Multicolinealidad

X2 X3 Y

10 19 51

11 21 56

12 23 61

13 25 66

14 27 71

15 29 76

MULTICOLLINEARITY

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Suppose that Y = 2 + 3X2 + X3 and that X3 = 2X2 – 1. There is no disturbance term in the equation for Y, but that is not important. Suppose that we have the six observations shown.

MULTICOLLINEARITY

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The three variables are plotted as line graphs above. Looking at the data, it is impossible to tell whether the changes in Y are caused by changes in X2, by changes in X3, or jointly by changes in both X2 and X3.

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10 19 51 1 2 5

11 21 56 1 2 5

12 23 61 1 2 5

13 25 66 1 2 5

14 27 71 1 2 5

15 29 76 1 2 5

MULTICOLLINEARITY

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Numerically, Y increases by 5 in each observation. X2 changes by 1.

MULTICOLLINEARITY

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Hence the true relationship could have been Y = 1 + 5X2.

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However, it can also be seen that X3 increases by 2 in each observation.

Change from previous observation

X2 X3 Y X2 X3 Y

10 19 51 1 2 5

11 21 56 1 2 5

12 23 61 1 2 5

13 25 66 1 2 5

14 27 71 1 2 5

15 29 76 1 2 5

MULTICOLLINEARITY

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Hence the true relationship could have been Y = 3.5 +2.5X3.

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These two possibilities are special cases of Y = 3.5 – 2.5p + 5pX2 + 2.5(1 – p)X3, which would fit the relationship for any value of p.

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Y = 3.5 – 2.5p + 5pX2 + 2.5(1 – p)X3

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There is no way that regression analysis, or any other technique, could determine the true relationship from this infinite set of possibilities, given the sample data.

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What would happen if you tried to run a regression when there is an exact linear relationship among the explanatory variables?

MULTICOLLINEARITY

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We will investigate, using the model with two explanatory variables shown above. [Note: A disturbance term has now been included in the true model, but it makes no difference to the analysis.]

MULTICOLLINEARITY

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The expression for the multiple regression coefficient b2 is shown above. We will substitute for X3 using its relationship with X2.

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First, we will replace the terms highlighted.

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We have made the replacement.

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Next, the terms highlighted now.

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We have made the replacement.

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Finally this term.

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Again, we have made the replacement.

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It turns out that the numerator and the denominator are both equal to zero. The regression coefficient is not defined.

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It is unusual for there to be an exact relationship among the explanatory variables in a regression. When this occurs, it s typically because there is a logical error in the specification.

. reg EARNINGS S EXP EXPSQ

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 45.57 Model | 22762.4472 3 7587.48241 Prob > F = 0.0000 Residual | 89247.7839 536 166.507059 R-squared = 0.2032-------------+------------------------------ Adj R-squared = 0.1988 Total | 112010.231 539 207.811189 Root MSE = 12.904

------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.754372 .2417286 11.39 0.000 2.279521 3.229224 EXP | -.2353907 .665197 -0.35 0.724 -1.542103 1.071322 EXPSQ | .0267843 .0219115 1.22 0.222 -.0162586 .0698272 _cons | -22.21964 5.514827 -4.03 0.000 -33.05297 -11.38632------------------------------------------------------------------------------

MULTICOLLINEARITY

20

However, it often happens that there is an approximate relationship. For example, when relating earnings to schooling and work experience, it if often reasonable to suppose that the effect of work experience is subject to diminishing returns.

uEXPSQEXPSEARNINGS 4321




MULTICOLLINEARITY

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A standard way of allowing for this is to include EXPSQ, the square of EXP, in the specification. According to the hypothesis of diminishing returns, 4 should be negative.





MULTICOLLINEARITY

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We fit this specification using Data Set 21. The schooling component of the regression results is not much affected by the inclusion of the EXPSQ term. The coefficient of S indicates that an extra year of schooling increases hourly earnings by $2.75.


. reg EARNINGS S EXP


------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.678125 .2336497 11.46 0.000 2.219146 3.137105 EXP | .5624326 .1285136 4.38 0.000 .3099816 .8148837 _cons | -26.48501 4.27251 -6.20 0.000 -34.87789 -18.09213------------------------------------------------------------------------------

MULTICOLLINEARITY

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In the specification without EXPSQ it was 2.68, not much different.





