Oil vs Gold vs S&P500
Ushnik
November 18, 2016
Data
Loading data & exploration
gold <- read.csv("C:/Users/Ushnik/Downloads/Gold (2).csv", header=TRUE, stringsAsFactors=FALSE)
View(gold)
names(gold)## [1] "Date" "Value"str(gold)## 'data.frame': 322 obs. of 2 variables:
## $ Date : chr "1/1/1990" "2/1/1990" "3/1/1990" "4/1/1990" ...
## $ Value: num 410 417 393 374 369 ...head(gold)## Date Value
## 1 1/1/1990 410.1
## 2 2/1/1990 416.8
## 3 3/1/1990 393.1
## 4 4/1/1990 374.2
## 5 5/1/1990 369.1
## 6 6/1/1990 352.3summary(gold)## Date Value
## Length:322 Min. : 256.1
## Class :character 1st Qu.: 344.3
## Mode :character Median : 392.6
## Mean : 673.8
## 3rd Qu.:1096.9
## Max. :1771.9Date<-as.Date(gold$Date, "%m/%d/%Y")
gold$Date<-Date
str(gold)## 'data.frame': 322 obs. of 2 variables:
## $ Date : Date, format: "1990-01-01" "1990-02-01" ...
## $ Value: num 410 417 393 374 369 ...View(gold)
oil <- read.csv("C:/Users/Ushnik/Downloads/Oil (2).csv", header=TRUE, stringsAsFactors=FALSE)
View(oil)
names(oil)## [1] "DATE" "VALUE"str(oil)## 'data.frame': 322 obs. of 2 variables:
## $ DATE : chr "1/1/1990" "2/1/1990" "3/1/1990" "4/1/1990" ...
## $ VALUE: num 22.9 22.1 20.4 18.4 18.2 ...head(oil)## DATE VALUE
## 1 1/1/1990 22.86
## 2 2/1/1990 22.11
## 3 3/1/1990 20.39
## 4 4/1/1990 18.43
## 5 5/1/1990 18.20
## 6 6/1/1990 16.70summary(oil)## DATE VALUE
## Length:322 Min. : 11.35
## Class :character 1st Qu.: 20.40
## Mode :character Median : 32.23
## Mean : 46.63
## 3rd Qu.: 71.02
## Max. :133.88DATE<-as.Date(oil$DATE, "%m/%d/%Y")
oil$DATE<-DATE
str(oil)## 'data.frame': 322 obs. of 2 variables:
## $ DATE : Date, format: "1990-01-01" "1990-02-01" ...
## $ VALUE: num 22.9 22.1 20.4 18.4 18.2 ...View(oil)The two data set has 322 observations with columns representing Date and Monthly Values of Gold and Oil in USD respectively.
Time Series Objects
We then store both the data sets in two time series objects in R. To store the data in a time series object, we use the ts() function in R.
ts.gold<-ts(gold$Value, start=c(1990,1), frequency = 12)
ts.gold## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 1990 410.1 416.8 393.1 374.2 369.1 352.3 362.5 394.7 389.3 380.7
## 1991 383.6 363.8 363.3 358.4 357.0 366.7 367.7 356.3 348.7 358.7
## 1992 354.5 353.9 344.3 338.6 337.2 340.8 352.7 343.1 345.4 344.4
## 1993 329.0 329.3 330.1 342.2 367.2 371.9 392.2 378.8 355.3 364.2
## 1994 386.9 381.9 384.1 377.3 381.4 385.6 385.5 380.4 391.6 389.8
## 1995 378.6 376.6 382.1 391.0 385.2 387.6 386.2 383.7 383.1 383.1
## 1996 399.5 404.8 396.2 392.9 391.9 385.3 383.5 387.3 383.2 381.1
## 1997 355.1 346.6 351.8 344.5 343.8 340.8 324.1 324.0 322.8 324.9
## 1998 289.1 297.5 295.9 308.3 299.1 292.3 292.9 284.1 289.0 295.9
## 1999 287.1 287.3 286.0 282.6 276.4 261.3 256.1 256.7 264.7 310.7
## 2000 284.3 299.9 286.4 279.7 275.2 285.7 281.6 274.5 273.7 270.0
## 2001 265.5 261.9 263.0 260.5 272.4 270.2 267.5 272.4 283.4 283.1
## 2002 281.5 295.5 294.1 302.7 314.5 321.2 313.3 310.3 319.1 316.6
## 2003 356.9 359.0 340.6 328.2 355.7 356.4 351.0 359.8 379.0 378.9
## 2004 413.8 404.9 406.7 403.3 383.8 392.4 398.1 400.5 405.3 420.5
## 2005 424.0 423.4 434.3 429.2 421.9 430.7 424.5 437.9 456.1 469.9
## 2006 549.9 555.0 557.1 610.7 675.4 596.2 633.7 632.6 598.2 585.8
## 2007 631.2 664.7 654.9 679.4 666.9 655.5 665.3 665.4 712.7 754.6
## 2008 889.6 922.3 968.4 909.7 888.7 889.5 939.8 839.0 829.9 806.6
## 2009 858.7 943.2 