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# Class Note for EMSE 388 at GW (3)

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Date Created: 02/07/15

EMSE 388 Quantitative Methods in Cost Engineering CHAPTER 6 USING MULTIPLE REGRESSION There are many situations in which one wants to predict the value the dependent variable from the value of one or more independent variables Typically 0 independent variables are easily measurable 0 Collection of independent variables explains the dependent variable Example The return of most stocks is closely tied to the return on the market In finance it is important to try and predict the return on a stock dependent variable from the return on the market eg SampP500 Index Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 106 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Note that for an individual stock Annual Return 2 End of Year39s Price Dividend Last Year39 s Price Last Year39 s Price Closing Stock Annual Return Annual Regression Fit Market Return on Annual Return Year Price Dividends on Lilly Year Return Lilly on Lilly 1994 1650 075 1994 627 1995 2788 080 7382 1995 3216 7382 4868 1996 3713 090 3641 1996 1847 3641 2668 1997 3900 100 773 1997 523 773 540 1998 4275 115 1256 1998 1681 1256 2401 1999 6850 135 6339 1999 3149 6339 4760 2000 7325 164 933 2000 317 933 810 2001 8350 200 1672 2001 3055 1672 4609 2002 6075 220 2461 2002 767 2461 932 2003 5938 242 173 2003 999 173 1305 2004 6563 250 1474 2004 131 1474 090 2005 2000 2914 Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 6 Page 107 EMSE 388 Quantitative Methods in Cost Engineering Plotting Annual Return against Market Return in a Scatter Plot and including the best linear fit we get Lilly Return vs Market n nnO JUII Iu 3000 W 1000 a 2000 a Lilly Return 5C0 0 500 1000 1500 2000 2500 3000 350 2000 W 1 n nnO vvv Iu Market Return Linear Annual Return on Lilly 9 Annual Return on Lilly Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 108 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering CONCLUSION There appears to be a linear relationship but the wide dispersion about the best fitting line indicates we do not have a perfect linear relationship Possibly other factors besides market return determine the annual return on the Lilly stock INTERMEZZO ANALYSIS OF VARIANCE PROBLEM DESCRIPTION 0 Suppose you have a linear function fx19 9xk a19 9ak9balox1Hoakoxkb k Zai xl b i1 were the parameters a1 akb unknown Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 109 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 1 I I I n 0 You have N data points 9 9 y generated by this linear function but the data points contain some measurement error 0 Together with each datapoint yj of this linear function you have the input variables x119 39 399 xi 0 If the measurement error were to be zero and you knew the values of the parameters a1 akb then k y 2a or 19 i1 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 110 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering OBJECTIVE 1 Obtain a bestguess for the parameters 31 3k b 2 Estimate the uncertainty in the measurement error MEETING THE FIRST OBJECTIVE 0 Calculate demadka such that the sum of squares error s SSE is minimized N k A N k A 2 o 2 20 Zal or b Mle 20 Zal or b j1 i1 a1quotquotak j1 i1 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 111 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering MEETING THE SECOND OBJECTIVE 0 Introduce the random variable Y representing a measurement of the function k fx19 9xk ala39naakab Zai xi 19 i1 and assume that k 139 EYamp19 399ampk9bZclixijb i1 YEYa1akb8 where 8 does not depend on the inputvariables or independent variables x19 9xk Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 112 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 0 Assume that error term 5 is normal distributed with mean 0 and standard deviation 6 or variance 62 PROBLEM THAT REMAINS TO BE SOLVED Estimation of the variance 62 THE ANALYSIS OF VARIANCE Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 113 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering ANALYSIS OF VARIANCE Simplest Approach Suppose fUmumlw Hence we have data 319 39 39a yquot and we need to represent this entire set of data by one bestguess b As it turns the leastsquares fit for b equals l3l yjf We calculate nobserved error terms 8jyj Z II a lt 1 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 114 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering and estimate the variance for the errorterm using 1 j21n j 321n j 2 Z8 Zy Zy y n j1 n j1 n i1 BUT We know estimator of variance above is BIASED An unbiased estimator of 1 n yy2 Conclusion An unbiased estimator for the variance of the error term is 1 0902 n 11 the variance is Informally we divide by n1 instead of dividing by n as we used 1 parameter b to represent the entire data set yl 9 9 yn