Richard Yung richard.yung@gmail.com VEE regression analysis spring 9 (51)915188 Paid loss triangle: Parameter Stability Introduction: As mentioned in the background description of the paid loss triangle, it is mainly affected by three factors: volume of business growth, inflation, and loss payment patterns. Regression analysis can use past or simulated data conveniently to estimate the coefficients of these three factors. Since business growth, inflation and loss payment patterns are very unlikely to remain constant from year to year; one must understand changes that are needed in regression analysis when any or all of these factors changed. In this project, regression analysis is used to generate residual plots of different simulations based on how inflation rate changes and which variables are regressed while volume of business growth and loss payment patterns of the regression equation are constant. Simulated random errors were also introduced to see if the generated residual plots will reflect how the calculated regression equations are affected by stochasticity. Simulation Setup: A VBA macro MS Excel was written in a way to simulate paid loss triangle, perform regression analysis using its Data Analysis Toolpak, and generate residual plots against development and calendar. The macro utilized parameters entered in the worksheet named parameters and simulates 5 scenarios: Simulation 1: constant inflation rate with CY and DY regressed Simulation : discretely changed inflation rate with only CY and DY regressed Simulation 3: discretely changed inflation rate with all variables regressed Simulation 4: continuously changed inflation rate with only CY and DY regressed Simulation 5: continuously changed inflation rate with all variables regressed The only difference of parameter usage is that β b was used as the final inflation rate in simulation 4 and 5. Consequently, β continuously shifted towards β b at a constant rate each consecutive year. The rate of change was calculated from β, β b, and the year when inflation rate started to change. The parameters names and the method of simulating errors were the same ones listed in illustrative worksheet provided by NEAS. In this project, different levels of stochasticity via different values of σ were tested in all simulations. Also, the -test method was used to compare simulations and 3 and simulations 4 and 5. Proposed Equations: In order to test if a proposed regression equation fits well, sigma is set to in the simulations. Since simulation 1 assumes all parameters (α, β 1, β, are constant), the proposed equation is E ( Y ) = α + β1( DY) + β ( CY ) + ε. Simulation and 3, on the contrary have to use a modified equation from simulation 1 because of the changed inflation rate. By using a dummy variable and introducing another variable k, where the
end of k th year is when inflation rate changed. Thus, the E(Y) should be equations: E Y ) = α + β ( DY ) + β ( k) + β b ( CY ) + ε when inflation rate changed, and ( 1 k ( Y ) = α + β1( DY) + β ( CY ) + ε E before the inflation rate changed. Simulation 4 and 5 also use a modified equation from simulation 1 also because of the changed inflation rate. However, the inflation rate is changed continuously instead. Another estimator β a and variable k are introduced here as well. Let r be the continuous rate of inflation rate change and k is where the end of k th year is when inflation rate β b β changed. Thus, β a =.Thus the E(Y) should be equations: CY k ( CY k)( CY k+ 1) E( Y ) = α + β1( DY ) + β ( CY ) + β a + ε when inflation rate changed, and E ( Y ) = α + β1( DY) + β ( CY ) + ε before the inflation rate changed. (See appendix for the derivations of these equations) Results of 1 st Simulation Run: The importance of this run was that all simulations used the very same low sigma of.1, thus testing the accuracy of each simulation. The other parameters were also the same in each simulation. Parameters for Run #1: Sigma.1 Alpha 15 beta1 -.15 beta.5 beta1b -.15 betab. CY DY x th year inflation changed? 1 betaa.1875 Simulation 1:.5..15.1.5 -.5 -.1 1 3 4 5 11 1 13 14 15 16 17 18 19
CY residuals: 1 3 4 5.17 -.755.731 -.78.15.54 -.14.156 -.414 -.1 11 1 13 14 15.58 -.137 -.368.1311.77 16 17 18 19.151 -.17 -.43.33.3 Simulation 1 statistics: Multiple R.99986 R Square.99974 Square.9997 Error.1675 Observations 1 df SS MS Regression 85.5894 4.75447 375187.8 Residual 7.3589.114 Total 9 85.5353 Coefficients Error t Stat P-value Lower Alpha 15.7.7 74.894 14.99664 15.481 Beta1 -.1515.177-85.673 -.155 -.14981 Beta.585.177 83.7458 3.8E-7.49737.5433 As expected, the adjusted R is approximately 1, the standard error is close to.1, and the coefficient α, β 1, β, were also very close to their respective inputted values. of -stat is, indicating that hypothesis of β 1, β are can be rejected. The plotted against calendar showed a relatively horizontal line proving that the equation E ( Y ) = α + β1( DY) + β ( CY ) + ε is a very good fit if the coefficients are assumed to be constant throughout the development and calendar.
