1.6 TERMINOLOGY AND NOTATION ( Damodar N. Gujarati )

Before we proceed to a formal analysis of regression theory,let us dwell briefly on the matter of terminology and notation. In the literature the terms briefly on the matter of terminology and notation . In the literature the terms dependent variable and explanatory variable are described variously. A representative list is:

 

Althought it is matter of personal taste and tradition, in this text we will uses the dependent variable/explanatory variable or the more neutral, regressand and the regressor terminology.

If We are studying the dependence  of variable on only a single explanatory variable, such as that of consumption expenditure on real income,such a study is know as simple, or two-variable, regression analysis. However, if we are studying the dependence of one variable on more than on explanatory variable, as in the crop-yield , rainfall, temperature, sunshine, and fertilizer example, it is known as multiple-regression analysis. In the other word, in two-variable regression there is more than one explanatory variable.

The term random is a synonym for the term stochastic. As noted earlier, a random or stochastic variable is a variable that can take on any set of values, positive or negative,  with a given probability.9

Unless stated otherwise, the later Y will denote the dependent variable and X's ( X1, X2,...,Xk ) will denote the explanatory variables, Xk being the kth explanatory variable. The subscript  i or t will denote  the ith or the tth observation or value. Xki (or Xkt ) wiil denote the ith ( or tth ) observation on variable Xk. N( or T ) will denote the total number of observation or values in population, and n( or t ) the total number of observations in a sample.

As a matter of convention, the observation subscript i will be used for cross- sectional data (i.e., data collected at one point time  ) and the subscript t will used for time series data ( i.e.,data collected over a period of time).

The nature of cross-sectional and time series data, was well as the important topic of the nature and sources data for empirical analysis, is discussed in following section.

___________________________________

9See App.A for formal definition an further detail.

1.5. REGRESSION VERSUS CORRELATION ( Damodar N. Gujarati)

Closely related to but conceptually very much different from regression analysis is correlation analysis, where the primary objective  is to measure the strength or degree of linear association between two variables. The correlation  coefficient, which we shall study in detail in Chapter 3, measures this strength of ( linear ) association. For Example, we may be interested in finding the correlation (coefficient) between smoking and lung cancer,between scores on statistics and mathematics examination,between high school grades and college grades, and so on. In regression analysis, as already noted, we are not primarily interested in such a measure. Instead, we try to estimated or predict the average value of want to know whether we can predict the average score on statistics examination by knowing a student's score on mathematics examination.

Regression and correlation have same fundamental differences that are worth mentioning . In regression analysis there is an asymmetry in the way the dependent and explanatory variables are are treated. The dependent variable is assumed to be statistical ,random, or stochastic,that is , to have a probability distribution. The explanatory variables, on the other hand , are assumed to have fixed values ( in repeated sampling)7 which was made explicit in the definition of regression given in Section 1.2.Thus , in figure 1.2 we assumed the the variable age was fixed at given levels and height measurements were obtained at these levels. In Correlation analysis, on the other hand, we treat any (two) variable symmetrically; there is no distinction between the dependent and explanatory variable. After all, the correlation between scores on mathematics and  statistic  examinations is the same as that between scores on statistics and mathematics examination.
Moreover,both variables are assumed to be random. As we shall see, most of the correlation theory is based on the assumption of the randomness of variables, whereas most of the regression theory to be expounded in this book is conditional upon the assumption that the dependent variable is stochastic but the explanatory variables are fixed or nonstochastic.8 
_______________________________
7It is crucial to note the the explanatory variables may be intrinsically stochastic, but for the purpose of regression analysis we assume that their values are fixed in repeated sampling ( that is, X assumes the same values in various samples),thus rendering them in effect nonrandom or nonstochastic. But more on this in Chap.3, Sec.3.2
8In advanced treatment of economics, one can relax the assumption that the explanatory variables are nonstochastic ( see introduction to part II ).

