The Nature Of Heteroscedasticity
As noted in Chapter 3, one of the important assumptions of the classical linear regression model is that the variance of each disturbance term ui, conditional on the chosen values of the explanatory variables, is some constant number equal to a2. This is the assumption of homoscedasticity, or equal (homo) spread (scedasticity), that is, equal variance. Symbolically,
Diagrammatically, in the twovariable regression model homoscedasticity can be shown as in Figure 3.4, which, for convenience, is reproduced as
388 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
388 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
Figure 11.1. As Figure 11.1 shows, the conditional variance of Yi (which is equal to that of ui), conditional upon the given Xi, remains the same regardless of the values taken by the variable X.
In contrast, consider Figure 11.2, which shows that the conditional variance of Yi increases as X increases. Here, the variances of Yi are not the same. Hence, there is heteroscedasticity. Symbolically,
Notice the subscript of o2, which reminds us that the conditional variances of ui (= conditional variances of Yi) are no longer constant.
To make the difference between homoscedasticity and heteroscedasticity clear, assume that in the twovariable model Yi = fa + faXi + ui, Y represents savings and X represents income. Figures 11.1 and 11.2 show that as income increases, savings on the average also increase. But in Figure 11.1
CHAPTER ELEVEN: HETEROSCEDASTICITY 389
the variance of savings remains the same at all levels of income, whereas in Figure 11.2 it increases with income. It seems that in Figure 11.2 the higherincome families on the average save more than the lowerincome families, but there is also more variability in their savings.
There are several reasons why the variances of ui may be variable, some of which are as follows.1
1. Following the errorlearning models, as people learn, their errors of behavior become smaller over time. In this case, a2 is expected to decrease. As an example, consider Figure 11.3, which relates the number of typing errors made in a given time period on a test to the hours put in typing practice. As Figure 11.3 shows, as the number of hours of typing practice increases, the average number of typing errors as well as their variances decreases.
2. As incomes grow, people have more discretionary income2 and hence more scope for choice about the disposition of their income. Hence, a2 is likely to increase with income. Thus in the regression of savings on income one is likely to find a2 increasing with income (as in Figure 11.2) because people have more choices about their savings behavior. Similarly, companies with larger profits are generally expected to show greater variability in their dividend policies than companies with lower profits. Also, growthoriented companies are likely to show more variability in their dividend payout ratio than established companies.
3. As data collecting techniques improve, a2 is likely to decrease. Thus, banks that have sophisticated data processing equipment are likely to
'See Stefan Valavanis, Econometrics, McGrawHill, New York, 1959, p. 48. 2As Valavanis puts it, "Income grows, and people now barely discern dollars whereas previously they discerned dimes,'' ibid., p. 48.
390 PART TWO: RELAXING THE ASSUMPTIONS OF THE CLASSICAL MODEL
commit fewer errors in the monthly or quarterly statements of their customers than banks without such facilities.
4. Heteroscedasticity can also arise as a result of the presence of outliers. An outlying observation, or outlier, is an observation that is much different (either very small or very large) in relation to the observations in the sample. More precisely, an outlier is an observation from a different population to that generating the remaining sample observations.3 The inclusion or exclusion of such an observation, especially if the sample size is small, can substantially alter the results of regression analysis.
As an example, consider the scattergram given in Figure 11.4. Based on the data given in exercise 11.22, this figure plots percent rate of change of stock prices (Y) and consumer prices (X) for the postWorld War II period through 1969 for 20 countries. In this figure the observation on Y and X for Chile can be regarded as an outlier because the given Y and X values are much larger than for the rest of the countries. In situations such as this, it would be hard to maintain the assumption of homoscedasticity. In exercise 11.22, you are asked to find out what happens to the regression results if the observations for Chile are dropped from the analysis.
Rules Of The Rich And Wealthy
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