The Significance Of The Stochastic Disturbance Term

As noted in Section 2.4, the disturbance term ui is a surrogate for all those variables that are omitted from the model but that collectively affect Y. The obvious question is: Why not introduce these variables into the model explicitly? Stated otherwise, why not develop a multiple regression model with as many variables as possible? The reasons are many.

1. Vagueness of theory: The theory, if any, determining the behavior of Y may be, and often is, incomplete. We might know for certain that weekly income X influences weekly consumption expenditure Y, but we might be ignorant or unsure about the other variables affecting Y. Therefore, ui may be used as a substitute for all the excluded or omitted variables from the model.

2. Unavailability of data: Even if we know what some of the excluded variables are and therefore consider a multiple regression rather than a simple regression, we may not have quantitative information about these

8See App. A for a brief discussion of the properties of the expectation operator E. Note that E(Y I Xi), once the value of Xi is fixed, is a constant.

9As a matter of fact, in the method of least squares to be developed in Chap. 3, it is assumed explicitly that E(m I Xi) = 0. See Sec. 3.2.

46 PART ONE: SINGLE-EQUATION REGRESSION MODELS

variables. It is a common experience in empirical analysis that the data we would ideally like to have often are not available. For example, in principle we could introduce family wealth as an explanatory variable in addition to the income variable to explain family consumption expenditure. But unfortunately, information on family wealth generally is not available. Therefore, we may be forced to omit the wealth variable from our model despite its great theoretical relevance in explaining consumption expenditure.

3. Core variables versus peripheral variables: Assume in our consumption-income example that besides income X1, the number of children per family X2, sex X3, religion X4, education X5, and geographical region X6 also affect consumption expenditure. But it is quite possible that the joint influence of all or some of these variables may be so small and at best nonsystematic or random that as a practical matter and for cost considerations it does not pay to introduce them into the model explicitly. One hopes that their combined effect can be treated as a random variable ui .10

4. Intrinsic randomness in human behavior: Even if we succeed in introducing all the relevant variables into the model, there is bound to be some "intrinsic" randomness in individual Y's that cannot be explained no matter how hard we try. The disturbances, the us, may very well reflect this intrinsic randomness.

5. Poor proxy variables: Although the classical regression model (to be developed in Chapter 3) assumes that the variables Y and X are measured accurately, in practice the data may be plagued by errors of measurement. Consider, for example, Milton Friedman's well-known theory of the consumption function.11 He regards permanent consumption (Yp) as a function of permanent income (Xp). But since data on these variables are not directly observable, in practice we use proxy variables, such as current consumption (Y) and current income (X), which can be observable. Since the observed Y and X may not equal Yp and Xp, there is the problem of errors of measurement. The disturbance term u may in this case then also represent the errors of measurement. As we will see in a later chapter, if there are such errors of measurement, they can have serious implications for estimating the regression coefficients, the p's.

6. Principle of parsimony: Following Occam's razor,12 we would like to keep our regression model as simple as possible. If we can explain the behavior of Y "substantially" with two or three explanatory variables and if

10A further difficulty is that variables such as sex, education, and religion are difficult to quantify.

11Milton Friedman, A Theory of the Consumption Function, Princeton University Press, Princeton, N.J., 1957.

12"That descriptions be kept as simple as possible until proved inadequate," The World of Mathematics, vol. 2, J. R. Newman (ed.), Simon & Schuster, New York, 1956, p. 1247, or, "Entities should not be multiplied beyond necessity," Donald F. Morrison, Applied Linear Statistical Methods, Prentice Hall, Englewood Cliffs, N.J., 1983, p. 58.

CHAPTER TWO: TWO-VARIABLE REGRESSION ANALYSIS: SOME BASIC IDEAS 47

our theory is not strong enough to suggest what other variables might be included, why introduce more variables? Let ui represent all other variables. Of course, we should not exclude relevant and important variables just to keep the regression model simple.

