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UW-Madison ECON 310 - Simple Regression, Residuals, and Confidence or Prediction Intervals

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Economics 310 Menzie D. Chinn Fall 2003 Social Sciences 7418 University of Wisconsin-Madison Handout for Lecture of 3 December 1. Simple Regression, Residuals, and Confidence/Prediction Intervals The graph below left is of the actual series (y) and the predicted ($y), where the OLS regression is: $..yx=+×0049855 0514719 Below right are some actual and fitted values, and resulting “residuals” (what the textbook calls “errors”), for observations i=2001q1 to 2003q1. These columns correspond to the variables xi , yi $yi, and yyii−$ obs USINFL1Y USIGB_ USIGB_HAT USIGB_-USIGB_HAT2001:1 0.033895 0.050500 0.067302 -0.0168022001:2 0.033718 0.052700 0.067210 -0.0145102001:3 0.026937 0.049800 0.063720 -0.0139202001:4 0.018562 0.047700 0.059410 -0.0117102002:1 0.012576 0.050800 0.056328 -0.0055282002:2 0.012961 0.051000 0.056527 -0.0055272002:3 0.015944 0.042600 0.058062 -0.0154622002:4 0.022006 0.040100 0.061182 -0.0210822003:1 0.028608 0.039200 0.064580 -0.025380 This graph presents the Fitted values, Confidence Interval (narrow band) for mean of y at x = x p , and the Prediction Interval (broader band) for individual value of y at x=x p . Dependent Variable: USIGB_ Method: Least Squares Date: 12/02/03 Time: 14:52 Sample: 1961:1 2003:1 Included observations: 169 Variable Coefficient Std. Error t-Statistic Prob. C 0.049855 0.002720 18.33019 0.0000USINFL1Y 0.514719 0.050905 10.11129 0.0000R-squared 0.379732 Mean dependent var 0.072622Adjusted R-squared 0.376017 S.D. dependent var 0.025109S.E. of regression 0.019834 Akaike info criterion -4.991068Sum squared resid 0.065696 Schwarz criterion -4.954027Log likelihood 423.7452 F-statistic 102.2383Durbin-Watson stat 0.070167 Prob(F-statistic) 0.000000.02.04.06.08.10.12.14.1660 65 70 75 80 85 90 95 00USIGB_ USIGB_HATactualpredicted.00.04.08.12.16.20.00 .04 .08 .12 .1695% upper limitfor mean95% lower limitfor mean95% upperlimit for apred. value95% lowerlimit for apred. valueFittedINFL1YUSIGB_ , USIGB_HAT-.08-.04.00.04.08.12.1665 70 75 80 85 90 95 00USIGB_ USBUSGDPUS 10 yearconstant maturitybond yieldsUS gov't budgetsurplus (NIPA).02.04.06.08.10.12.14.1665 70 75 80 85 90 95 00USIGB_ USIGB_HAT USIGB_HAT2ActualSimple Regression(Inflation)Mult. Regression(Inflation, BudgetSurplus)2. Multiple Regression Consider augmenting the regression with an additional independent variable. Economic theory suggests that interest rates depend upon government borrowing. One can exam whether the prediction of this theory is borne out by the data. Dependent Variable: USIGB_ Method: Least Squares Date: 12/02/03 Time: 21:26 Sample(adjusted): 1961:1 2003:1 Included observations: 169 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 0.045264 0.002250 20.11792 0.0000USINFL1Y 0.463224 0.041480 11.16733 0.0000USBUSGDP -0.556245 0.058666 -9.481509 0.0000R-squared 0.597636 Mean dependent var 0.072622Adjusted R-squared 0.592788 S.D. dependent var 0.025109S.E. of regression 0.016023 Akaike info criterion -5.412028Sum squared resid 0.042617 Schwarz criterion -5.356468Log likelihood 460.3164 F-statistic 123.2808Durbin-Watson stat 0.164138 Prob(F-statistic) 0.000000 The figure on the left is the 10 year government T-bill rate, and the US government budget surplus expressed as a ratio to GDP (National Income and Product Accounts definition). The figure on the right shows the actual interest rate, the prediction from the simple regression and the prediction from the multiple regression. Below are columns for the actual series, the residuals from the simple and multiple regression. obs USIGB_ USIGB_-USIGB_HAT USIGB_-USIGB_HAT2 2001:1 0.050500 -0.016802 -0.001740 2001:2 0.052700 -0.014510 -0.001978 2001:3 0.049800 -0.013920 -0.013334 2001:4 0.047700 -0.011710 -0.006650 2002:1 0.050800 -0.005528 -0.011163 2002:2 0.051000 -0.005527 -0.013170 2002:3 0.042600 -0.015462 -0.024090 2002:4 0.040100 -0.021082 -0.031590 2003:1 0.039200 -0.025380


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UW-Madison ECON 310 - Simple Regression, Residuals, and Confidence or Prediction Intervals

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