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34

For all of the four data sets, we have the following statistics: /3 = 11 X = 9.0 = 7.5 S„ = 110.0 The regression equation is

Syy = 41.25

5 = 55.0

= 3.0 + 0.5x = 0.667

{0 8)

regression sum of squares = 27.50 (1 d.f.)

residual sum of squares = 13.75 (9 d.f.)

Although the regression equations are identical, the four data sets exhibit widely different characteristics. This is revealed when we see the plots of the four data sets. These plots are shown in Figure 3.6.

10 -

20

(1) Data Sel I

(II) Data Set 2

10 -

(III) Data Set 3

10 -

10 X

(iv) Data Set 4

Figure 3.6 Identical regression lines for four different data sets.



Table 3.6 Per Capita Personal Consumption Expenditures (C) and Per Capita Disposable Personal Income ( ) (in 1972 Dollars) for the United States, 1929-1984

Year

Year

Year

1929

1765

1883

1957

2416

2660

1977

3924

4280

1933

1356

1349

1958

2400

2645

1978

4057

4441

1939

1678

1754

1959

2487

1979

4121

4512

1940

1740

1847

1960

2501

27Q9

1980

4093

4487

1941

2083

1961

2511

2742

1981

4131

4561

1942

1788

2354

1962

2583

2813

1982

4146

4555

1943

1815

2429

1963

2644

2865

1983

4303

4670

1944

1844

2483

1964

2751

3026

1984

4490

4941

1945

1936

2416

1965

2868

3171

1946

2129

2353

1966

2979

3290

1947

2122

2212

1967

3032

3389

1948

2129

2290

1968

3160

3493

1949

2140

2257

1969

3245

3564

1950

2224

2392

1970

3277

3665

1951

2214

2415

1971

3355

3752

1952

2230

2441

1972 •

3511

3860

1953

227/

2501

1973

3623

1954

2278

2483

1974

3566

4009

1955

2384

2582

1975

3609

4051

1956

2410

2653

1976

3774

4158

Source: Economic Report of the President. 19M, p. 261.

set 1 shown in Figure 3.6(i) shows no special problems. Figure 3.6(ii) shows that the regression line should not be linear. Figure 3.6(iii) shows how a single outlier has twisted the regression line slightly. If we omit this one observation (observation 3), we would have obtained a slightly different regression line. Figure 3.6(iv) shows hov. an outlier can produce an entirely different picture. If that one observation (observation 8) is omitted, we would have a vertical regression line.

We have shown graphically how outliers can produce drastic changes in the regression estimates. Next we give some real-world examples.

Some Ulustrative Examples

Example 1

This example consists of the estimation of the consumption function for the United States for the period 1929-1984. Table 3.6 gives the data on per capita disposable income (F) and per capita consumption expenditures (C) both in constant dollars for the United States. The data are not continuous. They are for 1929, 1933. and then continuous from 1939. The continuous data for 1929-1939 can be obtained from an earlier Presidents Economic Report (say, for



Table 3.7 Residuals for the Consumption Function Estimated from the Data in Table 3.6 (Rounded to the First Decimal)

Observation

Residual

Observation

Residual

Observation

Residual

75.0

24.2

24.1

152.4

41.6

-35.9

105.5

57.4

-37.1

82.8

18.8

20.5

-46.1

21 •

18.4

-67.9

-330.9

16.0

-60.2

-372.2

44.8

-55.4

-392.4

58.8

12.1

-239.4

38.7

51.0

11.0

46.0

37.4

132.4

59.7

36.7

68.4

20.1

31.5

109.4

70.5

22.5

39.5

-29.5

74.8

31.8

32 •

15.0

1972). But these data are in 1958 dollars and thus we have to Unk the two series.* We have not attempted this.

We will estimate a regression of on Y. It was estimated by the SAS regression package. The results are as follows:

= -24.944 + 0.911 = 0.9823

(58 124) (0 018)

The /-value for p is 0.911/0.018 = 50.47 and it can be checked that r = tV [t + (n - 2)] and n = 48. (The figures have all been rounded to three decimals from the computer output.) The next step is to examine the residuals. These are presented in Table 3.7. One can easily notice the large negative residuals for observations 6, 7, 8, and 9. These observations are outliers. They correspond to the war years 1942-1945 with strict controls on consumer expenditures. We therefore discard these observations and reestimate the equation. The equation now is

= 85.725 + 0.885 r = 0.9975

(22 353) (0 007)

Earlier the intercept was not significantly different from zero. Now it is significantly positive. Further, the estimate of the marginal propensity to consume

The data for 1929-1970 at 1958 prices have been analyzed in G. S Maddala, Econometrics (New York McGraw-Hill, 1977), pp. 84-86. These data are given as an exercise.



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