7. Wrong functional form: Even if we have theoretically correct variables explaining a phenomenon and even if we can obtain data on these variables, very often we do not know the form of the functional relationship between the regressand and the regressors. Is consumption expenditure a linear (invariable) function of income or a nonlinear (invariable) function? If it is the former, Yi = j + B2Xi + ui is the proper functional relationship between Y and X,but if it is the latter, Yi = j + j2Xi + j3X2 + ui may be the correct functional form. In two-variable models the functional form of the relationship can often be judged from the scattergram. But in a multiple regression model, it is not easy to determine the appropriate functional form, for graphically we cannot visualize scattergrams in multiple dimensions.

For all these reasons, the stochastic disturbances ui assume an extremely critical role in regression analysis, which we will see as we progress.

By confining our discussion so far to the population of Y values corresponding to the fixed X's, we have deliberately avoided sampling considerations (note that the data of Table 2.1 represent the population, not a sample). But it is about time to face up to the sampling problems, for in most practical situations what we have is but a sample of Y values corresponding to some fixed X's. Therefore, our task now is to estimate the PRF on the basis of the sample information.

As an illustration, pretend that the population of Table 2.1 was not known to us and the only information we had was a randomly selected sample of Y values for the fixed X's as given in Table 2.4. Unlike Table 2.1, we now have only one Y value corresponding to the given X's; each Y (given Xi) in Table 2.4 is chosen randomly from similar Y's corresponding to the same Xi from the population of Table 2.1.

The question is: From the sample of Table 2.4 can we predict the average weekly consumption expenditure Y in the population as a whole corresponding to the chosen X's? In other words, can we estimate the PRF from the sample data? As the reader surely suspects, we may not be able to estimate the PRF "accurately" because of sampling fluctuations. To see this, suppose we draw another random sample from the population of Table 2.1, as presented in Table 2.5.

Plotting the data of Tables 2.4 and 2.5, we obtain the scattergram given in Figure 2.4. In the scattergram two sample regression lines are drawn so as

2.6 THE SAMPLE REGRESSION FUNCTION (SRF)

48 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Was this article helpful?

+6 -1
Rules Of The Rich And Wealthy

Rules Of The Rich And Wealthy

Learning About The Rules Of The Rich And Wealthy Can Have Amazing Benefits For Your Life And Success. Discover the hidden rules and beat the rich at their own game. The general population has a love / hate kinship with riches. They resent those who have it, but spend their total lives attempting to get it for themselves. The reason an immense majority of individuals never accumulate a substantial savings is because they don't comprehend the nature of money or how it works.