Simulation :.8.6.4. -. -.4 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5.59351.566.48761.33994.53796.18753.941.155 -.684 -.155 11 1 13 14 15 -.388 -.31816 -.4597 -.189 -.11684 16 17 18 19 -.541.577.8785.15937.798 Simulation statistics: SUMMARY OUTPUT Multiple R.96897 R Square.97171 Square.96468 Error.1931 Observations 1 df SS MS Regression 98.8578 49.1489 1317.64 1.8E-118 Residual 7 7.788.3796 Total 9 16.61
Coefficients Error t Stat P-value Lower Alpha 14.414.37479 384.4855.1E-97 14.33615 14.48393 Beta1 -.14993.3193-46.9514.5E-11 -.1563 -.14363 Beta.13368.3193 41.45174 3.4E-1.167.138663 The adjusted R and the standard error were calculated to around.93 and.19 respectively. urthermore, even though the coefficient α and β 1 are close to the respective inputted values, β was found to be no where near.5. The plotted against calendar showed a V shape, proving that the equation E ( Y ) = α + β1( DY) + β ( CY ) + ε was not a good fit for a linear model. Simulation 3:.1.8.6.4. -. -.4 -.6 -.8 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5.7984 -.591.14.68.6154 -.41-1.1E-6 -.373-6.1E-5.97 11 1 13 14 15 -.3.3744 -.14 -.37.546 16 17 18 19 -.189 -.76 -.16 -.154.1939
Simulation 3 statistics: Multiple R.99998 R Square.999817 Square.999814 Error.974 Observations 1 Df SS MS Regression 3 16.1614 35.38714 3746.8 Residual 6.19478 9.46E-5 Total 9 16.189 Coefficients Error t Stat P-value Lower Alpha 15.684.86 5347.173 15.13 15.137 Beta1 -.157.161-933.36 -.1539 -.14975 Beta.49135.33 148.189.9E-11.48481.49788 Betab.95.86 699.5937.199731.86 By including an extra variable which indicated the year when inflation rate changed in this regression analysis, the adjusted R became very close to 1, and the coefficient α, β 1, β, and β b were also very close to their respective inputted values. -test would be used to compare simulation and 3. Since simulation s regression equation was E( Y ) = α + β1( DY) + β ( CY ) + ε and simulation 3 s regression is E( Y ) = α + β1( DY ) + β ( k) + β b ( CY k) + ε, simulation s regression equation would be restricted and simulation 3 s regression equation would be unrestricted. So - test would be: ( RUR RR ) / q (.999817.97171) /1 q, k = 1,1 3 = 8173.34 (1 R ) /( k) (1.999817) /(1 3) UR The -test statistic was greater than the critical values for 1,1 3 between 3.9 and 3.84 for 5% significance, and between 6.85 and 6.63 for 1% significance. Either way, this showed that variable k introduced in simulation 3 was statistically significant.
