Case of Modelling M3 Time Series for Forecasting

7/31/2019 Case of Modelling M3 Time Series for Forecasting

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Case of Modelling M3 Time Series for forecasting

Data for M3 time series is available in RBI website. Consider model for forecasting monthly M3.Though the series has long history, for regime uniformity sake, let us take the sample period of 1991:01 to 2012:01, a total of 250 observations. The pursuit is to forecast the level of M3 for

oncoming months from Feb 2012 onwards.

Stage 1: Plot Graph of the given series

Plot the series to get intuitive idea of the model that the series is following. Is the series stationary?Is there a trend? If yes, what kind of trend?

The above picture of this sort suggests that the series is growing at an exponential rate. The firststep in dealing with exponential series is to take logarithms. Logs of the series make it linear

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8e+006

1990 1995 2000 2005 2010

M3, Rs Crs


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Stage 2: Test the stationarity of the series l_M3

Is the l_M3 stationary? Obviously, not it not as the series is growing over time and mean at everypoint of time is higher. One of the following two is possible.

1) The series of l_M3 has time trend in it2) The series of l_M3 is drift

The best way to identify the series is to undertake a Dickey-Fuller test. The following is the result of the D-F test on l_M3

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1990 1995 2000 2005 2010

log(M3)

Augmented Dickey-Fuller test for l_M3including 12 lags of (1-L)l_M3 (max was 15)sample size 237unit-root null hypothesis: a = 1

test with constantmodel: (1-L)y = b0 + (a-1)*y(-1) + ... + e1st-order autocorrelation coeff. for e: -0.040lagged differences: F(12, 223) = 9.539 [0.0000]estimated value of (a - 1): 0.000139036test statistic: tau_c(1) = 0.223199asymptotic p-value 0.9741

with constant and trendmodel: (1-L)y = b0 + b1*t + (a-1)*y(-1) + ... + e1st-order autocorrelation coeff. for e: -0.013lagged differences: F(13, 220) = 8.897 [0.0000]estimated value of (a - 1): -0.0411438test statistic: tau_ct(1) = -2.27374asymptotic p-value 0.4478


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Stage 3: Modelling the ld_M3 in ARMA terms

Now that we know first differences of log of M3 is stationary, we can try to model it in terms of

ARMA process. The correlogram on the series would suggest the lag structure.

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lag

ACF for ld_M3

+- 1.96/T^0.5

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lag

PACF for ld_M3

+- 1.96/T^0.5

Autocorrelation function for ld_M3

LAG ACF PACF Q-stat. [p-value]

1 -0.0092 -0.0092 0.0212 [0.884]2 -0.0815 -0.0816 1.7002 [0.427]3 -0.1779 *** -0.1807 *** 9.7414 [0.021]4 -0.1323 ** -0.1511 ** 14.2073 [0.007]5 0.0203 -0.0213 14.3129 [0.014]6 0.2778 *** 0.2343 *** 34.1546 [0.000]7 -0.1171 * -0.1603 ** 37.6931 [0.000]8 -0.1117 * -0.1135 * 40.9272 [0.000]9 -0.2095 *** -0.1740 *** 52.3590 [0.000]

10 -0.0836 -0.1001 54.1880 [0.000]11 0.2002 *** 0.1201 * 64.7150 [0.000]12 0.4701 *** 0.4031 *** 122.9808 [0.000]13 -0.0115 0.0747 123.0160 [0.000]14 -0.0388 0.0587 123.4169 [0.000]15 -0.1460 ** 0.0425 129.1134 [0.000]16 -0.1360 ** -0.0937 134.0716 [0.000]17 0.0312 -0.1395 ** 134.3345 [0.000]18 0.2298 *** 0.0651 148.6179 [0.000]19 -0.0752 0.0458 150.1560 [0.000]20 -0.1280 ** -0.0336 154.6262 [0.000]21 -0.1601 ** 0.0330 161.6548 [0.000]22 -0.0701 -0.0173 163.0087 [0.000]23 0.2554 *** 0.0814 181.0435 [0.000]24 0.2728 *** 0.0220 201.7227 [0.000]


