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BUAD 300Prof. RobinsonThe χ2 test for Normality of the Random Errors in OLS EstimationIf a random variable is normally distributed, then its variance is chi-square (χ2 ) distributed. Knowing this, we can use a calculated χ2 as a test statistic to see if the data actually is normal. Is the assumption justified that the error term in a regression is normally distributed? The answer concerns, our use of the t-distribution for testing hypotheses about our regression coefficients (β0 and β1),our use of the t-distribution for computing confidence intervals about our predictions, andthe use of an F distribution in our ANOVA analysis. (The importance of this F distribution is explored later in the course.)Here we explore how to test the hypothesis that our sample data is from a normal distribution. To do this we calculate a χ2 test statistic from a table that divides our sample data into 10 equally probable cells, or ranges. These ranges are calculated with the assumption that our data is normally distributed. We then count how many of our sample observations actually fit within each range. Since these cells are arranged so that if our data is normally distributed, then 10% of our sample of observations should fit within each cell, then the differences between the numbers we observe and the hypothetical 10% form the basis for a χ2 test statistic. For example, if our data has 100 observations, then if the data is perfectly normal, we should have 10 observations in each cell. Of course, we might have some deviations from this hypothetical 10, but these deviations should be spurious and small if our sample data is from a normal distribution. We then count how many of our observations actually are within each range, and then we calculate the differences between the actual number and the hypothetical 10. The larger these differences, the lower the probability that our sample data is actually from a normal distribution. Of course, if our sample size were not 100, but say for example 80, then we should expect to observe 8 within each of our equally probable cells assuming our sample is from a normal distribution.In our OLS regression analysis, we assumed that our random error term is normally distributed. Our third theorem from our previous handout (handout 3) asserted that any linear function of a normally distributed random variable is itself normally distributed. For example, in model equation (1) below, if “u” is normal, then “sales” must also be normally distributed. Knowing this allows us to use t-tests on our sample estimates for our regression coefficients of β0 and β1 and also our F statistic for our ANOVA. To see if our claims of normality for the residual errors (u) justify the use of these test statistics, then when we calculate our regression, we could have Minitab save our residual errors, and we could use this sample of errors to calculate a χ2 test for normality. As an illustration, we shall proceed as follows:Given the sales data sent to you, estimate model equation (1) by OLS regression using Minitab. When you calculate your regression, save your standardized errors by using the “Storage” button, and clicking on “standardized errors.”.Salest = β0 + β1 Time in Quarterst + ut(1)Standardized errors are merely calculated t-statistics, which when we have a large number of observations (and 80 observations should be sufficient) are approximately z-statistics. We can then use these calculated z-statistics to compute our χ2 table below. (This test for normality is reviewed in your text, pp. 530-534.) Save the standardized errors from your regression estimate of model equation (1). Then “sort” these errors. “Sort” is a command under the “Data” tool in Minitab. Fill in Table 1 below. (See the attached example spreadsheet.) Test the hypothesis H0: The random errors are normally distributed. (Note that this χ2 test is presented in section 12.3 of your text - 12th edition.)Decision rule: If χ2calculated > χ2critical, then reject H0!Critical Value of Test Statistic: