Exercise session 4 1. Your aim is to estimate how the number of prenatal examinations and several other characteristics influence the birth weight of a baby. Your initial hypothesis is that more responsible pregnant women visit the doctor more often and this leads to healthier and thus also bigger babies. (a) In your first specification, you run the following model: bwght = β[0] + β[1] npvis + β[2] npvis^2 + β[3] monpre + β[4] male + ε , where bwght is birth weight of the baby (in grams), npvis is the number of prenatal doctor’s visits, monpre is the month on pregnancy in which the prenatal care began and male is a dummy, equal to one if the baby is a boy and zero if it is a girl. You obtain the following results from Stata[1]: Textové pole: Source SS df MS Model 12848047.5 4 3212011.87 RESIDUAL 570003184 1721 331204.639 TOTAL 582851231 1725 337884.772 Textové pole: Number of obs = 1726 F( 4, 1721) = 9.70 Prob > F = 0.0000 R-SQUARED = 0.0220 Adj R-SQUARED = 0.0198 Root MSE = 575.5 bwght Coef. Std. Err. t P>|t| [95% Conf. Interval] npvis 53.50974 11.41313 4.69 0.000 31.12468 75.8948 npvissq -1.173175 .3591552 -3.27 0.001 -1.877601 -.4687481 monpre 30.47033 12.40794 2.46 0.014 6.134091 54.80657 male 76.69243 27.76083 2.76 0.006 22.24391 131.141 _cons 2853.196 101.3073 28.16 0.000 2654.498 3051.895 i. Is there strong evidence that npvissq (stands for npvis^2) should be included in the model? ii. How do you interpret the negative coefficient of npvissq? iii. Holding npvis and monpre fixed, test the hypothesis that newborn boys weight by 100 grams more than newborn girls (at 95% confidence level). (b) A friend of yours, student of medicine, reminds you of the fact that the age of the parents (especially of the mother) might be a decisive factor for the health and for the weight of the baby. Therefore, in your second specification, you decide to include in your model also the age of the mother (mage) and of the father (fage). The results of your estimation are now the following: Textové pole: Source SS df MS Model 16270165.8 6 2711694.3 RESIDUAL 563258231 1713 328813.912 TOTAL 579528396 1719 337131.121 Textové pole: Number of obs = 1720 F( 6, 1713) = 8.25 Prob > F = 0.0000 R-SQUARED = 0.0281 Adj R-SQUARED = 0.0247 Root MSE = 573.42 bwght Coef. Std. Err. t P>|t| [95% Conf. Interval] npvis 52.43859 11.40558 4.60 0.000 30.06826 74.80891 npvissq -1.138545 .3585648 -3.18 0.002 -1.841816 -.4352743 monpre 34.35661 12.69477 2.71 0.007 9.457725 59.2555 male 74.45482 27.75247 2.68 0.007 20.02252 128.8871 mage .5285275 4.218069 0.13 0.900 -7.744582 8.801637 fage 8.697342 3.465973 2.51 0.012 1.899357 15.49533 _cons 2592.813 139.6173 18.57 0.000 2318.974 2866.651 i. Comment on the significance of the coefficients on mage and fage sepa- rately: are they in line with your friend’s claim? ii. Test the hypothesis that mage and fage are jointly significant (at 95% confidence level). Is the result in line with your friend’s claim? iii. How can you reconcile you findings from the two previous questions? (c) In your third specification, you decide to drop fage and you get the following results: Textové pole: Source SS df MS Model 14451685.6 5 2890337.13 RESIDUAL 568399545 1720 330464.852 TOTAL 582851231 1725 337884.772 Textové pole: Number of obs = 1726 F( 5, 1720) = 8.75 Prob > F = 0.0000 R-SQUARED = 0.0248 Adj R-SQUARED = 0.0220 Root MSE = 574.86 bwght Coef. Std. Err. t P>|t| [95% Conf. Interval] npvis 52.27885 11.41406 4.58 0.000 29.89196 74.66575 npvissq -1.142647 .3590214 -3.18 0.001 -1.846811 -.4384821 monpre 35.25912 12.58328 2.80 0.005 10.57898 59.93927 male 79.38175 27.75667 2.86 0.004 24.94136 133.8221 mage -6.91257 3.137972 -2.20 0.028 -13.06721 -.757928 _cons 2648.851 137.2778 19.30 0.000 2379.602 2918.1 Comment on the significance of the coefficient on mage, compared to the results from part (b). Is your finding in line with your reasoning in part (b)? Does it confirm your friend’s claim? (d) Having regained trust in your friend, you consult your results once more with him. Together, you come up with an interesting question: whether smoking during pregnancy can affect the weight of the baby. Fortunately, you have at your disposition the variable cigs, standing for the average number of cigarettes each woman in your sample smokes per day during the pregnancy, and so you can include it in your model. However, your friend warns you that women who smoke during pregnancy are in general less responsible than those who do not smoke, and that these women also tend to visit the doctor less often. (In other words, the more the women smokes, the less prenatal doctor’s visits she has). This is an important fact that you have to take into consideration while interpreting your final results, which are: Textové pole: Source SS df MS Model 14560828.9 6 2426804.81 RESIDUAL 523281374 1615 324013.235 TOTAL 537842203 1621 331796.547 Textové pole: Number of obs = 1622 F( 6, 1615) = 7.49 Prob > F = 0.0000 R-SQUARED = 0.0271 Adj R-SQUARED = 0.0235 Root MSE = 569.22 bwght Coef. Std. Err. t P>|t| [95% Conf. Interval] npvis 42.43442 11.59582 3.66 0.000 19.68999 65.17885 npvissq -.8948737 .3624432 -2.47 0.014 -1.605782 -.1839653 monpre 31.77658 12.78156 2.49 0.013 6.706395 56.84676 male 82.39438 28.34937 2.91 0.004 26.78897 137.9998 mage -6.980738 3.227181 -2.16 0.031 -13.31064 -.6508356 cigs -10.209 3.398309 -3.00 0.003 -16.87456 -3.54344 _cons 2748.856 141.868 19.38 0.000 2470.591 3027.12 i. Interpret the coefficient on cigs. ii. What evidence do you find that cigs really should be included in the model? List at least two arguments. iii. Compare the coefficient on npvis with the one you obtained in part (c). Do you think there was a bias? If yes, explain where it came from and interpret its sign. ________________________________ [1] Stata is a statistical software, which can be used to for econometric purposes. The Stata output is quite similar to the Gretl output you are familiar with. In particular, Coef. denotes the estimated coefficients, Std.Err. denotes the standard errors of these coefficients, t denotes the t-statistic of the test of significance of the coefficients, P > |t| denotes the corresponding p-value.