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Unformatted text preview: o Ec0220 Exercise Set 9/10. In the first ﬁve questions let yi = 7x, + el , i = 1,..,n where yi and Xi are observed variables, 7
is an unobserved parameter and the ei i = 1,..,n are unobserved random variables with a zero mean and variance 6’ > 0 for all i.
1) Derive the ordinary least squares estimator of y. 2) Stating any necessary additional assumptions prove that the estimator in 1) is unbiased and
‘ derive its variance. 3) Stating any ﬁirther additional assumptions derive a test statistic and its critical value for H0 7
21 against HIV <1. 4)T he following set of data (yi, xi) corresponds to logarithms of expenditures on a commodity and
income of individual i respectively so that in the above model 7 corresponds to the income
elasticity of demand for that comodity. Test the hypothesis that income elasticity of demand is
ainst the alternative that it is inelastic. Set the size of the test at .05. 5). When in as 0 an alternative to the OLS estimator you derived in question 1 is given by Zyi/in,
show that this is unbiased and derive its variance. Comment on its efﬁciency relative to your OLS estimator (hint 2x1.2 > n"(2xi)2). In the next set of questions let y| = C + 7x, + ei , i = 1,..,n where yi and 1!:i are observed
variables, C and 7 are unobserved parameters and the ei i = 1,..,n are unobserved random
variables with a zero mean and variance 0" > 0 for all i. 6) Derive the ordinary least squares estimators of C and y. 7) Stating any necessary additional assumptions prove that the estimators in 6) are unbiased and
derive their variances. 8) Stating any further additional assumptions derive a test statistic and its critical value for H(] y I
21 against H17 < 1. 9) Stating any further additional assumptions derive a test statistic and its critical value for Ho C
20 against Hl C < 0. 10) Using the data in question 4) test the same hypothesis for this model. 11). Given n observations on the pairs (ybxi) i = 1,..,n with means y and x, three alternative
representations of the OLS estimator of the slope of a regression function follow, show that they are all the same estimator. n (xi—x)y,- (yi—y)x,. E x yi—njg/ i=1 12. 20 identical wheat ﬁelds were randomly allocated to one of three fertilizer treatments, the
yields (in bushels) in the ﬁelds were as follows: Fields under treatment 1: 40, 42, 39, 39, 37, 43 Fields under treatment 2: 43, 44, 44, 45, 44, 42, 46. Fields under treatment 3: 35, 37, 35, 38, 38, 32, 36.
Test the hypothesis that the different treatments had no distinguishable effects at the 5% level. 13 For the following data test HO: [52 0 against H1: [3 < 0 in the regression Yi = (1 + BXi + ei , state
clearly what assumptions underlay your testing procedure. 14. A regression of the log of earnings (Y) ON AGE (X1), AGE2 (X2) and the number of children
(X3 ) yielded the following results (standard errors of the parameter estimates in brackets): For women with no schooling
Y = 1.8294 + 0.1063 X1, - 0.0010 X2 - 0.0562 X3 (0.6006) (0.3389) (0.0003) (0.0494)
n=93, R2 = .2123
For women with post graduate experience
Y = 3.4360 + 0.2226 X1, - 0.0023 X2 - 0.2267 X3 (1.6348) (0.0485) (0.0006) (0.0668)
n=228, R2 = .1632 Comment on the signiﬁcance of the impact of children on the respective earnings proﬁles and the
joint signiﬁcance of the explanatory variables X1, X2 and X3. What do the slopes of the earnings
proﬁles look like at age 25 for the two groups of women? When would you expect their earnings capacity to peak? ...
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- Spring '11