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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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