Sharp increases in both the average level and the variability of inflation in the 1970s focused the attention of economists on the inflationary process. Outside the mainstream of academic research, a number of economists have proposed leading indicators of inflation. These indicators, it is hoped, will warn of impending significant changes in the rate of inflation. An underlying assumption is that inflation, like economic output, is cyclical, with peaks and troughs in the rate of inflation defining inflation cycles. The inflation indicators are designed to “predict” as accurately as possible past peaks and troughs in inflation. Thus, the development of leading indicators of inflation extends the indicator approach long associated with business cycle research to inflation forecasting.

This chapter evaluates five leading indicators of inflation. Three are composite indexes that use the methodology employed by the Department of Commerce in computing the composite indexes for the business cycle. Another is the growth rate of M1. The fifth is based on the ratio of capacity utilization to the foreign exchange value of the dollar. In general, the indicators show promise, but it is too early to embrace them wholeheartedly. The indicators are new, and while they “predict” past turning points in inflation quite well, there is no guarantee that they can warn of future turning points in inflation – the task for which they were developed.

This chapter is structured as follows. Section 16.1 describes the five leading indicators of inflation, section 16.2 assesses how useful the indicators may be in predicting inflation, and section 16.3 summarizes the study.

The basic idea of scatter plot smoothing can be extended to higher dimensions in a straightforward way. Theoretically, the regression smoothing for a d -dimensional predictor can be performed as in the case of a one-dimensional predictor. The local averaging procedure will still give asymptotically consistent approximations to the regression surface. However, there are two major problems with this approach to multiple regression smoothing. First, the regression function m(x) is a high dimensional surface and since its form cannot be displayed for d > 2, it does not provide a geometrical description of the regression relationship between X and Y. Second, the basic element of nonparametric smoothing - averaging over neighborhoods - will often be applied to a relatively meager set of points since even samples of size n ≥ 1000 are surprisingly sparsely distributed in the higher dimensional Euclidean space. The following two examples by Werner Stuetzle exhibit this “curse of dimensionality.”

A possible procedure for estimating two-dimensional surfaces could be to find the smallest rectangle with axis-parallel sides containing all the predictor vectors and to lay down a regular grid on this rectangle. This gives a total of one hundred cells if one cuts each side of a twodimensional rectangle into ten pieces. Each inner cell will have eight neighboring cells. If one carried out this procedure in ten dimensions there would be a total of 1010 = 10,000,000,000 cells and each inner cell would have 310 — 1 = 59048 neighboring cells. In other words, it will be hard to find neighboring observations in ten dimensions!

The problem of deciding how much to smooth is of great importance in nonparametric regression. Before embarking on technical solutions of the problem it is worth noting that a selection of the smoothing parameter is always related to a certain interpretation of the smooth. If the purpose of smoothing is to increase the “signal to noise ratio” for presentation, or to suggest a simple (parametric) models, then a slightly “oversmoothed” curve with a subjectively chosen smoothing parameter might be desirable. On the other hand, when the interest is purely in estimating the regression curve itself with an emphasis on local structures then a slightly “undersmoothed” curve may be appropriate.

However, a good automatically selected parameter is always a useful starting (view)point. An advantage of automatic selection of the band-width for kernel smoothers is that comparison between laboratories can be made on the basis of a standardized method. A further advantage of an automatic method lies in the application of additive models for investigation of high-dimensional regression data. For complex iterative procedures such as projection pursuit regression (Friedman and Stuetzle 1981) or ACE (Breiman and Friedman 1985) it is vital to have a good choice of smoothing parameter for one-dimensional smoothers that are elementary building blocks for these procedures.