924.3 890.2 928.6 945.7 934.2 949.4 996.6 1043.2
## 2010 1118.0 1095.4 1113.3 1148.7 1205.4 1232.9 1193.0 1215.8 1271.0 1342.0
## 2011 1356.4 1372.7 1424.0 1473.8 1510.4 1528.7 1572.8 1755.8 1771.9 1665.2
## 2012 1656.1 1742.6 1673.8 1650.1 1585.5 1596.7 1593.9 1626.0 1744.5 1747.0
## 2013 1671.0 1627.6 1592.9 1485.1 1413.5 1342.4 1286.7 1347.1 1348.8 1316.2
## 2014 1244.8 1301.0 1336.1 1299.0 1287.5 1279.1 1311.0 1296.0 1238.8 1222.5
## 2015 1251.9 1227.2 1178.6 1197.9 1199.1 1181.5 1130.0 1117.5 1124.5 1159.3
## 2016 1097.4 1199.9 1246.3 1242.3 1259.4 1276.4 1337.3 1341.1 1326.0 1266.6
## Nov Dec
## 1990 381.7 377.0
## 1991 360.2 361.7
## 1992 335.0 334.8
## 1993 373.8 383.3
## 1994 384.4 379.3
## 1995 385.3 387.4
## 1996 377.9 369.0
## 1997 306.0 288.7
## 1998 294.1 291.7
## 1999 293.2 283.1
## 2000 266.0 271.5
## 2001 276.2 275.9
## 2002 319.1 331.9
## 2003 389.9 407.0
## 2004 439.4 442.1
## 2005 476.7 510.1
## 2006 627.8 629.8
## 2007 806.3 803.2
## 2008 760.9 816.1
## 2009 1127.0 1134.7
## 2010 1369.9 1390.6
## 2011 1739.0 1652.3
## 2012 1721.1 1688.5
## 2013 1275.8 1225.4
## 2014 1176.3 1202.3
## 2015 1085.7 1068.3
## 2016ts.oil<-ts(oil$VALUE, start=c(1990,1), frequency = 12)
ts.oil## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 1990 22.86 22.11 20.39 18.43 18.20 16.70 18.45 27.31 33.51 36.04
## 1991 25.23 20.48 19.90 20.83 21.23 20.19 21.40 21.69 21.89 23.23
## 1992 18.79 19.01 18.92 20.23 20.98 22.38 21.78 21.34 21.88 21.69
## 1993 19.03 20.09 20.32 20.25 19.95 19.09 17.89 18.01 17.50 18.15
## 1994 15.03 14.78 14.68 16.42 17.89 19.06 19.65 18.38 17.45 17.72
## 1995 18.04 18.57 18.54 19.90 19.74 18.45 17.33 18.02 18.23 17.43
## 1996 18.85 19.09 21.33 23.50 21.17 20.42 21.30 21.90 23.97 24.88
## 1997 25.13 22.18 20.97 19.70 20.82 19.26 19.66 19.95 19.80 21.33
## 1998 16.72 16.06 15.12 15.35 14.91 13.72 14.17 13.47 15.03 14.46
## 1999 12.51 12.01 14.68 17.31 17.72 17.92 20.10 21.28 23.80 22.69
## 2000 27.26 29.37 29.84 25.72 28.79 31.82 29.70 31.26 33.88 33.11
## 2001 29.59 29.61 27.24 27.49 28.63 27.60 26.42 27.37 26.20 22.17
## 2002 19.71 20.72 24.53 26.18 27.04 25.52 26.97 28.39 29.66 28.84
## 2003 32.95 35.83 33.51 28.17 28.11 30.66 30.75 31.57 28.31 30.34
## 2004 34.31 34.68 36.74 36.75 40.28 38.03 40.78 44.90 45.94 53.28
## 2005 46.84 48.15 54.19 52.98 49.83 56.35 59.00 64.99 65.59 62.26
## 2006 65.49 61.63 62.69 69.44 70.84 70.95 74.41 73.04 63.80 58.89
## 2007 54.51 59.28 60.44 63.98 63.45 67.49 74.12 72.36 79.91 85.80
## 2008 92.97 95.39 105.45 112.58 125.40 133.88 133.37 116.67 104.11 76.61
## 2009 41.71 39.09 47.94 49.65 59.03 69.64 64.15 71.04 69.41 75.72
## 2010 78.33 76.39 81.20 84.29 73.74 75.34 76.32 76.60 75.24 81.89
## 2011 89.17 88.58 102.86 109.53 100.90 96.26 97.30 86.33 85.52 86.32
## 2012 100.27 102.20 106.16 103.32 94.65 82.30 87.90 94.13 94.51 89.49
## 2013 94.76 95.31 92.94 92.02 94.51 95.77 104.67 106.57 106.29 100.54
## 2014 94.62 100.82 100.80 102.07 102.18 105.79 103.59 96.54 93.21 84.40
## 2015 47.22 50.58 47.82 54.45 59.27 59.82 50.90 42.87 45.48 46.22
## 2016 31.68 30.32 37.55 40.76 46.71 48.76 44.65 44.72 45.18 49.78
## Nov Dec
## 1990 32.33 27.28
## 1991 22.46 19.50
## 1992 20.34 19.41
## 1993 16.61 14.51
## 1994 18.07 17.16
## 1995 17.99 19.03
## 1996 23.71 25.23
## 1997 20.19 18.33
## 1998 13.00 11.35
## 1999 25.00 26.10
## 2000 34.42 28.44
## 2001 19.64 19.39
## 2002 26.35 29.46
## 2003 31.11 32.13
## 2004 48.47 43.15
## 2005 58.32 59.41
## 2006 59.08 61.96
## 2007 94.77 91.69
## 2008 57.31 41.12
## 2009 77.99 74.47
## 2010 84.25 89.15
## 2011 97.16 98.56
## 2012 86.53 87.86
## 2013 93.86 97.63
## 2014 75.79 59.29
## 2015 42.44 37.19
## 2016Time Series Plot
We then plot the price changes of Gold and Oil from 1990 until 2016
