CLASS9 HW GSB420 Let's consinder a mortgage application using HMDA (The Home Mortgage Disclosure Act). Here is a sample from 30 mortgage applications (excel file is available in D2L) : ID 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 loanam t 109 185 121 125 119 153 380 100 110 41 115 248 126 260 90 50 125 125 158 130 204 30 114 188 187 84 450 108 100 53 loanamt = Amount of Mortgage Loan Application (in $1000) income = Applicant's Annual Income (in $1000) hprice = House Price to buy (in $1000) Use Minitab to answer the following questions. income 63 137 53 78 37 65 188 58 78 31 54 117 60 192 40 36 45 55 62 29 77 28 60 91 85 44 265 49 53 24 hprice 155 264 128 125 149 171 484 125 158 116.5 128 280 157.5 325 145 230 125 145 175 209 260 150 143 253 285 105 650 120 125 66 1. Descriptive statistics 1) Make a scatter plot between loanamt and income, and explain the relationship between them. 2) Calculate the covariance and correlation coefficient between loanamt and income, and explain the linear relationship between them. 2. Regression Analyis Let's consider the following regression model. Estimate the model using Minitab. Loanamti = b0 + b1 * incomei + et Write the equations for the following statistics, find or calculate them from the Minitab output, and explain the meanings of the statistics. 1) Estimated intercept 2) Estimated slope coefficient 3) TSS (Total Sum of Square), RSS (Regression Sum of Square), and ESS (Error Sum of Square) 4) R2 5) Variance of b, Standard Error of b 6) Variance of et 7) According to the model what are the predicted loan amount if applicants have annual income of $50,000, $100,000, and $200,000? 8) What are the income elasticities on loan applications if applicants have annual income of $50,000, $100,000, and $200,000? 9) Perform a t test on the coefficient of income, and state if it is significant at 1% and 5%. 10) Carefully explain what is the OLS (ordinary least square) estimator. GSB420 CLASS 9 STATISTICS (CH13) CLASS 9 Chapters 13. Simple Regression Analysis Page 1 of 14 GSB420 CLASS 9 STATISTICS (CH13) Page 2 of 14 Example 1) Consumption and Income income consumption 10000 25000 35000 32000 30000 27000 25000 31000 70000 59000 65000 64000 83000 95000 Scatterplot of consumption vs income 100000 Person7 90000 80000 consumption obs Person1 Person2 Person3 Person4 Person5 Person6 Person7 70000 Person6 Person5 60000 50000 40000 30000 Person1 Person4 Person2 Person3 20000 10000 0 Obs 1 2 3 4 5 6 7 income 10000 35000 30000 25000 70000 65000 83000 consumption 25000 32000 27000 31000 59000 64000 95000 Fit 15474 38124 33594 29064 69832 65302 81610 Residual 9526 -6124 -6594 1936 -10832 -1302 13390 10000 20000 30000 40000 50000 60000 income 70000 80000 90000 GSB420 CLASS 9 STATISTICS (CH13) Page 3 of 14 Example 2) Wealth and Salary Obs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Wealth Salary 1 100 50 2 200 75 3 150 45 4 300 100 5 205 80 6 100 30 7 150 35 8 270 50 9 280 55 10 700 90 11 500 30 12 300 25 13 650 110 14 600 50 15 600 30 Salary 50 75 45 100 80 30 35 50 55 90 30 25 110 50 30 Wealth 100.0 200.0 150.0 300.0 205.0 100.0 150.0 270.0 280.0 700.0 500.0 300.0 650.0 600.0 600.0 Fit 324.3 381.7 312.8 439.1 393.1 278.3 289.8 324.3 335.7 416.1 278.3 266.9 462.0 324.3 278.3 Scatterplot of Wealth vs Salary 10 700 13 15 600 14 11 500 Wealth Person 400 12 300 8 4 9 2 200 7 20 Residual -224.3 -181.7 -162.8 -139.1 -188.1 -178.3 -139.8 -54.3 -55.7 283.9 221.7 33.1 188.0 275.7 321.7 3 6 100 30 5 1 40 50 60 70 Salary 80 90 100 110 GSB420 CLASS 9 STATISTICS (CH13) Page 4 of 14 A simple