1.4 REGRESSION VERSUS CAUSATION (Damodar N. Gujarati)

Although regression analysis deals with the dependence of one variable on other variable, it does not necessarily imply causation. the words of Kendall and Stuart,"A statistical relationship, however strong an however suggestive, can never establish causal connection: our ideas of causation must com from outside statistics, ultimately from theory or other."5
In the corp-yield example cited previously, there is no statistical reason to assume that rainfall does not depend on corp yield. The fact that we treat crop yield considerationas dependent on rainfall ( among other things ) is due to non statistical consideration: Common sense suggests that the relationship cannot be reversed, for we cannot control rainfall by vaying crop yield.
In all the example cited in Section1.2 the point to note  is that a statistical relationship in self cannot logically imply causation. To ascribe causality , one must appeal to a priori or theoretical consideration. thus,in the third example cited, one can invoke economic theory in saying that consumption expenditure depends on real income.6
_____________________
5 M.G.Kendall an A. Stuart, The Advanced Theory of Statistics, Charles Griffin Publishers. New York,1961, vol. 2. chap.26,p.279.
6But as we shall see in Chap.3, classical regression analysis is based on the assumption that the model used in the analysis is the correct model. Therefore, the direction of causality may be implicit in the model postulated

STATISTICAL VERSUS DETERMINISTIC RELATIONSHIPS (Damodar N. Gujarati )

1.3

THE MODERN INTERPRETATION OF REGRESSION

The modern interpretation of regression is , however, quite different, Broadly speaking, we may say

Regression analysis is concerned whit the study of the dependence of one variable , the dependent variable , on one or more other variable , the explanatory variable, with a view to estimating and / or predicting ( 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.

1. Reconsider Galton's law universal regression. Galton was interested in finding pot 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 father's height. In other words, our concern is with predicting the average height of sons knowing the height of their fatherts, To see how this can be done, consider Figure 1.1 , Wich is a scatter diagram, or scatter gram.

 This figure shows the distribution of height of son in a hypothetical population corresponding to given or fixed vaues 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 given value of father's height. the average show this clearly, the circled crosses in the figure indicate the average height average , of son corresponding to a given height of the father's.Connecting these average, 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 Figure1.2 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 given age are likely to have identical height. But height on the average increases with age ( of corse, up to a certain age ), which can be seen clearly if we draw aline ( the regression line ) though the circled point that represent the average height at the given ages. thus, knowing the age , we may by able to predict from the regression line the average height corresponding that age.
 

FIGURE 1.2

___________________

3. At this stage of the  development of the subject matter, we shall call this regression line simply line connecting the mean, or average , value  of the dependent variable ( son's height ) corresponding to given value of the explanatory variable ( father's height ). Note that this line has a positive slope is less than 1, which is in conformity with Galton's regression to mediocrity. ( Why ? )

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 helppfuk in estimating the marginal propensity to consume ( MPC ) , that is, average change in consumption exspenditure  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 mos 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 Philips curve relating change 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 rat. 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 price.

FIGURE  1.3

FIGURE1.4

 

 7. The marketing director of a company may want to know how the demand for company's product is related to,say ,advertising expenditure. Such study will be of considerable help in finding out elasticity of demand with respect to advertising expenditure, that is, the percent change in demand in response to ,say, a1 percent change in advertising budget.

This  Knowledge may be helpful in determining the " optimum" advertising budget.

8. Finally, an agronomist may be interested in studying the dependence of crop tied, say,of wheat, on temperature, rainfall, amount of sunshine, and fertilizer. Such a dependence analysis may enable the prediction  or forecasting of the average crop yield, given information about the explanatoy variable.

The reader can supply scores of such examples  of the dependence of one variable on one or more other variable. The techniques of regression analysis discussed in this text are specially designed to study such dependence among variables.

THE NATURE OF REGRESSION ANALYSIS

As Mentioned in the Itroduction,regession is main tool of econometrics, and in this chapter we consider very brefly the nature of this tool.

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 mover or " regress" toward the average height in the populations as a whole.1 In other words , the height of the children of unusually tall or unusually short parents tends to move toward thr average height of the population. Gaton's law universal regression was confirmed by his friend Karl Pearson. Who collected more than a thousand records of height of members of family groups.2 He Found that the average height of sons of agruop of tall fathers was less than their father's height an average height of sons of group of short  sons alike toward the avarage height of all men. In the words of Galton, this was "regression to mediocrity ".

