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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 determined by the coin and the way it is tossed. As a model of this, we imagine a symmetric coin is tossed fairly such that there is an equal chance of the outcome being heads or tails, so the probability of each is 1/2. It is convenient to think in terms of a random variable X describing the outcome of this coin-flipping experiment, so X takes values 0 and 1 if the outcome is tails and heads, respectively. The distribution of the outcomes can be described in terms of an underlying probability measure, P say, giving rise to the imagined population frequencies:

f (0) = P (X = 0) = 1/2 and f (1) = P (X = 1) = 1/2.

Here f appears without a "hat" as it is a population quantity, and P(X = 0) is read as "the probability of the event X = 0". We think of the frequency f as related to the random variable X, although that aspect is suppressed in the notation. In contrast, the probability measure P is more generic. We could introduce a new random variable Y = 1 - X, which takes the value 0 and 1 for heads and tails, rather than tails and heads, and write P(Y = 1) = P(X = 0) = 1/2.

We can sample from this population distribution as many times as we want. If, for instance, we toss the coin n = 27 times, we may observe 12 heads, so the sample frequency of heads is [??] (1) = 12/27. In fact, when sampling an odd number of times, we can never observe that_f(1) = f(1) = 1/2. One important difference between f(x) and [??] (x) is that f is a deterministic function describing the distribution of possible outcomes for a random variable X, whereas [??] is a random function describing the observed frequency of the outcomes in a sample of random variables X₁, ..., X_n; another sample of tosses would lead to different values of [??], but not f.

1.2 DISTRIBUTION FUNCTIONS AND DENSITIES

We need a structured way of thinking about distributions in order to build appropriate models. From probability theory, we can use the concepts of distribution functions and densities.

1.2.1 Distribution functions and random variables

Distribution theory is centered around cumulative distribution functions or just distribution functions. This is the function F(x) = P(X ≤ x), which is well defined regardless of the type of random variable. For the coin example, the distribution function is plotted in panel (a) of Figure 1.1. It has the defining property of any distribution function: it starts at zero on the far left and increases toward unity, reaching one at the far right. The probability is zero of observing an outcome less than zero, jumps to 1/2 at 0 (tails), then to unity at 1 (heads), and stays there.

If we consider the inverse of the distribution function, we get the quantiles of a distribution. The 50% quantile, also called the median, is the smallest value of x such that P(X ≤ x) = 0.5. For the coin-flipping example considered in Figure 1.1, the median is 0.

When dealing with two random variables X and Y that could, for instance, describe the outcomes of two coin tosses, we have the joint distribution function:

F(x, y) = P(X ≤ x and Y ≤ y).

We get the marginal distribution function of Y by allowing X to take any value:

F (y) = P (Y ≤ y) = P (X < ∞ and Y ≤ y).

For example, when X and Y refer respectively to whether the mother is young/old and the child is boy/girl, then the marginal distribution of the sex of the child is so called because it refers to the distribution in the margin of the 2 x 2 table of possible outcomes irrespective of the mother's age: see Table 4.2 below.

If we flip a coin twice and let X and Y describe the two outcomes, we do not expect any influence between the two outcomes, and hence obtain:

P(X ≤ x and Y ≤ y) = P(X ≤ x)P(Y ≤ y). (1.2.1)

If so, we say that the variables X and Y are independent. More generally, the variables X₁, ..., X_n are said to be independent if their joint distribution function equals the product of the marginal distribution functions:

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For example, the probability of 3 heads in a row is 1/2 x 1/2 x 1/2 = 1/8.

1.2.2 Density functions

In an econometric analysis, it is often convenient to consider the rate of increase of the distribution function rather than the distribution function itself.

For the birth and the coin-flipping experiments, the jumps of the distribution function determine the distribution uniquely. This is generally the case for any distribution function that is piecewise constant and therefore associated with a discrete distribution. The jumps are the probability mass function or the density:

f(x) = P (X = x).

Since the distribution function is piecewise constant, we can find the size of the jump at a point x as the value of the distribution function at x minus the value immediately before x, which we could write as:

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As an example, compare the distribution function and the corresponding density shown in Figure 1.1. Here it is seen that the density is 0.5 when x = 0 or x = 1 and otherwise 0.