MULTICOLLINEARITY

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The standard error, 0.23 in the specification without EXPSQ, is also little changed and the coefficient remains highly significant.




MULTICOLLINEARITY

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By contrast, the inclusion of the new term has had a dramatic effect on the coefficient of EXP. Now it is negative, which makes little sense, and insignificant.

MULTICOLLINEARITY

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Previously it had been positive and highly significant.







MULTICOLLINEARITY

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The coefficient of EXPSQ is also strange. It is positive, suggesting increasing returns to experience. However, it is not significant.




MULTICOLLINEARITY

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The reason for these problems is that EXPSQ is highly correlated with EXP. This makes it difficult to discriminate between the individual effects of EXP and EXPSQ, and the regression estimates tend to be erratic.

. cor EXP EXPSQ(obs=540)

| EXP EXPSQ------+------------------ EXP | 1.0000EXPSQ | 0.9812 1.0000





MULTICOLLINEARITY

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The high correlation causes the standard error of EXP to be larger than it would have been if EXP and EXPSQ had been less highly correlated, warning us that the point estimate is unreliable.





MULTICOLLINEARITY

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When high correlations among the explanatory variables lead to erratic point estimates of the coefficients, large standard errors and unsatisfactorily low t statistics, the regression is said to said to be suffering from multicollinearity.





MULTICOLLINEARITY

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Note that the coefficients remain unbiased and the standard errors remain valid.





MULTICOLLINEARITY

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Multicollinearity may also be caused by an approximate linear relationship among the explanatory variables. When there are only 2, an approximate linear relationship means there will be a high correlation, but this is not always the case when there are more than 2.

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY

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What can you do about multicollinearity if you encounter it? We will discuss some possible measures, looking at the model with two explanatory variables.

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Before doing this, two important points should be emphasized. First, multicollinearity does not cause the regression coefficients to be biased.


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The problem is that they have unsatisfactorily large variances.


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Second, the standard errors and t tests remain valid. The standard errors are larger than they would have been in the absence of multicollinearity, warning us that the regression estimates are erratic.


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Since the problem of multicollinearity is caused by the variances of the coefficients being unsatisfactorily large, we will seek ways of reducing them.


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(1) Reduce by including further relevant variables in the model.2u

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We will focus on the slope coefficient and look at the various components of its variance. We might be able to reduce it by bringing more variables into the model and reducing u

2, the variance of the disturbance term.


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The estimator of the variance of the disturbance term is the residual sum of squares divided by n – k, where n is the number of observations (540) and k is the number of parameters (4). Here it is 166.5.





8

. 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------------------------------------------------------------------------------

We now add two new variables that are often found to be determinants of earnings: MALE, sex of respondent, and ASVABC, the composite score on the cognitive tests in the Armed Services Vocational Aptitude Battery.


9




MALE is a qualitative variable and the treatment of such variables will be explained in Chapter 5.


10




Both MALE and ASVABC have coefficients significant at the 0.1% level.






11

However they account for only a small proportion of the variance in earnings and the reduction in the estimate of the variance of the disturbance term is likewise small.






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As a consequence the impact on the standard errors of EXP and EXPSQ is negligible.






13

Note how unstable the coefficients are. This is often a sign of multicollinearity.






14

Note also that the standard error of the coefficient of S has actually increased. This is attributable to the correlation of 0.58 between S and ASVABC.

. cor S ASVABC(obs=540) | S ASVABC--------+------------------ S | 1.0000 ASVABC | 0.5810 1.0000






15

This is a common problem with this approach to attempting to reduce the problem of multicollinearity. If the new variables are linearly related to one or more of the variables already in the equation, their inclusion may make the problem of multicollinearity worse.

. cor S ASVABC(obs=540) | S ASVABC--------+------------------ S | 1.0000 ASVABC | 0.5810 1.0000


16

The next factor to look at is n, the number of observations. If you are working with cross-section data (individuals, households, enterprises, etc) and you are undertaking a survey, you could increase the size of the sample by negotiating a bigger budget.


(2) Increase the number of observations.

