(1)Combine the correlated variables. 1 In this sequence, we look at four possible indirect methods for alleviating a problem of multicollinearity. POSSIBLE

(1) Combine the correlated variables.

1

In this sequence, we look at four possible indirect methods for alleviating a problem of multicollinearity.

POSSIBLE INDIRECT MEASURES FOR ALLEVIATING MULTICOLLINEARITY

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First, if the correlated variables are similar conceptually, it may be reasonable to combine them into some overall index.


(1) Combine the correlated variables.

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That is precisely what has been done with the three cognitive ASVAB variables. ASVABC has been calculated as a weighted average of scores on subtests: ASVAB02 (arithmetic reasoning), ASVAB03 (word knowledge), and ASVAB04 (paragraph comprehension).

. reg EARNINGS S EXP EXPSQ MALE ASVABC

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 5, 534) = 37.24 Model | 28957.3532 5 5791.47063 Prob > F = 0.0000 Residual | 83052.8779 534 155.529734 R-squared = 0.2585-------------+------------------------------ Adj R-squared = 0.2516 Total | 112010.231 539 207.811189 Root MSE = 12.471

------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.031419 .296218 6.86 0.000 1.449524 2.613315 EXP | -.0816828 .6441767 -0.13 0.899 -1.347114 1.183748 EXPSQ | .0130223 .021334 0.61 0.542 -.0288866 .0549311 MALE | 5.762358 1.104734 5.22 0.000 3.592201 7.932515 ASVABC | .2447687 .0714294 3.43 0.001 .1044516 .3850858 _cons | -26.18541 5.452032 -4.80 0.000 -36.89547 -15.47535------------------------------------------------------------------------------


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The three components are highly correlated and by combining them as a weighted average, rather than using them individually, one avoids a potential problem of multicollinearity.


. reg EARNINGS S EXP EXPSQ MALE ASVABC


------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.031419 .296218 6.86 0.000 1.449524 2.613315 EXP | -.0816828 .6441767 -0.13 0.899 -1.347114 1.183748 EXPSQ | .0130223 .021334 0.61 0.542 -.0288866 .0549311 MALE | 5.762358 1.104734 5.22 0.000 3.592201 7.932515 ASVABC | .2447687 .0714294 3.43 0.001 .1044516 .3850858 _cons | -26.18541 5.452032 -4.80 0.000 -36.89547 -15.47535------------------------------------------------------------------------------

(2) Drop some of the correlated variables.

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Dropping some of the correlated variables, if they have insignificant coefficients, may alleviate multicollinearity.


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However, this approach to multicollinearity is dangerous. It is possible that some of the variables with insignificant coefficients really do belong in the model and that the only reason their coefficients are insignificant is because there is a problem of multicollinearity.


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If that is the case, their omission may cause omitted variable bias, to be discussed in Chapter 6.


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A further way of dealing with the problem of multicollinearity is to use extraneous information, if available, concerning the coefficient of one of the variables.

(3) Empirical restriction


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For example, suppose that Y in the equation above is the demand for a category of consumer expenditure, X is aggregate disposable personal income, and P is a price index for the category.


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To fit a model of this type you would use time series data. If X and P are highly correlated, which is often the case with time series variables, the problem of multicollinearity might be eliminated in the following way.


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Obtain data on income and expenditure on the category from a household survey and regress Y' on X'. (The ' marks are to indicate that the data are household data, not aggregate data.)


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This is a simple regression because there will be relatively little variation in the price paid by the households.


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Now substitute b' for 2 in the time series model. Subtract b' X from both sides, and regress Z = Y – b' X on price. This is a simple regression, so multicollinearity has been eliminated.

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There are some problems with this technique. First, the 2 coefficients may be conceptually different in time series and cross-section contexts.


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Second, since we subtract the estimated income component b' X, not the true income component 2X, from Y when constructing Z, we have introduced an element of measurement error in the dependent variable.

2


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The last, but by no means least, indirect method for alleviating multicollinearity is the use of a theoretical restriction, which is defined as a hypothetical relationship among the parameters of a regression model.

(4) Theoretical restriction


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It will be explained using an educational attainment model as an example. Suppose that we hypothesize that highest grade completed, S, depends on ASVABC, and highest grade completed by the respondent's mother and father, SM and SF, respectively.


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A one-point increase in ASVABC increases S by 0.13 years.

. reg S ASVABC SM SF


------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SM | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .1076825 .0309522 3.48 0.001 .04688 .1684851 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------


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S increases by 0.05 years for every extra year of schooling of the mother and 0.11 years for every extra year of schooling of the father.





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Mother's education is generally held to be at least, if not more, important than father's education for educational attainment, so this outcome is unexpected.





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It is also surprising that the coefficient of SM is not significant, even at the 5 percent level, using a one-sided test.








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However assortive mating leads to correlation between SM and SF and the regression appears to be suffering from multicollinearity.

. cor SM SF(obs=540) | SM SF--------+------------------ SM | 1.0000 SF | 0.6241 1.0000


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Suppose that we hypothesize that mother's and father's education are equally important. We can then impose the restriction 3 = 4.


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uSPASVABC

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This allows us to rewrite the equation as shown.


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Defining SP to be the sum of SM and SF, the equation may be rewritten as shown. The problem caused by the correlation between SM and SF has been eliminated.


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. g SP=SM+SF

. reg S ASVABC SP


------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1253106 .0098434 12.73 0.000 .1059743 .1446469 SP | .0828368 .0164247 5.04 0.000 .0505722 .1151014 _cons | 5.29617 .4817972 10.99 0.000 4.349731 6.242608------------------------------------------------------------------------------

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The estimate of 3 is now 0.083.


. g SP=SM+SF

. reg S ASVABC SP


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Not surprisingly, this is a compromise between the coefficients of SM and SF in the previous specification.




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The standard error of SP is much smaller than those of SM and SF. The use of the restriction has led to a large gain in efficiency and the problem of multicollinearity has been eliminated.


. g SP=SM+SF

. reg S ASVABC SP




. g SP=SM+SF

. reg S ASVABC SP




29

The t statistic is very high. Thus it would appear that imposing the restriction has improved the regression results. However, the restriction may not be valid. We should test it. Testing theoretical restrictions is one of the topics in Chapter 6.


Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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2012.10.28

Documents

(1)Combine the correlated variables. 1 In this sequence, we look at four possible indirect methods for alleviating a problem of multicollinearity. POSSIBLE