We can recover the distribution function from the density by summation:

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In particular, the sum over all possible outcomes is always unity.

We can also work with joint densities. It follows from (1.2.1) that two discrete random variables X, Y are independent if and only if:

f (x, y) = P(X = x and Y = y) = P(X = x) P(Y = y) = f (x) f (y), (1.2.2)

whereas the marginal density of X is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.2.3)

where the sum is taken over all possible outcomes for Y.

1.2.3 A discrete distribution: the Bernoulli distribution

If a variable X takes the values 0 and 1, as with the variable for sex, it is said to be binary or dichotomous. It then has what is called a Bernoulli. The probability of a unit outcome:

θ = P (X = 1)

is the success probability. The parameter θ takes a value in the range [0, 1]. It follows that the probability of a failure is P (X = 0) = 1 - θ. In short, we write X [??] Bernoulli[θ] to indicate that X is Bernoulli-distributed with parameter θ. Figure 1.1 shows the distribution function and the density for a Bernoulli distribution with success parameter θ = 0.5.

The density for the Bernoulli distribution can be written in a compact form:

f (x) = θ^x (1 - θ^1-x for x = 0, 1. (1.2.4)

In (1.2.4), it holds that P(X = 0) = f(0) = (1 - θ) and P(X = 1) = f(1) = θ.

1.3 THE BERNOULLI MODEL

We are now ready to develop our first statistical model. Using the above distribution theory, a statistical model and its associated likelihood function can be defined for the birth data. The likelihood function can then be used to find the specific member of the statistical model that is most likely to have generated the observed data.

1.3.1 A statistical model

Reconsider the birth data summarized in Table 1.2, where we are interested in learning about the population frequency of girls among newborn children. To do this, we will build a simple statistical model for the sex of newborn children. The objective is to make a good description of the sample distribution, which will eventually allow us to make inferences about, in this case, the frequency of girl births in the population of possible births. Here we will concentrate on describing the distribution with a view toward checking any assumptions we make.

Let Y_i denote the sex for child i, and consider n random variables Y₁, ..., Y_n representing the data. We will make four assumptions:

(i) independence: Y₁, ..., Y_n are mutually independent;

(ii) identical distribution: all children are drawn from the same population;

(iii) Bernoulli distribution: Y_i [??] Bernoulli[θ];

(iv) parameter space: 0 < θ < 1, which we write as θ ε &PHI; = (0, 1).

We need to think about whether these assumptions are reasonable. Could they be so wrong that all inferences we draw from the model are misleading? In that case, we say the model is mis-specified. For example, the assumptions of independence or an identical distribution could well be wrong. In cases of identical twins, the independence assumption (i) is indeed not correct. Perhaps young and old mothers could have different chances of giving birth to girls, which could be seen as a violation of (ii). Is that something to worry about? In this situation, no more data are available, so we either have to stick to speculative arguments, turn to an expert in the field, or find more detailed data. We will proceed on the basis that any violations are not so large as to seriously distort our conclusions. Assumption (iii), however, is not in question in this model as the Bernoulli distribution is the only available distribution for binary data. Assumption (iv) is also not problematic here, even though the parameter space is actually restrictive in that it is chosen as 0 < θ < 1 as opposed to 0 ≤ θ ≤ 1. The resolution is that since we have observed both girls and boys, it is not possible that θ = 0 or θ = 1. These two points can therefore be excluded.

1.3.2 The likelihood function

Based on the statistical model, we can analyze how probable different outcomes y₁, ..., y_n of Y₁, ..., Y_n are for any given choice of the parameter θ. This is done by writing down the joint density of Y₁, ..., Y_n. Using the notation f_θ (y₁, ..., y_n) for the joint density and the rules from §1.2.2, we get:

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This expression can be reduced further using the fact that:

θ^aθ^b = θ^a+b, (1.3.1)

which is the functional equation for power functions. Then (1.3.1) implies that:

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(Continues...)

Excerpted from Econometric Modelingby David F. Hendry Bent Nielsen Copyright © 2007 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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