Statistician refers to this as loosing a degree of freedom Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 115 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Simple Linear Regression Suppose fxp39 axkba139x139b Hence we have data 3 19quot 399 yquot and we need to represent this entire set of A data by one linear line specified by two bestguesses a1 b and the independent variable x1 Simple linear regression gives us the these least A squares parameter estimates 671 and b We calculate nobserved error terms 1 jA A 3 y a1 x1 bj1n and estimate the variance for the errorterm using 1 Zoe n lH Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 116 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering BUT As it turns out the estimate for the variance of the errorterm above is BIASED An unbiased estimator of the variance in of the error term in this case 1 Z 8 2 n 2 11 lnformally we divide by n2 instead of dividing by n as we used 2 parameters is a1 9 to represent the entire data set yl y We lost again a degree of freedom HOW MUCH BETTER DOES THE SIMPLE LINEAR REGRESSION EXPLAIN THE VARIANCE IN THE DATA COMPARED TO THE SIMPLEST APPROACH OF ONE POINT ESTIMATE Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 117 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering ANSWER IS GIVEN BY RSQUARED I l 39 2 SS 20 y j1 A SSE 291 5z1x1j b2 j1 SS SSE SSE Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 118 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering n Variance Residuals Sample Variance Sample Variance 0 One says that RSquared metric favors the more complicated approach as n2n1 is less than 1 and does not account for the fact that simple linear regression uses one more parameter to represent the data Multiple Linear Regression Suppose fx1xk a1akba1x1akxkb Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 119 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 1 n Hence we have data y 9 9y and we need to represent this entire set of data by one hyperplane line specified by k1 bestguesses alan39aaknb A A Multiple linear regression gives us the these leastsquares fits a1 39 39 39aaka b We calculate nobserved error terms k 812321 21 flixJ b i1 J1n and estimate the variance for the errorterm using 1 Zgjz n 2 121 BUT As it turns out the estimate for the variance of the errorterm above is BIASED Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 120 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering An unbiased estimator of the variance in of the error term is 1 n J 2 n k18 lnformally we divide by nk1 instead of dividing by n as we used k1 1 o O I n parameters to represent the entire data set y a a y We lost k1 degrees of freedom HOW MUCH BETTER DOES THE MULTIPLE LINEAR REGRESSION APPROACH EXPLAIN THE VARIANCE IN THE DATA COMPARED TO THE SIMPLEST APPROACH OF ONE POINT ESTIMATE Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 121 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering ANSWER IS GIVEN BY RSQUARED SS 20quot W j1 n k SSE 201 Z 51 x b2 j1 i1 1 R2 SS SSE 1SSE SS SS n j k A j bAZ 1 n j k A j bAZ y 2 ai 39xl Z ai 39xl 1 11 i1 1 n j1 i1 n 1 n y y2 Zy y2 11 1j1 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Ll Variance Residuals n Sample Variance 0 One says that RSquared favors the more complicated approach considerably as nk1n1 is less than 1 and does not account for the fact that multiple linear regression uses k more parameters to represent the data 0 One says if you want to have a measure that compares the variance in the multiple regression fit compared to the simplest approach one should be comparing variances and not just the SS and SSE Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 123 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering ADJUSTED RSQUARED Adjusted R2 2 1 Variance Residuals 1 n39k11 Sample Variance ISS n Practical Implications 0 One can always improve the Rsquared value by including more independent variables It appears that we are obtaining a better fit 0 However with the same number of data points the number of data points per fitted parameter decreases and we intuitively feel a loss of accuracy in each fitted parameter Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 124 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 0 In the extreme case if we take as many independent variables and parameters as the original data we obtain a perfect fit RSquared1 We basically get out data back Is this a perfect fit 0 No you need to account for the complexity of the model 0 You can increase the number of independent variables as long as the ADJUSTED RSQUARED increases STRIVE FOR THE SIMPLEST MODEL THAT BEST EXPLAINS THE VARIANCE IN THE DATA Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 125 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering MULTIPLE LINEAR REGRESSION FORECASTING AUTO SALES Suppose you want to Forecast AutoSales