Get My Free Ebook


Responses

  • ERIC
    What are the 5 reasons why we introduce a stochastic disturbance term?
    4 years ago
  • magnus
    What are the reasons for introducing a stochastice term in the regression model?
    4 years ago
  • Nairn
    What are the reasons for introducing a stochastik diaturbance term in the regression model?
    4 years ago
  • hildigard
    What is the significance of Stochastic distrubance term?
    4 years ago
  • duenna
    What are the importance of d stochastic terms in a regression model analysis?
    3 years ago
  • Lavinia
    What is the significance of the stocastic term?
    3 years ago
  • Fredrik
    Why the stochastic term is important?
    3 years ago
  • donna
    Why stchastic term is important?
    3 years ago
  • Italo
    What is the significance of the stochastic disturbance term?
    3 years ago
  • federico
    What is the role of the stochastic disturbance term ui in regression analysis?
    3 years ago
  • Neftalem
    Why distrubance error not introduce these variables into the model explicity?
    3 years ago
  • catherine millar
    Why not introduce these variable in the model explicity?
    3 years ago
  • kaitlin
    Why we not introduce the disturbance term to our model?
    3 years ago
  • Isto
    Why no introduce many variables in to simple regression model explicitly?
    3 years ago
  • senait
    Why does the stochastic disturbance term exist?
    3 years ago
  • natsnet
    Why stochastic term is included in regession?
    3 years ago
  • sonia
    What are the reasons for introducing a stochastic disturbance term in regression model?
    3 years ago
  • Liliana
    What are the relevance of stochastic term in an econometric model?
    3 years ago
  • pupa
    Why is a stochastic disturbance introduced?
    3 years ago
  • augusto
    Why is stochatsic disturbance introduced in two variable linesr regression model?
    3 years ago
  • medoro
    What are the reason of inclusion of stochastic term in econometrics?
    3 years ago
  • ferumbras rumble
    What is stochastic disturbance term in econometric?
    3 years ago
  • anssi
    What are the significance of stochastic disturbances term?
    3 years ago
  • wegahta
    Why introduce a stochastic term in econometrics?
    3 years ago
  • Ville
    Why we include stochastic term in regression?
    3 years ago
  • Senja
    Why we include the distrubance Ui in the regression model?
    3 years ago
  • william
    What is stochastic disturbance in econometrics?
    2 years ago
  • Chanelle
    What is the significance or justification of stochastic disturbance term.?
    2 years ago
  • jukka
    What is stochastic disturbance term or error term?
    2 years ago
  • amanda
    Why there is stochastic distrurbance in economatrics analysis?
    2 years ago
  • hollie
    Why there is stochastic disturbance in econometrics analysis?
    2 years ago
  • Elizabeth
    What is the significance of the disturbance term in econometrics?
    2 years ago
  • Belladonna Maggot
    What is four reason stochastic in term regression?
    2 years ago
  • Hannes
    Why does the error or stochastic term exist?
    2 years ago
  • TAMZIN
    What is the significance of stocostic disterbance term in economic analysis?
    2 years ago
  • Jana Maier
    What is significant of stochastic disturbance term in econometrics?
    2 years ago
  • TYTTI KYT
    Why do we include stochastic disturbance in regression model?
    2 years ago
  • sigismond
    What are the nature of stochastic error term?
    2 years ago
  • Benjamin Kruger
    What is the role of stochastic error term?
    2 years ago
  • Judy
    What is the roles of stochastic error terms?
    2 years ago
  • Kristin Koch
    Why do we introduce a stochatic dusturbance term in an economic model?
    2 years ago
  • Beau Hill
    What is the role of stochastic error term in regression analysis?
    2 years ago
  • pontus m
    What is the significance of the stochastic disturbance term briefly?
    1 year ago
  • lea
    Why we include disterbance term?
    1 year ago
  • abdullah hamid
    What is the disturbance term in a regressiob?
    12 months ago
  • laura
    Why including the stochastic error in regression?
    11 months ago
  • Lukas
    What is the role of the stochastic error term in econometrics?
    11 months ago
  • Sointu
    What is the siginificant of stochastic disturbance term?
    11 months ago
  • samuli
    What is the significance of stochastic disturbance error term?
    11 months ago
  • mike
    Why do we include the stochastic error term?
    10 months ago
  • ELISA
    Why introduce a stochastic disturbance term in econometrics.?
    8 months ago
  • Belba
    Why does the stochastic error term include the effects of any omitted variables?
    8 months ago
  • cedivar
    What is the significance of the disturbance term in logit model?
    6 months ago
  • florian
    What are the reasons for the addition of ui term in the model?
    6 months ago
  • raimondo toscani
    What is the reason the insertion of stochastic disturbance in the model?
    6 months ago
  • billye
    Is the error term and disturbances the same economics?
    5 months ago
  • fre-weini
    What are the reason for the inclusion of stochastic error in simple regression model?
    4 months ago
  • bernd
    Why introduce stochastic distubance?
    3 months ago
  • paige russell
    Why do we introduce a stochastic disturbance term (ui) to the regression model?
    3 months ago
  • Adam Dickson
    What are the significants of stochastic error trem?
    29 days ago
  • SEMOLINA
    What is the ultimate role of a stochastic error term?
    11 days ago

Post a comment