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Unformatted text preview: Duke University, practice problems for Introduction to Econometrics February 16, 2011 1 Properties of Expectations, Variances and Covariances, Distributions 1. (5 points) You know that income per head in Italy (denoted by y ), expressed in Euros, is normally distributed, that is, y & N & & y ; 2 y You also know that the mean & y is Euros 16,000, and the standard deviation of y is 2000. What is the distribution of income per head in Italy, in US$, if the exchange rate is 1.10 US$ per one Euro? Solution: Denote income per head in US$ as ~ y . Then ~ y = 1 : 10 y . Given that y is normally distributed and ~ y is just a linear transformation of y , ~ y also follows a normal distribution with mean ~ & y : ~ & y = 1 : 10 & y = 1 : 10 16 ; 000 = 17 ; 600 and variance ~ 2 y : ~ 2 y = V ar (1 : 10 y ) = 1 : 10 2 V ar ( y ) = (1 : 10 SD ( y )) 2 = (1 : 10 2 ; 000) 2 = 4 ; 840 ; 000 Thus: ~ y & N (17 ; 600 ; 4 ; 840 ; 000) Here, notice that, by convention, the second argument of N is the variance, and not the standard deviation! 2. (6 points) Let Y have a uniform distribution over the interval (0 ; ) . Show that 2 Y is an unbiased estimator for . (Recall that the pdf of a uniform distribution over the interval ( a;b ) is given by f ( x ) = 1 b & a ). Answer: Here Y i & U [0 ; ] : The pdf of this uniform distribution is f ( y ) = 1 & : Therefore, E ( Y i ) = 1 R &1 yf ( y ) dy = & R y 1 dy = y 2 2 j & = 2 2 = 2 Now, E & 2 Y = 2 E & Y = 2 E 1 n n P i =1 Y i = 2 1 n n P i =1 E ( Y i ) = 2 1 n n P i =1 2 = 2 2 = Since E & 2 Y = ; 2 Y is an unbiased estimator of : Note: Several people showed that E (2 Y ) = ; which is also true, but not what the question was asking so you only got partial credit for that. 3. (7 points total) Suppose that a mutual fund is investing in three di/erent asset categories. Each asset category includes many di/erent stocks or bonds. Let the variable X represent the asset category, and let R indicate the one-year expected (predicted) percentage return for a particular asset (one particular bond, or one particular stock). The following table shows the asset allocation of the fund, together with the one-year 1 percentage expected return for each asset category . The expected returns have been calculated using some forecasting model, but how the forecast has been done has no relevance in this problem. Proportion of assets invested One-year expected return for in asset type X assets in asset category X Asset Category ( X ) X = 1 ; Domestic Stock .30 0.10 X = 2 ; International Stock .20 0.15 X = 3 ; Bonds .50 0.00 (a) (5 points) Calculate the expected return of a dollar invested in the mutual fund. Which property of expectations is useful to solve this problem? Explain. 0.10(0.3) + 0.15(0.20) + 0.00(0.5) = 0.06 L.I.E. : E ( Return ) = P ( DS ) E ( Return j DS ) + P ( IS ) E ( Return j IS ) + P ( B ) E ( Return j B ) (b) (2 points) Calculate the predicted one-year percentage return of the fraction of your investment invested...
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This note was uploaded on 08/03/2011 for the course ECON 139 taught by Professor Alessandrotarozzi during the Spring '08 term at Duke.
- Spring '08