Simulation 4:.4.3..1 -.1 -. 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5.3196.8391.34673.17619.144736.113365.677.3745 -.864 -.499 11 1 13 14 15 -.8957 -.141 -.153 -.14776 -.13878 16 17 18 19 -.155 -.4748.1979.117356.7844 Simulation 4 statistics: Multiple R.97969 R Square.959793 Square.95944 Error.19966 Observations 1 df SS MS Regression 83.4647 41.7335 47.66 3.5E-145 Residual 7 3.49647.16891 Total 9 86.96117
Coefficients Error t Stat P-value Lower Alpha 14.68983.5 58.4159 14.641 14.73955 Beta1 -.158.149-69.837 7.3E-146 -.1543 -.14584 Beta.8994.149 41.85183 5.7E-13.8573.94177 The adjusted R and the standard error were calculated to around.95 and.13 respectively. Also, only the coefficient β 1 are close to the respective inputted value. α was found to be 14.69 and β was found to be near.9. The plotted against calendar showed a rounded v shape, proving that the equation E ( Y ) = α + β1( DY ) + β ( CY ) + ε was not a good fit for a linear regression model. Simulation 5:.1.8.6.4. -. -.4 -.6 -.8 -.1 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5 -.539 -.63 -.816.8431 -.6.575 -.8891699.38.441.493 11 1 13 14 15 -.36.738 -.157 -.45 -.93 16 17 18 19 -.14.1417959.37.197 -.96
Simulation 5 statistics: Multiple R.999881 R Square.99976 Square.999758 Error.14 Observations 1 df SS MS Regression 3 86.79378 8.9316 87941 Residual 6.698.1 Total 9 86.81448 Coefficients Error t Stat P-value Lower Alpha 14.99691.561 5855.644 14.99187 15.196 Beta1 -.14999.166-94.96 -.153 -.14967.7E- Beta.538.71 185.7 31.49793.586 9.E- Betaa.18589.11 184.596 31.1839.18787 Just like simulation 3, by including an extra variable which indicated the year when inflation rate changed in this regression analysis, the adjusted R became very close to 1, and the coefficient α, β 1, β, and β b were also very close to their respective inputted values. -test would also be used to compare simulation 4 and 5. Since simulation 4 s regression equation was E( Y ) = α + β1( DY) + β ( CY ) + ε and simulation 5 s regression ( CY k)( CY k+ 1) equation was E( Y ) = α + β1( DY ) + β ( CY ) + β a + ε, simulation 4 s regression equation would be restricted and simulation 5 s regression equation would be unrestricted. So -test would be: ( RUR RR ) / q (.99976.959793) /1 q, k = 1,1 3 = 3476.95 (1 R ) /( k) (1.99976) /(1 3) UR The -test statistic was greater than the critical values for 1,1 3 between 3.9 and 3.84 for 5% significance, and between 6.85 and 6.63 for 1% significance. Either way, this showed that variable k introduced in simulation 5 was statistically significant. Results of nd Simulation Run: After using low stochasticity to find out the accuracies and patterns of the five simulations, a larger value of sigma could be used to see if the same patterns would be masked by stochasticity in each of the simulations. This run was that all simulations used a higher sigma of.5. The other parameters were also the same as the ones in previous each simulation run.
Parameters for Run #: Sigma.5 Alpha 15 beta1 -.15 beta.5 beta1b -.15 betab. CY DY x th year inflation changed? 1 betaa.1875 Simulation 1:.4.3..1 -.1 -. 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5.347449 -.6357.9899 -.677 -.6485.13869 -.3644 -.548 -.1779 -.934 11 1 13 14 15.99659.7175 -.4.9337 -.116 16 17 18 19.1793.345 -.18.1757 -.6999 Simulation 1 statistics: Multiple R.94187 R Square.8871 Square.88599 Error.4661 Observations 1
df SS MS Regression 94.1888 47.5444 81.4351 9.8E-99 Residual 7 11.98898.57918 Total 9 16.979 Coefficients Error t Stat P-value Lower Alpha 14.9675.4675 319.171 1.1E-8 14.81467 14.99883 Beta1 -.1588.3979-39.91 3.6E-99 -.16666 -.1597 Beta.59894.3979 15.516 4.5E-35.548.67739 Because of the larger error values due to higher sigma, the adjusted R is now approximately.89 and the standard error is about.4. Similarly, the coefficient α, β 1, β also deviated slightly from their respective inputted values. The plotted against calendar also showed a noisier horizontal line, proving that it was affected by higher value of sigma. Simulation :.7.6.5.4.3..1 -.1 -. -.3 -.4 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5.1461.47354.56853.471.54844.6916.454.516 -.3673 -.1568 11 1 13 14 15 -.4336 -.3143 -.1684 -.3547 -.746 16 17 18 19 -.149.3717.86841.119568.6531
