Introduction ② The McG econometrics. Fourth Chapter 5 is devoted to the topic of hypothesis testing. In this chapter, we ry to find out whether the estimated regression coefficients are compatible with the hypothesized values of such coefficients, the hypothesized values being suggested by theory and/or prior empirical work Chapter 6 considers some extensions of the two-variable regression model. In particular, it discusses topics such as(1) regression through the origin, (2)scaling and units of measurement, and (3) functional forms of regression models such as double- log, semilog, and reciprocal models. In Chapter 7, we consider the multiple regression model, a model in which there is more than one explanatory variable, and show how the method of OLS can be extended to estimate the parameters of such models. In Chapter 8, we extend the concepts introduced in Chapter 5 to the multiple regression model and point out some of the complications arising from the introduction of several explanatory variables Chapter 9 on dummy, or qualitative, explanatory variables concludes Part I of the text. This chapter emphasizes that not all explanatory variables need to be quantitative(i.e, ratio scale). Variables, such as gender, race, re ligion, nationality, and region of residence, cannot be readily quantified, yet they play a valuable role in explaining many an economic phenomenon
Gujarati: Basic Econometrics, Fourth Edition I. Single−Equation Regression Models Introduction © The McGraw−Hill Companies, 2004 16 Chapter 5 is devoted to the topic of hypothesis testing. In this chapter, we try to find out whether the estimated regression coefficients are compatible with the hypothesized values of such coefficients, the hypothesized values being suggested by theory and/or prior empirical work. Chapter 6 considers some extensions of the two-variable regression model. In particular, it discusses topics such as (1) regression through the origin, (2) scaling and units of measurement, and (3) functional forms of regression models such as double-log, semilog, and reciprocal models. In Chapter 7, we consider the multiple regression model, a model in which there is more than one explanatory variable, and show how the method of OLS can be extended to estimate the parameters of such models. In Chapter 8, we extend the concepts introduced in Chapter 5 to the multiple regression model and point out some of the complications arising from the introduction of several explanatory variables. Chapter 9 on dummy, or qualitative, explanatory variables concludes Part I of the text. This chapter emphasizes that not all explanatory variables need to be quantitative (i.e., ratio scale). Variables, such as gender, race, religion, nationality, and region of residence, cannot be readily quantified, yet they play a valuable role in explaining many an economic phenomenon
② The McG econometrics. Fourth Regression Analysis THE NATURE OF REGRESSION ANALYSIS As mentioned in the Introduction, regression is a main tool of econometrics and in this chapter we consider very briefly the nature of this tool 1.1 HISTORICAL ORIGIN OF THE TERM REGRESSION The term regression was introduced by Francis Galton. In a famous paper, Galton found that, although there was a tendency for tall parents to have tall children and for short parents to have short children, the average height f children born of parents of a given height tended to move or"regress"to- ward the average height in the population as a whole. In other words,the height of the children of unusually tall or unusually short parents tends to move toward the average height of the population. Galtons law of universal regression was confirmed by his friend Karl Pearson, who collected more than a thousand records of heights of members of family groups. He found that the average height of sons of a group of tall father rs was le ss than their fathers height and the average height of sons of a group of short fathers was greater than their fathers height, thus"regressing tall and short sons alike toward the average height of all men In the words of Galton, this was Francis Galton, "Family Likeness in Stature, "Proceedings of Royal Society, London, vol 40 1886,pp.42-72 K. Pearson and A. Lee. "On the Laws of Inheritance. "Biometrika, vol. 2. Now. 1903
Gujarati: Basic Econometrics, Fourth Edition I. Single−Equation Regression Models 1. The Nature of Regression Analysis © The McGraw−Hill Companies, 2004 17 1Francis Galton, “Family Likeness in Stature,” Proceedings of Royal Society, London, vol. 40, 1886, pp. 42–72. 2K. Pearson and A. Lee, “On the Laws of Inheritance,’’ Biometrika, vol. 2, Nov. 1903, pp. 357–462. 1 THE NATURE OF REGRESSION ANALYSIS As mentioned in the Introduction, regression is a main tool of econometrics, and in this chapter we consider very briefly the nature of this tool. 1.1 HISTORICAL ORIGIN OF THE TERM REGRESSION The term regression was introduced by Francis Galton. In a famous paper, Galton found that, although there was a tendency for tall parents to have tall children and for short parents to have short children, the average height of children born of parents of a given height tended to move or “regress” toward the average height in the population as a whole.1 In other words, the height of the children of unusually tall or unusually short parents tends to move toward the average height of the population. Galton’s law of universal regression was confirmed by his friend Karl Pearson, who collected more than a thousand records of heights of members of family groups.2 He found that the average height of sons of a group of tall fathers was less than their fathers’ height and the average height of sons of a group of short fathers was greater than their fathers’ height, thus “regressing” tall and short sons alike toward the average height of all men. In the words of Galton, this was “regression to mediocrity