in thousands of cars What are some variables that might influence auto sales in a particular quarter of the year 0 Previous Quarter s GNP Gross National Product 0 Previous Quarter s Interest Rate File Auto Multiple Linear Regressionxls contains AutoSales Data for 1979 1986 STEP 1 Let GNPx be Gross National Product in Quarter x LagGNPxk GNPxk ie GNP k quarters before Create lag variables from the original data Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 126 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Historical data Interest Year Quarter Sales GNP Rate LagGNP1 Laglnt1 79 1 2709 2541 940 Interest Year Quarter Sales GNP Rate LagGNP1 Laglnt1 79 2 2910 2640 940 2541 940 79 3 2562 2595 970 2640 940 79 4 2385 2701 1190 2595 970 80 1 2520 2785 1340 2701 1190 80 2 2142 2509 960 2785 1340 80 3 2130 2570 920 2509 960 80 4 2190 2667 1360 2570 920 81 1 2370 2878 1440 2667 1360 81 2 2208 2835 1530 2878 1440 81 3 2196 2897 1510 2835 1530 81 4 1758 2744 1180 2897 1510 STEP 2 Use Data Analysis Toolpack Regression Analysis to perform Multiple Linear Regression Analysis Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 6 Page 127 EMSE 388 Quantitative Methods in Cost Engineering STRATEGY Use Years 1979 1984 5 Years for the regression analysis and use Years 19851986 for validation of the regression analysis Note Have to use Data Analysis Toolpack because of multiple independent variables INTERCEPT and SLOPE only work in the case of SINGLE LINEAR REGRESSION Coef cients Intercept 2156914576 LagGNP1 0247308278 Laglnt1 5202313996 Sqx Sales in Quarter q and Year x Sqx02473LagGNPx1520231LaglNTx1215691 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 128 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering INTEPRETATION 0 When GNP increases AUTO SALES INCREASE o 1 Billion Dollars increase in GNP will result in approximately 247 more cars being sold 0 a onepercentage point increase in interest say from 6 to 7 will result in a reduction of car sales of 520231 1 OOO001 52023 PREDICTION Interest Rate in Current Quarter 88 GNP 3919 billion Predicted Auto Sales Next Quarter O247339195202310088215691 266831 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 129 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering UNCERTAINTY OF PREDICTION MAKE NORMAL ASSUMPTION OF RESIDUALS Y EYamp1amp2I X1X2g 2amp1X1Ez2 XZ 13g STEP 5A Model the Error Term Residuals as normal distributed random variable with mean u0 and standard deviation s8 such that LOST THREE DEGREES OF FREEDOM Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 130 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering INTERMEZZO THE NORMAL DISTRIBUTION Many biological phenomena height weight length follow a bellshaped curve that can be represented by a normal distribution Consider the production of men shoes You want to offer these shoes in many different sizes However you need to decide the percentage of shoes to produce in each size Let Y be the length of men s feet 1 ltx u22 YNltrrcr Mylrzagt e 20 EY u 7 o VarY 02 0 Some handy rules of thumb Pru 039 ltY lt y039 z 068 Pru 2039 ltY lty20 z 095 Pru 3039 ltY lty30 z 099 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 131 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 50 400 Probability Density Function N205 09 08 07 06 05 04 03 02 01 0 000 050 150 150 200 250 330 3 z68 I z95 I I z99 Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 6 Page 132 EMSE 388 Quantitative Methods in Cost Engineering RETURNING TO UNCERTAINTY OF PREDICTION Interest Rate in Current Quarter 88 GNP 3919 billion Predicted Auto Sales Next Quarter O247339195202310088215691 266831 0 68 Credibility Interval 266831 25464 0 95 Credibility Interval 266831i 225464 266831i 50928 0 99 Credibility Interval 266831i 325464 266831i 76392 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 133 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering WHAT ABOUT THE NORMALITY ASSUMPTION OF THE RESIDUALS 1 Normal Distribution Assumption of Residuals Residuals are normal distributed random variable with mean u0 and standard deviation s8 2 Rescale Residuals to Standardized Residuals by dividing by 58 Hence with the assumption above Standardized Residuals are Standard Normal Distributed with mean 0 and standard deviation 1 Le NO1 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 134 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 3 Order the Standardized Residuals such that 51 52 504 1 gm lt ltlt lt s s s s 8 8 8 8 4 calculate a a an QUZF J QaZF 7 gltngtF 0 S8 S8 S8 where F is the standard normal cumulative distribution function 5 Plot the points 1 1 2 1 n l l n l 2 7g17727 2I3977 n 177 n I l If Standardized Residuals are in fact NO1 distributed these points should all be one the line yx Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 135 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering NORMAL PROBABILITY PLOT 100 