Simulation statistics: Multiple R.91567 R Square.84987 Square.84783 Error.9381 Observations 1 df SS MS Regression 1.6961 5.3486 583.337 8.68E-86 Residual 7 17.8694.8636 Total 9 118.5655 Coefficients Error t Stat P-value Lower Alpha 14.44135.57 53.71 5.8E-6 14.3893 14.55376 Beta1 -.1581.4858-31.4545 8.46E-81 -.1639 -.1434 Beta.13394.4858 7.5159.1E-7.1816.14197 Unlike previous simulation, the adjusted R dropped to.85 and the standard error increased to.9. Not only that, the coefficients α, and β 1 also drifted slightly away from their respective inputted values. The plotted against calendar showed a V shape,, but it was not a perfect V shape as seen in run 1 simulation. Thus, this also showed that the regression was affected by higher value of sigma. Simulation 3:.15.1.5 -.5 1 3 4 5 11 1 13 14 15 16 17 18 19 -.1
CY residuals: 1 3 4 5 -.779 -.6883.876 -.6591.57834 -.951.4398 -.69.5484 -.7156 11 1 13 14 15 -.3414.1647.389.158.89 16 17 18 19 -.845 -.5381.45447 -.8917.811 Simulation 3 statistics: Multiple R.93837 R Square.8888 Square.878545 Error.54188 Observations 1 df SS MS Regression 3 97.87334 3.6445 54.933 1.1E-94 Residual 6 13.3995.64611 Total 9 111.1833 Coefficients Error t Stat P-value Lower Alpha 14.97439.73364 4.199 1.1E-39 14.8975 15.1193 Beta1 -.14879.43-35.45 1.59E-89 -.1578 -.145 Beta.56737.8667 6.54634 4.61E-1.39649.7386 Betab.1784.7484 3.8395 3.8E-61.163664.193175 As noted earlier, this was the more accurate regression of discretely changed inflation rate. The adjusted R was.88 opposed to.85 from previous simulation. Just like the previous run, -test would be used to compare simulation and 3, which is: ( RUR RR ) / q (.8888.84987) /1 q, k = 1,1 3 = 53.65 (1 RUR ) /( k) (1.8888) /(1 3) The -test statistic was a lot smaller than the one calculated previously, but it was still greater than the critical values for 1,1 3 between 3.9 and 3.84 for 5% significance, and between 6.85 and 6.63 for 1% significance. Either way, this showed that variable k introduced in simulation 3 was still statistically significant. Also, just like -test statistic, the residuals plot against calendar showed that the seemingly horizontal line was affected more by higher value of sigma.
Simulation 4:.5.4.3..1 -.1 -. -.3 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5.19313.367315.43788.4536.18161.8648.146636 -.14176 -.6763.759 11 1 13 14 15 -.1768 -.16458 -.8689 -.1754 -.1817 16 17 18 19 -.9856 -.9633.331.13546.6766 Simulation 4 statistics: Multiple R.91764 R Square.833138 Square.83156 Error.84537 Observations 1 df SS MS Regression 83.677 41.8385 516.773 3.7E-81 Residual 7 16.75894.8961 Total 9 1.4359
Coefficients Error t Stat P-value Lower Alpha 14.641.5519 65.143 4.6E- 64 14.53114 14.74887 Beta1 -.14986.475-31.8518 9.8E-8 -.15913 -.1458 Beta.9694.475 19.717.31E- 49.83418.11969 The adjusted R was.83 and the standard error was.8. The plotted against calendar showed a rounded, but noisier V shape, unlike the one seen in run 1 simulation 4. Thus, this showed that the regression was affected by the higher value of sigma. Simulation 5:.15.1.5 -.5 -.1 -.15 -. -.5 -.3 -.35 1 3 4 5 11 1 13 14 15 16 17 18 19 CY residuals: 1 3 4 5 -.3164 -.443 -.1891.661.318 -.3533.1148.66 -.588.7538 11 1 13 14 15 -.1163.5486.111 -.1313.145 16 17 18 19 -.49.484 -.3941.5639 -.48
Simulation 5 statistics: Multiple R.9957 R Square.863146 Square.861153 Error.437 Observations 1 df SS MS Regression 3 77.16388 5.719 433.849 1.16E-88 Residual 6 1.345.59391 Total 9 89.3984 Coefficients Error t Stat P-value Lower Alpha 14.9644.667 39.718 5E-54 14.8367 15.49 Beta1 -.1461.43-34.8937.3E-88 -.14855 -.1366 Beta.561.6589 7.98319 9.85E-14.39611.6559 Betaa.17435.448 7.1147 1.75E-11.168.6 Also noted earlier, this was the more accurate regression of continuously changed inflation rate. The adjusted R was.86 opposed to.83 from previous simulation. -test would also be used to compare simulation 4 and 5, which would be: ( RUR RR ) / q (.863146.833138) /1 q, k = 1,1 3 = 45.389 (1 RUR ) /( k) (1.863146) /(1 3) The -test statistic was a lot smaller than the one calculated previously, but it was still greater than the critical values for 1,1 3 between 3.9 and 3.84 for 5% significance, and between 6.85 and 6.63 for 1% significance. Either way, this showed that variable k introduced in simulation 5 was still statistically significant. Also, just like -test statistic, the residuals plot against calendar showed that the seemingly horizontal line was affected more by the higher value of sigma. Results of 3 rd Simulation Run: Previously, a moderate value of sigma was used to find out that the simulated random errors were not large enough to mask the significance of new variables introduced in simulation 3 and 5. Now, a larger value of sigma would be used to see if the same patterns would show in each of the simulations. This run was that all simulations used an even higher sigma of.6. The other parameters were also the same as the ones in previous each simulation run.
