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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 oneyear 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 oneyear 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 Oneyear 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 oneyear percentage return of the fraction of your investment invested...
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 Spring '08
 ALESSANDROTAROZZI
 Econometrics, Regression Analysis, Standard Deviation, Variance, Yi Yi

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