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## Inhaltsverzeichnis

### 1. Introduction

Abstract
Consider a continuous random variable and its probability density function (pdf). The pdf tells you “how the random variable is distributed”. From the pdf you cannot only calculate the statistical characteristics as mean and variance, but also the probability that this variable will take on values in a certain interval.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 2. Histogram

Abstract
Let X be a continuous random variable and f its probability density function (pdf). The pdf tells you “how X is distributed”. From the pdf you can calculate the mean and variance of X (if they exist) and the probability that X will take on values in a certain interval. The pdf is, thus, very useful to characterize the distribution of the random variable X.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 3. Nonparametric Density Estimation

Abstract
Contrary to the treatment of the histogram in statistics textbooks we have shown that the histogram is more than just a convenient tool for giving a graphical representation of an empirical frequency distribution. It is a serious and widely used method for estimating an unknown pdf. Yet, the histogram has some shortcomings and hopefully this chapter will persuade you that the method of kernel density estimation is in many respects preferable to the histogram.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 4. Nonparametric Regression

Abstract
An important question in many fields of science is the relationship between two variables, say X and Y. Regression analysis is concerned with the question of how Y (the dependent variable) can be explained by X (the independent or explanatory or regressor variable). This means a relation of the form
$$Y = m(X)$$
, where m(●) is a function in the mathematical sense. In many cases theory does not put any restrictions on the form of m(●), i.e. theory does not say whether m(●) is linear, quadratic, increasing in X , etc.. Hence, it is up to empirical analysis to use data to find out more about m(●).
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 5. Semiparametric and Generalized Regression Models

Abstract
In the previous part of this book we found the curse of dimensionality to be one of the major problems that arises when using nonparametric multivariate regression techniques. For the practitioner, a further problem is that for more than two regressors, graphical illustration or interpretation of the results is hardly ever possible. Truly multivariate regression models are often far too flexible and general for making detailed inference.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 6. Single Index Models

Abstract
A single index model (SIM) summarizes the effects of the explanatory variables X1, ..., Xd within a single variable called the index. As stated at the beginning of Part II, the SIM is one possibility for generalizing the GLM or for restricting the multidimensional regression E(Y|X) to overcome the curse of dimensionality and the lack of interpretability. For more examples of motivating the SIM see Ichimura (1993). Among others, this reference mentions duration, truncated regression (Tobit) and errors-in-variables modeling.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 7. Generalized Partial Linear Models

Abstract
As indicated in the overview in Chapter 5, a partial linear model (PLM) consists of two additive components, a linear and a nonparametric part:
$$E(Y|U,T) = U^ \top \beta + m(T)$$
where β=(β1,...,β p )is a finite dimensional parameter and m(●) a smooth function. Here, we assume again a decomposition of the explanatory variables into two vectors, U and T. The vector U denotes a p-variate random vector which typically covers categorical explanatory variables or variables that are known to influence the index in a linear way. The vector T is a q-variate random vector of continuous explanatory variables which is to be modeled in a nonparametric way. Economic theory or intuition should guide you as to which regressors should be included in U or T, respectively.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

### 8. Additive Models and Marginal Effects

Abstract
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(●) 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.
Wolfgang Härdle, Axel Werwatz, Marlene Müller, Stefan Sperlich

$$E(Y|X) = c + \sum\limits_{\alpha = 1}^d {g_\alpha (x_\alpha )} .$$