plot.ts(ts.gold, ylab= "Values of Gold")
plot.ts(ts.oil, ylab= "Values of Oil")
Correlation between Gold and Oil
par(mar=c(2,8,4,8)+0.1)
plot(ts.gold, col="blue", ylim=c(0,2000), axes=F, ylab="")
box()
axis(2, col="blue")
mtext("Value of Gold", side=2, line=3)
par(new=T)
plot(ts.oil, col="red", ylim=c(0,250), axes=F, ylab="")
axis(4, col="red")
mtext("Value of Oil", side=4, line=3)
axis(1, xlim=c(1990,2016))
mtext("Gold vs Oil values from 1990 to 2016", line=3)
cor(ts.gold, ts.oil)## [1] 0.8219019Therefore from the comparison above, we can see that there is a general upward trend for both, but Gold has a steeper rise than that of Oil. There is an overall positive correlation but there seems to be an inverse relationship of the value trends at times. For instance, in 2002 and 2009 Gold seems to have a decreasing value whereas Oil has an increasing value.
S&P500 data
Loading and exploring
sp <- read.csv("C:/Users/Ushnik/Downloads/S&P500.csv", header=TRUE, stringsAsFactors=FALSE)
View(sp)
names(sp)## [1] "Date" "Open" "High" "Low" "Close" "Volume"
## [7] "Adj.Close"str(sp)## 'data.frame': 322 obs. of 7 variables:
## $ Date : chr "1/2/1990" "2/1/1990" "3/1/1990" "4/2/1990" ...
## $ Open : num 353 329 332 340 331 ...
## $ High : num 361 336 344 347 362 ...
## $ Low : num 320 322 331 328 331 ...
## $ Close : num 329 332 340 331 361 ...
## $ Volume : num 1.81e+08 1.66e+08 1.56e+08 1.46e+08 1.71e+08 ...
## $ Adj.Close: num 329 332 340 331 361 ...head(sp)## Date Open High Low Close Volume Adj.Close
## 1 1/2/1990 353.40 360.59 319.83 329.08 181041300 329.08
## 2 2/1/1990 329.08 336.09 322.10 331.89 165598400 331.89
## 3 3/1/1990 331.89 344.49 331.08 339.94 155573600 339.94
## 4 4/2/1990 339.94 347.30 327.76 330.80 146198500 330.80
## 5 5/1/1990 330.80 362.26 330.80 361.23 171016800 361.23
## 6 6/1/1990 361.26 368.78 351.23 358.02 160561400 358.02summary(sp)## Date Open High Low
## Length:322 Min. : 304.0 Min. : 319.7 Min. : 294.5
## Class :character 1st Qu.: 674.8 1st Qu.: 696.7 1st Qu.: 660.8
## Mode :character Median :1132.4 Median :1170.3 Median :1095.3
## Mean :1098.0 Mean :1133.4 Mean :1060.2
## 3rd Qu.:1378.4 3rd Qu.:1420.0 3rd Qu.:1325.3
## Max. :2173.1 Max. :2193.8 Max. :2147.6
## Close Volume Adj.Close
## Min. : 304.0 Min. :1.462e+08 Min. : 304.0
## 1st Qu.: 691.8 1st Qu.:4.440e+08 1st Qu.: 691.8
## Median :1133.7 Median :1.491e+09 Median :1133.7
## Mean :1103.4 Mean :2.128e+09 Mean :1103.4
## 3rd Qu.:1379.1 3rd Qu.:3.666e+09 3rd Qu.:1379.1
## Max. :2173.6 Max. :7.633e+09 Max. :2173.6Date<-as.Date(sp$Date, "%m/%d/%Y")
sp$Date<-Date
str(sp)## 'data.frame': 322 obs. of 7 variables:
## $ Date : Date, format: "1990-01-02" "1990-02-01" ...
## $ Open : num 353 329 332 340 331 ...
## $ High : num 361 336 344 347 362 ...
## $ Low : num 320 322 331 328 331 ...
## $ Close : num 329 332 340 331 361 ...
## $ Volume : num 1.81e+08 1.66e+08 1.56e+08 1.46e+08 1.71e+08 ...
## $ Adj.Close: num 329 332 340 331 361 ...View(sp)Time Series Objects
We then store the S&P500 data set in a time series object in R. To store the data in a time series object, we use the ts() function in R.
