# Why does the upper prediction interval region gets larger than lower prediction interval when data are transformed?

I present here two examples one with transformed data and the other without any transformation. In the transformed data case, the upper interval gets enormous large, whereas not in the untransformed case. (Function in forecast package in R)

    dmnd=c(8.6,9.8,11.2,12.4,13.5,15.7,18.6,21.1,22.3,23.6,24.6,26.3,28.3,29.6,33.3,36.4,40.5,44.9,48.4,52.6,56.8,60.5,67.2,73.4,77.8,85.6,94.8,105.5,114.0,118.5,128.3,126.9,132.6,141.2,150.0,160.8,174.6,190.0,198.1,194.1,210.4,230.3,242.4,246.4,255.5)
#with transformation
fit <- Arima(dmnd[11:45], order=c(1,2,0), lambda=-0.25)
prg=forecast(fit,h=16,level=c(95),fan=FALSE,lambda=-0.25)
prg

Point Forecast     Lo 95      Hi 95
36       262.8665 241.34010   286.8475
37       271.4097 231.18466   320.7765
38       279.9162 216.75679   367.8499
39       288.9224 201.26616   429.9376
40       298.2242 184.91488   513.2452
41       307.9328 168.56079   626.1724
42       318.0273 152.59291   782.3793
43       328.5442 137.35984  1003.7047
44       339.4971 123.06285  1326.8115
45       350.9112 109.82630  1815.8213
46       362.8087  97.70297  2589.1252
47       375.2149  86.69722  3879.8347
48       388.1558  76.77777  6185.0627
49       401.6596  67.89002 10677.4213
50       415.7557  59.96484 20513.6022
51       430.4756  52.92545 45881.0816

#without transformation
fit <- Arima(dmnd[11:45], order=c(1,2,0))
prg=forecast(fit,h=16,level=c(95),fan=FALSE)
prg

36       263.5024 252.64559 274.3592
37       271.7410 249.52972 293.9523
38       279.9288 243.87448 315.9832
39       288.1275 236.23983 340.0153
40       296.3239 226.81344 365.8344
41       304.5208 215.76687 393.2747
42       312.7176 203.22448 422.2107
43       320.9144 189.28707 452.5417
44       329.1112 174.03721 484.1851
45       337.3079 157.54444 517.0715
46       345.5047 139.86827 551.1412
47       353.7015 121.06045 586.3426
48       361.8983 101.16647 622.6302
49       370.0951  80.22673 659.9635
50       378.2919  58.27746 698.3063
51       386.4887  35.35135 737.6260

• (+1) In a nutshell, this is the entire point of a nonlinear transformation. If it did not have that property, then in the end it would have accomplished nothing. – whuber May 29 '15 at 22:27
• @whuber, I did not get what u mean. Then how can R fit an Arima model, if the transformation cannot accomplish anything? The transformation is necessary for normality and non-autocorrelated residuals. In this sense, it worked. – Dirk May 30 '15 at 10:33