ECON301_Handout_10_1213_02

If there are one or more near linear dependences in

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If there are one or more near-linear dependences in the data, then one or more characteristic roots will be small. One or more small eigenvalues imply that there are near-linear dependences among the columns of X. Some analysts prefer to examine the condition number of, defined as: max min (13) Generally: If the condition number is less than 100, there is no serious problem with multicollinearity.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 13 Condition number between 100 and 1000 imply moderate to strong multicollinearity, If it exceeds 1000, severe multicollinearity is indicated. Suppose that for a model with 4 explanatory variables, the eigenvalues are as follows: λ 1 =4.2048, λ 2 =2.1626, λ 3 =1.1384, λ 4 =0.0001.The condition number, max min , equals 4.2048/0.0001=42048 which exceeds 1000. This indicates presence of severe multicollinearity. Example As an illustration of the problems introduced by multicollinearity, consider the consumption equation: t C = a 0 + a 1 t Y + a 2 t W +u t where t C is consumption expenditures at time t , t Y is income at time t and t W is wealth at time t . Economic theory suggests that the coefficient on income should be slightly less than one and the coefficient on wealth should be positive. The time-series data for this relationship are given in the following table:
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 14 t C t Y t W 70 80 810 65 100 1009 90 120 1273 95 140 1425 110 160 1633 115 180 1876 120 200 2052 140 220 2201 155 240 2435 150 260 2686 Table 2 Consumption Data Applying least squares to this equation and data yields ˆ t C = 24.775 + 0.942 t Y - 0.042 t W t (3.67) (1.14) (-0.53) R 2 =0.9635, SSR = 324.446, F = 92.40 High R 2 shows that income and wealth together explain about 96% of the variation in consumption. The coefficient estimate for the marginal propensity to consume seems to be a reasonable value however it is not significantly different from either zero or one. And the coefficient on wealth is negative, which is not consistent with economic theory. Wrong signs and insignificant coefficient estimates on a priori important variables are the classic symptoms of multicollinearity. As an indicator of the possible multicollinearity the squared correlation between t Y and t W is .9979, which suggests near extreme multicollinearity among the explanatory variables.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 15 The two variables are so highly correlated that it is impossible to isolate the individual effects. (1) If regress t W on t Y we get: ˆ t W = 7.54 + 10.19 t Y t (0.26) ( 62.04) R 2 = .99 There is almost perfect multicollinearity between t Y and t W (2) Regress ˆ t C on income only: ˆ t C = 24.45 + 0.51 t Y t (3.81) (14.24) R 2 = 0.96 Income variable is highly significant, whereas before it was insignificant.
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If there are one or more near linear dependences in the...

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