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ac18781 - THE UNIVERSITY OF HONG KONG School of Economics...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG School of Economics 81. Finance 2001-2002 1st Semester Examination Economics: ECON0701 Introductory Econometrics Dr P Lau December 2'7, 2001 9:30-11:30 AM Candidates may use any self—contained, silent, battery-operated and pocket-sized calculator. The calcu- lator should have numeral-display facilities only and should be used only for the purposes of calculation. It is the candidate’s responsibility to ensure that his calculator operates satisfactorily. Candidates must record the name and type of their calculators on the front page of their examination scripts. Some critical values for the t—distribution and the F-distribution are given in the last two pages. (If the critical value you want is not shown, use the closest one.) Answer any four questions. (If you answer all five questions, only the first four questions answered will be graded.) All parts of a question are equally weighted unless specified otherwise. Please give logical, concise and precise answers. Marks will be deducted for inadequacy as well as redundancy or irrelevancy. Failure to answer one part will generally not prevent you from able to answer subsequent parts. 1. (25%) Give a brief answer (of a few sentences) to the following short questions: a What is the distinction between a ‘ o ulation regression line’ and a ‘sample re ession line’ ? P P 8T (b) Do we typically observe significant t-values but an insignificant analue when colinearity is present? (o) Is the slope term in a simple regression model more precisely estimated (by the least squares method) when the values of the explanatory variable are less spread out? (d) According to the econometric software output for a two-sided test of Ho : ,62 = 0 against H1 : fig # 0, the p value is 0.082. What is the meaning of p value? Based on the above p value, how would you decide, at a 5% level of significance, regarding the one-sided test of H0 : 32 = 0 against H1 '. fig :3 0? (e) Give an exarnple of a regression equation which is ‘non-linear in parameter’ and an example of a regression equation which is ‘non-linear in variable but linear in parameter’. Does the term ‘linear regression model’ refer to ‘Iinearity in parameter’ or ‘linearity in variable’ ? 2. (25%) For each of the following statements, indicate whether it is true or false, and give a brief explanation. (a) The lease squares estimate for the slope coefficient of a simple regression model is b2 = 0.25 for a particular sample. Since it is obtained by the least squares method, the above estimate is unbiased. (b) The requirement for a test statistic is that its probability distributions under both the null and alternative hypotheses must be known. (c) The slope of the simple regression model 111(y) = 61 + £32 In (a) + s (where 1n represents natural log) indicates the percentage change of the dependent variable when the explanatory variable changes by one percent. (d) It is easy to identify the collinearity problem by checking the pairwise correlation between any two explanatory variables. (e) John and Tom work together on a simple regression model involving an intercept dummy variable. John is interested in the slope coeficient and Tom in the coeficient of the dununy variable. Since the variables that John and Tom focus are different (one is a continuous variable and the other is a discrete variable), the estimation and testing procedures they use are also different. 3,. A simple econometric model relating the quantity demanded of chocolate (q) to the price of chocolate (p), the price of other goods and services (r) and income (is) is as follows: 1M4)=fi1+52111®l+33111(rl+541niwl+51 (1) where in represents natural log. a is the random error term, and the standard assumrations of the multiple regression model are satisfied. The estimation result based on a sample of 30 observations is as follows: __...