Surveys: increase the budget, use clustering

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Alternatively, you could make a fixed budget go further by using a technique known as clustering. You divide the country geographically by zip code or postal area.




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You select a number of these randomly, perhaps using stratified random sampling to make sure that metropolitan, other urban, and rural areas are properly represented.




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You then confine the survey to the areas selected. This reduces the travel time and cost of the fieldworkers, allowing them to interview a greater number of respondents.




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Time series: use quarterly instead of annual data

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If you are working with time series data, you may be able to increase the sample by working with shorter time intervals for the data, for example quarterly or even monthly data instead of annual data.


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------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.312461 .135428 17.08 0.000 2.046909 2.578014 EXP | -.3270651 .308231 -1.06 0.289 -.9314569 .2773268 EXPSQ | .023743 .0101558 2.34 0.019 .0038291 .0436569 MALE | 5.947206 .5221755 11.39 0.000 4.923303 6.971108 ASVABC | .2086846 .0336869 6.19 0.000 .1426301 .2747392 _cons | -27.40462 2.579435 -10.62 0.000 -32.46248 -22.34676------------------------------------------------------------------------------

21

Here is the result of running the regression with all 2,714 observations in the EAEF data set.


. reg EARNINGS S EXP EXPSQ MALE ASVABC Number of obs = 540------------------------------------------------------------------------------ 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------------------------------------------------------------------------------. reg EARNINGS S EXP EXPSQ MALE ASVABC Number of obs = 2714------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.312461 .135428 17.08 0.000 2.046909 2.578014 EXP | -.3270651 .308231 -1.06 0.289 -.9314569 .2773268 EXPSQ | .023743 .0101558 2.34 0.019 .0038291 .0436569 MALE | 5.947206 .5221755 11.39 0.000 4.923303 6.971108 ASVABC | .2086846 .0336869 6.19 0.000 .1426301 .2747392 _cons | -27.40462 2.579435 -10.62 0.000 -32.46248 -22.34676------------------------------------------------------------------------------

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Comparing this result with that using Data Set 21, we see that the standard errors are much smaller, as expected.



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As a consequence, the t statistics of the variables are higher. However the correlation between EXP and EXPSQ is as high as in the smaller sample and the increase in the sample size has not been large enough to have much impact on the problem of multicollinearity.



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The coefficients of EXP and EXPSQ both still have unexpected signs since we expect the coefficient of EXP to be positive and that of EXPSQ to be negative, reflecting diminishing returns.



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The EXPSQ coefficient has a rather large t statistic, which is a matter of concern. We could assume that this has occurred as a matter of chance. Alternatively, it might be an indication that the model is misspecified.



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As we will see in the next and subsequent chapters, there are good reasons for supposing that the dependent variable in an earnings function should be the logarithm of earnings, rather than earnings in linear form.


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A third possible way of reducing the problem of multicollinearity might be to increase the variation in the explanatory variables. This is possible only at the design stage of a survey.


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(3) Increase MSD(X2).

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For example, if you were planning a household survey with the aim of investigating how expenditure patterns vary with income, you should make sure that the sample included relatively rich and relatively poor households as well as middle-income households.


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(3) Increase MSD(X2).

(4) Reduce .32 ,XXr

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Another possibility might be to reduce the correlation between the explanatory variables. This is possible only at the design stage of a survey and even then it is not easy.


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(5) Combine the correlated variables.

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


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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 ASVAB02 (arithmetic reasoning), ASVAB03 (word knowledge), and ASVAB04 (paragraph comprehension).





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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.





(6) 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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(7) Empirical restriction

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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.


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uPXY 321

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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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Last, but by no means least, is the use of a theoretical restriction, which is defined as a hypothetical relationship among the parameters of a regression model.


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(8) 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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uSFSMASVABCS 4321

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





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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% 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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uSFSMASVABCS 4321

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This allows us to rewrite the equation as shown.


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uSFSMASVABCS 4321

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uSPASVABC

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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------------------------------------------------------------------------------

54

The estimate of 3 is now 0.083.


. g SP=SM+SF

. reg S ASVABC SP




55

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


. g SP=SM+SF

. reg S ASVABC SP




56

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




57

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.


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Multicolinealidad