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Reading the correlogram gives interesting insights into to the process AR and MA lag structures. Thestars indicate that the correlations are statistically significant. Three starts indicate the significanceat 99 %. Taking correlations of 99% significance as bench mark, we have several autocorrelations

and several partial autocorrelations significant. First looking at the PACF, the correlations at order 3,6, 9 and 12 are significant. This means that the process AR of corresponding lags. Similarly on theACF side, lag structures of 3,6,9, 11 and 12 are significant. (further lag could be ignored, thoughsignificant). Therefore the model we try is AR(3,6,9,12) and MA(3,6,9,11,12). The following is theoutput of the model estimated.

Reading the correlogram gives interesting insights into to the lag structure of the process. Going bythe three stars indication, the ACF suggests that autocorrelations are significant at lags 3, 6, 9, 11and 12. The process could have the corresponding MAs. The PACF indicates significantautocorrelations at lags 3, 6 ,9 and 12. The process could be tried for ARs of corresponding lags.

Therefore, the model we try for ld_M3 is AR(3,6,9,12) MA(3,6,9,11,12). The following is the modeloutput.

It is rather difficult to say anything about the adequacy of the above model by simply looking at theoutput. This output could be useful to compare models estimated for different ARMA structures. Weare not trying here different versions for the model, which you could do at your leisure. For now, wehave to examine the residual terms or the innovation terms to whether they are whitenoise or not. If we find them white noise, then the model could be accepted for forecasting. If not, then model

needs to be refined further. The best test is to examine the correlogram of innovation terms.

Model 8: ARIMA, using observations 1990:05-2011:01 (T = 249)Dependent variable: (1-L) l_M3

Standard errors based on HessianCoefficient Std. Error z p-value

const 0.0131024 0.00104836 12.4980


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lag

Residual ACF

+- 1.96/T^0.5

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lag

Residual PACF

+- 1.96/T^0.5

Residual autocorrelation function


1 -0.1674 *** -0.1674 *** 7.0611 [0.008]2 -0.0422 -0.0722 7.5114 [0.023]3 0.0110 -0.0090 7.5419 [0.056]4 -0.0053 -0.0082 7.5491 [0.110]5 0.0579 0.0579 8.4087 [0.135]6 -0.0477 -0.0289 8.9944 [0.174]7 -0.1231 * -0.1350 ** 12.9095 [0.074]8 0.0043 -0.0489 12.9143 [0.115]9 -0.0426 -0.0678 13.3871 [0.146]

10 -0.0358 -0.0635 13.7217 [0.186]11 0.0096 -0.0103 13.7461 [0.247]12 0.0765 0.0879 15.2889 [0.226]13 -0.0926 -0.0775 17.5597 [0.175]14 0.0263 -0.0117 17.7430 [0.219]15 0.0494 0.0370 18.3938 [0.243]16 -0.0399 -0.0480 18.8200 [0.278]17 0.0058 -0.0304 18.8292 [0.338]18 0.1341 ** 0.1526 ** 23.6917 [0.165]19 -0.0078 0.0537 23.7082 [0.208]20 -0.0333 -0.0406 24.0107 [0.242]21 0.0068 0.0176 24.0234 [0.292]22 -0.0426 -0.0397 24.5222 [0.320]23 0.0752 0.0260 26.0857 [0.297]24 -0.0487 -0.0192 26.7449 [0.316]


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The residual correlogram as it was plotted above shows that there is very significant correlation atlag 1, 7 and 18. As such this autocorrelation does not qualify the residual as a white noise. Thereforemodel needs to be refined. The way to eliminate these autocorrelations is to corresponding AR andMA lags in the process and re-estimate. So, the new model would now be is AR(1,3,6,7, 9,12, 18) andMA(1,3,6,7,9,11,12,18).

The correlogram of the residuals from the new model are plotted blow. It suggests that the residualsare whitenoise as there is no significant autocorrelation.

