Deliverable #3

install.packages("vars")

library(ggfortify)

Warning: package ‘ggfortify’ was built under R version 4.1.2
Loading required package: ggplot2
Warning: package ‘ggplot2’ was built under R version 4.1.2

library(tseries)

Warning: package ‘tseries’ was built under R version 4.1.2
Registered S3 method overwritten by 'quantmod':
  method            from
  as.zoo.data.frame zoo 

    ‘tseries’ version: 0.10-49

    ‘tseries’ is a package for time series analysis and computational
    finance.

    See ‘library(help="tseries")’ for details.

library(forecast)

Warning: package ‘forecast’ was built under R version 4.1.2
Registered S3 methods overwritten by 'forecast':
  method                 from     
  autoplot.Arima         ggfortify
  autoplot.acf           ggfortify
  autoplot.ar            ggfortify
  autoplot.bats          ggfortify
  autoplot.decomposed.ts ggfortify
  autoplot.ets           ggfortify
  autoplot.forecast      ggfortify
  autoplot.stl           ggfortify
  autoplot.ts            ggfortify
  fitted.ar              ggfortify
  fortify.ts             ggfortify
  residuals.ar           ggfortify

library(vars)

Warning: package ‘vars’ was built under R version 4.1.2
Loading required package: MASS
Loading required package: strucchange
Warning: package ‘strucchange’ was built under R version 4.1.2
Loading required package: zoo

Attaching package: ‘zoo’

The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric

Loading required package: sandwich
Warning: package ‘sandwich’ was built under R version 4.1.2
Loading required package: urca
Loading required package: lmtest

homes = read.csv(file ="C:/CS-637-Time-Series-and-Forecasting--main/EHSv2.csv")
homes.ts <- ts(homes, start=c(1999, 1), frequency=12)

# Plot the raw data using the base plot function
plot(homes.ts,xlab="Date", ylab = "Existing Homes (1000's)",main="Existing Home Sales from 1999 to 2020")

decomposehomes <- decompose(homes.ts,"multiplicative")
autoplot(decomposehomes)

#Holtwinters

H.hw <- HoltWinters(homes.ts, alpha = 1, gamma = 0)
plot(H.hw)

acf(resid(H.hw))

plot (H.hw$fitted)

H.hw$coef

           a            b           s1           s2           s3           s4 
169.59722222  -0.11413878 -72.34722222 -62.72222222  -7.18055556   0.06944444 
          s5           s6           s7           s8           s9          s10 
 32.27777778  63.19444444  52.02777778  52.86111111  -1.59722222 -11.18055556 
         s11          s12 
-23.80555556 -21.59722222 

H.hw$SSE

[1] 101053.3

#AR Model

acf(diff(homes.ts))

homes.ar <- ar(homes.ts)
paste0("Mean: ", mean(homes.ts))

[1] "Mean: 93.0416666666667"

paste0("Order: ", homes.ar$order)

[1] "Order: 18"

paste0("Alpha: ", homes.ar$ar)

 [1] "Alpha: 0.74092284126783"     "Alpha: -0.00064557182983364"
 [3] "Alpha: 0.0695373768056502"   "Alpha: -0.135294734973541"  
 [5] "Alpha: -0.0143891027438956"  "Alpha: 0.0917129578437509"  
 [7] "Alpha: 0.0022692678372882"   "Alpha: 0.0324558909377331"  
 [9] "Alpha: -0.00593466469575808" "Alpha: -0.0322397889430825" 
[11] "Alpha: 0.0885102471552561"   "Alpha: 0.696120587845432"   
[13] "Alpha: -0.58257269720213"    "Alpha: 0.120507164069272"   
[15] "Alpha: -0.172167110327466"   "Alpha: 0.151507269552817"   
[17] "Alpha: 0.000165208083174551" "Alpha: -0.131012461122949"  

paste0("95% CI: ", homes.ar$ar + c(-2, 2) * sqrt(homes.ar$asy.var))