--h--.. ln(q) = —2.339 - 1.130 x 1n(p) + 0.195 x 111(r) + 0.733 x 111(11):), (3.053) (0.215) (0.079) (0.374) with residual sum of squares = 0.0938, and standard deviation of dependent variable = 0.133. (a) (4%) What is the interpretation of 62 and [34 in (I). (b) (5%) Construct a 95% interval estimate for fig. (c) (5%) Find R2, the coefficient of determination. (d) (5%) Test the joint hypothesis H0 : [32 = (:33 = ii; = 0. What is your conclusion? (e) (6%) Describe how to test the null hypothesis of no money illusion versus the alternative hypothesis that money illusion is present. 4. Consider the following multiple linear regression model: y=fi1+fl2$2+fi3$3+5v (1) where the standard assmnptions are satisfied. (a) Let be and b3 denote respectively the least squares estimators of [32 and 83. Express ear (b2 -- 4b3) in terms of the variances of ()2 and 63 and the covariance between them. (b) What is the t-statistic for testing the null hypothesis H0 : )6; —-— 433 = 1 against H1 : fig —4fi3 at 1? In what way is the test statistic related to your answer in part (a)? (0) Define 6 = fig —- 4:33. Transform Equation (1) into a regression equation involving parameter 6', [31 and {33. (d) Describe how you would use the transformed regression equation in part (c) to directly test the null hypothesis Ho : ,8? -—— 4fi3 = 1 against H; : A92 -- 4,63 at 1. Specify the t—statistic and the rejection region (at 5% level of significance) for this hypothesis in terms of '5 (the least squares estimator of 9) and its standard error. (e) Describe an alternative procedure to test the hypothesis in part (d). 5. It is common to observe that new releases of compact discs are priced lower than older CDs. One way to represent the above relationship is: Price = a. + agree + a. where Price retail price of CD, Age = age of the recording. We have information on variables Price and Age of CDs sold through interact and through traditional ways. For 46 observations of ODs sold through traditional ways, we have the following estimation result: P323 = 15.076 + 0.493 x Ago (0.221) (0.098) where T = 46, R2 = 0.367, and residual sum of squares (RSSNI) = 41.71. For 72 observations of CDs sold through internet, we have the following estimation result: FEE = 13.343 + 0.130 x Age (0.284) (0.125) where T = 72, R2 = 0.015, and standard error of regression (3) = 1.574. (a) (4%) Calculate the residual sum of squares for the second sample (R551). (b) (6%) We are interested to know whether there is any systematic djfi‘erence in the pricing rela- tionship of CDs sold through internet versus traditional ways. If you are allowed to run only one more regression, describe how you would do. State the null hypothesis and give the test statistic. (c) (5%) The hypothesis in part (b) can also be tested by a dummy variable approach. Describe how you would carry out such a test. (d) (4%) Are the test procedures suggested in parts (b) and (c) equivalent? Ehcplain. (e) (6%) Test, at a 10% significance level, the null hypothesis of equal error variances for the two samples against the alternative that the error variances are difl'erent. In light of your result, comment on the method you suggested in part (b). ***** END OF PAPER ##3##“le Introductory Econometrics 2001-2002 1. Probability and statistics 5 if X is a discrete random variable that can take the values m1, ..., so, with probability density f (ml), f (m2), ..., f (33“) respectively, then the expected value of X: EUQ = 2113f (fl?) = 2155f ($21) = 931f($1) +$2f($2) + +$nf($a) 11’! i=1 9. if X is a discrete random variable and g (X) is a function of it, then E[9(X)l=29(fl=)f(w) a. variance = expected value of the squared deviations around the expected value of X nor (X) = E [X — E (.74012 y if X is a normally distributed random variable with mean a and variance a”, symbolized as X m N (a, :72), then its p.d.f. is expressed