Model 11: ARIMA, using observations 1990:05-2011:01 (T = 249)Dependent variable: (1-L) l_M3

Standard errors based on Outer Products matrixCoefficient Std. Error z p-value

const 0.013183 0.000581906 22.6549


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Residual ACF

+- 1.96/T^0.5

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lag

Residual PACF

+- 1.96/T^0.5

Residual autocorrelation function


1 0.0504 0.0504 0.6394 [0.424]2 -0.0017 -0.0043 0.6402 [0.726]3 0.0032 0.0035 0.6428 [0.887]4 -0.0009 -0.0012 0.6430 [0.958]5 0.0572 0.0575 1.4819 [0.915]6 -0.0114 -0.0173 1.5150 [0.958]7 -0.0573 -0.0558 2.3640 [0.937]8 -0.0187 -0.0135 2.4542 [0.964]9 -0.0272 -0.0257 2.6473 [0.977]

10 -0.0413 -0.0421 3.0938 [0.979]11 -0.0277 -0.0226 3.2952 [0.986]12 0.0449 0.0543 3.8279 [0.986]13 -0.0879 -0.0937 5.8729 [0.951]14 0.0258 0.0354 6.0503 [0.965]15 0.0393 0.0387 6.4633 [0.971]16 -0.0105 -0.0156 6.4927 [0.982]17 0.0633 0.0536 7.5738 [0.975]18 0.1089 * 0.1137 * 10.7849 [0.903]19 0.0606 0.0481 11.7843 [0.895]20 -0.0351 -0.0585 12.1213 [0.912]21 -0.0015 0.0098 12.1220 [0.936]22 -0.0369 -0.0418 12.4960 [0.946]23 0.0254 0.0121 12.6747 [0.959]24 -0.0321 -0.0367 12.9617 [0.967]


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We therefore accept the second model as the one for forecasting. Below graph shows the actual v/sforecast series using the second model.

As one can see, the forecast is more or less overlapping the original series. This suggests that modelis well fit.

Stage 4: Forecasting over sample period

Model estimation range: 1990:05 - 2011:01

Standard error of residuals = 0.00764511l_M3 fitted residual

1990:05 12.5342 12.5276 0.006548291990:06 12.5367 12.5466 -0.00992013

1990:07 12.5398 12.5511 -0.01135891990:08 12.5458 12.5530 -0.00720549

1990:09 12.5600 12.5618 -0.001709711990:10 12.5885 12.5755 0.01301731990:11 12.6131 12.6043 0.008865531990:12 12.6233 12.6234 -6.063e-005

1991:01 12.6381 12.6337 0.00437008

1991:02 12.6545 12.6479 0.006566031991:03 12.6668 12.6681 -0.00132128

1991:04 12.6896 12.6842 0.005435351991:05 12.7121 12.7031 0.009060241991:06 12.7191 12.7153 0.00381296

1991:07 12.7292 12.7232 0.006035651991:08 12.7298 12.7358 -0.00592299

1991:09 12.7348 12.7464 -0.01167071991:10 12.7597 12.7604 -0.0006900071991:11 12.7664 12.7810 -0.01457381991:12 12.7690 12.7824 -0.0133511

1992:01 12.7801 12.7857 -0.00558868

1992:02 12.7876 12.7927 -0.005107741992:03 12.8050 12.8027 0.00227100

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1990 1995 2000 2005 2010

Actual and fitted l_M3

fittedactual


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1992:04 12.8452 12.8260 0.0191821 *1992:05 12.8524 12.8565 -0.004138811992:06 12.8524 12.8622 -0.00979035

1992:07 12.8613 12.8612 0.0001958071992:08 12.8683 12.8672 0.00115955

1992:09 12.8783 12.8812 -0.002912291992:10 12.8920 12.9000 -0.007942441992:11 12.9077 12.9046 0.003156801992:12 12.9214 12.9162 0.00517396

1993:01 12.9411 12.9333 0.007833891993:02 12.9571 12.9520 0.005050951993:03 12.9741 12.9742 -0.0001655701993:04 13.0099 12.9970 0.01290841993:05 13.0209 13.0182 0.002694121993:06 13.0261 13.0272 -0.00113425

