// File New-Lec2-03-01-01-04.txt. Edition 7/21/2010. // Lab Specs // Window Title Estimating_Variances // Option (0 Normal, 1 Auto, 2 Heter, 3 X-e correlation, 4 Measurement, 5 Consistency (Is this used?) 0 // Constant List 40 60 10 50 // Coefficient List -2 6 2 2 // iListVarErrors 50 500 150 500 // iListSamples 3 250 1 5 // iListSamplesAutocorrelation 3 50 1 40 // iListSamplesMeasurement 3 50 1 5 // iListXMinima 0 10 10 0 // iListXMaxima 20 30 10 30 // strListFromCoef -1.0 4.0 .5 1.0 // strListToCoef 1.0 6.0 .5 3.0 // strListRho -.9 .9 .3 .0 // strListHeter -2.0 2.0 1.0 .0 // strListCoefXAndError -.9 .9 .3 .0 // iListXMeasErrVar 0 100 50 0 // Problem Specs // SampleSize // PauseCheckbox (-1 checked, 0 cleared) // EstType (0 Error, 1 Constant, 2 Coefficient) // ErrorVar // ConstValue // CoefValue // xMinValue // xMaxValue // ErrTermCheckbox (0 not visible and unchecked, -1 visible and checked, 1 visible and unchecked)) // VarEstAndFromTo (-1 VarEst visible, 0 nothing visible, 1 From-To visible) ` 3 -1 0 500 50 2 0 30 0 -1 Objective: Illustrate our first estimation procedure for the variance of the error term's probability distribution: SSE divided by T, the sample size. Show that this estimation procedure is unbiased. _ Several new fields have now appeared in the simulation: ___SSE: Reports the sum of squared errors from each repetition ___SSR: Reports the sum of squared residuals from each repetition ___Use: We can select Err, the errors, or Res, the residuals. ___Divide by: We can select divide by T, the sample size, or T-2. ` 3 -1 0 500 50 2 0 30 0 -1 Note that in the Use line the Err radiobutton is selected; consequently, the values of the actual constant and coefficient are used to calculate the value of the error term. Next, note the the T radiobutton is selected. The sum of squared errors is divided by T, the sample size. _ In other words, after each repetition of the experiment, the variance of error term's probability distribution is estimated by summing the squared errors and dividing this sum by the sample size. ` 3 -1 0 500 50 2 0 30 0 -1 Each repetition's estimate for the variance of the error term's probability distribution is reported on the Error Var Est line. The mean (average) of these estimates is reported on the Mean line immediately beneath the Error Var Est line. _ 1. Click Start. What is the sum of squared errors for the first repetition? Based on this sum, calculate the estimate for the variance of the error's term probability distribution. Record its value. Focus on the Error Var Est line. Is the simulation calculating the variance estimate correctly? ` 2a. Click Continue to run the second repetition. What is the sum of squared errors for the second repetition? Based on this sum, calculate the estimate for the variance of the error's term probability distribution. Record its value. Focus on the Error Var Est line. Is the simulation calculating the variance estimate for the second repetition correctly? _ 2b. Next, focus the Mean line immediately below the Error Var Est line. Is the simulation calculating the mean (average) of the variance estimates from the first two repetitions correctly? ` 3. Click Continue a several more times. _ 3a. Convince yourself that the simulation is calculating the estimates of the variance and their mean correctly. _ 3b. Explain why the estimate is a random variable. _ 3c. The actual value of the variance for the error term's probability distribution equals 500. Are the variance estimates always less than the actual value? Are the estimates always greater than the actual value? What is the best we can hope for? ` 3 0 0 500 50 2 0 30 0 -1 4. Note that the Pause checkbox is now cleared. Click Continue and then, after many, many repetitions click Stop. What is the mean (average) of the estimates for the variance of the error term's probability distribution? What is the actual variance of the error term's probability distribution? ` 3 0 0 200 50 2 0 30 0 -1 5. Note that the actual variance of the error term's probability distribution has been reduced to 200. Click Continue and then, after many, many repetitions click Stop. What is the mean (average) of the estimates for the variance of the error term's probability distribution? What is the actual variance of the error term's probability distribution? ` 3 0 0 50 50 2 0 30 0 -1 6. Note that the actual variance of the error term's probability distribution has been reduced to 50. Click Continue and then, after many, many repetitions click Stop. What is the mean (average) of the estimates for the variance of the error term's probability distribution? What is the actual variance of the error term's probability distribution? ` 7. Review your answers to questions 4, 5, and 6. Does this estimation procedure for the variance of the error term's probability distribution appear to be unbiased? What problems would arise if you tried to apply this procedure in practice?