EC220 - Introduction to econometrics (chapter 3) Christopher Dougherty EC220 - Introduction to econometrics (chapter 3) Slideshow: alleviation of multicollinearity   Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 3). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/129/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/   http://learningresources.lse.ac.uk/

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY What can you do about multicollinearity if you encounter it? We will discuss some possible measures, looking at the model with two explanatory variables. 1

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY Before doing this, two important points should be emphasized. First, multicollinearity does not cause the regression coefficients to be biased. 2

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY The problem is that they have unsatisfactorily large variances. 3

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY 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. 4

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY Since the problem of multicollinearity is caused by the variances of the coefficients being unsatisfactorily large, we will seek ways of reducing them. 5

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (1) Reduce by including further relevant variables in the model. 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 su2, the variance of the disturbance term. 6

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 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. 7

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 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. 8

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 MALE is a qualitative variable and the treatment of such variables will be explained in Chapter 5. 9

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 Both MALE and ASVABC have coefficients significant at the 0.1% level. 10

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 . reg EARNINGS S EXP EXPSQ MALE ASVABC -------------+------------------------------ 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 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. 11

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg EARNINGS S EXP EXPSQ ------------------------------------------------------------------------------ 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 . reg EARNINGS S EXP EXPSQ MALE ASVABC 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 As a consequence the impact on the standard errors of EXP and EXPSQ is negligible. 12

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg EARNINGS S EXP EXPSQ ------------------------------------------------------------------------------ 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 . reg EARNINGS S EXP EXPSQ MALE ASVABC 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 Note how unstable the coefficients are. This is often a sign of multicollinearity. 13

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg EARNINGS S EXP EXPSQ ------------------------------------------------------------------------------ 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 . reg EARNINGS S EXP EXPSQ MALE ASVABC 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 . cor S ASVABC (obs=540) | S ASVABC --------+------------------ S | 1.0000 ASVABC | 0.5810 1.0000 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. 14

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg EARNINGS S EXP EXPSQ ------------------------------------------------------------------------------ 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 . reg EARNINGS S EXP EXPSQ MALE ASVABC 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 . cor S ASVABC (obs=540) | S ASVABC --------+------------------ S | 1.0000 ASVABC | 0.5810 1.0000 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. 15

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (2) Increase the number of observations. Surveys: increase the budget, use clustering 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. 16

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (2) Increase the number of observations. Surveys: increase the budget, use clustering 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. 17

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (2) Increase the number of observations. Surveys: increase the budget, use clustering 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. 18

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (2) Increase the number of observations. Surveys: increase the budget, use clustering 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. 19

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (2) Increase the number of observations. Surveys: increase the budget, use clustering Time series: use quarterly instead of annual data 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. 20

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg EARNINGS S EXP EXPSQ MALE ASVABC Source | SS df MS Number of obs = 2714 -------------+------------------------------ F( 5, 2708) = 183.99 Model | 161795.573 5 32359.1147 Prob > F = 0.0000 Residual | 476277.268 2708 175.877869 R-squared = 0.2536 -------------+------------------------------ Adj R-squared = 0.2522 Total | 638072.841 2713 235.190874 Root MSE = 13.262 ------------------------------------------------------------------------------ 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 Here is the result of running the regression with all 2,714 observations in the EAEF data set. 21

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 Number of obs = 2714 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 Comparing this result with that using Data Set 21, we see that the standard errors are much smaller, as expected. 22

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 Number of obs = 2714 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 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. 23

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 Number of obs = 2714 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 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. 24

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 Number of obs = 2714 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 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. 25

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 Number of obs = 2714 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 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. 26

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (3) Increase MSD(X2). 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. 27

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (3) Increase MSD(X2). 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. 28

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (4) Reduce . 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. 29

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (5) Combine the correlated variables. If the correlated variables are similar conceptually, it may be reasonable to combine them into some overall index. 30

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 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). 31

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 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. 32

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (6) Drop some of the correlated variables. Dropping some of the correlated variables, if they have insignificant coefficients, may alleviate multicollinearity. 33

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (6) Drop some of the correlated variables. 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. 34

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (6) Drop some of the correlated variables. If that is the case, their omission may cause omitted variable bias, to be discussed in Chapter 6. 35

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction 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. 36

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction 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. 37

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction 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. 38

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction 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.) 39

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction This is a simple regression because there will be relatively little variation in the price paid by the households. 40

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction Now substitute b' for b2 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. 2 2 2 41

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction There are some problems with this technique. First, the b2 coefficients may be conceptually different in time series and cross-section contexts. 42

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (7) Empirical restriction Second, since we subtract the estimated income component b' X, not the true income component b 2X, from Y when constructing Z, we have introduced an element of measurement error in the dependent variable. 2 43

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (8) Theoretical restriction 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. 44

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (8) Theoretical restriction 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. 45

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ 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 A one-point increase in ASVABC increases S by 0.13 years. 46

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ 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 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. 47

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ 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 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. 48

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ 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 It is also surprising that the coefficient of SM is not significant, even at the 5% level, using a one-sided test. 49

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . reg S ASVABC SM SF Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686 -------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943 ------------------------------------------------------------------------------ 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 . cor SM SF (obs=540) | SM SF --------+------------------ SM | 1.0000 SF | 0.6241 1.0000 However assortive mating leads to correlation between SM and SF and the regression appears to be suffering from multicollinearity. 50

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (8) Theoretical restriction Suppose that we hypothesize that mother's and father's education are equally important. We can then impose the restriction b3 = b4. 51

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (8) Theoretical restriction This allows us to rewrite the equation as shown. 52

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY (8) Theoretical restriction 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. 53

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . g SP=SM+SF . reg S ASVABC SP Source | SS df MS Number of obs = 540 -------------+------------------------------ F( 2, 537) = 156.04 Model | 1177.98338 2 588.991689 Prob > F = 0.0000 Residual | 2026.99996 537 3.77467403 R-squared = 0.3675 -------------+------------------------------ Adj R-squared = 0.3652 Total | 3204.98333 539 5.94616574 Root MSE = 1.9429 ------------------------------------------------------------------------------ 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 The estimate of b3 is now 0.083. 54

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 . reg S ASVABC SM SF 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 Not surprisingly, this is a compromise between the coefficients of SM and SF in the previous specification. 55

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 . reg S ASVABC SM SF 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 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. 56

POSSIBLE MEASURES FOR ALLEVIATING MULTICOLLINEARITY . 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 . reg S ASVABC SM SF 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 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. 57

Copyright Christopher Dougherty 2011. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 3.4 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lse. 11.07.25