ts.sp<-ts(sp$Open, start=c(1990,1), frequency = 12)
ts.sp## Jan Feb Mar Apr May Jun Jul Aug
## 1990 353.40 329.08 331.89 339.94 330.80 361.26 358.02 356.15
## 1991 330.20 343.91 367.07 375.22 375.35 389.81 371.18 387.81
## 1992 417.03 408.79 412.68 403.67 414.95 415.35 408.20 424.19
## 1993 435.70 438.78 443.38 451.67 440.19 450.23 450.54 448.13
## 1994 466.51 481.60 467.19 445.66 450.91 456.50 444.27 458.28
## 1995 459.21 470.42 487.39 500.70 514.76 533.40 544.75 562.06
## 1996 615.93 636.02 640.43 645.50 654.17 669.12 670.63 639.95
## 1997 740.74 786.16 790.82 757.12 801.34 848.28 885.14 954.29
## 1998 970.43 980.28 1049.34 1101.75 1111.75 1090.82 1133.84 1120.67
## 1999 1229.23 1279.64 1238.33 1286.37 1335.18 1301.84 1372.71 1328.72
## 2000 1469.25 1394.46 1366.42 1498.58 1452.43 1420.60 1454.60 1430.83
## 2001 1320.28 1366.01 1239.94 1160.33 1249.46 1255.82 1224.42 1211.23
## 2002 1148.08 1130.20 1106.73 1147.39 1076.92 1067.14 989.82 911.62
## 2003 879.82 855.70 841.15 848.18 916.92 963.59 974.50 990.31
## 2004 1111.92 1131.13 1144.94 1126.21 1107.30 1120.68 1140.84 1101.72
## 2005 1211.92 1181.27 1203.60 1180.59 1156.85 1191.50 1191.33 1234.18
## 2006 1248.29 1280.08 1280.66 1302.88 1310.61 1270.05 1270.06 1278.53
## 2007 1418.03 1437.90 1406.80 1420.83 1482.37 1530.62 1504.66 1455.18
## 2008 1467.97 1378.60 1330.45 1326.41 1385.97 1399.62 1276.69 1269.42
## 2009 902.99 823.09 729.57 793.59 872.74 923.26 920.82 990.22
## 2010 1116.56 1073.89 1105.36 1171.23 1188.58 1087.30 1031.10 1107.53
## 2011 1257.62 1289.14 1328.64 1329.48 1365.21 1345.20 1320.64 1292.59
## 2012 1258.86 1312.45 1365.90 1408.47 1397.86 1309.87 1362.33 1379.32
## 2013 1426.19 1498.11 1514.68 1569.18 1597.55 1631.71 1609.78 1689.42
## 2014 1845.86 1782.68 1857.68 1873.96 1884.39 1923.87 1962.29 1929.80
## 2015 2058.90 1996.67 2105.23 2067.63 2087.38 2108.64 2067.00 2104.49
## 2016 2038.20 1936.94 1937.09 2056.62 2067.17 2093.94 2099.34 2173.15
## Sep Oct Nov Dec
## 1990 322.56 306.10 303.99 322.23
## 1991 395.43 387.86 392.46 375.11
## 1992 414.03 417.80 418.66 431.35
## 1993 463.55 458.93 467.83 461.93
## 1994 475.49 462.69 472.26 453.55
## 1995 561.88 584.41 581.50 605.37
## 1996 651.99 687.31 705.27 757.02
## 1997 899.47 947.28 914.62 955.40
## 1998 957.28 1017.01 1098.67 1163.63
## 1999 1320.41 1282.71 1362.93 1388.91
## 2000 1517.68 1436.52 1429.40 1314.95
## 2001 1133.58 1040.94 1059.78 1139.45
## 2002 916.07 815.28 885.76 936.31
## 2003 1008.01 995.97 1050.71 1058.20
## 2004 1104.24 1114.58 1130.20 1173.78
## 2005 1220.33 1228.81 1207.01 1249.48
## 2006 1303.80 1335.82 1377.76 1400.63
## 2007 1473.96 1527.29 1545.79 1479.63
## 2008 1287.83 1164.17 968.67 888.61
## 2009 1019.52 1054.91 1036.18 1098.89
## 2010 1049.72 1143.49 1185.71 1186.60
## 2011 1219.12 1131.21 1251.00 1246.91
## 2012 1406.54 1440.90 1412.20 1416.34
## 2013 1635.95 1682.41 1758.70 1806.55
## 2014 2004.07 1971.44 2018.21 2065.78
## 2015 1970.09 1919.65 2080.76 2082.93
## 2016 2171.33 2164.33Time Series Plot
We then plot the S&P500 values from 1990 until 2016
plot.ts(ts.sp, ylab= "S&P500 Values")
Decomposing, Holt Smoothening and forcasting for Gold, Oil and S&P500
library(forecast)## Warning: package 'forecast' was built under R version 3.3.2## Loading required package: zoo## Warning: package 'zoo' was built under R version 3.3.2##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: timeDate## Warning: package 'timeDate' was built under R version 3.3.2## This is forecast 7.3##
## Attaching package: 'forecast'## The following object is masked _by_ '.GlobalEnv':
##
## goldts.gold.d <- decompose(ts.gold)
plot(ts.gold.d)
ts.gold.holt <- HoltWinters(ts.gold, gamma=TRUE)
plot(ts.gold.holt, ylab="Gold Values", xlab="Year")
ts.gold.forecasts <- forecast.HoltWinters(ts.gold.holt, h=14)
ts.gold.forecasts## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 1268.054 1225.857 1310.251 1203.5196 1332.589