Suppose that one observes data such as those in Figure 6.1: the main body of the data lies in a strip around zero and a few observations, governing the scaling of the scatter plot, lie apart from this region. These few data points are obviously outliers. This terminology does not mean that outliers are not part of the joint distribution of the data or that they contain no information for estimating the regression curve. It means rather that outliers look as if they are too small a fraction of the data to be allowed to dominate the small-sample behavior of the statistics to be calculated. Any smoother (based on local averages) applied to data like that in Figure 6.1 will exhibit a tendency to “follow the outlying observations.” Methods for handling data sets with outliers are called robust or resistant.

From a data-analytic viewpoint, a nonrobust behavior of the smoother is sometimes undesirable. Suppose that, a posteriori, a parametric model for the response curve is to be postulated. Any erratic behavior of the nonparametric pilot estimate will cause biased parametric formulations. Imagine, for example, a situation in which an outlier has not been identified and the nonparametric smoothing method has produced a slight peak in the neighborhood of that outlier. A parametric model which fitted that “nonexisting” peak would be too high-dimensional.

A regression curve describes a general relationship between an explanatory variable X and a response variable Y. Having observed X , the average value of Y is given by the regression function. It is of great interest to have some knowledge about this relation. The form of the regression function may tell us where higher Y -observations are to be expected for certain values of X or whether a special sort of dependence between the two variables is indicated. Interesting special features are, for instance, monotonicity or unimodality. Other characteristics include the location of zeros or the size of extrema. Also, quite often the regression curve itself is not the target of interest but rather derivatives of it or other functionals.

One is often interested not only in the curve itself but also in special qualitative characteristics of the smooth. The regression function may be constrained to simple shape characteristics, for example, and the smooth should preferably have the same qualitative characteristics. A quite common shape characteristic is a monotonic or unimodal relationship between the predictor variable and the response variable. This a priori knowledge about the qualitative form of the curve should be built into the estimation technique. Such qualitative features do not necessarily lead to better rates of convergence but help the experimenter in interpretation of the obtained curves.

In economic applications involving demand, supply and price, functions with prescribed shape (monotonicity, convexity, etc.) are common. Lipsey, Sparks and Steiner (1976, chapter 5) present a number of convex decreasing demand curves and convex increasing supply curves (in both cases, price as a function of quality). They also give an example for quantity demanded as a function of household income. A more complex procedure could be applied to the potato Engel curve in Figure 1.2. The nonparametric fit shows a partially increasing and a decreasing segment. This curve could be estimated by a unimodal regression technique.

For a pragmatic scientist the conclusion of Fisher (1922), to “confine ourselves to those forms that we know how to handle,” must have an irresistible attractive power. Indeed, we know that the nonparametric smoothing task is hard, especially in high dimensions. So why not come back to parametrics, at least partially? A parametric together with a nonparametric component may handle the model building even better than just the nonparametric or the parametric approach! In this chapter I present approaches from both views. The discussed models incorporate both parametric and nonparametric components and are therefore called semiparametric models.

Three topics are addressed. First, the estimation of parameters in a partial linear model. Second, the comparison of individual curves in a shape-invariant context. Third, a method is proposed to check the appropriateness of parametric regression curves by comparison with a nonparametric smoothing estimator.

If the smoothing parameter is chosen as a suitable function of the sample size n , all of the above smoothers converge to the true curve if the number of observations increases. Of course, the convergence of an estimator is not enough, as Kendall and Stuart in the above citation say. One is always interested in the extent of the uncertainty or at what speed the convergence actually happens. Kendall and Stuart (1979) aptly describe the procedure of assessing measures of accuracy for classical parametric statistics: The extent of the uncertainty is expressed in terms of the sampling variance of the estimator which usually tends to zero at the speed of the square root of the sample size n.

In contrast to this is the nonparametric smoothing situation: The variance alone does not fully quantify the convergence of curve estimators. There is also a bias present which is a typical situation in the context of smoothing techniques. This is the deeper reason why up to this chapter the precision has been measured in terms of pointwise mean squared error (MSE), the sum of variance and squared bias. The variance alone doesn't tell us the whole story if the estimator is biased.