② The McG econometrics. Fourth Regression Analysis 18 PART ONE: SINGLE-EQUATION REGRESSION MODELS 1.2 THE MODERN INTERPRETATION OF REGRESSION The modern interpretation of regression is, however, quite different Broadly spe Regression analysis is concerned with the study of the dependence of one vari- able, the dependent variable, on one or more other variables, the explanatory vari ables, with a view to estimating and/or predicting the (population) mean or aver- age value of the former in terms of the known or fixed (in repeated sampling) values of the latter. The full import of this view of regression analysis will become clearer as we progress, but a few simple examples will make the basic concept quite clear Examples 1. Reconsider Galtons law of universal regression Galton was in ested in finding out why there was a stability in the distribution of heights in a population. But in the modern view our concern is not with this expla- nation but rather with finding out how the average height of sons changes given the fathers height. In other words, our concern is with predicting the average height of sons knowing the height of their fathers. To see how this can be done, consider Figure 1.1, which is a scatter diagram, or scatter 75F Mean value xxxx8 FIGURE 1.1 Hypothetical distribution of sons heights corresponding to given heights of fathers
Gujarati: Basic Econometrics, Fourth Edition I. Single−Equation Regression Models 1. The Nature of Regression Analysis © The McGraw−Hill Companies, 2004 18 PART ONE: SINGLE-EQUATION REGRESSION MODELS Son's height, inches Father's height, inches × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × 75 70 65 60 60 65 70 75 × × × × × Mean value FIGURE 1.1 Hypothetical distribution of sons’ heights corresponding to given heights of fathers. 1.2 THE MODERN INTERPRETATION OF REGRESSION The modern interpretation of regression is, however, quite different. Broadly speaking, we may say Regression analysis is concerned with the study of the dependence of one variable, the dependent variable, on one or more other variables, the explanatory variables, with a view to estimating and/or predicting the (population) mean or average value of the former in terms of the known or fixed (in repeated sampling) values of the latter. The full import of this view of regression analysis will become clearer as we progress, but a few simple examples will make the basic concept quite clear. Examples 1. Reconsider Galton’s law of universal regression. Galton was interested in finding out why there was a stability in the distribution of heights in a population. But in the modern view our concern is not with this explanation but rather with finding out how the average height of sons changes, given the fathers’ height. In other words, our concern is with predicting the average height of sons knowing the height of their fathers. To see how this can be done, consider Figure 1.1, which is a scatter diagram, or scatter-
1. The Nature of ② The McG econometrics. Fourth Regression Analysis CHAPTER ONE: THE NATURE OF REGRESSION ANALYSIS 19 gram. This figure shows the distribution of heights of sons in a hypothetical population corresponding to the given or fixed values of the father's height Notice that corresponding to any given height of a father is a range or dis tribution of the heights of the sons. However, notice that despite the vari ability of the height of sons for a given value of father's height, the average height of sons generally increases as the height of the father increases. To show this clearly, the circled crosses in the figure indicate the average height of sons corresponding to a given height of the father. Connecting these averages, we obtain the line shown in the figure. This line as we shall see, is known as the regression line. It shows how the average height of sons increases with the fathers height. 2. Consider the scattergram in Figure 1. 2, which gives the distribution in a hypothetical population of heights of boys measured at fixed ages Corresponding to any given age, we have a range, or distribution, of heights Obviously, not all boys of a given age are likely to have identical heights But height on the average increases with age(of course, up to a certain age), which can be seen clearly if we draw a line(the regression line) through the 70区 Mean value FIGURE 1.2 Hypothetical distribution of heights corresponding to selected ages. At this stage of the development of the subject matter, we shall call this regressi ply the line connecting the or average, value of the dependent variable(sons sponding to the given value of the explanatory variable (father's height ). Note that thi positive slope but the slope is less than 1, which is in conformity with Galtons regression to mediocrity.(Why?)