regression model tries to establish a statistically significant relationship between two variables. One of them, called an independent variable, is believed to explain (or determine or influence) the other variable, called a dependent variable. This causal relationship must be established prior to data collection on the basis of economic theory, business intuition, life experience, etc. Given the population simple regression equation of: Yt X t t where Yt = dependent variable = actual, observed value of Y at time period t Xt = independent variable = actual, observed value of X at time period t = population intercept term = population slope or coefficient term t = population error term at time period t we are implicitly saying that X causes or explains Y. Furthermore, we must estimate (or guess) this population regression equation by: Yt a b X t et where a = intercept term calculated from a sample data an estimate for b = slope (or coefficient) term calculated from a sample data an estimate for et = error term at time period t based on a sample data an estimate for t GSB420 CLASS 9 STATISTICS (CH13) 1. Estimation of simple regression model 2. Meaning of Regression Equation Page 5 of 14 GSB420 CLASS 9 STATISTICS (CH13) Page 6 of 14 3. Regression Output Example in Minitab MTB > Regress 'Wealth' 1 'Salary'; SUBC> Constant; SUBC> Brief 2. Regression Analysis: Wealth versus Salary The regression equation is Wealth = 209 + 2.30 Salary Predictor Constant Salary Coef 209.5 2.296 S = 208.953 SE Coef 127.7 2.030 R-Sq = 9.0% T 1.64 1.13 P 0.125 0.279 R-Sq(adj) = 2.0% Analysis of Variance Source Regression Residual Error Total DF 1 13 14 SS 55828 567595 623423 MS 55828 43661 F 1.28 P 0.279 GSB420 CLASS 9 STATISTICS (CH13) Page 7 of 14 4. Decomposition of the ANOVA The regression analysis provides a very convenient way to understand how much the independent variable explains the variability (=movement) of the dependent variable. This can be accomplished by decomposing the relationship and then organizing it into an ANalysis Of VAriance (ANOVA) table as follows: Given Yt Yt et , we note that the following relationships hold: (Yt Y ) (Yt Y ) et (Y Y ) 2 (Y Y ) 2 e 2 t t t Note that these sums of squares can be named as follows: TSS = RSS + ESS where Y )2 RSS = Regression Sum of Squares = (Yt Y ) 2 TSS = Total Sum of Squares = (Y t ESS = Error (or Residual) Sum of Squares1 = e 2 t That is, we can construct the following ANOVA table for regression analysis: Source of Variation Due to Regression Sum of Squares RSS Degrees of Freedom k Due to Error (or Residual) ESS n-k-1 Total TSS n-1 Mean Square (=Variance) RSS k ESS MSE= n k 1 TSS Var(Y) = n 1 MSR= Goodness to Fit measurement R2 1 RSS 1 ESS ESS 1 TSS TSS TSS Note that we called this the SSE (sum of squared errors) in Section 1 of this Chapter 13. Calculated F Value = Fc Fc= MS r MSe GSB420 CLASS 9 STATISTICS (CH13) 5. Statistical Properties of Regression Equation 1) Gauss-Markov Theorem Page 8 of 14 GSB420 CLASS 9 STATISTICS (CH13) 2) Confidence Interval of coefficients 3) Single Coefficient Hypothesis Test ( t test) Regression Analysis: Wealth versus Salary The regression equation is Wealth = 209 + 2.30 Salary Predictor Constant Salary Coef 209.5 2.296 S = 208.953 SE Coef 127.7 2.030 R-Sq = 9.0% T 1.64 1.13 P 0.125 0.279 R-Sq(adj) = 2.0% Page 9 of 14 GSB420 CLASS 9 STATISTICS (CH13) Page 10 of 14 WEEK 9 Example: Gold price vs DJIA YM GOLD DJIA 2006M01 568.75 10865 2006M02 556 10993 2006M03 582 11109 2006M04 644 11367 2006M05 653 11168 2006M06 613.5 11150 2006M07 632.5 11186 2006M08 