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Semester – IV Question Paper 2015 Introductory Econometrics Duration : 3 Hours 1. 2. Maximum Marks : 75 State whether the following statements are true or false. Give reasons for your answer. (a) In a two-variable PRF, if the slope coefficient β 2 is zero, the intercept β1 is estimated by the sample mean. (b) In regression through origin models the conventionally computed R 2 may not be meaningful. (c) In the presence of heteroscedasticity OLS estimators are biased as well as inefficient. (d) The Durbin-Watson d test can also be applied to models that include the lagged value of the dependent variable as one of the explanatory variables. (e) In a double log model, the slope and elasticity coefficients are the same. (5 x 3 = 15) (a) You are given the following data based on 10 pairs of observation on Y and X. X X i 1700, Yi 1110, Xi Yi 205, 500 2 i 322,000, Yi2 132,100 Suppose the assumptions of the simple two variable CLRM are fulfilled, obtain (b) (i) OLS estimators, b1 and b 2 . (ii) Standard errors of these estimators. (iii) What is the value of r 2 ? (9) Consider the following model: GNPt B1 B2 M t B3M t 1 B4 M t M t 1 u t where GNPt GNP at time t. M t money supply at time t. M t 1 money supply at time (t – 1) and M t M t 1 = change in the money supply between time t and time (t – 1). The model thus postulates that the level of GNP at time t is a function of the money supply at time t and time (t – 1) as well as the change in the money supply between these time periods. Assuming you have data to estimate the preceding model, can you estimate all the coefficients of this model? Why or why not? If not, what coefficients can be estimated? (6) Bliss Point Studies 9811343411 1 Amit 9891555578 3. (a) Regression results for Korean savings-income data are presented for the period 1970-1995, Ŷt 1.0161 152.4786Dt 0.0803Xt 0.0655 Dt Xt t = (0.0504) (4.6090) (5.5413) R 2 0.8819 (–4.0963) where Yt savings X t income D t 1 for observations in 1982–1995 = 0 otherwise (b) 4. (a) (i) Interpret the regression results and obtain the regressions for the two time periods, that is, 1970–1981 and 1982–1995. (ii) What do you infer by the statistical significance of the differential intercept and the differential slope coefficients? (6) (i) What are the practical consequences of in perfect multicollinearity? (ii) Outline the White’s heteroscedasticity. test to detect for the presence of (9) Suppose the true model is : Yi β1 β2X2i u i But we add an irrelevant variable X 3 to the model and estimate Yi A1 A 2 X 2i A3X3i vi (i) Are the estimates of β1 and β 2 in the first regression unbiased? (ii) Does inclusion of “irrelevant” variable X 3 affect the variances of A 2 and A3 in the second regression? (iii) (b) Would R 2 for the second regression be larger than that for the first regression? (9) To study the rate of growth of population in an economy over the period 1970-1992, the following models were estimated: Model I: ln pop t = 4.73 + 0.024t t = (781.25) (54.71) Model II: ln pop t t= = 4.77 + 0.015t - 0.075 D t + 0.011 D t t (2477.92) (34.01) (–17.03) (25.54) where pop = population in millions t = trend variable D t 1 for 1970–1979 = 0 otherwise Bliss Point Studies 9811343411 2 Amit 9891555578 5. (a) (i) In model I, what is the rate of growth of population over the sample period? (ii) Are the population growth rates statistically different pre and post 1980? (iii) If they are different, then what are the growth rates for 1970–79 and 1980–1992? (6) Based on the data on annual percentage change in wages (Y) and percent annual unemployment rate (X) for the years 1950 to 1966, following regression was obtained: 1 Ŷ 1.4282 8.7243 Xt se 2.0675 R 2 0.3849 (b) 2.8478 F1,15 9.39 (i) Interpret the above regression (ii) What would be the slope of the regression? What would be some likely shapes of the curve corresponding to the above regression? (iii) What is the elasticity of Y with respect to X at mean values of Y 4.8% and X 1.5% . (7) Consider the following two regressions: Ct 26.19 0.6248GNPt 0.4398D t se 2.73 0.0060 0.0736 R 2 0.99 Dt 1 C 0.6246 0.4315 25.92 GNPt GNPt GNP se = (2.22) (0.0068) (0.0597) R 2 0.875 where C = aggregate private consumption expenditure GNP = gross national product D = national defence expenditure t = time 6. (a) (i) What might be the reason for transforming the first equation into the second equation? (ii) What assumption has been made about the error variance? (iii) According to (i) above, how do you know that the problem has been corrected in the transformed regression? (8) State and prove the Gauss Markov theorem for the slope coefficient in the classical linear regression model. (6) Bliss Point Studies 9811343411 3 Amit 9891555578 (b) Consider the following Cobb Douglas production function estimated for Taiwan for the period 1955–1974. lnGDP t 1.6524 0.3397lnL t 0.8460lnK t t = (–2.725) (1.8295) (9.0625) R 2 0.9951 RSS UR 0.0136 where GDPt GDP at time t, L t = labour at time t, K t = capital at time t, ln = natural logarithms. (i) Interpret the coefficients of the regression and comment on their individual significance. (ii) Comment on the returns to scale experienced by the Taianese economy. (iii) By imposing the restriction of constant returns to scale, the GDP K 0.4947 1.0153ln L t L t following regression was obtained : ln T= (–4.0612) (28.1056) R 2 0.9777 RSSR 0.0166 Interpret the above regression. (iv) 7. (a) Use a test statistic to see whether the economy is characterized by constant returns to scale. (9) For the two variable regression model Yi B1 B2 Xi u i , show that. y ŷ y ŷ x y x 2 (b) (i) r (ii) β̂ 2 2 i 2 i i i 2 i i 2 i Consider the following model of Indian imports estimated using data for 40 years for the period 1945–1985. (Standard errors are given in parentheses) lnY t 1.5495 0.9972lnX 2t 0.3315lnX 3t 0.5284lnY t 1 se = (0.0903) (0.0191) (0.0243) (0.024) R 2 0.994 d 1.8 where Y = imports X 2 GDP, X3 CPI (i) Does the model suffer from first-order autocorrelation? Which test statistic do you use and why? Bliss Point Studies 9811343411 4 Amit 9891555578 (ii) Outline the steps of the test used. Compute the test statistic and test the hypotheses that the preceding regression does not suffer from first-order autocorrelation. (iii) If the general model is given Yi B1 B2 X 2i B3X3i u i where errors follow AR(1) scheme, that is u t ρu t 1 ε t and where ε t is a white noise error term. Then how would you transform the model to correct for the problem of autocorrelation. (9) Bliss Point Studies 9811343411 5 Amit 9891555578