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# Introduction to Econometrics ECO 4305

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This 5 page Class Notes was uploaded by Kaden Orn on Thursday October 22, 2015. The Class Notes belongs to ECO 4305 at Texas Tech University taught by Summers in Fall. Since its upload, it has received 32 views. For similar materials see /class/226392/eco-4305-texas-tech-university in Economcs at Texas Tech University.

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Date Created: 10/22/15

Chapter 2 Review of Prob ability I Solutions to Exercises 1 a Probability distribution function for Y Outcome Y 0 Y 1 Y 2 number of heads 39 39 quot39 025 050 025 b Cumulative probability distribution function for Y Outcome Ylt0 0 Yltl 1 Ylt2 Y22 number of heads Probability 0 025 075 10 c y E 1 0 x 025 1x 050 2x 025 100 Using Key Concept 23 var Y E Y 7 E Y2 and 191 02 x 025 12 x 050 22 x 025 150 so that varY E Y E 12 150 71002 050 We know from Table 22 that Pr Y 0 022 Pr Y 1 078 PrX 0 030 PrX 1070 So a y EY0xPrY 0 1xPrY 1 0 x 0221x 078 078 0X EX0xPrX 0 lerX 1 0 x 0301x 070 070 b 0 EX 592 0 0702 x PrX 0 1 0702 x PrX 1 0702 x 030 0302 x 070 021 0 E Y 021 00782 x PrY 0 10782 x PrY 1 0782 x 022 0222 x 078 01716 c Table 22 shows PrX 0 Y 0 015 PrX 0 Y 1 015 PrX 1 Y 0 007 PrX 1 Y 1 063 So an covXY E1XXY 40 0 0700 078PrX 0 Y 0 0 0701 078PrX 0Y 1 1 0700 078PrX 1Y 0 1 0701 078PrX 1Y 1 070 x 078 x 015 070 x 022 x 015 030 x 078 x 007 030 x 022 x 063 0084 cor X Y XV 04425 0X0 021x 01716 The table shows that PrX 0 Y 0 0045 PrX 0 Y 1 0709 PrX 1 Y 0 0005 PrX 1 Y 1 0241 PrX 0 0754 PrX 1 0246 Pr Y 0 0050 Pr Y 1 0950 a EYuy0xPrY01xPrY1 0 x 0050 1x 0950 0950 b Unemployment Rate W labor force PrY0005010950 1EY 0 Calculate the conditional probabilities rst my 01X 0 w 00597 PrX 0 0754 PrX 0 Y 1 7 0709 Pr Y llX 0 7 09403 PrX 0 0754 Pr Y 01X 1 PrX 1 Y 0 Loos 00203 PrX 1 0246 HY1 X1W o9797 PrX 1 0246 The conditional expectations are EHX 1 0 x PrY 01X11xPrY11X1 0 x 002031x 09797 09797 EHX 0 0 x PrY 01X 01xPrY11X 0 0 x 005971x 09403 09403 d Use the solution to part b Unemployment rate for college grads 1 EquotX 1 1 09797 00203 Unemployment rate for noncollege grads 1 E XX 01 09403 00597 e The probability that a randomly selected worker who is reported being unemployed is a college graduate is PrX Y07M7 PrX11 Y 0 Pr Y 0 0050 01 The probability that this worker is a noncollege graduate is PrX 01 Y 01PrX 11 Y 0101 09 F 39 39 39 39 39 r39 status are not independent because they do not satisfy that for all values of xand y PrY th X PrY y For example PrY 01X 0 00597 PrY 0 0050 Using obvious notation C M F thus uC uM uF and 7 fig 039 2covMF This implies a uC 40 45 85000 per year b corMF C le so that CotMP o MO39FcorMF Thus GilFF CotMP 12 x 18 x 080 17280 where the units are squared thousands of dollars per year c aquot fig 039 2covMF so that 139239122 182 2 x 17280 81360 and 039 081360 28524 thousand dollars per year C d First you need to look up the current Eurodollar exchange rate in the Wall Street Journal the Federal Reserve web page or other nancial data outlet Suppose that this exchange rate is 5 say 5 080 euros per dollar each 1 is therefore with 5E The mean is therefore egg in units of thousands of euros per year and the standard deviation is 505 in units of thousands of euros per year The correlation is unitfree and is unchanged Distribution a The probability distribution is given in the table above EY14x 021 22x 023 30 x 030 40x 015 65 x 011 3015 1912142 x 021 222 x 023 302 x 030 402 x 015 652 x 011 112723 VarY 1912EY2 21821 cry 1477 b Conditional Probability of Yl X 8 is given in the table below I Value of Y I 22 30 40 65 002039 003039 015039 010039 009039 E 8X 8 14x 002039 22x 003039 30 x 015039 40 x 010039 65 x 009039 3921 H YlX 8 142 x 002039 222 x 003039 302 x 015039 402 x 010039 652 x 009039 17787 Var H 17787 39212 24165 cm 1554 c EXY 1x 14x 002 1x 22 005 8 x 65 x 009 1717 CovXY EXYEXEY 1717 533 x 3015 110 CorrXY CovXYoX039Y110546 x 1477 0136 Update There s an error in this last line The standard deviation of X is 26 not 546 the variance is 676 so the correlation is 0286 instead of 0136 We ve given everyone 2 additional points for this 14 The central limit theorem suggests that when the sample size 11 is large the distribution of the sample average 7 is approximately NEly 039 with aquot Given uY 100 Y 7 430 a 1100 o i043 and PrY 101Pr Y loogw zqgt152509364 I043 043 b 1165 of7if02606 and 7 7 37100 987100 PrYgt9817PrY9817Pr J02606 J02606 m 1 739178 39178 1000 rounded to four decimal places air c 11 64 7 a 3 06719 and 7 1017100 37100 1037100 Pr101 Y103Pr 7g 7 06719 06719 J06719 m 36599 q312200 09999 08888 01111 18 Pr Y 0 095 Pr Y 20000 005 a The mean of Yis uY 0 xPrY 0 20000 xPrY 20000 1000 The variance of Y is 0 E Y M 0 710002 x Pr Y 0 20000 710002 x Pr Y 20000 710002 x 095190002 x 005 19x107 so the standard deviation of Y1s 03 19 x 107 1 4359 b i Em y 1000 0 7133307 19x105 ii Using the central limit theorem PrYgt 2000 17PrYg 2000 liPr Y71000 g 200071000 J19x105 ll9X105 m1 q3229421 09891 00109

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