interpreting log transformed regression/outlier variable in PROC ARIMA

[Forecaster]

How do I interpret the outlier effects such as level shift on log transformed data, clearly exponentiating the coefficients is not correct.

As an example, see below is what I get. how to interpret Nov 2008 level shift value of -0.09102?

below is a reproducible example to produce the above coefficients.

data retail(label="Retail Sales");
   attrib 
           Date length=8 label="Date" format=MONYY7. informat=MONYY7.
           sales length=8 label="Retail Sales" 
          ;
   input Date sales;
datalines;
Jan1992	146376
Feb1992	147079
Mar1992	159336
Apr1992	163669
May1992	170068
Jun1992	168663
Jul1992	169890
Aug1992	170364
Sep1992	164617
Oct1992	173655
Nov1992	171547
Dec1992	208838
Jan1993	153221
Feb1993	150087
Mar1993	170439
Apr1993	176456
May1993	182231
Jun1993	181535
Jul1993	183682
Aug1993	183318
Sep1993	177406
Oct1993	182737
Nov1993	187443
Dec1993	224540
Jan1994	161349
Feb1994	162841
Mar1994	192319
Apr1994	189569
May1994	194927
Jun1994	197946
Jul1994	193355
Aug1994	202388
Sep1994	193954
Oct1994	197956
Nov1994	202520
Dec1994	241111
Jan1995	175344
Feb1995	172138
Mar1995	201279
Apr1995	196039
May1995	210478
Jun1995	211844
Jul1995	203411
Aug1995	214248
Sep1995	202122
Oct1995	204044
Nov1995	212190
Dec1995	247491
Jan1996	185019
Feb1996	192380
Mar1996	212110
Apr1996	211718
May1996	226936
Jun1996	217511
Jul1996	218111
Aug1996	226062
Sep1996	209250
Oct1996	222663
Nov1996	223953
Dec1996	258081
Jan1997	200389
Feb1997	197556
Mar1997	225133
Apr1997	220329
May1997	234190
Jun1997	227365
Jul1997	231521
Aug1997	235252
Sep1997	222807
Oct1997	232251
Nov1997	228284
Dec1997	271054
Jan1998	207853
Feb1998	203863
Mar1998	230313
Apr1998	234503
May1998	245027
Jun1998	244067
Jul1998	241431
Aug1998	240462
Sep1998	231243
Oct1998	244234
Nov1998	240991
Dec1998	288969
Jan1999	218126
Feb1999	220650
Mar1999	253550
Apr1999	250783
May1999	262113
Jun1999	260918
Jul1999	262051
Aug1999	265089
Sep1999	253905
Oct1999	258040
Nov1999	264106
Dec1999	317659
Jan2000	236422
Feb2000	250580
Mar2000	279515
Apr2000	264417
May2000	283706
Jun2000	281288
Jul2000	271146
Aug2000	283944
Sep2000	269155
Oct2000	270899
Nov2000	276507
Dec2000	319958
Jan2001	250746
Feb2001	247772
Mar2001	280449
Apr2001	274925
May2001	296013
Jun2001	287881
Jul2001	279098
Aug2001	294763
Sep2001	261924
Oct2001	291596
Nov2001	287537
Dec2001	326202
Jan2002	255598
Feb2002	253086
Mar2002	285261
Apr2002	284747
May2002	300402
Jun2002	288854
Jul2002	295433
Aug2002	307256
Sep2002	273189
Oct2002	287540
Nov2002	290705
Dec2002	337006
Jan2003	268328
Feb2003	259051
Mar2003	293693
Apr2003	294251
May2003	312389
Jun2003	300998
Jul2003	309923
Aug2003	317056
Sep2003	293890
Oct2003	304036
Nov2003	301265
Dec2003	357577
Jan2004	281460
Feb2004	282444
Mar2004	319077
Apr2004	315191
May2004	328408
Jun2004	321044
Jul2004	328000
Aug2004	326317
Sep2004	313524
Oct2004	319726
Nov2004	324259
Dec2004	387155
Jan2005	293261
Feb2005	295062
Mar2005	339141
Apr2005	335632
May2005	345348
Jun2005	350945
Jul2005	351827
Aug2005	355701
Sep2005	333289
Oct2005	336134
Nov2005	343798
Dec2005	405608
Jan2006	318546
Feb2006	314051
Mar2006	361993
Apr2006	351667
May2006	373560
Jun2006	366615
Jul2006	362203
Aug2006	375795
Sep2006	346214
Oct2006	348796
Nov2006	356928
Dec2006	417991
Jan2007	328877
Feb2007	323162
Mar2007	374142
Apr2007	358535
May2007	391512
Jun2007	376639
Jul2007	372354
Aug2007	388016
Sep2007	353936
Oct2007	368681
Nov2007	377802
Dec2007	426077
Jan2008	342639
Feb2008	343893
Mar2008	372907
Apr2008	368920
May2008	397956
Jun2008	378507
Jul2008	383726
Aug2008	382862
Sep2008	350542
Oct2008	349847
Nov2008	335545
Dec2008	384236
Jan2009	310188
Feb2009	299412
Mar2009	328526
Apr2009	329854
May2009	347728
Jun2009	344405
Jul2009	348071
Aug2009	353392
Sep2009	324646
Oct2009	338610
Nov2009	339363
Dec2009	400281
Jan2010	314557
Feb2010	310925
Mar2010	360679
Apr2010	356333
May2010	365605
Jun2010	358604
Jul2010	361939
Aug2010	362627
Sep2010	345999
Oct2010	355209
Nov2010	365828
Dec2010	426663
Jan2011	335587
Feb2011	337314
Mar2011	387088
Apr2011	380775
May2011	391999
Jun2011	388700
Jul2011	384682
Aug2011	394609
Sep2011	375025
Oct2011	379482
Nov2011	391220
Dec2011	451821
Jan2012	355184
Feb2012	372401
Mar2012	414149
Apr2012	392949
May2012	418608
Jun2012	400975
Jul2012	396026
Aug2012	417922
Sep2012	385609
Oct2012	399400
Nov2012	411065
Dec2012	462102
Jan2013	375587
Feb2013	373987
Mar2013	421719
Apr2013	408544
May2013	437188
Jun2013	414714
Jul2013	422410
Aug2013	435005
Sep2013	396213
Oct2013	415700
Nov2013	423786
Dec2013	476910
Jan2014	383054
Feb2014	380019
Mar2014	432651
Apr2014	431396
May2014	459231
Jun2014	433282
Jul2014	443281
Aug2014	451366
Sep2014	421424
Oct2014	438457
Nov2014	439165
Dec2014	502330
Jan2015	398027
Feb2015	388063
Mar2015	445970
Apr2015	439637
May2015	464785
Jun2015	449794
Jul2015	459533
Aug2015	457905
Sep2015	432782
Oct2015	446593
Nov2015	446773
Dec2015	519625
Jan2016	401982
Feb2016	415189
Mar2016	461198
Apr2016	451468
May2016	470430
Jun2016	465023
Jul2016	462162
Aug2016	471996
Sep2016	448749
Oct2016	453088
Nov2016	467765
Dec2016	540273
Jan2017	421560
Feb2017	417983
Mar2017	483059
Apr2017	466058
May2017	495264
Jun2017	482456
Jul2017	475984
Aug2017	491090
Sep2017	470406
Oct2017	476925
Nov2017	499446
Dec2017	560379
Jan2018	445484
Feb2018	437005
Mar2018	510380
Apr2018	482412
May2018	530082
Jun2018	510029
Jul2018	508010
Aug2018	523933
Sep2018	481094
Oct2018	506360
Nov2018	522804
Dec2018	563497
Jan2019	459143
Feb2019	444794
Mar2019	518304
Apr2019	510176
May2019	547036
Jun2019	517984
Jul2019	533058
;
run;