## Dec 2016 1265.386 1206.135 1324.637 1174.7690 1356.003
## Jan 2017 1274.440 1200.985 1347.895 1162.1002 1386.780
## Feb 2017 1279.063 1192.796 1365.330 1147.1293 1410.997
## Mar 2017 1260.586 1162.319 1358.853 1110.2997 1410.872
## Apr 2017 1246.917 1137.165 1356.669 1079.0658 1414.768
## May 2017 1248.752 1127.856 1369.648 1063.8575 1433.647
## Jun 2017 1248.922 1117.111 1380.733 1047.3345 1450.510
## Jul 2017 1250.799 1108.226 1393.372 1032.7523 1468.846
## Aug 2017 1261.130 1107.894 1414.365 1026.7758 1495.483
## Sep 2017 1268.686 1104.847 1432.524 1018.1161 1519.255
## Oct 2017 1264.817 1090.405 1439.229 998.0770 1531.557
## Nov 2017 1266.271 1080.472 1452.071 982.1152 1550.427
## Dec 2017 1263.603 1067.270 1459.936 963.3374 1563.869We would therefore want to invest in gold in April 2017 when the point forecast is $1246.917 and sell gold in Sep 2017 at $1268.686. The forecast above lies within a 80% and 95% confidence level intervals as indicated above.
We plot the forecast of gold by usinn the plot.forecast function:
plot.forecast(ts.gold.forecasts, ylab="Gold Values", xlab="Year")
ts.oil.d <- decompose(ts.oil)
plot(ts.oil.d)
ts.oil.holt <- HoltWinters(ts.oil, gamma=TRUE)
plot(ts.oil.holt, ylab="Oil Values", xlab="Year")
ts.oil.forecasts <- forecast.HoltWinters(ts.oil.holt, h=14)
ts.oil.forecasts## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 45.90922 38.50265 53.31578 34.581855 57.23658
## Dec 2016 45.25659 35.78074 54.73244 30.764534 59.74865
## Jan 2017 46.78734 35.61926 57.95543 29.707230 63.86746
## Feb 2017 52.54851 39.91282 65.18420 33.223888 71.87312
## Mar 2017 57.32470 43.37497 71.27444 35.990424 78.65898
## Apr 2017 60.26042 45.11018 75.41065 37.090134 83.43070
## May 2017 59.75044 43.48809 76.01279 34.879321 84.62156
## Jun 2017 56.79770 39.49457 74.10084 30.334838 83.26057
## Jul 2017 53.43384 35.14907 71.71862 25.469694 81.39799
## Aug 2017 51.03212 31.81579 70.24845 21.643275 80.42097
## Sep 2017 49.98313 29.87836 70.08791 19.235533 80.73073
## Oct 2017 47.50486 26.54928 68.46044 15.456063 79.55366
## Nov 2017 43.63408 21.40811 65.86005 9.642395 77.62576
## Dec 2017 42.98145 19.98301 65.97989 7.808377 78.15453We would therefore want to invest in oil in Jan 2017 when the point forecast is $46.78 and sell oil in Apr 2017 at $60.26. The forecast above lies within a 80% and 95% confidence level intervals as indicated above.
We plot the forecast of gold by usinn the plot.forecast function:
plot.forecast(ts.oil.forecasts, ylab="Oil Values", xlab="Year")
ts.sp.d <- decompose(ts.sp)
plot(ts.sp.d)
ts.sp.holt <- HoltWinters(ts.sp, gamma=TRUE)
plot(ts.sp.holt, ylab="S&P500 Values", xlab="Year")
ts.sp.forecasts <- forecast.HoltWinters(ts.sp.holt, h=14)
ts.sp.forecasts## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 2173.259 2111.134 2235.385 2078.247 2268.272
## Dec 2016 2171.768 2085.641 2257.895 2040.048 2303.488
## Jan 2017 2181.598 2076.831 2286.365 2021.370 2341.826
## Feb 2017 2178.730 2058.171 2299.289 1994.351 2363.109
## Mar 2017 2202.175 2067.666 2336.685 1996.461 2407.890
## Apr 2017 2223.679 2076.536 2370.822 1998.643 2448.715
## May 2017 2231.856 2073.081 2390.631 1989.031 2474.681
## Jun 2017 2229.472 2059.862 2399.083 1970.075 2488.869
## Jul 2017 2227.815 2048.021 2407.610 1952.843 2502.787
## Aug 2017 2243.395 2053.963 2432.826 1953.684 2533.105
## Sep 2017 2220.808 2022.206 2419.409 1917.072 2524.543
## Oct 2017 2205.367 1998.001 2412.734 1888.227 2522.507
## Nov 2017 2214.296 1997.824 2430.769 1883.230 2545.363
## Dec 2017 2212.805 1988.264 2437.347 1869.399 2556.212We would therefore want to invest in S&P500 stocks in Jan 2017 when the point forecast is $2181.59 and sell the stocks in Sep 2017 at $2220.8. The forecast above lies within a 80% and 95% confidence level intervals as indicated above.
We plot the forecast of gold by usinn the plot.forecast function:
plot.forecast(ts.sp.forecasts, ylab="S&P500 Values", xlab="Year")
Arima model for Gold
We use the Arima model to further consolidate our Forecast Models achieved as above. We use the auto.arima function to predict the (p,d,q) variables and use the returned values for an optimum forecast model.
auto.arima(ts.gold)## Series: ts.gold
## ARIMA(0,1,1)
##
## Coefficients:
## ma1
## 0.2071
## s.e. 0.0576
##
## sigma^2 estimated as 944.4: log likelihood=-1554.51
## AIC=3113.02 AICc=3113.05 BIC=3120.56gold.arima<-arima(ts.gold, c(0,1,1))
gold.arima.forecasts <- forecast.Arima(gold.arima, h=14)
gold.arima.forecasts## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 1254.873 1215.551 1294.194 1194.7357 1315.010
## Dec 2016 1254.873 1193.236 1316.509 1160.6073 1349.138
## Jan 2017 1254.873 1177.078 1332.667 1135.8960 1373.849
## Feb 2017 1254.873 1163.741 1346.004 1115.4993 1394.246
## Mar 2017 1254.873 1152.121 1357.624 1097.7280 1412.017
## Apr 2017 1254.873 1141.688 1368.057 1081.7717 1427.974
## May 2017 1254.873 1132.139 1377.607 1067.1670 1442.578
## Jun 2017 1254.873 1123.280 1386.465 1053.6193 1456.126
## Jul 2017 1254.873 1114.982 1394.764 1040.9278 1468.818
## Aug 2017 1254.873 1107.149 1402.597 1028.9481 1480.797
## Sep 2017 1254.873 1099.710 1410.035 1017.5724 1492.173
## Oct 2017 1254.873 1092.613 1417.132 1006.7177 1503.028
## Nov 2017 1254.873 1085.813 1423.932 996.3182 1513.427
## Dec 2017 1254.873 1079.276 1430.469 986.3212 1523.424plot(gold.arima.forecasts)
The ??forecast errors?? are calculated as the observed values minus predicted values, for each time point. We can only calculate the forecast errors for the time period covered by our original time series, which is 1990 to 2016 for the exchange rate data. As mentioned above, one measure of the accuracy of the predictive model is the sum-of-squared-errors (SSE) for the in-sample forecast errors.
The in-sample forecast errors are stored in the named element ??residuals?? of the list variable returned by forecast.HoltWinters(). If the predictive model cannot be improved upon, there should be no correlations between forecast errors for successive predictions. In other words, if there are correlations between forecast errors for successive predictions, it is likely that the simple exponential smoothing forecasts could be improved upon by another forecasting technique.
To figure out whether this is the case, we can obtain a correlogram of the in-sample forecast errors for lags 1-20. We can calculate a correlogram of the forecast errors using the acf() function in R. To specify the maximum lag that we want to look at, we use the lag.max parameter in acf().
For example, to calculate a correlogram of the forecast errors for the values of Gold for lags 1-20, we type:
acf(gold.arima.forecasts$residuals, lag.max=20)
We can see from the correlogram that the autocorrelation at lag 4 and lag 9 are outside the significance bounds and lag 7 is just touching the significance bounds. To test whether there is significant evidence for non-zero correlations at lags 1-20, we can carry out a Ljung-Box test. This can be done in R using the Box.test() function. The maximum lag that we want to look at is specified using the ??lag?? parameter in the Box.test() function. For example, to test whether there are non-zero autocorrelations at lags 1-20, for the in-sample forecast errors for exchange rate data, we type:
Box.test(gold.arima.forecasts$residuals, lag=20, type="Ljung-Box")##
## Box-Ljung test
##
## data: gold.arima.forecasts$residuals
## X-squared = 32.274, df = 20, p-value = 0.04046Here the Ljung-Box test statistic is 32.2, and the p-value is 0.04, so there is little evidence of non-zero autocorrelations in the in-sample forecast errors at lags 1-20.
To be sure that the predictive model cannot be improved upon, it is also a good idea to check whether the forecast errors are normally distributed with mean zero and constant variance. To check whether the forecast errors have constant variance, we can make a time plot of the in-sample forecast errors:
plot.ts(gold.arima.forecasts$residuals) # make time plot of forecast errors
The plot shows that the in-sample forecast errors do not seem to have a constant variance over time, although the size of the fluctuations in the start of the time series may be slightly less than that at later dates.