1993:07 13.0408 13.0345 0.00624381

1993:08 13.0409 13.0457 -0.00484384

1993:09 13.0727 13.0512 0.0214356 *1993:10 13.0919 13.0869 0.004942901993:11 13.0994 13.1040 -0.004671311993:12 13.1054 13.1126 -0.00723581

1994:01 13.1118 13.1233 -0.01149931994:02 13.1238 13.1285 -0.004664881994:03 13.1761 13.1500 0.0260867 *1994:04 13.1709 13.1895 -0.01857371994:05 13.1812 13.1857 -0.004575601994:06 13.1844 13.1878 -0.00338099

1994:07 13.1883 13.1946 -0.00638287

1994:08 13.1966 13.2011 -0.004489151994:09 13.2187 13.2196 -0.0009409591994:10 13.2316 13.2284 0.003169781994:11 13.2329 13.2433 -0.01037821994:12 13.2401 13.2420 -0.00183619

1995:01 13.2549 13.2555 -0.0005886801995:02 13.2647 13.2738 -0.009108141995:03 13.3033 13.2963 0.007051881995:04 13.3200 13.3129 0.007108861995:05 13.3233 13.3270 -0.003701821995:06 13.3367 13.3292 0.00752042

1995:07 13.3410 13.3416 -0.000653104

1995:08 13.3467 13.3500 -0.003292181995:09 13.3667 13.3722 -0.005526791995:10 13.3712 13.3771 -0.005979811995:11 13.3827 13.3816 0.001075711995:12 13.3902 13.3916 -0.00143499

1996:01 13.4142 13.4033 0.01089751996:02 13.4245 13.4286 -0.004133211996:03 13.4531 13.4531 1.18039e-0051996:04 13.4709 13.4675 0.003434541996:05 13.4815 13.4805 0.000995487

1996:06 13.4930 13.4883 0.00471634

1996:07 13.4956 13.5011 -0.005465071996:08 13.4993 13.5012 -0.00191476

1996:09 13.5201 13.5192 0.0009175821996:10 13.5316 13.5305 0.001114421996:11 13.5458 13.5427 0.00307324

1996:12 13.5519 13.5573 -0.005328491997:01 13.5652 13.5673 -0.00201404

1997:02 13.5816 13.5819 -0.0003799461997:03 13.6187 13.6069 0.01173371997:04 13.6375 13.6314 0.006037421997:05 13.6482 13.6487 -0.000457018

1997:06 13.6582 13.6525 0.005693391997:07 13.6687 13.6655 0.003172951997:08 13.6914 13.6739 0.01748941997:09 13.7114 13.7063 0.005111671997:10 13.7267 13.7215 0.005162731997:11 13.7313 13.7358 -0.00452418

1997:12 13.7361 13.7442 -0.00801593

1998:01 13.7539 13.7524 0.00147791

1998:02 13.7655 13.7764 -0.01089861998:03 13.7963 13.7937 0.002597001998:04 13.8097 13.8115 -0.001826711998:05 13.8199 13.8208 -0.000987070

1998:06 13.8261 13.8252 0.0009422451998:07 13.8386 13.8392 -0.0005675571998:08 13.8468 13.8479 -0.001078451998:09 13.8647 13.8667 -0.001957321998:10 13.8760 13.8756 0.0003444921998:11 13.8833 13.8844 -0.00109987

1998:12 13.9067 13.8945 0.0122103

1999:01 13.9057 13.9169 -0.01116811999:02 13.9209 13.9294 -0.008550811999:03 13.9326 13.9477 -0.01509991999:04 13.9588 13.9539 0.004907651999:05 13.9651 13.9689 -0.00381567

1999:06 13.9789 13.9741 0.004789691999:07 13.9794 13.9852 -0.005802571999:08 13.9858 13.9886 -0.002819121999:09 14.0005 14.0002 0.0003573801999:10 14.0170 14.0156 0.001380711999:11 14.0413 14.0271 0.0141791

1999:12 14.0564 14.0523 0.00416814

2000:01 14.0595 14.0604 -0.0008157462000:02 14.0707 14.0768 -0.006060742000:03 14.0880 14.0936 -0.005609742000:04 14.1161 14.1116 0.004489672000:05 14.1299 14.1292 0.000758723