Parameters for Run #3: Sigma.6 Alpha 15 beta1 -.15 beta.5 beta1b -.15 betab. CY DY x th year inflation changed? 1 betaa.1875 Simulation 1:.8.6.4. -. -.4 -.6 1 3 4 5 11 1 13 14 15 16 17 18 19 CV Residuals: 1 3 4 5 -.1379.44396.5899 -.9137 -.88 -.36587 -.3937 -.637.1473 -.5361 11 1 13 14 15.61649.364 -.7574.17693 -.1479 16 17 18 19.1614 -.3149 -.847 -.1691.15585 Simulation 1 statistics: Multiple R.767396 R Square.588897 Square.58495 Error.549473 Observations 1
df SS MS Regression 89.565 44.7635 148.616 1.11E-4 Residual 7 6.49759.3191 Total 9 15.41 Coefficients Error t Stat P-value Lower Alpha 15.1453.16635 14.89 1.7E-7 14.843 15.476 Beta1 -.1573.986-16.81 1.47E-4 -.1764 -.1348 Beta.46987.986 5.171533 5.46E-7.974.64899 Since an even higher sigma was used, the adjusted R is now approximately.59 and the standard error is about.55. The plotted against calendar also showed a higher absolute residual averages, proving that it was affected by the higher value of sigma. Simulation :.6.5.4.3..1 -.1 -. -.3 -.4 -.5 1 3 4 5 11 1 13 14 15 16 17 18 19 CV Residuals: 1 3 4 5.3359.5336.1393.8176.89645.18156.143 -.14 -.166.395 11 1 13 14 15 -.3748 -.311 -.6566.117487 -.19634 16 17 18 19.17887 -.11374 -.868.13488.1951
Simulation statistics: Multiple R.783955 R Square.614586 Square.6186 Error.5557 Observations 1 df SS MS Regression 1.774 5.3851 165.44 1.39E-43 Residual 7 63.19431.3587 Total 9 163.9647 Coefficients Error t Stat P-value Lower Alpha 14.4334.178 134.511 1.9E-3 14.1194 14.63474 Beta1 -.1583.9136-16.781.64E-4 -.1784 -.1348 Beta.1354.9136 14.5333.35E-33.11449.15516 Just as the sigma increased, the adjusted R dropped even lower to.61 and the standard error increased to.55. urthermore, it was harder to tell whether the average residuals plotted against calendar showed a V shape, or a noisy horizontal trend as in simulation 1. Therefore, this also showed that the regression was affected by the higher value of sigma. Simulation 3:.6.4. -. -.4 -.6 1 3 4 5 11 1 13 14 15 16 17 18 19
CV Residuals: 1 3 4 5 -.557.5791 -.13461.5355.9498 -.4481 -.51 -.113.8991 -.3451 11 1 13 14 15.69486 -.996.739 -.15393.9663 16 17 18 19 -.11868 -.19671.94367 -.679.113597 Simulation 3 statistics: Multiple R.74436 R Square.553589 Square.54788 Error.5964 Observations 1 df SS MS Regression 3 9.8654 3.8847 85.1579 7.18E-36 Residual 6 73.733.355696 Total 9 164.1387 Coefficients Error t Stat P-value Lower Alpha 14.9335.17135 86.75188 3.8E-164 14.59368 15.743 Beta1 -.14734.986-14.941 1.11E-34 -.16679 -.179 Beta.6315.336 3.18469.146.311.1331 Betab.15777.1756 8.9893 1.7E-16.1386.1938 The adjusted R was.55 opposed to.61 from simulation and the standard error was.6. Just like the previous runs, -test would be used to compare simulation and 3, which is: ( RUR RR ) / q (.553589.614586) /1 q, k = 1,1 3 = 8.84 (1 R ) /( k) (1.553589) /(1 3) UR The -test statistic was less than the critical values for 1,1 3 between 3.9 and 3.84 for 5% significance, and between 6.85 and 6.63 for 1% significance. The introduced variable k in simulation 3 was no longer statistically significant. Also, it was much more difficult to tell if the residuals plot against calendar showed even a seemingly horizontal line. Both these pointed out that stochasticity was large enough to mask the significance of the newly introduced variable.
