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SOLUTIONS TO PROBLEMS, 2 1 i Income age and family background such as number of siblings are just a few. possibilities It seems that each of these could be correlated with years of education Income. and education are probably positively correlated age and education may be negatively correlated. because women in more recent cohorts have on average more education and number of siblings. and education are probably negatively correlated, ii Not if the factors we listed in part i are correlated with educ Because we would like to. hold these factors fixed they are part of the error term But if u is correlated with educ then. E u educ 0 and so SLR 4 fails, 2 2 In the equation y 0 1x u add and subtract 0 from the right hand side to get y 0. 0 1x u 0 Call the new error e u 0 so that E e 0 The new intercept is 0. 0 but the slope is still 1, 2 3 i Let yi GPAi xi ACTi and n 8 Then x 25 875 y 3 2125 xi x yi y. 5 8125 and xi x 2 56 875 From equation 2 9 we obtain the slope as 1. 5 8125 56 875 1022 rounded to four places after the decimal From 2 17 0 y 1 x. 3 2125 1022 25 875 5681 So we can write,GPA 5681 1022 ACT.
The intercept does not have a useful interpretation because ACT is not close to zero for the. population of interest If ACT is 5 points higher GPA increases by 1022 5 511. ii The fitted values and residuals rounded to four decimal places are given along with. the observation number i and GPA in the following table. i GPA GPA u,1 2 8 2 7143 0857,2 3 4 3 0209 3791,3 3 0 3 2253 2253. 4 3 5 3 3275 1725,5 3 6 3 5319 0681,6 3 0 3 1231 1231. 7 2 7 3 1231 4231,8 3 7 3 6341 0659, You can verify that the residuals as reported in the table sum to 0002 which is pretty close to. zero given the inherent rounding error,5681 1022 20 2 61. iii When ACT 20 GPA, iv The sum of squared residuals u i2 is about 4347 rounded to four decimal places.
and the total sum of squares yi y 2 is about 1 0288 So the R squared from the regression. R2 1 SSR SST 1 4347 1 0288 577, Therefore about 57 7 of the variation in GPA is explained by ACT in this small sample of. 2 4 i When cigs 0 predicted birth weight is 119 77 ounces When cigs 20 bwght 109 49. This is about an 8 6 drop, ii Not necessarily There are many other factors that can affect birth weight particularly. overall health of the mother and quality of prenatal care These could be correlated with. cigarette smoking during birth Also something such as caffeine consumption can affect birth. weight and might also be correlated with cigarette smoking. iii If we want a predicted bwght of 125 then cigs 125 119 77 524 10 18 or. about 10 cigarettes This is nonsense of course and it shows what happens when we are trying. to predict something as complicated as birth weight with only a single explanatory variable The. largest predicted birth weight is necessarily 119 77 Yet almost 700 of the births in the sample. had a birth weight higher than 119 77, iv 1 176 out of 1 388 women did not smoke while pregnant or about 84 7 Because we. are using only cigs to explain birth weight we have only one predicted birth weight at cigs 0. The predicted birth weight is necessarily roughly in the middle of the observed birth weights at. cigs 0 and so we will under predict high birth rates. 2 5 i The intercept implies that when inc 0 cons is predicted to be negative 124 84 This of. course cannot be true and reflects that fact that this consumption function might be a poor. predictor of consumption at very low income levels On the other hand on an annual basis. 124 84 is not so far from zero, ii Just plug 30 000 into the equation cons 124 84 853 30 000 25 465 16 dollars. iii The MPC and the APC are shown in the following graph Even though the intercept is. negative the smallest APC in the sample is positive The graph starts at an annual income level. of 1 000 in 1970 dollars,1000 10000 20000 30000, 2 6 i Yes If living closer to an incinerator depresses housing prices then being farther away.
increases housing prices, ii If the city chose to locate the incinerator in an area away from more expensive. neighborhoods then log dist is positively correlated with housing quality This would violate. SLR 4 and OLS estimation is biased, iii Size of the house number of bathrooms size of the lot age of the home and quality of. the neighborhood including school quality are just a handful of factors As mentioned in part. ii these could certainly be correlated with dist and log dist. 2 7 i When we condition on inc in computing an expectation inc becomes a constant So. E u inc E inc e inc inc E e inc inc 0 because E e inc E e 0. ii Again when we condition on inc in computing a variance inc becomes a constant So. Var u inc Var inc e inc inc 2Var e inc e2 inc because Var e inc e2. iii Families with low incomes do not have much discretion about spending typically a. low income family must spend on food clothing housing and other necessities Higher income. people have more discretion and some might choose more consumption while others more. saving This discretion suggests wider variability in saving among higher income families. 2 8 i From equation 2 66,Plugging in yi 0 1xi ui gives. 1 xi 0 1 xi ui xi2, After standard algebra the numerator can be written as. 0 xi 1 x 2 xi ui i,i 1 i 1 i 1, Putting this over the denominator shows we can write 1 as.