2000:06 14.1408 14.1387 0.002121432000:07 14.1431 14.1438 -0.0006862542000:08 14.1509 14.1498 0.001130322000:09 14.1576 14.1616 -0.004028202000:10 14.1668 14.1787 -0.0119405

2000:11 14.1796 14.1850 -0.00542591

2000:12 14.1874 14.1961 -0.008664612001:01 14.1929 14.1946 -0.00170972


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2001:02 14.2038 14.2075 -0.003726082001:03 14.2199 14.2243 -0.004433532001:04 14.2487 14.2419 0.00679992

2001:05 14.2888 14.2631 0.0256491 *2001:06 14.2914 14.2902 0.00125317

2001:07 14.2943 14.2970 -0.002701922001:08 14.3045 14.3009 0.003590272001:09 14.3104 14.3112 -0.0007224922001:10 14.3217 14.3279 -0.00624010

2001:11 14.3305 14.3415 -0.01101402001:12 14.3355 14.3449 -0.009350222002:01 14.3425 14.3450 -0.002522502002:02 14.3504 14.3559 -0.005454212002:03 14.3566 14.3717 -0.01509442002:04 14.3868 14.3860 0.000793297

2002:05 14.3948 14.4063 -0.0115729

2002:06 14.4046 14.4025 0.00209143

2002:07 14.4070 14.4107 -0.003707202002:08 14.4140 14.4156 -0.001508912002:09 14.4207 14.4187 0.002038602002:10 14.4387 14.4339 0.00476436

2002:11 14.4441 14.4502 -0.006072892002:12 14.4554 14.4615 -0.006065012003:01 14.4692 14.4647 0.004498442003:02 14.4882 14.4794 0.008714732003:03 14.5115 14.5029 0.008622582003:04 14.5373 14.5337 0.00367927

2003:05 14.5364 14.5500 -0.0136134

2003:06 14.5397 14.5473 -0.007533762003:07 14.5395 14.5494 -0.009891602003:08 14.5494 14.5540 -0.004559332003:09 14.5525 14.5583 -0.005815912003:10 14.5638 14.5660 -0.00225925

2003:11 14.5668 14.5683 -0.001514482003:12 14.5785 14.5839 -0.005354242004:01 14.6068 14.5923 0.01453312004:02 14.6108 14.6204 -0.009546792004:03 14.6245 14.6313 -0.006826322004:04 14.6620 14.6474 0.0145892

2004:05 14.6614 14.6686 -0.00724331

2004:06 14.6685 14.6687 -0.0002107232004:07 14.6735 14.6776 -0.004050872004:08 14.6849 14.6829 0.002035382004:09 14.7196 14.6934 0.0262296 *2004:10 14.7201 14.7272 -0.00701084

2004:11 14.7273 14.7229 0.004375482004:12 14.7380 14.7446 -0.006556532005:01 14.7409 14.7540 -0.01317202005:02 14.7586 14.7597 -0.001027352005:03 14.8160 14.7836 0.0323340 *

2005:04 14.8320 14.8245 0.00750977

2005:05 14.8345 14.8405 -0.006002422005:06 14.8360 14.8367 -0.000657781

2005:07 14.8548 14.8461 0.008759162005:08 14.8750 14.8691 0.005910322005:09 14.8945 14.8941 0.000350471

2005:10 14.8929 14.9021 -0.009239142005:11 14.9082 14.8966 0.0115940

2005:12 14.9161 14.9162 -0.0001497722006:01 14.9363 14.9306 0.005661672006:02 14.9594 14.9580 0.001357452006:03 15.0125 14.9927 0.0198008 *

2006:04 15.0114 15.0165 -0.005066452006:05 15.0161 15.0259 -0.009775432006:06 15.0337 15.0208 0.01292532006:07 15.0556 15.0455 0.01006312006:08 15.0642 15.0650 -0.0007833832006:09 15.0923 15.0851 0.00721480

2006:10 15.1007 15.0955 0.00519625

2006:11 15.1185 15.1055 0.0130635

2006:12 15.1253 15.1267 -0.001384602007:01 15.1517 15.1445 0.007199872007:02 15.1718 15.1743 -0.002480412007:03 15.2063 15.2059 0.000322465