80 Standard Normal CDF Values 20 40 60 Empirical CDF Values 80 100 ARE RESIDUALS NORMALLY DISTRIBUTED Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 6 Page 136 EMSE 388 Quantitative Methods in Cost Engineering VALIDATION OF REGRESSION STRATEGY Use Years 1979 1984 5 Years for the regression analysis and use Years 19851986 for validation of the regression analysis Regression Regression Coefficient Coefficient Intercept 0247 5202314 2156915 Interest Year Quarter Sales GNP Rate LagGNP1 Laglnt1 PREDICTION RESIDUALS 84 4 2460 3919 880 85 1 2646 4040 820 3919 880 266831 2231 85 2 2988 4133 750 4040 820 272945 25855 85 3 2967 4303 710 4133 750 278887 17813 85 4 2439 4393 720 4303 710 285172 41272 86 1 2598 4560 890 4393 720 286877 27077 86 2 3045 4587 770 4560 890 282163 22337 86 3 3213 4716 740 4587 770 289074 32226 86 4 2685 4796 740 4716 740 293825 25325 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 137 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering PLOT OF ADDITIONAL RESIDUALS IN NORMAL PDF WITH MEAN 0 and STANDARD DEVIATION 25463 NORMAL DISTRIBUTION RESIDUALS 800 600 400 200 0 200 400 600 800 RESIDUALS NORMAL PDF 25 BOUND 975 BOUND O VALIDATION SET Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 138 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering CONCLUSION All additional observations fall well within a 95 credibility interval Only one observation does falls on the boundary of this 95 credibility interval out of 8 observations Normal Probability Plot Graph with additional observations support the linear fit and the uncertainty model for predictions CONCERN RSQUARED 3094 Sample Variance 8536418 Variance Residuals 6484083 ADJUSTED RSQUARED 2404 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 139 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering NEXT ATTEMPT Available information contains QUARTER OF THE YEAR and UNEMPLOYMENT RATE Suggestion Use QUARTER OF THE YEAR and UNEMPLOYMENT RATE in previous quarter as additional independent variable in multiple linear regression Existing Independent Variables 0 Previous Quarter s GNP Gross National Product 0 Previous Quarter s Interest Rate New Independent Variables 0 Previous Quarter s Unemployment rate 0 Quarter of the Year Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 140 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering MODELING QUARTER OF THE YEAR AS A INDEPENDENT VARIABLE o GNP Interest Rate and Unemployment Rate are measurable variables with a natural attribute scale GNP is measured in the attribute Dollars Interest Rate is measured in Percentage Points Unemployment Rate is measured in Percentage Points A higher GNP Value in terms of dollars is better A lower Interest Rate in terms of is better A Lower Unemployment Rate in terms of is better Original Data assigns 1 to first quarter 2 to second quarter 3 to third quarter 4 to fourth quarter Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 141 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Year Quarter Sales GNP Int Unemp LagGNP1Lagnt1LagUnEMp1 79 2 2910 2640 940 570 2541 940 590 79 3 2562 2595 970 590 2640 940 570 79 4 2385 2701 1190 600 2595 970 590 80 1 2520 2785 1340 620 2701 1190 600 80 2 2142 2509 960 730 2785 1340 620 80 3 2130 2570 920 770 2509 960 730 80 4 2190 2667 1360 740 2570 920 770 81 1 2370 2878 1440 740 2667 1360 740 81 2 2208 2835 1530 740 2878 1440 740 81 3 2196 2897 1510 740 2835 1530 740 81 4 1758 2744 1180 830 2897 1510 740 82 1 1944 2582 1280 880 2744 1180 830 82 2 2094 2613 1240 940 2582 1280 880 82 3 1911 2529 930 1000 2613 1240 940 82 4 2031 2544 790 1070 2529 930 1000 83 1 2046 2633 780 1040 2544 790 1070 83 2 2502 2878 840 1010 2633 780 1040 83 3 2238 3051 910 940 2878 840 1010 83 4 2394 3274 880 850 3051 910 940 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 142 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering HOWEVER We can not assign an interpretation of Better or Worse to a higher quarter number Modeling this variable as Q 1234 would do that Better Approach Introduce Three Binary Dummy Variables 01 02 QB 01 1 QZOQ 3O First Quarter January March QlOQZ1Q 3O Second Quarter April June 01 OQZOQ 31 Third Quarter July September 01 OQZOQ 3O Fourth Quarter October December Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 143 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering Sqx Sales in Quarter q and Year x Coef cients Intercept 3181 770327 LagGNP1 024245406 Laglnt1 8157270787 LagUnEMp11044193445 Q1 1716881431 QZ 31 16366643 QB 7597226941 Sqx O2424LagGNPx 1 81 5727LagNTx1 1044193LagUnEMP117108Q1 31164027597Qs318177 Lecture Notes by Dr J Rene van Dorp Source Financial Models Using Simulation and Optimization by Wayne Winston Chapter 6 Page 144 EMSE 388 Quantitative Methods in Cost Engineering INTEPRETATION o 1 Billion Dollars increase in GNP will result in approximately 242 more cars being sold 0 A