Tut sol week11 - BES Tutorial Sample Solutions S1/10 WEEK...

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1 BES Tutorial Sample Solutions, S1/10 WEEK 11 TUTORIAL EXERCISES (To be discussed in the week starting May 17) 1. Use a calculator to compute the sample least squares regression line for the model ݕ ൌ ߚ ൅ߚ ݔ ൅ ߝ, given the following six observations. y 2 8 6 12 9 11 x 1 4 3 10 10 8 ݔҧ ൌ 1൅4൅3൅10൅10൅8 6 ൌ6 തൌ 2൅8൅6൅12൅9൅11 6 ൌ8 ෍ሺݔ െݔҧሻሺݕ െݕ തሻ ൌ ሺ1 െ 6ሻ ሺ2 െ 8ሻ ൅ ڮ ൅ ሺ8 െ 6ሻ ሺ11 െ 8ሻ ൌ 62 ෍ሺݔ െݔҧሻ ൌ ሺ1െ6ሻ ൅ڮ൅ ሺ8െ6ሻ ൌ 74 TN: Dividing these 2 terms by ( n ‐1) = 5 gives the sample covariance ܾ ݏ ௫௬ ݏ ∑ሺݔ െݔҧሻሺݕ െݕ തሻ ∑ሺݔ െݔҧሻ 62 74 ൎ0 .8378 ܾ ൌݕ തെܾ ݔҧ ൌ 8 െ 0.8378 ൈ 6 ൌ 2.9732 Thus the sample regression line is ݕ ො ൌ 2.9732 ൅ 0.8378ݔ 2. Suppose the relationship between the dependent variable weekly household consumption expenditure in dollars ( y ) and the independent variable weekly household income in dollars ( x ) is represented by the simple regression model ( i refers to the i th observation or household): ݕ ൌߚ ൅ߚ ݔ ൅ߝ Suppose a sample of observations yields least squares estimates of b 0 = ‐32 and b 1 = 0.82. (a) What does ߝ represent in the model? It is the random disturbance term. It includes any purely random factors or
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2 (b) State the basic (classical) assumptions made about the ‘s in this model. Explain in words what the assumptions mean. (i) ܧሺߝ ሻൌ0 for all observations. The conditional mean of the disturbance does not depend on x and is normalized to zero. Note this is different from Keller who only mentions the normalization to zero. That the conditional mean of the disturbances does not depend on x ensures unbiasedness of the OLS estimator and so is the much more important component of this assumption. Relating back to the previous part of the question it implies that omitted factors that might affect expenditure but appear in the disturbance are assumed to be uncorrelated with x. (ii)
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Tut sol week11 - BES Tutorial Sample Solutions S1/10 WEEK...

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