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? The p-value on the coefficient on npvissq is very small, and hence the variable is strongly significant and should be included in the model. ii. How do you interpret the negative coefficient of npvissq? The negative coefficient on npvissq signals a concave form of the impact of the number of prenatal doctor’s visits, meaning that there are decreasing returns to visiting the doctor. A possible explanation is that some number of visits is beneficiary for all pregnant women, but higher necessity of visits could mean that the pregnancy is risky for some reasons and the woman has to go to the doctor more often than usually. Such woman is also more likely to have smaller baby. iii. Holding npvis and monpre fixed, test the hypothesis that newborn boys weight by 100 grams more than newborn girls (at 95% confidence level). Such hypothesis can be stated as Test statistic =-1.96. Therefore, we failed to reject the null hypothesis that newborn boys weigh 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 separately: are they in line with your friend’s claim? When we look on the p-values of the corresponding coefficients, we see that whereas fage is significant at 99% confidence level, mage is insignificant. This is not in line with our friend’s claim, who says that especially the age of the mother should be an important factor. 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? To test joint significance, we need restricted and unrestricted models. In the regression in part (b) we have included mage and fage while they are not included in the regression in part (a). Therefore, we can use SSR from both regression outputs in order to judge the joint significance of the mage and fage variables. According to output in part (a) SSR[r]=570003184, According to output in part (b) SSR[ur]=563258231. We construct F test based on the formula: , where q is the number of restrictions in this case q=2 (mage and fage) and df is degrees of freedom. Df=n-k-1=1720-7 Therefore, in the F-table we will find a critical value at 5% it will be . 10.36>3, hence, we can reject the null hypothesis and we conclude that mage and fage are jointly significant. iii. How can you reconcile you findings from the two previous questions? The finding about the joint significance from the second question is not surprising, since we know already from the first question that fage is individually significant. If a variable is significant, then the H[A] of the test of the joint significance has to be valid and so the variables have to be jointly significant. 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? Now, the p-value of the coefficient on mage is very low and so the coefficient is strongly significant. When we compare this finding to part (b), we realize that the insignificance of this coefficient in that part was probably given by a strong correlation between mage and fage, leading to the multicollinearity problem, which increases the standard errors and decreases thus the significance of the coefficients. When we drop fage, the multicollinearity problem is solved and we see that our friend’s claim was true. 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. The coefficient on cigs tells us that with each additional cigarette smoked by the pregnant woman on average per day, the weight of the baby is smaller by 10 grams, ceteris paribus. ii. What evidence do you find that cigs really should be included in the model? List at least two arguments. We can see from the p-value that the coefficient on cigs is strongly significant. We can also see that the R^2 as well as the adjusted R^2 are higher than in the model without this variable (in part (c)). Moreover, we see that the coefficient on npvis has changed quite a lot once we included cigs, which is a signal of an omitted variable bias in part (c) and a proof that cigs indeed should be included in the model. 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. In part (c), the coefficient on npvis was approximatively equal to 52, now it is equal to 42. This shows there was a positive bias in part (c): the coefficient was overestimated there. We know that the sign of this bias is the sign of the product of two correlations: the correlation between the omitted variable cigs and the variable npvis and the correlation between cigs and the dependent variable bwght. The correlation between cigs and the dependent variable bwght is negative as we can see from the negative coefficient on cigs in the model estimated in part (d), the correlation between cigs and npvis is negative as we learn from our friend (women who smoke tend to visit the doctor less often). The product of these two correlations is thus positive and so is the bias in part (c). Intuitively, we can say that when cigs was omitted, everything that could measure the degree of responsibility of pregnant women in our model was the variable npvis. Once we included cigs, we can measure separately the responsibility of going to the doctor and the responsibility of not smoking, and so the coefficient on npvs is reflecting only the correct part of this influence and it is not overestimated. ________________________________ [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.