onepercentage point increase in interest say from 6 to 7 will result in a reduction of car sales of 815727100000181572 Approx 82000 0 a onepercentage point increase in unemployment rate say from 6 to 7 will result in a reduction of car sales of 10441931000001104419 Approx 104000 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 145 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering 0 During January March car sales run 171000 higher than October December 0 During April June car sales run 312000 higher than October December 0 During July September car sales run 76000 higher than October December PREDICTION In Fourth Quarter of the year Interest Rate 88 GNP 3619 billion Unemployment Rate 72 Predicted Auto Sales First Quarter 0242436198157270088 10441930072171091 31164075970318177283338 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 146 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering UNCERTAINTY OF PREDICTION MAKE NORMAL ASSUMPTION OF RESIDUALS STEP 5A A YEYamp1Damp2amp3a 4a amp b X1X22X3Q12Q2Q3 5 6 a 2611Xla2X2a3X3a4Qla5Q2a Q3b8 Model the Error Term Residuals as normal distributed random variable with mean u0 and standard deviation s8 such that WE LOST SEVEN DEGREES OF FREEDOM Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 147 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering SSE 49685271 VARIANCE RESIDUALS 3105329 STANDARD ERROR 17622 2STANDARD ERROR 35244 3STANDARD ERROR 52866 SS 187801200 SAMPLE VARIANCE 8536418 RSquared 7354 Adjusted RSquared 6362 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 148 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering RETURNING TO UNCERTAINTY OF PREDICTION Predicted Auto Sales First Quarter O242436198157270088 10441930072171091 31164O7597O318177283338 o 68 Credibility Interval 283338 17622 0 95 Credibility Interval 283338i 217622 283338i 35244 0 99 Credibility Interval 283338i 317622 283338i 52866 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 149 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering WHAT ASSUMPTION WAS USED ABOVE N ORMAL PROBABILITY PLOT Standard Normal CDF Values 0000 0200 0400 0600 0800 1000 Empirical CDF Values Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 150 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering VALIDATION OF REGRESSION STRATEGY Use Years 1979 1984 5 Years for the regression analysis and use Years 19851986 for validation of the regression analysis PREDICTED Year Quarter Sales GNP Int Unemp SALES RESIDUALS 85 1 2646 4040 820 740 283338 18738 85 2 2988 4133 750 730 303132 4332 85 3 2967 4303 710 710 288575 8125 85 4 2439 4393 720 700 290451 46551 86 1 2598 4560 890 710 309970 50170 86 2 3045 4587 770 710 313162 8662 86 3 3213 4716 740 690 300039 21261 86 4 2685 4796 740 680 300105 31605 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 151 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering PLOT OF ADDITIONAL RESIDUALS IN NORMAL PDF WITH MEAN 0 and STANDARD DEVIATION 17622 NORMAL DISTRIBUTION RESIDUALS 100EO3 500EO4 W O i 600 400 200 O 200 400 600 RESIDUALS NORMAL PDF 25 BOUND 975 BOUND O VALIDATION SET Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 152 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering SUMMARY 0 We included more information to predict Auto Sales ie Unemployment Rate and Quarter of the Year and were able to achieve a higher RSquared Value and consequently a smaller Standard Error 0 The uncertainty in the prediction is less as a result of the smaller standard error 0 The model is LESS VALID as more residuals of the validation set fall outside the 25 or 975 credibility bounds SO DID WE IMPROVE OUR MODEL Answer Check the Adjusted RSquared Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 153 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering THREE PARAMETER MODEL RSQUARED 3094 ADJUSTED RSQUARED 2404 SEVEN PARAMETER MODEL RSQUARED 7354 ADJUSTED RSQUARED 6362 SIX PARAMETER MODEL RSQUARED 7263 ADJUSTED RSQUARED 6457 FIVE PARAMETER MODEL RSQUARED 6937 ADJUSTED RSQUARED 6257 Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 154 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering VALIDATION SIX PARAMETER MODEL LagGNP1 Laglnt1 LagUnEMp1 Q1 02 INTERCEPT NORMAL DISTRIBUTION RESIDUALS 100E03 500E04 600 400 200 0 200 400 600 RESIDUALS NORMAL PDF 25 BOUND 975 BOUND O VALIDATION SET Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 155 Source Financial Models Using Simulation and Optimization by Wayne Winston EMSE 388 Quantitative Methods in Cost Engineering VALIDATION FIVE PARAMETER MODEL LagGNP1 Laglnt1 LagUnEMp1 oz INTERCEPT NORMAL DISTRIBUTION RESIDUALS 100EO3 500E04 600 400 200 O 200 400 600 RESIDUALS NORMAL PDF 25 BOUND 975 BOUND O VALIDATION SET Lecture Notes by Dr J Rene van Dorp Chapter 6 Page 156 Source Financial Models Using Simulation and Optimization by Wayne Winston

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