623.5 11381 2006M09 599.25 11679 2006M10 603.75 12081 2006M11 646.7 12222 2006M12 635.7 12463 2007M01 650.5 12622 2007M02 664.2 12269 2007M03 661.75 12354 2007M04 677 13063 2007M05 659.1 13628 2007M06 650.5 13409 2007M07 665.5 13212 2007M08 672 13358 2007M09 743 13896 15000 800 14000 750 DJIA 13000 700 12000 650 11000 600 10000 550 8000 500 2006M01 2006M02 2006M03 2006M04 2006M05 2006M06 2006M07 2006M08 2006M09 2006M10 2006M11 2006M12 2007M01 2007M02 2007M03 2007M04 2007M05 2007M06 2007M07 2007M08 2007M09 9000 DJIA MTB > Regress 'GOLD' 1 'DJIA'; SUBC> Constant; GOLD GSB420 CLASS 9 STATISTICS (CH13) SUBC> Page 11 of 14 Brief 2. Regression Analysis: GOLD versus DJIA The regression equation is GOLD = 232 + 0.0334 DJIA Predictor Constant DJIA Coef 232.39 0.033357 S = 26.9785 SE Coef 75.51 0.006188 R-Sq = 60.5% Analysis of Variance Source DF SS Regression 1 21150 Residual Error 19 13829 Total 20 34979 T 3.08 5.39 P 0.006 0.000 R-Sq(adj) = 58.4% MS 21150 728 F 29.06 P 0.000 1) Write the estimated equation and explain the economic meanings. 2) Find TSS (Total Sum of Square), RSS (Regression Sum of Square), ESS (Error Sum of Square), R square, Correlation Coefficient between GOLD and DJIA. 3) Perform a t test if DJIA influences GOLD Price. 4) The DJIA was 13,896 in September, 2007. What was the predicted Gold price and the error according to this regression model? 5) Today's DJIA was 17600, what is the predicted GOLD price according to the model? The actual 1 oz of gold price was $1151 at the market. What was the error from the regression model? GSB420 CLASS 9 STATISTICS (CH13) Page 12 of 14 GSB 420 WEEK 9 Example: Gold price vs DJIA OBS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 YM GOLD DJIA 2008.01 923.25 12650.36 2008.02 971.5 12266.39 2008.03 933.5 12262.89 2008.04 871 12820.13 2008.05 885.75 12638.32 2008.06 930.25 11350.01 2008.07 918 11378.02 2008.08 833 11543.55 2008.09 884.5 10850.66 2008.1 730.75 9325.01 2008.11 814.5 8829.04 2008.12 865 8776.39 2009.01 919.5 8000.86 2009.02 952 7062.93 2009.03 916.5 7608.92 2009.04 883.23 8168.12 2009.05 975.5 8500.33 2009.06 934.5 8447 2009.07 939 9171.61 2009.08 955.5 9496.28 2009.09 995.75 9712.28 2009.1 1040 9712.73 2009.11 1175.75 10344.84 2009.12 1104 10428.05 2010.01 1078.5 10067.33 2010.02 1108.25 10325.26 2010.03 1115.5 10856.63 2010.04 1179.25 11008.61 14000 1400 12000 1200 10000 1000 8000 800 6000 600 4000 400 2000 200 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 DJIA GOLD GSB420 CLASS 9 STATISTICS (CH13) Page 13 of 14 Regression Analysis: GOLD versus DJIA The regression equation is GOLD = 900 + 0.0057 DJIA Predictor Constant DJIA Coef 900.4 0.00572 S = 109.983 SE Coef 133.4 0.01301 R-Sq = 0.7% T 6.75 0.44 P 0.000 0.664 R-Sq(adj) = 0.0% Analysis of Variance Source Regression Residual Error Total DF 1 26 27 SS 2342 314505 316847 MS 2342 12096 F 0.19 P 0.664 2) Write the estimated equation and explain the economic meanings. 2) Find TSS (Total Sum of Square), RSS (Regression Sum of Square), ESS (Error Sum of Square), R square, Correlation Coefficient between GOLD and DJIA. 3) Perform a t test if DJIA influences GOLD Price. 4) The DJIA was 11008.61 in April, 2010. What was the predicted Gold price and the error? 5) Today's DJIA was 17484, what is the predicted GOLD price according to the model? The 1 oz of gold was $1140 at the market. What was the error from the regression model? GSB420 CLASS 9 STATISTICS (CH13) Page 14 of 14 Required Steps to explain a simple regression model 1. 2. 3. 4. 5. 6. 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