Über die Autorin bzw. den Autor

David F. Hendry is Professor of Economics at the University of Oxford and a Fellow of Nuffield College. Bent Nielsen is Reader in Econometrics at the University of Oxford and a Fellow of Nuffield College

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"Hendry and Nielsen's Econometric Modeling is a well-thought-out alternative to other introductory econometric textbooks. I especially like the decision to treat time-series and cross-section analysis simultaneously, since the dichotomy between them, which arises in most other texts, is artificial."--Douglas Steigerwald, University of California, Santa Barbara

"This textbook is concise, up-to-date, and largely self-contained. The models it presents are just complicated enough to set out the main econometric ideas."--Marius Ooms, Free University, Amsterdam

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Econometric Modeling

PRINCETON UNIVERSITY PRESS

Contents

Preface......................................................................ixData and software............................................................xiChapter 1. The Bernoulli model...............................................1Chapter 2. Inference in the Bernoulli model..................................14Chapter 3. A first regression model..........................................28Chapter 4. The logit model...................................................47Chapter 5. The two-variable regression model.................................66Chapter 6. The matrix algebra of two-variable regression.....................88Chapter 7. The multiple regression model.....................................98Chapter 8. The matrix algebra of multiple regression.........................121Chapter 9. Mis-specification analysis in cross sections......................127Chapter 10. Strong exogeneity................................................140Chapter 11. Empirical models and modeling....................................154Chapter 12. Autoregressions and stationarity.................................175Chapter 13. Mis-specification analysis in time series........................190Chapter 14. The vector autoregressive model..................................203Chapter 15. Identification of structural models..............................217Chapter 16. Non-stationary time series.......................................240Chapter 17. Cointegration....................................................254Chapter 18. Monte Carlo simulation experiments...............................270Chapter 19. Automatic model selection........................................286Chapter 20. Structural breaks................................................302Chapter 21. Forecasting......................................................323Chapter 22. The way ahead....................................................342References...................................................................345Author index.................................................................357Subject index................................................................359

Chapter One

In this chapter and in Chapter 2, we will consider a data set recording the number of newborn girls and boys in the UK in 2004 and investigate whether the distribution of the sexes is even among newborn children. This question could be of interest to an economist thinking about the wider issue of incentives facing parents who are expecting a baby. Sometimes the incentives are so strong that parents take actions that actually change basic statistics like the sex ratio.

When analyzing such a question using econometrics, an important and basic distinction is between sample and population distributions. In short, the sample distribution describes the variation in a particular data set, whereas we imagine that the data are sampled from some population about which we would like to learn. This first chapter describes that distinction in more detail. Building on that basis, we formulate a model using a class of possible population distributions. The population distribution within this class, which is the one most likely to have generated the data, can then be found. In Chapter 2, we can then proceed to question whether the distribution of the sexes is indeed even.

1.1 SAMPLE AND POPULATION DISTRIBUTIONS

We start by looking at a simple demographic data set showing the number of newborn girls and boys in the UK in 2004. This allows us to consider the question whether the chance that a newborn child is a girl is 50%. By examining the frequency of the two different outcomes, we obtain a sample distribution. Subsequently, we will turn to the general population of newborn children from which the data set has been sampled, and establish the notion of a population distribution. The econometric tools will be developed with a view toward learning about this population distribution from a sample distribution.

1.1.1 Sample distributions

In 2004, the number of newborn children in the UK was 715996, see Office for National Statistics (2006). Of these, 367586 were boys and 348410 were girls. These data have come about by observing n = 715996 newborn children. This gives us a cross-sectional data set as illustrated in Table 1.1. The name cross-section data refers to its origins in surveys that sought to interview a cross section of society. In a convenient notation, we let i = 1, ..., n be the child index, and for each child we introduce a random variable Y_i, which can take the numerical value 0 or 1 representing "boy" or "girl," respectively. While the data set shows a particular set of outcomes, or observations, of the random variables Y₁, ..., Y_n, the econometric analysis will be based on a model for the possible variation in the random variables Y₁, ..., Y_n. As in this example, random variables always take numerical values.

To obtain an overview of a data set like that reported Table 1.1, the number of cases in each category would be counted, giving a summary as in Table 1.2. This reduction of the data, of course, corresponds to the actual data obtained from the Office of National Statistics.

The magnitudes of the numbers in the cells in Table 1.2 depend on the numbers born in 2004. We can standardize by dividing each entry by the total number of newborn children, with the result shown in Table 1.3. Each cell of Table 1.3 then shows:

[??] (y) = "frequency of sex y among n = 715996 newborn children."

We say that Table 1.3 gives the frequency distribution of the random variables Y₁, ..., Y_n. There are two aspects of the notation [??] (y) that need explanation. First, the argument y of the function [??] represents the potential outcomes of child births, as opposed to the realization of a particular birth. Second, the function [??], said as "f-hat", is an observed, or sample, quantity, in that it is computed from the observations Y₁, ..., Y_n. The notation [??], rather than f, is used to emphasize the sample aspect, in contrast to the population quantities we will discuss later on.

The variables Y₁, ..., Y_n (denoted Y_i in shorthand) take the values 0 or 1. That is, Y_i takes J = 2 distinct values for j = 1, ..., J. Thus, the sum of the cell values in Table 1.3 is unity:

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1.1.2 Population distributions

We will think of a sample distribution as a random realization from a population distribution. In the above example, the sample is all newborn children in the UK in 2004, whereas the population distribution is thought of as representing the biological causal mechanism that determines the sex of children. Thus, although the sample here is actually the population of all newborn children in the UK in 2004, the population from which that sample is drawn is a hypothetical one.

The notion of a population distribution can be made a little more concrete with a coin-flipping example. The outcome of a coin toss is...