With 7 degrees of freedom (i.e. when you have 10 cells), χ2critical =14.07Decision result: The cells of the table below are arranged by z statistics so that the probability is 10% for each cell, i.e. 10% of the observations (8 observations) should be in each cell for the sample data to be perfectly normal. This is not likely to occur, so that the χ2 is based on the differences between the numbers observed (f) and the numbers expected (e). The χcalc2 is the sum of the last column of the Table. If χcalc2 > 14.07, then we reject the hypothesis that the data is normal. χ2 test for Normality of the Random Errors in OLS Estimation of Equation (1)Cells for Standardized ResidualsObserved Cell Numbers: fExpected Cell Numbers: e(f – e)2(f – e)2/eBelow -1.282836-.84 to -1.2817881-.52 to -.8416864-.25 to -.5238250.00 to -.25684+.25 to 0.00781+.52 to +.25880+.52 to +.843825+.84 to +1.28880Above +1.281084χ2 calc. = 30 H0: The random errors are normally distributed.Significance level is 95%.Decision rule: If χ2calculated > χ2critical, then reject H0!Critical Value of Test Statistic: With 7 degrees of freedom (i.e. when you have 10 cells), χ2critical =14.07Decision result: Since χ2calculated > χ2critical, then reject H0!The Minitab spreadsheets for these calculations are attached. Review question: Repeat the χ2 analysis above for the residual errors from estimating model equation (2)?Ln (salesi) = β0 + β1 (Time in Quartersi) + ui(2)χ2 test for Normality of the Random Errors in OLS Estimation of Equation (2)Cells for Standardized ResidualsObserved Cell Numbers: fExpected Cell Numbers: e(f – e)2(f – e)2/eBelow -1.28-.84 to -1.28-.52 to -.84-.25 to -.520.00 to -.25+.25 to 0.00+.52 to +.25+.52 to +.84+.84 to +1.28Above +1.28χ2 calc. = H0: The random errors are normally distributed.Significance level is 95%.Decision rule: If χ2calculated > χ2critical, then reject H0!Critical Value of Test Statistic: With 7 degrees of freedom (i.e. when you have 10 cells), χ2critical =14.07Decision result: The Minitab spreadsheets for these calculations should be attached. Time in QuartersSalesSRESSorted SRES131.7501.41759-1.50370245.5611.49663-1.28589343.4361.34108-1.25471477.3761.71600-1.13544548.9841.17485-1.12404655.0631.14049-1.12196759.6141.08382-1.10301873.0401.15729-1.09519958.5450.82194-1.087241060.1300.72234-1.065851136.9980.26163-1.065411293.9530.97137-1.046491358.1980.32689-1.025271472.9350.41982-1.024171571.7670.28078-0.9807916106.6990.66786-0.930491779.3340.14740-0.876281886.4740.12970-0.873001992.1250.09034-0.8544720106.6890.18061-0.835972193.372-0.13436-0.822792296.176-0.21487-0.788652374.306-0.65354-0.7123624132.5660.07101-0.709822598.162-0.54926-0.6535426114.298-0.43619-0.6499627114.578-0.55305-0.6450728151.009-0.14597-0.6409229125.194-0.64092-0.6350530133.940-0.63505-0.6269231141.252-0.64996-0.6141332157.535-0.53499-0.5530533145.997-0.82279-0.5492634150.643-0.87628-0.5350835130.680-1.28589-0.5349936190.913-0.53508-0.4521137158.551-1.12404-0.4361938176.801-0.98079-0.4171839179.269-1.06585-0.2330140217.963-0.62692-0.2148741194.492-1.08724-0.1845142205.663-1.04649-0.1459743215.485-1.02527-0.1343644234.367-0.87300-0.0093645225.518-1.121960.0710146232.947-1.135440.0903447215.864-1.503700.1297048279.079-0.709820.1474049249.803-1.254710.1494950271.246-1.065410.1510351277.020-1.103010.1806152319.136-0.614130.2616353299.205-1.024170.2761454314.042-0.930490.2807855327.657-0.854470.3268956350.465-0.645070.3468357345.679-0.835970.4198258357.314-0.788650.4487359344.584-1.095190.4879560412.305-0.233010.6678661387.691-0.712360.7223462413.960-0.452110.8219463424.729-0.417180.8663264472.0150.149490.9713765457.435-0.184511.0838266477.809-0.009361.1404967497.1570.151031.1572968525.8970.448731.1748569527.2510.346831.2366470545.2410.487951.2450771539.0890.276141.3410872613.6171.245071.3838873596.0490.866321.4175974629.6111.236641.4966375647.9281.383881.7160076703.0252.071591.8551977696.5311.855192.0715978725.2732.156882.1568879753.2822.448372.4483780790.9872.883402.88340 ................
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