mathematically as: 1 - (a: — :02] f(m)=W9xP[ 2J2 for—oo<s:<co 2. Simple linear regression model 9 = 131 + 132:1: + e AOLS AOLS I. the least squares estimators of £31 and fig, denoted by {3, and ,62 (or simply, b1 and b2), are given by: T T T b T (2:31 55:31:) — (2&1 mt) (2&1 gt) 231:1 (mt — E) (3;; ~— ":9") 2 = W'_‘ "—2 = T -- 2 T (2:1 11:?) -* (2;, 33;) 21:1 (3t — 35') _ 23:1 yt 23:1 flit __ .. _,. bl * T 52 T — y ‘- I32:17 I if the regression model assumptions (SLRI) to (SLR4) are correct, then the variances and covariance of .51 and be are: W) __. as [__.1_.._] 23:1 (“it ‘* —)2 5 an unbiased estimator of 02 is o. if we make the additional assumption that the random errors at are normally distributed, then in and b2 are normal: bj~N( j,ver(bj)), j: 1,2 c the random variable t bi " fij 36’ (ha) t = N It(tr—52);; where as (bi) is the standard error of hi I a (1 — or) x 100% confidence interval for 55,- is [bj — tcse (bi) ,le + $536 lbs“ I: the least squares predictor of go is so = b1 + 5255's a the variance of the forecast error is 1 ($0 — E 2 wen—yo) =03 1+——+—~s—-~——~—:——§ T Zt=1(mt-m) :- a (1 — or) x 100% confidence interval (or prediction interval) for ya is [is '— taBB (ifs - so) as?) + We (fie - '90)] 3. Multiple linear regression model a coefficient of determination (R2): where RSS is residual (or unexplained) sum of squares, ESS is explained sum of squares and TSS(= ESS -|- RS S) is total sum of squares u an unbiased estimator of :72 is T 2 32 = Eta-1 (fit) -—-_...__I-—n- T --— K where K is the number of parameters being estimated I the random variable t bj " X35 . ts- 360,3.) N t(T——K): J —- 1121*“:K e the Futest statistic is F __ (R353 —- RSSU) /J N F “ RSSU/ (T -— K) ”'T‘K) where J is the number of restrictions, RSSU and R853 are the unrestricted residual sum of squared deviation and restricted residual sum of squared deviation respectively o Goldfeld and Quandt test statistic: under the normality assumption and Ho, the ratio of the two variance estimators follows an F-distrihution with (T1 —— K1,,T2 - K 2) degrees of freedom A 2 1 GQ = E3212 N F(T1“K1.Ta-Ke) 53.3% .3 33mm“. 335505.“. u «a “E233“ B 3356. hBEOEE n 5 #2E conga Qmfim 2: 9:3 uflflufim 3...... 03.: ”EM. “8.5%. 8. 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Ed can a: 34. S... w: 24.... mm... mm... mm 1 E... 36 mm. m a fin v 2... v a...» 2% on. v n3. mm... 3...... mm... 15. 2..» «3. mm... 3.... new 2 n a...“ . and E m m 8. n 8. w aw w mm.“ mm. m R.“ 3.“ 8.“ an 36 8.... Ed 36 .36 and mm. a and 3.» EH w mm m and and and and Ed and a; 33 Ed 3d mam 3w 3d 3m fl m. mm... mud 2.2 m am. 2 01$ mw 3 E 2 . 3.3 9.3 mu. 3 9.3 3.2 3.2 mg“ 5.2 3.3 33 and.” 3.3 2.“: 35“ Ema a. Rena madam 5.3.... 3.3” Edna. 3. man 8 m3 363 3 new mm r.” em Ea mm mmn R. wan am. mmu Edna mwéwm 3.2m on?“ 932 H E :3 an cw an en an 3 NH 2 m m h. m. m w m m a 5? £23m fin Ema—D afifirfifli 3: E 323., 13:6 muhéma m. u... 9 ~33... Table Right-tail Critical Values-for the t-distn'hufion DF a=.05 a=.025 a=.005 1 6.314 12.706 63.657 2 2.920 4.303 9.925 3 2.353 3.132 5.341 4 2.132 2.776 4.604 5 2.015 2.571 4.032 6 1.943 2.447 3.707 7 1.395 2.365 3.499 3 1.360 2.306 3.355 9 1.833 2.262 3.250 10 1.312 2.223 3.169 11 1.796 2.201 3.106 12 1.732 2.179 3.055 13 1.771 2.160 3.012 14 1.761 2.145 2.977 15 1.753 2.131 2.947 16 1.746 2.120 2.921 17 1.740 2.110 2.393 13 1.734 2.101 2.393 19 1.729 2.093 2.361 20 1.725 2.036, 2.345 21 1.721 2.030 2.331 22 1.717 2.074 2.319 23 1.714 2.069 2.307 24 1.711 2.064 2.797 25 1.703 2.060 2.737 26 1.706 2.056 2.779 27 1.703 2.052 2.771 23 1.701 2.043 2.763 .29 1.699 2.045 2.756 30 1.697 2.042 2.750 40 1.634 2.021 2.704 50 1-676 2.009 2.673 60 1.671 2.000 2.660 70 1.667 1.994 2.643 30 1-6‘64 1.990 2.639 90 1:662 1.937 2.632 100 1.660 1.934 2.626 110 1.659 1.932 2.621 120 1.653 1.980 2.617 m 1.645 1.960 2.576 Source: This table. was generated using the 313.30 function TINV ...
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