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This code below shows my Discussion Week 2 Assignment. I worked with the Financial Times Index leading equity prices (quarterly) 1960-1971.

library(forecast)
## Warning: package 'forecast' was built under R version 3.4.2
## Warning in as.POSIXlt.POSIXct(Sys.time()): unknown timezone 'zone/tz/2018c.
## 1.0/zoneinfo/America/New_York'
library(fpp)
## Loading required package: fma
## Loading required package: expsmooth
## Loading required package: lmtest
## Loading required package: zoo
## 
## Attaching package: 'zoo'
## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric
## Loading required package: tseries
library(car)
## Warning: package 'car' was built under R version 3.4.3
library(MASS)
## 
## Attaching package: 'MASS'
## The following objects are masked from 'package:fma':
## 
##     cement, housing, petrol
library(readr)
library(tseries)
mydata <- read_csv("~/Desktop/financial-times-index-leading-eq.csv")
## Parsed with column specification:
## cols(
##   Quarter = col_character(),
##   `Financial Times index leading equity prices (quarterly) 1960 ? 1971` = col_double()
## )
View(mydata)
attach(mydata)
index=`Financial Times index leading equity prices (quarterly) 1960 ? 1971`

Look at data

Plotted data.

Decomposition

Build an additive and multiplicative model.

#decompose
decA<-decompose(myts,type="additive")  
decM<-decompose(myts,type="multiplicative")
#plot
plot(decA)

plot(decM)