Simulation 4: 1.5 1.5 -.5-1 1 3 4 5 11 1 13 14 15 16 17 18 19 CV Residuals: 1 3 4 5 1.63677 -.1947.1581 -.1946.359813.35798.98883 -.145 -.169.15414 11 1 13 14 15 -.15645 -.4719.13869 -.8754 -.4835 16 17 18 19.14117 -.139 -.15633.16635.5385 Simulation 4 statistics: Multiple R.787 R Square.51646 Square.496831 Error.616375 Observations 1 df SS MS Regression 79.165 39.5816 14.1838 4.96E-3 Residual 7 78.6496.379918 Total 9 157.855
Coefficients Error t Stat P-value Lower Alpha 14.685.119619 1.747.6E-195 14.4463 14.91788 Beta1 -.1468.119-14.434 4.85E-33 -.16689 -.167 Beta.81814.119 8.7417 7.36E-14.6171.1197 The adjusted R was.5 and the standard error was.6. The difference here was that the plotted against calendar did not show a V shape. This showed that the V shape of the regression had less impact than the stochasticity introduced in this simulation. Simulation 5:.6.5.4.3..1 -.1 -. -.3 -.4 -.5 1 3 4 5 11 1 13 14 15 16 17 18 19 CV Residuals: 1 3 4 5 -.4361 -.9875.4681 -.5165.38879.5468 -.3413.77769 -.3484 -.483 11 1 13 14 15.9954 -.14466.88647.96968.6386 16 17 18 19 -.344 -.9481.9796 -.9387.43833
Simulation 5 statistics: Multiple R.716379 R Square.513199 Square.5611 Error.651 Observations 1 df SS MS Regression 3 78.83771 6.794 7.3943 5.18E-3 Residual 6 74.783.3631 Total 9 153.6 Coefficients Error t Stat P-value Lower Alpha 14.87595.153943 96.6359 1.5E-173 14.5744 15.17946 Beta1 -.14434.9963-14.4879.91E-33 -.16398 -.147 Beta.6515.169 3.997854 8.9E-5.339.974 Betaa.1135.653 1.85613.64863 -.7.3168 The adjusted R was.51 opposed to.5 from simulation 4 and the standard error was.6. Just like the previous runs, -test would be used to compare simulation 4 and 5, which is: ( RUR RR ) / q (.513199.51646) /1 q, k = 1,1 3 = 4.913 (1 R ) /( k) (1.513199) /(1 3) UR The -test statistic was greater than the critical values for 1,1 3 between 3.9 and 3.84 for 5% significance, but less than the range between 6.85 and 6.63 for 1% significance. The introduced variable k in simulation 5 was statistically significant enough at level of confidence. Also, it was much more difficult to tell if the residuals plot against calendar showed even a seemingly horizontal line. These also pointed out that stochasticity was large enough to mask the significance of the newly introduced variable. Conclusion: These simulations have shown that when a change in the inflation rate was introduced in the paid loss triangle simulation, two factors would affect the regression analysis: the inclusion of the extra variable being the year when the change occurred, and the severity of stochasticity affecting the data points. As long as the effect of stochasticity was small enough, the regression equation including the extra variable would be the better fit.
Appendix: