Sample Exam for EC 306 BRIEF SKETCHES OF ASWERS GIVE I BOLD Ths exam contans four questons. Please do all questons. 1. Emprcal Practce (0%) In a study of costs n the bankng ndustry, data has been collected for 85 banks n Amerca. Some banks specalze n lendng money to households, other banks specalze n lendng money to busnesses. Smlar ssues hold wth respect to ther depostors. I am nterested n nvestgatng whether these bank characterstcs affect ther costs. Accordngly, I defne my dependent and explanatory varables as follows: Y total costs per employee (n thousands of dollars per year). X 1 proporton of total loans whch go to busnesses (measured as a percentage so that a value of, say, 0 means 0% of loans are made to busnesses). X proporton of total deposts whch come from households (measured as a percentage so that a value of, say, 0 means 0% of deposts come from households). D a dummy varable whch equals 1 f the bank s a bg bank (has more than 100 employees), 0 otherwse. I also constructed another varable, Z X D. a) I ran a regresson of Y on X 1, X, D and Z. Results from ths regresson are gven below n the followng ftted regresson lne: Y 960-109 X 1 + 10 X - 1.49 D -3 Z (6 10-7 ) (0.008) (0.04) (0.03) (0.00) where the numbers n parentheses are P-values for testng the hypothess that the coeffcent equals zero. ) How would you nterpret (n words) the estmated coeffcents n ths model? What s the OLS estmate of the margnal effect of X on Y? The nterpretaton of any coeffcent s: f the explanatory varable changes by one unt, then the dependent varable tends to change by [nsert coeffcent here] unts, holdng other explanatory varables constant (gettng unts rght and ceters parbus dea s mportant for frst class grade) For dummy varable (D) ths can be refned n the usual way: ndvduals wth D1 have a regresson wth ntercept 1.49 less than ndvduals wth D0 or n the lectures I used expected value operator to wrte out expected Y for D1 and 0.
The presence of the nteracton varable means the margnal effect of X on Y (ceters parbus) s dfferent for companes wth D0 versus D1. To be precse, ths margnal effect s 97 for bg banks and 10 for small banks. ) Whch of the statements you have just made are statstcally sgnfcant at the 5% level? Whch are sgnfcant at the 1% level? All explanatory varables are sgnfcant at the 5% level, only ntercept X1 and Z are sgnfcant at 1% level. b) I then dd a Whte test (usng X as the ndependent varable to explan the heteroskedastcty) and found a test statstc value of 5.0 (wth a p-value of.05). I re-ran the prevous regresson usng a heteroskedastcty consstent estmator (HCE) and obtaned: Y 960-109 X 1 + 10 X - 10,449 D -03 Z (4 10-5 ) (0.03) (0.067) (0.04) (0.005) where the numbers n parentheses are P-values for testng the hypothess that the coeffcent equals zero. ) When presentng fnal results n a project, would you use my OLS results of part a) or my HCE results of part b)? Why? Snce the p-value for the heteroskedastcty test s less than.05, we accept the hypothess that heteroskedastcty s present. Therefore, the varance of the OLS estmator n part a) was ncorrect (and, although estmates were unbased, p- values were wrong). So HCE (whch uses the correct formula for the varance) wll be better and should be used.
. Econometrc Theory: Dervatons and Proofs (30%) a) Consder the smple regresson model wth a sngle explanatory varable under the assumptons (for 1,..,): y βx + ε E ε var ( ) 0 ( ε ) σ ω and the errors are uncorrelated wth one another. X s not a random varable. The OLS estmator for β s gven by: ˆβ 1 1 X y X ) Calculate the expected value of the OLS estmator. Is ths estmator unbased? THIS IS STADARD DERIVATIO DOE I LECTURES. WRITE ESTIMATOR AS ˆ β 1 β + 1 X ε X TAKE EXPECTED VALUE OF BOTH SIDES AD OTE THAT, SICE X IS OT RADOM, THE EXPECTED VALUE OF THE LAST TERM IS ZERO. HECE OLS IS UBIASED (EVE UDER HETEROSKEDASTICITY). ) Calculate the varance of ths estmator. What does the Gauss-Markov theorem tell us about the sze of ths varance relatve to the sze of the varance of the generalsed least squares (GLS) estmator? TAKE VARIACE OPERATOR OF BOTH SIDES OF EQUATIO ABOVE, YOU GET Var ˆ β 1 X σ 1 ( X ) GAUSS MARKOV THEOREM TELLS US (UDER THE ASSUMPTIOS OF THIS QUESTIO) THE GLS ESTIMATOR HAS THE SMALLEST VARIACE OF ALL LIEAR UBIASED ESTIMATORS. SICE OLS IS LIEAR AD UBIASED, IT MUST HAVE A LARGER VARIACE THA GLS.