2007:04 15.2112 15.2102 0.0009810722007:05 15.2255 15.2279 -0.002407522007:06 15.2282 15.2325 -0.004264642007:07 15.2384 15.2468 -0.008436032007:08 15.2568 15.2480 0.008841162007:09 15.2703 15.2763 -0.00600355

2007:10 15.2875 15.2809 0.00657004

2007:11 15.2946 15.2931 0.001518702007:12 15.3071 15.3038 0.003275212008:01 15.3361 15.3247 0.01139062008:02 15.3555 15.3576 -0.002091962008:03 15.3830 15.3831 -2.070e-005

2008:04 15.4051 15.3928 0.01226612008:05 15.4153 15.4160 -0.0007017992008:06 15.4166 15.4204 -0.003780052008:07 15.4346 15.4356 -0.0009840192008:08 15.4380 15.4430 -0.004961622008:09 15.4488 15.4562 -0.00738138

2008:10 15.4609 15.4646 -0.00371287

2008:11 15.4700 15.4693 0.0007636882008:12 15.4729 15.4815 -0.008672782009:01 15.4961 15.4921 0.004066122009:02 15.5125 15.5161 -0.003663272009:03 15.5388 15.5391 -0.000352605

2009:04 15.5471 15.5533 -0.006266042009:05 15.5565 15.5609 -0.004457042009:06 15.5578 15.5609 -0.003090212009:07 15.5807 15.5775 0.003122882009:08 15.5827 15.5844 -0.00168554

2009:09 15.5903 15.5966 -0.00629463

2009:10 15.6206 15.6034 0.01718202009:11 15.6223 15.6252 -0.00291684


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2009:12 15.6441 15.6333 0.01079512010:01 15.6491 15.6591 -0.01002952010:02 15.6666 15.6706 -0.00395995

2010:03 15.6872 15.6897 -0.002469752010:04 15.7100 15.7049 0.00513273

2010:05 15.7129 15.7179 -0.005008972010:06 15.7172 15.7222 -0.00491916

2010:07 15.7348 15.7307 0.004065042010:08 15.7390 15.7398 -0.0008219702010:09 15.7426 15.7526 -0.00996629

2010:10 15.7565 15.7620 -0.005545882010:11 15.7637 15.7663 -0.00267740

2010:12 15.7894 15.7764 0.01301992011:01 15.7839 15.7990 -0.0150310

Note: * denotes a residual in excess of 2.5 standard errorsForecast evaluation statistics

Mean Error -6.3271e-005Mean Squared Error 6.1595e-005Root Mean Squared Error 0.0078482Mean Absolute Error 0.0059259Mean Percentage Error -0.00048024

Mean Absolute Percentage Error 0.042073Theil's U 0.47617

The above statistics help in measuring the precision in historical forecasts.

Stage 5: Forecasting in to the future.

The sample period was 1991:04 to 2012: 01. The last observation at January 2012, indicatesthat the M3 was Rs. 7159465 Crs. The log value of the same is 15.7839. The model hasforecast the same as 15.7990. Using the actual M3 of January, what would be the Feb 2012forecast? In the same breath, can we also forecast, March, April, May, June and furtherforecasts of M3? The model gives the following results.

Converting the l_M3 forecasts into levels (by taking exp values), we get the forecasts interms of levels as given below

For 95% confidence intervals, z(0.025) = 1.96Obs l_M3 prediction std. error 95% interval

2010:12 15.789408 15.7763882011:01 15.783946 15.7989772011:02 15.806310 0.007645 15.7913 - 15.82122011:03 15.825254 0.009402 15.8068 - 15.843622011:04 15.845025 0.010777 15.8239 - 15.866172011:05 15.851928 0.011914 15.8285 - 15.87520

2011:06 15.866407 0.013081 15.8407 - 15.89206

M3 forecast M3

2011:01 7159465 72678922011:02 73213832011:03 74614062011:04 76103912011:05 7663104

2011:06 7774868


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Documents

Case of Modelling M3 Time Series for Forecasting