We therefore difference the given forecast once to obtain constant variance over time. This is also supported by the auto.arima function predicting a integrated variable “d” as 1.
gold.diff <- diff(gold.arima.forecasts$residuals, differences = 1)
plot.ts(gold.diff)
To check whether the forecast errors are normally distributed with mean zero, we can plot a histogram of the forecast errors, with an overlaid normal curve that has mean zero and the same standard deviation as the distribution of forecast errors. To do this, we can define an R function ??plotForecastErrors()??, below:
PlotForecastErrors <- function(forecasterrors)
{
mybinsize <- IQR(forecasterrors)/4
mysd <- sd(forecasterrors)
mymin <- min(forecasterrors)-mysd*5
mymax <- max(forecasterrors)+mysd*5
mynorm <- rnorm(10000,mean=0,sd=mysd)
mymin2 <- min(mynorm)
mymax2 <- max(mynorm)
if (mymin2<mymin)
{
mymin<-mymin2
}
if (mymax2>mymax)
{
mymax<-mymax2
}
mybins <- seq(mymin,mymax,mybinsize)
hist(forecasterrors, col="red", freq=FALSE, breaks=mybins)
myhist <- hist(mynorm, plot=FALSE, breaks=mybins)
points(myhist$mids, myhist$density, type="l", col="blue", lwd=2)
}
PlotForecastErrors(gold.arima.forecasts$residuals)
The plot shows that the distribution of forecast errors is roughly centred on zero, and is more or less normally distributed and so it is plausible that the forecast errors are normally distributed with mean zero.
The Ljung-Box test showed that there is little evidence of non-zero autocorrelations in the in-sample forecast errors, and the distribution of forecast errors seems to be normally distributed with mean zero. This suggests that the simple exponential smoothing method provides an adequate predictive model for the exchange rates, which probably cannot be improved upon. Furthermore, the assumptions that the 80% and 95% predictions intervals were based upon (that there are no autocorrelations in the forecast errors, and the forecast errors are normally distributed with mean zero and constant variance) are probably valid.
Arima model forecast for Oil
We carry out the same procedure to achieve forecast values using the Arima model and check for forecast errors.
auto.arima(ts.oil)## Series: ts.oil
## ARIMA(1,1,0)(0,0,2)[12]
##
## Coefficients:
## ar1 sma1 sma2
## 0.4332 0.0927 -0.1647
## s.e. 0.0514 0.0567 0.0599
##
## sigma^2 estimated as 17.16: log likelihood=-910.71
## AIC=1829.41 AICc=1829.54 BIC=1844.5oil.arima<-arima(ts.oil, c(1,1,0))
oil.arima.forecasts <- forecast.Arima(oil.arima, h=14)
oil.arima.forecasts## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 51.61879 46.23785 56.99972 43.389360 59.84822
## Dec 2016 52.35382 43.09726 61.61038 38.197139 66.51050
## Jan 2017 52.64764 40.15346 65.14182 33.539437 71.75584
## Feb 2017 52.76509 37.52003 68.01015 29.449779 76.08040
## Mar 2017 52.81204 35.17292 70.45116 25.835337 79.78874
## Apr 2017 52.83081 33.06107 72.60054 22.595600 83.06601
## May 2017 52.83831 31.13718 74.53944 19.649289 86.02733
## Jun 2017 52.84131 29.36381 76.31880 16.935572 88.74704
## Jul 2017 52.84251 27.71265 77.97236 14.409712 91.27530
## Aug 2017 52.84298 26.16244 79.52353 12.038619 93.64735
## Sep 2017 52.84318 24.69707 80.98928 9.797428 95.88892
## Oct 2017 52.84325 23.30415 82.38236 7.667086 98.01942
## Nov 2017 52.84328 21.97394 83.71263 5.632699 100.05387
## Dec 2017 52.84330 20.69871 84.98788 3.682395 102.00420plot(oil.arima.forecasts)
acf(oil.arima.forecasts$residuals, lag.max=20)
Box.test(oil.arima.forecasts$residuals, lag=20, type="Ljung-Box")##
## Box-Ljung test
##
## data: oil.arima.forecasts$residuals
## X-squared = 24.055, df = 20, p-value = 0.24plot.ts(oil.arima.forecasts$residuals) # make time plot of forecast errors
oil.diff <- diff(oil.arima.forecasts$residuals, differences = 1)
plot.ts(oil.diff)
PlotForecastErrors(oil.arima.forecasts$residuals)
mean(oil.arima.forecasts$residuals)## [1] 0.05615918The plot shows that the distribution of forecast errors is roughly centred on zero, and is more or less normally distributed and so it is plausible that the forecast errors are normally distributed with mean zero.
The Ljung-Box test showed that there is little evidence of non-zero autocorrelations in the in-sample forecast errors, and the distribution of forecast errors seems to be normally distributed with mean zero. This suggests that the simple exponential smoothing method provides an adequate predictive model for the exchange rates, which probably cannot be improved upon. Furthermore, the assumptions that the 80% and 95% predictions intervals were based upon (that there are no autocorrelations in the forecast errors, and the forecast errors are normally distributed with mean zero and constant variance) are probably valid.