1 0 xi xi2 1 xi ui xi2,i 1 i 1 i 1 i 1,Conditional on the xi we have. E 1 0 xi xi2 1, because E ui 0 for all i Therefore the bias in 1 is given by the first term in this equation. This bias is obviously zero when 0 0 It is also zero when x. i 0 which is the same as x, 0 In the latter case regression through the origin is identical to regression with an intercept. ii From the last expression for 1 in part i we have conditional on the xi. Var 1 xi2 Var xi ui xi2 xi2 Var ui,i 1 i 1 i 1 i 1. xi2 2 xi2 2 xi2,i 1 i 1 i 1, iii From 2 57 Var 1 2 xi x 2 From the hint xi2 x x i.
i 1 i 1 i 1, Var 1 Var 1 A more direct way to see this is to write xi x 2. i n x 2 which,is less than x,i unless x 0, iv For a given sample size the bias in 1 increases as x increases holding the sum of the. xi2 fixed But as x increases the variance of 1 increases relative to Var 1 The bias in 1. is also small when is small Therefore whether we prefer or on a mean squared error. basis depends on the sizes of 0 x and n in addition to the size of xi 1. 2 9 i We follow the hint noting that c1 y c1 y the sample average of c1 yi is c1 times the. sample average of yi and c2 x c2 x When we regress c1yi on c2xi including an intercept we. use equation 2 19 to obtain the slope,c2 xi c2 x c1 yi c1 y c1c2 xi x yi y. c22 xi x 2, From 2 17 we obtain the intercept as 0 c1 y 1 c2 x c1 y c1 c2 1 c2 x c1. y 1 x c1 0 because the intercept from regressing yi on xi is y 1 x. ii We use the same approach from part i along with the fact that c1 y c1 y and. c2 x c2 x Therefore c1 yi c1 y c1 yi c1 y yi y and c2 xi. c2 x xi x So c1 and c2 entirely drop out of the slope formula for the regression of c1. yi on c2 xi and The intercept is c y c x c1 y c2,1 1 0 1 1 2 1.
x y 1 x c1 c2 1 0 c1 c2 1 which is what we wanted to show. iii We can simply apply part ii because log c1 yi log c1 log yi In other words. replace c1 with log c1 yi with log yi and set c2 0. iv Again we can apply part ii with c1 0 and replacing c2 with log c2 and xi with log xi. If 0 and 1 are the original intercept and slope then 1 1 and. 0 0 log c2 1, 2 10 i This derivation is essentially done in equation 2 52 once 1 SSTx is brought inside. the summation which is valid because SSTx does not depend on i Then just define. wi di SSTx,ii Because Cov, 1 u E 1 1 u we show that the latter is zero But from part i. w E ui u Because the ui are pairwise uncorrelated, they are independent E ui u E ui2 n 2 n because E ui u h 0 i h Therefore. wi E ui u w 2 n,n i 1 wi 0, iii The formula for the OLS intercept is 0 y x and plugging in y 0 1 x u. gives 0 0 1 x u 1 x 0 u 1 1 x,iv Because 1 and u are uncorrelated.