b) Usng the setup, assumptons and defntons gven for part a) except that now there s no heteroskedastcty so that var(ε )σ. Suppose an estmator for β s gven by: ~ 1X y β 3 1 X ) Calculate the expected value of ths estmator. Is t an unbased estmator of β? DESPITE THE DIFFERET SETUP, THE PROOF IS ACTUALLY QUITE SIMILAR TO THAT TO PART A, ). I WILL LEAVE YOU TO WORK IT OUT (IT IS A UBIASED ESTIMATOR) ) Is ths estmator more effcent that the OLS estmator? O. THE GAUSS MARKOV THEOREM TELLS US THIS (SEE ASWER TO PART A,))
3. Understandng Econometrc Theory (5%) a) Defne the term multcollnearty and explan ts mportance for emprcal practce.. TEXTBOOK MATERIAL. WRITE A (VERY BRIEF) ESSAY SUMMARIZIG THIS MATERIAL. b) Defne the term nstrumental varable. Explan the mportance of ths concept for regresson analyss. TEXTBOOK MATERIAL. WRITE A (VERY BRIEF) ESSAY SUMMARIZIG THIS MATERIAL. 4. Tme Seres Econometrcs (5%) I have collected data on two tme seres varables, X t and Y t and run varous regressons usng ths data. Excel outputs for these regressons are below and labelled as OUTPUT 1, OUTPUT, OUTPUT 3 and OUTPUT 4. To be specfc: OUTPUT 1 contans results from a regresson of ΔY on one lag of Y. That s, ΔY t α + β Yt 1 + et. OUTPUT contans results from a regresson of ΔX on one lag of X. OUTPUT 3 contans results from the smple regresson of Y on X. OUTPUT 4 takes the resduals, e, from the regresson of Y on X (.e. the one n OUTPUT 3) and regresses Δe on one lag of e. ) Defne and descrbe the Dckey-Fuller test. Can ths test be done usng any of the OUPTPUTS above? If yes, what does the Dckey-Fuller test tell you about the propertes of Y and Y? You may assume that the 5% crtcal value for the Dckey-Fuller test s -.89. THE DICKEY FULLER TEST IS DESCRIBED O PAGES 77-88 OF THE TEXTBOOK. OUTPUTS 1 AD DO COTAI RELEVAT REGRESSIOS. SICE THE T-STATS ARE SMALL (SMALLER THA THE DICKEY FULLER CRITICAL VALUE METIOED O PAGE 80) I BOTH CASES WE CA COCLUDE THAT UIT ROOTS ARE PRESET I BOTH X AD Y. ) Defne and descrbe the Engle-Granger test for contegraton. Does contegraton seem to be present n ths data set? You may assume that the 5% crtcal value for the Engle-Granger test s -3.39.
COITEGRATIO TESTIG IS DICUSSED BEGIIG O PAGE 31 OF THE TEXTBOOK. OUTPUT 4 CA BE USED TO DO THE EGLE GRAGER TEST IS. COMPARIG -11.7749 TO THE EGLE-GRAGER CRITICAL VALUE OF -3.33 WE CA REJECT THE HYPOTHESIS THAT THE ERRORS HAVE A UIT ROOT. THUS COITEGRATIO IS PRESET ) Can you obtan an estmate of the long run multpler from any of these OUTPUTS? If yes, what s the estmate of the long multpler? SICE X AD Y ARE COITEGRATED, OUTPUT 3 CA BE USED TO GIVE US A MULTIPLIER OF 1.93891. OTE, HOWEVER, THAT IF X AD Y WERE OT COITEGRATED, THE OUTPUT 3 WOULD HAVE BEE A SPURIOUS REGRESSIO AD WE WOULD OT HAVE BEE ABLE TO USE IT TO CALCULATE THE MULTIPLIER.
OUTPUT 1 Regresson Statstcs Multple R 0.100336 R Square 0.010067 Adjusted 0.003957 R Square Standard 0.149963 Observat 164 ons AOVA df SS MS F Sgnfcance F Regresso n 1 0.03705 0.03705 1.647497 0.01133 Resdual 16 3.6430 0.0489 Total 163 3.68053 Coeffcent s Standard t Stat P-value Lower Lower Intercept 0.09069 0.148738 1.40565 0.16175-0.08465 0.50784-0.08465 0.50784 Y-lagged -0.01519 0.011833-1.8355 0.01133-0.03856 0.008179-0.03856 0.008179 Regresson Statstcs Multple R 0.071587 R Square 0.00515 Adjusted -0.0010 R Square Standard 0.010183 Observat 164 ons OUTPUT AOVA df SS MS F Sgnfcance F Regresso n 1 8.65E-05 8.65E-05 0.834485 0.36336 Resdual 16 0.016798 0.000104 Total 163 0.016885 Coeffcent s Standard t Stat P-value Lower Lower Intercept 0.011895 0.000 5.400638.33E-07 0.007545 0.01644 0.007545 0.01644 X-lagged -0.00148 0.0016-0.9135 0.36336-0.00468 0.001719-0.00468 0.001719
Regresson Statstcs Multple R 0.993897 R Square 0.987831 Adjusted 0.987755 R Square Standard 0.10946 Observat 164 ons OUTPUT 3 AOVA df SS MS F Sgnfcance F Regresso 1 157.4576 157.4576 13149.98 5.1E-157 n Resdual 16 1.939785 0.011974 Total 163 159.3974 Coeffcent s Standard t Stat P-value Lower Lower Intercept 9.99487 0.03857 418.9481 8.8E-48 9.947759 10.04198 9.947759 10.04198 X 1.93891 0.017434 114.6734 5.1E-157 1.964787.033641 1.964787.033641 Regresson Statstcs Multple R 0.6804 R Square 0.46705 Adjusted 0.459368 R Square Standard 0.10957 Observat 163 ons OUTPUT 4 AOVA df SS MS F Sgnfcance F Regresso 1 1.664557 1.664557 138.6493 1.75E-3 n Resdual 161 1.93888 0.01006 Total 16 3.597445 Coeffcent s Standard t Stat P-value Lower Lower Intercept -0.00013 0.008583-0.01468 0.988308-0.01708 0.01684-0.01708 0.01684 Resd(-1) -0.9397 0.079805-11.7749 1.75E-3-1.0973-0.781-1.0973-0.781