#show decomposition seasonal, time series, and random componants
decA$seasonal
##         Qtr1      Qtr2      Qtr3      Qtr4
## 1   2.322159  5.061932 -6.339205 -1.044886
## 2   2.322159  5.061932 -6.339205 -1.044886
## 3   2.322159  5.061932 -6.339205 -1.044886
## 4   2.322159  5.061932 -6.339205 -1.044886
## 5   2.322159  5.061932 -6.339205 -1.044886
## 6   2.322159  5.061932 -6.339205 -1.044886
## 7   2.322159  5.061932 -6.339205 -1.044886
## 8   2.322159  5.061932 -6.339205 -1.044886
## 9   2.322159  5.061932 -6.339205 -1.044886
## 10  2.322159  5.061932 -6.339205 -1.044886
## 11  2.322159  5.061932 -6.339205 -1.044886
## 12  2.322159  5.061932 -6.339205 -1.044886
decA$trend
##        Qtr1     Qtr2     Qtr3     Qtr4
## 1        NA       NA 317.9375 322.3250
## 2  325.4000 321.9375 316.9875 306.2375
## 3  293.4625 287.1000 284.8875 287.1375
## 4  293.0750 303.3750 315.6000 324.9125
## 5  337.4500 346.1625 347.4125 347.5875
## 6  342.3750 337.5625 338.5375 342.0500
## 7  344.0625 337.9375 328.3750 322.4500
## 8  325.4750 342.8125 366.3750 392.4500
## 9  423.5250 451.4375 473.2625 479.9875
## 10 462.3000 434.6375 410.2000 389.4625
## 11 375.3000 366.2750 352.5000 345.6375
## 12 356.1750 374.7250       NA       NA
decA$random
##            Qtr1         Qtr2         Qtr3         Qtr4
## 1            NA           NA   9.40170455  -8.38011364
## 2   -4.02215909  22.30056818  -0.24829545  -9.39261364
## 3    5.41534091  -6.36193182  -6.84829545  -2.49261364
## 4    0.30284091   0.86306818 -13.56079545  18.13238636
## 5   -4.67215909  -6.82443182  19.82670455  -0.04261364
## 6   -4.09715909  -2.32443182  -8.89829545   4.59488636
## 7    2.91534091  16.70056818  -2.03579545 -21.50511364
## 8   -9.29715909  -4.77443182   0.76420455   6.39488636
## 9  -16.74715909   4.60056818  24.47670455  11.55738636
## 10  26.37784091  -6.69943182 -25.86079545  -5.81761364
## 11  25.77784091 -16.63693182  -3.16079545   0.80738636
## 12 -28.09715909  -6.98693182           NA           NA
decM$seasonal
##         Qtr1      Qtr2      Qtr3      Qtr4
## 1  1.0063958 1.0161365 0.9811684 0.9962992
## 2  1.0063958 1.0161365 0.9811684 0.9962992
## 3  1.0063958 1.0161365 0.9811684 0.9962992
## 4  1.0063958 1.0161365 0.9811684 0.9962992
## 5  1.0063958 1.0161365 0.9811684 0.9962992
## 6  1.0063958 1.0161365 0.9811684 0.9962992
## 7  1.0063958 1.0161365 0.9811684 0.9962992
## 8  1.0063958 1.0161365 0.9811684 0.9962992
## 9  1.0063958 1.0161365 0.9811684 0.9962992
## 10 1.0063958 1.0161365 0.9811684 0.9962992
## 11 1.0063958 1.0161365 0.9811684 0.9962992
## 12 1.0063958 1.0161365 0.9811684 0.9962992
decM$trend
##        Qtr1     Qtr2     Qtr3     Qtr4
## 1        NA       NA 317.9375 322.3250
## 2  325.4000 321.9375 316.9875 306.2375
## 3  293.4625 287.1000 284.8875 287.1375
## 4  293.0750 303.3750 315.6000 324.9125
## 5  337.4500 346.1625 347.4125 347.5875
## 6  342.3750 337.5625 338.5375 342.0500
## 7  344.0625 337.9375 328.3750 322.4500
## 8  325.4750 342.8125 366.3750 392.4500
## 9  423.5250 451.4375 473.2625 479.9875
## 10 462.3000 434.6375 410.2000 389.4625
## 11 375.3000 366.2750 352.5000 345.6375
## 12 356.1750 374.7250       NA       NA
decM$random
##         Qtr1      Qtr2      Qtr3      Qtr4
## 1         NA        NA 1.0290103 0.9743652
## 2  0.9884537 1.0677632 0.9980126 0.9695049
## 3  1.0198435 0.9796636 0.9720144 0.9913489
## 4  1.0025446 1.0033398 0.9549283 1.0565010
## 5  0.9867251 0.9791090 1.0587609 1.0005742
## 6  0.9884934 0.9921005 0.9733194 1.0141317
## 7  1.0087706 1.0474951 0.9931991 0.9335214
## 8  0.9723508 0.9849450 1.0036843 1.0173975
## 9  0.9598019 1.0051837 1.0582530 1.0256975
## 10 1.0553312 0.9804120 0.9391881 0.9860286
## 11 1.0680424 0.9530196 0.9917254 1.0030248
## 12 0.9217386 0.9790642        NA        NA
#for additive, show proof summing all these components together will equal the original data set
pa.myts = decA$seasonal+decA$trend+decA$random
pa.myts
##     Qtr1  Qtr2  Qtr3  Qtr4
## 1     NA    NA 321.0 312.9
## 2  323.7 349.3 310.4 295.8
## 3  301.2 285.8 271.7 283.6
## 4  295.7 309.3 295.7 342.0
## 5  335.1 344.4 360.9 346.5
## 6  340.6 340.3 323.3 345.6
## 7  349.3 359.7 320.0 299.9
## 8  318.5 343.1 360.8 397.8
## 9  409.1 461.1 491.4 490.5
## 10 491.0 433.0 378.0 382.6
## 11 403.4 354.7 343.0 345.4
## 12 330.4 372.8    NA    NA
myts
##     Qtr1  Qtr2  Qtr3  Qtr4
## 1  323.8 314.1 321.0 312.9
## 2  323.7 349.3 310.4 295.8
## 3  301.2 285.8 271.7 283.6
## 4  295.7 309.3 295.7 342.0
## 5  335.1 344.4 360.9 346.5
## 6  340.6 340.3 323.3 345.6
## 7  349.3 359.7 320.0 299.9
## 8  318.5 343.1 360.8 397.8
## 9  409.1 461.1 491.4 490.5
## 10 491.0 433.0 378.0 382.6
## 11 403.4 354.7 343.0 345.4
## 12 330.4 372.8 409.2 427.6
#for multiplicative, show proof multiplying all these components together will equal the original data set
pm.myts = decM$seasonal*decM$trend*decM$random
pm.myts
##     Qtr1  Qtr2  Qtr3  Qtr4
## 1     NA    NA 321.0 312.9
## 2  323.7 349.3 310.4 295.8
## 3  301.2 285.8 271.7 283.6
## 4  295.7 309.3 295.7 342.0
## 5  335.1 344.4 360.9 346.5
## 6  340.6 340.3 323.3 345.6
## 7  349.3 359.7 320.0 299.9
## 8  318.5 343.1 360.8 397.8
## 9  409.1 461.1 491.4 490.5
## 10 491.0 433.0 378.0 382.6
## 11 403.4 354.7 343.0 345.4
## 12 330.4 372.8    NA    NA
myts
##     Qtr1  Qtr2  Qtr3  Qtr4
## 1  323.8 314.1 321.0 312.9
## 2  323.7 349.3 310.4 295.8
## 3  301.2 285.8 271.7 283.6
## 4  295.7 309.3 295.7 342.0
## 5  335.1 344.4 360.9 346.5
## 6  340.6 340.3 323.3 345.6
## 7  349.3 359.7 320.0 299.9
## 8  318.5 343.1 360.8 397.8
## 9  409.1 461.1 491.4 490.5
## 10 491.0 433.0 378.0 382.6
## 11 403.4 354.7 343.0 345.4
## 12 330.4 372.8 409.2 427.6

Mean Error

Using the mean error, evaluate if additive or multiplicative is the better model.

#additive
mea = mean(na.omit(decA$random))
mea
## [1] -0.5585227
# -0.5585227

#multiplicative
mem = mean(na.omit(decM$random))
mem
## [1] 0.9974633
# 0.9974633

Through these calculations, I believe that the additive model is the better model for the Financial Times Index leading equity prices. It has a mean error closer to 0.

When given the decision to use additive or multiplicative for forecasting, I would use additive. This is based on the low mean error.