8. Additive Models and Marginal Effects

Additive models have been proven to be very useful as they naturally generalize the linear regression model and allow for an interpretation of marginal changes, i.e. for the effect of one variable on the mean function $m(\bullet)$ when holding all others constant. This kind of model structure is widely used in both theoretical economics and in econometric data analysis. The standard text of Deaton & Muellbauer (1980) provides many examples in microeconomics for which the additive structure provides interpretability. In econometrics, additive structures have a desirable statistical form and yield many well known economic results. For instance, an additive structure allows us to aggregate inputs into indices; elasticities or rates of substitutions can be derived directly. The separability of the input parameters is consistent with decentralization in decision making or optimization by stages. In summary, additive models can easily be interpreted.

Additive models are also interesting from a statistical point of view. They allow for a componentwise analysis and combine flexible nonparametric modeling of multidimensional inputs with a statistical precision that is typical of a one-dimensional explanatory variable. Let $Y$ be the dependent variable and ${\boldsymbol{X}}$ the $d$-dimensional vector of explanatory variables. Consider the estimation of the general nonparametric regression function $m( {\boldsymbol{X}}) = E( Y \vert{\boldsymbol{X}})$. Stone (1985) showed that the optimal convergence rate for estimating $m(\bullet)$ is $n^{-\kappa/(2\kappa +d)}$ with $\kappa$ an index of smoothness of $m(\bullet)$. Note how high values of $d$ lead to a slow rate of convergence. An additive structure for $m(\bullet)$ is a regression function of the form

$$m({\boldsymbol{X}})=c+\sum_{\alpha =1}^d g_\alpha (X_\alpha ),$$ (8.1)

where $g_\alpha(\bullet)$ are one-dimensional nonparametric functions operating on each element of the predictor variables. Stone (1985) also showed that for the additive regression function the optimal rate for estimating $m(\bullet)$ is the one-dimensional $n^{-\kappa/(2\kappa +1)}$-rate. One speaks thus of dimension reduction through additive modeling.

Additive models of the form (8.1) were first considered in the context of input-output analysis by Leontief (1947a) who called them additive separable models. In the statistical literature, additive regression models were introduced in the early eighties, and promoted largely by the work of Buja et al. (1989) and Hastie & Tibshirani (1990). They proposed the iterative backfitting procedure to estimate the additive components. This method is presented in detail in Section 8.1.

More recently, Tjøstheim & Auestad (1994b) and Linton & Nielsen (1995) introduced a non-iterative method for estimating marginal effects. Note that marginal effects coincide with the additive components $g_\alpha(\bullet)$ if the true regression function $m(\bullet)$ is indeed additive. The idea of this method is to first estimate a multidimensional functional of $m(\bullet)$ and then use marginal integration to obtain the marginal effects. Under additive structure the marginal integration estimator yields the functions $g_\alpha(\bullet)$ up to a constant. This estimator will be introduced starting from Section 8.2. A comparison of backfitting and marginal integration is given in Section 8.3.