data xretail;
	set retail;
	lsales = log(sales);
run;

proc arima data=xretail;
   identify var= lsales(1,12);
   estimate q=(1)(12) noint method=ml;
   outlier type=(ao ls tc(5)) maxnum=10 id=Date;
run;

[rselukar]

Let's denote sales as Y_t, d1 the first difference, d12 the differencing of order 12, and N_t the MA noise associated with the specification q=(1)(12).  Then your initial model is d12(d1(log(Y_t))) = N_t.  A level shift at NOV2008 would imply a model d12(d1(log(Y_t))) = beta*LS_t + N_t where beta=-0.09102 and LS_t a dummy variable that is zero before t = NOV2008 and 1 on and after NOV2008.  We can expand both these models (with and without the level shift) in their multiplicative forms to understand the meaning of the "level shift".  Expanding the models fully one gets the following expressions:

Initial model:  Y_t = Y_(t-1) * (Y_(t-12) / Y_(t-13)) *  exp(N_t)

Level shift model:  Y_t = Y_(t-1) * (Y_(t-12) / Y_(t-13)) * M_t * exp(N_t) where M_t = 1 before t = NOV2008 and exp(beta) thereafter.

As you can see, calling (Y_(t-12) / Y_(t-13)) the previous year's growth rate, the level shift model says, after NOV2008, the growth rate should be damped by exp(-0.09102).

Does this help?

[rselukar]

I want to correct my earlier reply.  After checking the description of the outlier detection process in PROC ARIMA doc, I realized that the model for level shift is as follows (continuing with the earlier notation): d12(d1(log(Y_t))) = beta*d12(d1(LS_t)) + N_t.  That is, the level shift is differenced by the same differencing operator as the response variable.  Note that beta*d12(d1(LS_t)) is zero everywhere except at NOV2008, where it is 1, and at NOV2009, where it is -1.  This means that in the multiplicative form the model becomes

Y_t = Y_(t-1) * (Y_(t-12) / Y_(t-13)) * M_t * exp(N_t) where M_t is 1 everywhere except at NOV2008, where it is exp(beta), and at NOV2009, where it is exp(-beta).  That is, the growth rate gets a jolt of exp(-0.09102) on NOV2008 and another one (in a reverse direction) of exp(0.09102) on NOV2009.

Sorry for this mistake.