EX08 - 3 Summer 2008 examination EC220 Introduction to...

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Unformatted text preview: 3 Summer 2008 examination EC220 Introduction to Econometrics Suitable for all candidates Instructions to candidates Time allowed: 3 hours + 15 minutes reading time This paper contains NINE questions. Answer any FOUR questions. All questions will be given equal weight (25%). You are supplied with: Graph paper Statistical tables Logarithm tables (available on request). Calculators are NOT allowed in this examination. © LSE 2008/EC220 Page 1 of 11 1. A variable Y i is generated as Y i = β 1 + u i (1.1) where β 1 is a fixed parameter and u i is a disturbance term that is independently and identically distributed with expected value 0 and population variance . The least squares estimator of β 1 is 2 u σ Y , the sample mean of Y , with population variance n u 2 σ , where n is the number of observations in the sample. However a researcher believes that Y is a linear function of another variable X and uses ordinary least squares to fit the relationship (1.2) X b b Y 2 1 ˆ + = calculating b 1 as X b Y 2 − , where X is the sample mean of X . The population variance of b 1 is ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + ∑ 2 2 2 1 X X X n i u σ . X may be assumed to be a nonstochastic variable. (a) [4 marks] Given that (1.1) is the true relationship, demonstrate that Y is indeed the least squares estimator of β 1 . (b) [4 marks] What would be the value of R 2 in such a regression (a regression with only a constant and no explanatory variables)? Give a mathematical explanation. (c) [2 marks] Demonstrate that Y is an unbiased estimator of β 1 . (d) [2 marks] Demonstrate that the population variance of this estimator is n u 2 σ . (e) [5 marks] Determine whether the researcher’s estimator b 1 in equation (1.2) is biased or unbiased, and if biased, determine the direction of the bias. (f) [3 marks] Mathematically, the expression for the variance of the researcher’s estimator shows that it is an inverse function of the sum of the squared deviations of the observations on X around the sample mean X . Explain intuitively why this should be the case. (g) [2 marks] Mathematically, the expression for the variance of the researcher’s estimator shows that it is the same as that of Y for the special case X = 0. Explain why this should be the case. (h) [3 marks] Suppose that it is not known whether Y depends on X or not and that the coefficient of X in a regression of Y on X is not significant. Explain the potential advantages and disadvantages of using X b Y 2 − , rather than Y , as an estimator of β 1 . © LSE 2008/EC220 Page 2 of 11 2. A researcher investigating the determinants of juvenile delinquency has the following data for 2007 for a sample of 100 cities in a certain country: A , the number of arrests per 1,000 juveniles, defined as persons aged 14–18, in the city, P , the number of households per 1,000 in the city with incomes below the poverty line, and S , the number of single-parent households per 1,000 in the city. He is considering fitting the model...
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.

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EX08 - 3 Summer 2008 examination EC220 Introduction to...

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