Gujarati: Basic Econometrics, Fourth Edition I. Single−Equation Regression Models 1. The Nature of Regression Analysis © The McGraw−Hill Companies, 2004 CHAPTER ONE: THE NATURE OF REGRESSION ANALYSIS 19 Height, inches 40 50 60 70 Age, years 10 11 12 13 14 Mean value FIGURE 1.2 Hypothetical distribution of heights corresponding to selected ages. 3At this stage of the development of the subject matter, we shall call this regression line simply the line connecting the mean, or average, value of the dependent variable (son’s height) corresponding to the given value of the explanatory variable (father’s height). Note that this line has a positive slope but the slope is less than 1, which is in conformity with Galton’s regression to mediocrity. (Why?) gram. This figure shows the distribution of heights of sons in a hypothetical population corresponding to the given or fixed values of the father’s height. Notice that corresponding to any given height of a father is a range or distribution of the heights of the sons. However, notice that despite the variability of the height of sons for a given value of father’s height, the average height of sons generally increases as the height of the father increases. To show this clearly, the circled crosses in the figure indicate the average height of sons corresponding to a given height of the father. Connecting these averages, we obtain the line shown in the figure. This line, as we shall see, is known as the regression line. It shows how the average height of sons increases with the father’s height.3 2. Consider the scattergram in Figure 1.2, which gives the distribution in a hypothetical population of heights of boys measured at fixed ages. Corresponding to any given age, we have a range, or distribution, of heights. Obviously, not all boys of a given age are likely to have identical heights. But height on the average increases with age (of course, up to a certain age), which can be seen clearly if we draw a line (the regression line) through the
1. The Nature of ② The McG Econometrics. Fourth Regression Analysis 20 PART ONE: SINGLE-EQUATION REGRESSION MODELS circled points that represent the average height at the given ages. Thus, knowing the age, we may be able to predict from the regression line the average height corresponding to that age. 3. Turning to economic examples, an economist may be interested in studying the dependence of personal consumption expenditure on after- tax or disposable real personal income. Such an analysis may be helpful in estimating the marginal propensity to consume(MPC), that is, average change in consumption expenditure for, say, a dollars worth of change in real income(see Figure 1.3) 4. A monopolist who can fix the price or output(but not both) may want to find out the response of the demand for a product to changes in price Such an experiment may enable the estimation of the price elasticity (i.e price responsiveness)of the demand for the product and may help deter mine the most profitable price. 5. A labor economist may want to study the rate of change of money wages in relation to the unemployment rate. The historical data are shown in the scattergram given in Figure 1. 3. The curve in Figure 1.3 is an example of the celebrated Phillips curve relating changes in the money wages to the unemployment rate. Such a scattergram may enable the labor economist to predict the average change in money wages given a certain unemployment rate. Such knowledge may be helpful in stating something about the infla tionary process in an economy, for increases in money wages are likely to be reflected in increased pric Unemployment rate, FIGURE 1.3 Hypothetical Phillips curve
Gujarati: Basic Econometrics, Fourth Edition I. Single−Equation Regression Models 1. The Nature of Regression Analysis © The McGraw−Hill Companies, 2004 20 PART ONE: SINGLE-EQUATION REGRESSION MODELS circled points that represent the average height at the given ages. Thus, knowing the age, we may be able to predict from the regression line the average height corresponding to that age. 3. Turning to economic examples, an economist may be interested in studying the dependence of personal consumption expenditure on aftertax or disposable real personal income. Such an analysis may be helpful in estimating the marginal propensity to consume (MPC), that is, average change in consumption expenditure for, say, a dollar’s worth of change in real income (see Figure I.3). 4. A monopolist who can fix the price or output (but not both) may want to find out the response of the demand for a product to changes in price. Such an experiment may enable the estimation of the price elasticity (i.e., price responsiveness) of the demand for the product and may help determine the most profitable price. 5. A labor economist may want to study the rate of change of money wages in relation to the unemployment rate. The historical data are shown in the scattergram given in Figure 1.3. The curve in Figure 1.3 is an example of the celebrated Phillips curve relating changes in the money wages to the unemployment rate. Such a scattergram may enable the labor economist to predict the average change in money wages given a certain unemployment rate. Such knowledge may be helpful in stating something about the inflationary process in an economy, for increases in money wages are likely to be reflected in increased prices. Unemployment rate, % Rate of change of money wages 0 – + FIGURE 1.3 Hypothetical Phillips curve