inference for Marginal Effects

Marginal effects depend on estimated parameters, so they have sampling uncertainty even when evaluated at fixed covariates. Let \(\hat{\beta}\) be the fitted parameter vector and let \(\widehat V\) be its estimated covariance matrix. For fractional response QMLE, a sandwich covariance estimate is useful when the conditional mean is correctly specified but the variance need not equal the variance implied by a full likelihood model. The covariance estimator must also reflect the sampling dependence: an observation-level robust estimate is appropriate for independent observations, while clustered dependence calls for a covariance estimate robust to the specified clustering.

Robust covariance estimation changes the estimated uncertainty of \(\hat{\beta}\), not the marginal-effect formula or its point estimate. It does not repair a misspecified conditional mean, omitted variables, or a problematic sampling design. To obtain uncertainty for a marginal effect, apply the delta method to the effect as a function of \(\beta\).

Delta method for an effect: Let \(g(\beta)\) be a scalar marginal effect or APE, differentiable at the true parameter, and let \(\nabla_\beta g\) denote its gradient. Its estimated robust variance is \(\widehat{\operatorname{Var}}\{g(\hat{\beta})\}=\nabla_\beta g(\hat{\beta})'\widehat V\nabla_\beta g(\hat{\beta})\). The square root is the robust standard error. An approximate 95% confidence interval is \(g(\hat{\beta})\pm1.96\,SE\), under the usual large-sample normal approximation.

For a continuous regressor in a linear index, \(ME_{ij}(\beta)=f(x_i'\beta)\beta_j\). Its gradient includes both the direct appearance of \(\beta_j\) and the effect of changing \(\beta\) on the index. If \(e_j\) is the vector with a one in position \(j\) and zeros elsewhere, then

\nabla_\beta ME_{ij} = f'(\eta_i)\beta_j x_i+f(\eta_i)e_j.

For a discrete change, let \(x_i(1)\) and \(x_i(0)\) be the design vectors under the two settings. Its gradient is \(f(\eta_{i1})x_i(1)-f(\eta_{i0})x_i(0)\), where \(\eta_{id}=x_i(d)'\beta\). The gradient of an APE is the average of the individual gradients, because differentiation and a finite sample average commute.

Why the Delta Method Works

The justification is a first-order Taylor expansion around the true parameter \(\beta\). For a differentiable scalar function \(g\), the expansion is

g(\hat{\beta})-g(\beta) = \nabla_\beta g(\beta)'(\hat{\beta}-\beta) +o_p(n^{-1/2}).

If \(\hat{\beta}\) is asymptotically normal with covariance estimated by \(\widehat V\), the leading term is a linear combination of its parameter errors. Its variance is the gradient multiplied on both sides of the covariance matrix, giving \(\nabla_\beta g'\widehat V\nabla_\beta g\). Replacing the unknown gradient at \(\beta\) by the gradient at \(\hat{\beta}\) yields the plug-in standard error. This approximation is most reliable when the sample is large enough for the parameter estimates to be approximately normal and the effect is not at a point where the function is poorly approximated by a linear expansion.

Worked Example: Robust Inference for a Continuous APE

Suppose a hypothetical fractional logit fit has \(\hat{\beta}_0=-1\) and \(\hat{\beta}_1=0.8\), with \(x=0,1,2\) in the sample. The fitted logit densities at indices \(-1,-0.2,0.6\) are respectively \(0.1966,0.2475,0.2288\). The three individual marginal effects are \(0.8\) times these densities: \(0.1573,0.1980,0.1830\). Their average is

\widehat{APE} = \frac{0.1573+0.1980+0.1830}{3} =0.1794.

For this model, the individual gradient is \(\big(0.8f'(\eta_i),\,f(\eta_i)+0.8x_i f'(\eta_i)\big)'\), because the coefficient vector contains an intercept and a slope. Averaging these gradients over the three covariate values gives approximately \((0.01303,0.1953)'\). Suppose the fitted robust covariance matrix is

\widehat V= \begin{pmatrix} 0.0400 & -0.0100\\ -0.0100 & 0.0225 \end{pmatrix}.

The delta-method variance and standard error are

\widehat{\operatorname{Var}}(\widehat{APE}) = (0.01303,0.1953)\widehat V \begin{pmatrix}0.01303\\0.1953\end{pmatrix} \approx 0.0008144, \qquad SE\approx\sqrt{0.0008144}=0.02854.

An approximate 95% confidence interval is \(0.1794\pm1.96(0.02854)\), or approximately \([0.1235,0.2354]\) after rounding the endpoints to four significant figures. The interval is for the average change in the conditional mean fraction per one-unit increase in \(x\), in this model and sample. Its frequentist interpretation is that intervals constructed this way would cover the target APE in approximately 95% of repeated samples under the relevant large-sample conditions; it does not assign a 95% probability to this particular interval containing a fixed parameter.

Common Pitfall and Practical Interpretation

A frequent mistake is to report the coefficient on a regressor as its marginal effect. For example, in a logit model a slope of 0.8 means that a one-unit increase changes the index by 0.8, not that the conditional mean fraction rises by 0.8. The outcome-scale effect is \(f(\eta_i)\beta_j\), which varies with the covariates. The correction is to calculate effects on the conditional-mean scale, specify whether they are evaluated at selected covariates or averaged over observations, and then use the covariance matrix and delta method for uncertainty.

For a binary regressor, a second pitfall is using a derivative as though the variable could change infinitesimally. Calculate the finite difference between settings zero and one instead. Also distinguish the APE, which averages individual effects, from an effect evaluated at mean covariates; in a nonlinear model these generally differ.

Report units explicitly. If the response is a fraction, an effect of 0.05 is a five-percentage-point change in the conditional mean, not a five percent relative change. If the predictor is measured in tens of hours, the effect is per ten hours. Keep that scaling visible in the interpretation and in any table of results.

Check Your Understanding

Answer each question by identifying the relevant scale and calculation.

In a fractional logit model with index \(\eta_i=x_i'\beta\), write the continuous marginal effect for a regressor entering the index linearly with coefficient \(\beta_j\).
Why is a discrete change, rather than a derivative, the standard effect measure for a binary regressor?
Explain the difference between an APE and a marginal effect evaluated at the sample mean of the covariates.
What two ingredients are needed, in addition to the estimated effect, to calculate its delta-method standard error?
A fitted effect is 0.04 for a response measured as a fraction. Interpret it in percentage points, and state why it should not automatically be called a four percent relative increase.