Arima model forecast for S&P500
Similarly, we forecast values for S&P500 and check for forecast errors
auto.arima(ts.sp)## Series: ts.sp
## ARIMA(0,1,0) with drift
##
## Coefficients:
## drift
## 5.6415
## s.e. 2.6326
##
## sigma^2 estimated as 2232: log likelihood=-1692.51
## AIC=3389.02 AICc=3389.06 BIC=3396.57sp.arima<-arima(ts.sp, c(0,1,0))
sp.arima.forecasts <- forecast.Arima(sp.arima, h=14)
sp.arima.forecasts## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Nov 2016 2164.33 2103.453 2225.207 2071.227 2257.433
## Dec 2016 2164.33 2078.237 2250.423 2032.662 2295.998
## Jan 2017 2164.33 2058.888 2269.772 2003.070 2325.590
## Feb 2017 2164.33 2042.576 2286.084 1978.123 2350.537
## Mar 2017 2164.33 2028.205 2300.455 1956.145 2372.516
## Apr 2017 2164.33 2015.212 2313.448 1936.274 2392.386
## May 2017 2164.33 2003.265 2325.396 1918.002 2410.659
## Jun 2017 2164.33 1992.144 2336.516 1900.994 2427.666
## Jul 2017 2164.33 1981.699 2346.961 1885.020 2443.640
## Aug 2017 2164.33 1971.820 2356.840 1869.911 2458.749
## Sep 2017 2164.33 1962.424 2366.236 1855.541 2473.119
## Oct 2017 2164.33 1953.446 2375.214 1841.810 2486.850
## Nov 2017 2164.33 1944.835 2383.825 1828.641 2500.019
## Dec 2017 2164.33 1936.549 2392.111 1815.969 2512.691plot(sp.arima.forecasts)
acf(sp.arima.forecasts$residuals, lag.max=20)
Box.test(sp.arima.forecasts$residuals, lag=20, type="Ljung-Box")##
## Box-Ljung test
##
## data: sp.arima.forecasts$residuals
## X-squared = 20.791, df = 20, p-value = 0.4095plot.ts(sp.arima.forecasts$residuals) # make time plot of forecast errors
sp.diff <- diff(sp.arima.forecasts$residuals, differences = 1)
plot.ts(sp.diff)
PlotForecastErrors(sp.arima.forecasts$residuals)
The plot shows that the distribution of forecast errors is roughly centred on zero, and is more or less normally distributed and so it is plausible that the forecast errors are normally distributed with mean zero.
The Ljung-Box test showed that there is little evidence of non-zero autocorrelations in the in-sample forecast errors, and the distribution of forecast errors seems to be normally distributed with mean zero. This suggests that the simple exponential smoothing method provides an adequate predictive model for the exchange rates, which probably cannot be improved upon. Furthermore, the assumptions that the 80% and 95% predictions intervals were based upon (that there are no autocorrelations in the forecast errors, and the forecast errors are normally distributed with mean zero and constant variance) are probably valid.
Interpretations:
1. Gold
From the forecast plots, we see that investing in Gold in April 2017 ($1246.917) would yield a profit of 1.75% when sold in Sep 2017($1268.686).
These values lie at a confidence level of:
Buying price:
Holt's Forecasting:
80%: low of $1137.165; high of $1356.669
95%: low of $1079.0658; high of $1414.768
Arima model forecasting:
80%: low of $1141.688; high of $1368.057
95%: low of $1081.7717; high of $1427.974
Selling Price:
Holt's Forecasting:
80%: low of $1104.847; high of $1432.524
95%: low of $1018.1161; high of $1519.255
Arima model forecasting:
80%: low of $1085.813; high of $1423.932
95%: low of $996.3182; high of $1513.427
2. Oil
Similarly, we see that investing in Oil in January 2017 ($46.78) would yield a profit of 28.8% when sold in Apr 2017($60.26)
These values lie at a confidence level of:
Buying price:
Holt's Forecasting:
80%: low of $35.61926; high of $57.95543
95%: low of $29.707230; high of $63.86746
Arima model forecasting:
80%: low of $40.15346; high of $65.14182
95%: low of $33.539437; high of $71.75584
Selling Price:
Holt's Forecasting:
80%: low of $45.11018; high of $75.41065
95%: low of $37.090134; high of $83.43070
Arima model forecasting:
80%: low of $33.06107; high of $72.60054
95%: low of $22.595600; high of $83.06601
3. S&P500
We also see that investing in S&P500 in January 2017 ($2181.59) would yield a profit of 1.8% when sold in Sep 2017($2220.808)
Buying Price:
Holt's Forecasting:
80%: low of $2076.831; high of $2286.365
95%: low of $2021.370; high of $2341.826
Arima model forecasting:
80%: low of $2058.888; high of $2269.772
95%: low of $2003.07; high of $2325.590
Selling Price:
Holt's Forecasting:
80%: low of $2022.206; high of $2419.409
95%: low of $1917.072; high of $2524.543
Arima model forecasting:
80%: low of $1962.424; high of $2366.236
95%: low of $1855.541; high of $2473.119
Conclusion:
We use the arima model confidnce bands as a primary variation indicator as it removes the trend and seasonal components and works with the irregular component (noise) to model the forecasts. This, I feel, is more important in investing activities since it is safer to know what the maximum loss might be, incase of irregularities.
Therefore, within 80% and 95% confidence levels as mentioned above we should look to invest in Oil in January 2017 and sell our stocks in April 2017 to earn a profit of 28.8%