__________________________

1. Francuis Gakton, " Family Likeness In Stature. "Proceedings of Royal Sociaty, London,vol.40 1886,pp.42-72

2.K.Pearson and A.Lee."onm The Laws of Inherittance," Biometrika, vol.2 , Nov 1903. pp.357-462

 

 

Investors sell off French bonds, push up Italian borrowing rates

By Anthony Faiola and Michael Birnbaum, Published: November 15

ROME — Investors were threatening to open a worrisome new chapter in Europe’s debt crisis Tuesday, selling off French and Spanish bonds and sending Italian borrowing rates back into a danger zone that had prompted international bailouts in Greece, Ireland and Portugal.

Concern centered on France, whose borrowing rates for euro-denominated debt spiked to near record levels. Though the nation also posted figures that showed a slight rise in its economy of 0.4 percent in third quarter, investors remained nervous about the exposure of French banks to the troubled debt of Greece, Italy and other more heavily indebted European countries.

Should Paris be forced into a heavy intervention to prop up its banking system, the price tag could cost France its cherished AAA debt rating. That rating would also be in jeopardy should Europe move to dramatically boost the size of its rescue fund to help Italy. The price tag for France — Europe’s second-largest economy — could shake its solvency.

“Fundamentally, it’s Germany and France that are underwriting the euro zone,” said Gavan Nolan, a research analyst at Markit Group in London, “but France is in a much weaker position than Germany, and rumors are rife that it will lose its AAA credit rating.”

In late afternoon trading, France’s CAC stock index had dropped 1.2 percent; Spain’s IBEX was down 1.05 percent; Europe’s Stoxx was down 0.59 percent, and Germany’s DAX up 0.16 percent.

At the same time, in Italy, the honeymoon for Mario Monti, an economist who became the country’s premier designate on Sunday, was quickly ending. Monti was holding intense meetings with Italy’s notoriously divided political parties on Tuesday to win backing for a new cabinet. But after an initial recovery in Italian bonds, investors again drove Italy’s borrowing rates above the unsustainable 7 percent mark.

Monti on Tuesday was winning key support from the People of Liberty Party of Silvio Berlusconi, Italy’s longtime playboy prime minister who stepped down on Saturday. With that backing, Monti he could win backing for a new government of technical experts later this week.

But even if he does quickly establish a working government, Monti would still need support in the coming weeks to force through a package of politically unpopular economic reforms aimed at jump-starting Italy’s moribund economy.

“Clearly there is a market deterioration in Europe overall,” said Tito Boeri, an economist at Bocconi University in Milan. “Now, they are testing France, too. But in Italy, the task ahead is very difficult. Today, Monti managed to get more political support. But the key issue is still making sure the reforms can go through.”

Spain’s borrowing costs also spiked on Tuesday ahead of elections on Sunday that could speed economic reforms there. Spain did not raise its maximum target in an auction of 12-month bonds.

Those bonds yielded 5 percent on average, far higher than the 3.6 percent levels a month ago. Although countries can withstand a period of higher borrowing costs, trouble can come quickly if they have difficulty lining up any lenders. Spain will try to sell 10-year bonds on Thursday and may have to borrow at rates close to record highs in the euro era.

The leader that Spain chooses on Sunday will likely sweep in new reforms, analysts said, especially if the conservative People’s Party wins the absolute majority that polls predict. Unlike Greece and Italy’s new leaders, the Spanish government will have a popular mandate for its changes.

“In Spain you don’t have a sense that markets have twisted your arm,” despite the rising borrowing costs, said Jose Ignacio Torreblanca, who runs the Madrid office of the European Council on Foreign Relations.

In Germany on Tuesday, Chancellor Angela Merkel’s Christian Democrats were ending a party conference in which they agreed to try to make it easier for countries to leave the euro, sending further currents of worry through the markets.

Until earlier this month, euro zone leaders had treated the currency as sacrosanct. That ended when Merkel and French President Nicolas Sarkozy suggested that Greece would leave the euro if it rejected a European bailout plan. At the conference, Merkel’s party approved a goal to allow countries to leave the currency voluntarily without leaving the European Union.

In a speech, German finance minister Wolfgang Schaeuble warned of the potential consequences if the debt problems keep escalating.

“If Italy were to be in the same camp as Greece, we wouldn’t be able to hold on to the euro,” Schaeuble said. But he added that he did not think Italy would need to resort to Europe’s bailout fund.

Staff writer Michael Birnbaum reported from Berlin. Special correspondent Karla Adam contributed to this report from London.