Name: Studet Number: Midterm 2 ECO3151 Witer 2012 Istructios: 1. Prit your ame ad studet umber at the top of this midterm 2. No programmable calculators 3. You ca aswer i pecil or pe 4. This midterm cosists of 6 short aswer questios 5. Total marks: 50 1
Questio 1 a) Give the STATA output provided i Table 1, where prices is house price i thousads of dollars, bdrms is the umber of bedrooms, lotsize is the size of the lot i square feet, ad sqrft is the size of the house i square feet. a) Write the uderlyig populatio model. (2 marks) l(price) = β 0 + β 1 l(lotsize) + β 2 l(sqrt) + β 3 bdrms + u (1) b) Iterpret the bdrms coefficiet estimate? (3 marks) Addig oe bedroom (holdig other factors costat) will icrease the house price by 3.7% c) Is the lsqrf t coefficiet ecoomically sigificat? Justify your aswer. (3 marks) If the square footage icrease by 10%, the the price will go up by 7%. ecoomically sigificat, i.e. ecoomically importat. I would thik that this is d) Does lsqrf t have a statistically sigificat effect o lprice? Make sure to clearly state the ull ad alterative hypotheses ad the uderlyig idetifyig assumptios. (4 marks) H 0 : β 2 = 0 H a : β 2 > 0 Uder MLR1-6, we kow that sice =88 ad k=3. t stat t k 1 (2) t stat t 84 (3) t stat = 7.54 > 1.671 (5) therefore I reject H 0 i favor of H a at the 5% level of sigificace; square footage has a positive impact o the price of a home. (4) 2
0.5cm Questio 2 Let the true populatio model be represeted by the followig equatio y i = β 0 + β 1 x i + u i where β 1 is the causal parameter of iterest. Defie β 1 as β 1 = (z i z)y i (z where z i = l(1 + x 2 i ).Uder what coditio will β 1 cosistetly estimate β 1. Show your work. (5 marks) plim( β 1 ) = plim( (z i z)y (z ) (6) = plim( (z i z)(β 0 + β 1 x i + u i ) (z ) (7) = plim( (z i z)β 0 + (z i z)β 1 x i + (z i z)u (z ) (8) = plim( β 0 (z i z) + β 1 (z + (z i z)u (z ) (9) = plim(β 1 + (z i z)u (z ) (10) = plim(β 1 + (z i z)(u i ū)/ (z i z)(x i x)/ ) (11) = β 1 + E(z i E(z i ))(u i E(u i )) E(z i E(z i ))(x i E(x i )) (12) See class aswer for the ecessary coditios 3
Questio 3 Assume the followig populatio model log(wage) = β 0 + β 1 educ + β 2 exper + β 3 exper 2 + u (13) where wage is the hourly wage, educ years of educatio, ad exper years of work experiece. a) Educatio ad experiece are probably correlated. Is that problematic for carryig out a hypothesis test of whether educatio affects the wage? Explai your aswer (3 marks) As log as there is o perfect colliearity oe ca still carry out a t-test b) Give the STATA output i Table 2, test whether experiece matters (statistically speakig) i the model. Make sure to clearly state the ull ad alterative hypotheses ad the uderlyig idetifyig assumptios. (5 marks) H a : H 0 : β 2 = β 3 = 0 H 0 doesothold Uder MLR1-6, we kow that F stat F q, k 1 (14) F stat F 2,522 (15) (16) sice =526 ad k=3. F stat = ((0.3003.1858)/2)/(1 0.3003)/522) = 42.7 > 3.07 (17) therefore I reject H 0 i favor of H a at the 5% level of sigificace; experiece matters. 4
Questio 4 It was argued that havig a large sample ca be beeficial. explai. (5 marks) A large sample will mea that you will have small stadard errors. This is useful to carry out test. It also meas that if your estimator is cosistet, the large sample will mea that the odds of gettig a guess that is close to the true are good. Questio 5 Let the true populatio model be represeted by the followig equatio y i = β 0 + β 1 x i + u i (18) where β 1 is the causal parameter of iterest. For each idividual i (i the sample of size ) you ca observe x i, but ot y i. You ca, however, observe yi where y i = y i + ɛ i (19) ad where ɛ is a oise term that is uobserved. Uder what coditios would regressig y o x provide a cosistet estimator of β 1? (5marks) You would still the origial MLR1-4 (where MLR-4 is E(x u) = 0). You would also eed that the oise ot be correlated with the x. 5
Questio 6 Assume the followig model y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + u (20) where β 1, β 2 ad β 3 are the causal parameters of iterest. Evaluate the followig claims: i) The fact tha ˆβ 2 has a upward bias, i.e. E( ˆβ 2 ) > β 2 meas that the OLS coefficiet estimate will always be greater tha β 2. (5 marks) Not true. the actual guess could be above. below or equal to the true oe. All you kow is that o average your guesses will be high. ii) The presece of multicoliearity betwee x 2 ad x 3 (but ot perfectly correlated) meas that oe caot cosistetly estimate β 2. (5 marks) No, colliearity does ot ivalidate the MLR1-4 to hold. iii) If ˆβ 2 is cosistet (for β 2, it will geerate a accurate guess of β 2. (5 marks) o, cosistecy meas that the odds of beig close to the true populatio parameter get better ad better as gets large ad larger. So the odds of beig close are good if you have a large sample. Equatio Sheet I the simple liear regressio model where x is the explaatory variable, ad y the depedet variable, ˆβ 0 = ȳ ˆβ 1 = ˆβ ˆσ se( ˆβ 1 ) = SSTx 1 x (x i x)(y i ȳ) (x i x) 2 I the multiple liear regressio model R 2 = SSE/SST se( ˆβ j ) = ˆσ (SST j (1 R 2 j ))1/2 SST y = (y i ȳ) 2, SSR y = (y i ŷ) 2, SSE y = (ŷ i ȳ) 2 ˆσ 2 = ( û2 i )/( k 1) V ar(z) = E(z E(z))(z E(z)) = E(z 2 ) E(z) 2 Cov(z, y) = E(z E(z))(y E(y)) = E(zy) E(z)E(y) F = (R 2 ur R 2 u)/q (1 R 2 ur)/ k 1 6