Var u Var 1 x 2,2 n 2 SSTx x 2,2 n 2 x 2 SSTx,which is what we wanted to show. v Using the hint and substitution gives Var 0 2 SSTx n x 2 SSTx. x 2 SSTx 2 n 1 i 1 xi2 SSTx, 2 11 i We would want to randomly assign the number of hours in the preparation course so that. hours is independent of other factors that affect performance on the SAT Then we would. collect information on SAT score for each student in the experiment yielding a data set. sati hoursi i 1 n where n is the number of students we can afford to have in the study. From equation 2 7 we should try to get as much variation in hoursi as is feasible. ii Here are three factors innate ability family income and general health on the day of the. exam If we think students with higher native intelligence think they do not need to prepare for. the SAT then ability and hours will be negatively correlated Family income would probably be. positively correlated with hours because higher income families can more easily afford. preparation courses Ruling out chronic health problems health on the day of the exam should. be roughly uncorrelated with hours spent in a preparation course. iii If preparation courses are effective 1 should be positive other factors equal an. increase in hours should increase sat, iv The intercept 0 has a useful interpretation in this example because E u 0 0 is the. average SAT score for students in the population with hours 0. 2 12 i I will show the result without using calculus Let be the sample average of the and. y y 2 yi y y b0 y b0 2,yi y 2 2 y b0 yi y n y b0 2. where we use the fact see Appendix A that y y,0 always The first term does not.
depend on b0 and the second term n y b0 2 which is nonnegative is clearly minimized when. ii If we define u,y y and we already used the fact that this sum. is zero in the proof in part i,SOLUTIONS TO COMPUTER EXERCISES. C2 1 i The average prate is about 87 36 and the average mrate is about 732. ii The estimated equation is,prate 83 05 5 86 mrate. n 1 534 R2 075, iii The intercept implies that even if mrate 0 the predicted participation rate is 83 05. percent The coefficient on mrate implies that a one dollar increase in the match rate a fairly. large increase is estimated to increase prate by 5 86 percentage points This assumes of. course that this change prate is possible if say prate is already at 98 this interpretation makes. 83 05 5 86 3 5 103 59, iv If we plug mrate 3 5 into the equation we get prate.
This is impossible as we can have at most a 100 percent participation rate This illustrates that. especially when dependent variables are bounded a simple regression model can give strange. predictions for extreme values of the independent variable In the sample of 1 534 firms only. 34 have mrate 3 5, v mrate explains about 7 5 of the variation in prate This is not much and suggests that. many other factors influence 401 k plan participation rates. C2 2 i Average salary is about 865 864 which means 865 864 because salary is in thousands. of dollars Average ceoten is about 7 95, ii There are five CEOs with ceoten 0 The longest tenure is 37 years. iii The estimated equation is,salary 6 51 0097 ceoten. n 177 R2 013, We obtain the approximate percentage change in salary given ceoten 1 by multiplying the. coefficient on ceoten by 100 100 0097 97 Therefore one more year as CEO is predicted. to increase salary by almost 1,C2 3 i The estimated equation is.
sleep 3 586 4 151 totwrk,n 706 R2 103, The intercept implies that the estimated amount of sleep per week for someone who does not. work is 3 586 4 minutes or about 59 77 hours This comes to about 8 5 hours per night. ii If someone works two more hours per week then totwrk 120 because totwrk is. measured in minutes and so sleep 151 120 18 12 minutes This is only a few minutes. a night If someone were to work one more hour on each of five working days sleep. 151 300 45 3 minutes or about five minutes a night. C2 4 i Average salary is about 957 95 and average IQ is about 101 28 The sample standard. deviation of IQ is about 15 05 which is pretty close to the population value of 15. ii This calls for a level level model,wage 116 99 8 30 IQ. n 935 R2 096, An increase in IQ of 15 increases predicted monthly salary by 8 30 15 124 50 in 1980. dollars IQ score does not even explain 10 of the variation in wage. iii This calls for a log level model,wage 5 89 0088 IQ. n 935 R2 099, wage 0088 15 132 which is the approximate proportionate.
If IQ 15 then log, change in predicted wage The percentage increase is therefore approximately 13 2. C2 5 i The constant elasticity model is a log log model. log rd 0 1 log sales u, where 1 is the elasticity of rd with respect to sales. ii The estimated equation is,rd 4 105 1 076 log sales. n 32 R2 910, The estimated elasticity of rd with respect to sales is 1 076 which is just above one A one. percent increase in sales is estimated to increase rd by about 1 08. C2 6 i It seems plausible that another dollar of spending has a larger effect for low spending. schools than for high spending schools At low spending schools more money can go toward. purchasing more books computers and for hiring better qualified teachers At high levels of. spending we would expend little if any effect because the high spending schools already have. estimator is not unbiased I do not try to explain these subtleties in an introductory course but I have had instructors ask me about the difference Solutions Manual for Introductory Econometrics A Modern Approach 5th Edition by Wooldridge