Warning in sqrt(homes.ar$asy.var) : NaNs produced

  [1] "95% CI: 0.61424885927159"     "95% CI: NaN"                 
  [3] "95% CI: 0.051245006547405"    "95% CI: NaN"                 
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 [11] "95% CI: 0.0641015116521294"   "95% CI: NaN"                 
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 [25] "95% CI: -0.00754703718192579" "95% CI: 0.0367969495381682"  
 [27] "95% CI: -0.0292792230112198"  "95% CI: NaN"                 
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 [31] "95% CI: -0.666120903724099"   "95% CI: NaN"                 
 [33] "95% CI: -0.276755431181472"   "95% CI: NaN"                 
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 [39] "95% CI: -0.0874156684793713"  "95% CI: NaN"                 
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 [43] "95% CI: -0.0216322884626641"  "95% CI: NaN"                 
 [45] "95% CI: -0.0158468119813041"  "95% CI: -0.00858157958947119"
 [47] "95% CI: NaN"                  "95% CI: NaN"                 
 [49] "95% CI: -0.620417973687976"   "95% CI: 0.204304752412022"   
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 [61] "95% CI: -0.0167446066105613"  "95% CI: 0.0611378262395565"  
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 [65] "95% CI: 0.0637243863469893"   "95% CI: NaN"                 
 [67] "95% CI: -0.609695997200473"   "95% CI: 0.147533476546945"   
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 [83] "95% CI: 0.0794443888895956"   "95% CI: 0.716476532238231"   
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[101] "95% CI: 0.075677373625157"    "95% CI: 0.705229950459068"   
[103] "95% CI: -0.616086078867268"   "95% CI: NaN"                 
[105] "95% CI: -0.19929041032581"    "95% CI: 0.189352546038664"   
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[109] "95% CI: NaN"                  "95% CI: 0.00917073318938027" 
[111] "95% CI: 0.0456358205056979"   "95% CI: -0.116280860525691"  
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[119] "95% CI: 0.0700116927730463"   "95% CI: 0.712595510974366"   
[121] "95% CI: -0.591682059815766"   "95% CI: 0.140863108462071"   
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[125] "95% CI: -0.0406571155405534"  "95% CI: NaN"                 
[127] "95% CI: 0.72858297557404"     "95% CI: 0.00369548677060156" 
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[163] "95% CI: 0.731746874728714"    "95% CI: NaN"                 
[165] "95% CI: 0.0458791674520389"   "95% CI: -0.126644225424515"  
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[181] "95% CI: 0.716514105764703"    "95% CI: NaN"                 
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[185] "95% CI: -0.0234549610095559"  "95% CI: 0.10454583137385"    
[187] "95% CI: -0.0162292865449214"  "95% CI: NaN"                 
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[191] "95% CI: -0.0156591226536057"  "95% CI: NaN"                 
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[197] "95% CI: -0.00417585051726052" "95% CI: -0.118672595429159"  
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[203] "95% CI: -0.0347450471366948"  "95% CI: 0.100822320457388"   
[205] "95% CI: -0.0142056552916467"  "95% CI: 0.0509544453199427"  
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[217] "95% CI: NaN"                  "95% CI: 0.0829026346921353"  
[219] "95% CI: 0.0316921003198032"   "95% CI: -0.108171434975197"  
[221] "95% CI: NaN"                  "95% CI: 0.12522633950889"    
[223] "95% CI: -0.00684009477634864" "95% CI: 0.0452887644678323"  
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[237] "95% CI: -0.0142602115371001"  "95% CI: -0.108268422495868"  
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[241] "95% CI: -0.0180866765555111"  "95% CI: 0.0415217492033932"  
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[247] "95% CI: NaN"                  "95% CI: 0.27636292942125"    
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[253] "95% CI: NaN"                  "95% CI: 0.103942749024173"   
[255] "95% CI: NaN"                  "95% CI: -0.0529627894288909" 
[257] "95% CI: -0.0414154152215685"  "95% CI: 0.118836257842094"   
[259] "95% CI: NaN"                  "95% CI: 0.057241751746"      
[261] "95% CI: -0.0145851742447835"  "95% CI: NaN"                 
[263] "95% CI: 0.0598283118534327"   "95% CI: 0.715134462293281"   
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[267] "95% CI: -0.327824623510334"   "95% CI: NaN"                 
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[273] "95% CI: -0.0342341253782617"  "95% CI: NaN"                 
[275] "95% CI: -0.098186691086646"   "95% CI: 0.129558234329598"   
[277] "95% CI: NaN"                  "95% CI: NaN"                 
[279] "95% CI: -0.0295928740493695"  "95% CI: -0.0223276416575369" 
[281] "95% CI: NaN"                  "95% CI: 0.720022144145384"   
[283] "95% CI: NaN"                  "95% CI: 0.150165027360091"   
[285] "95% CI: NaN"                  "95% CI: 0.308460314837839"   
[287] "95% CI: NaN"                  "95% CI: -0.112720090864703"  
[289] "95% CI: NaN"                  "95% CI: 0.068581375440342"   
[291] "95% CI: NaN"                  "95% CI: -0.0307064141195342" 
[293] "95% CI: NaN"                  "95% CI: 0.17526116436572"    
[295] "95% CI: -0.0385530557864399"  "95% CI: NaN"                 
[297] "95% CI: NaN"                  "95% CI: -0.0088952306276206" 
[299] "95% CI: 0.0841691885548217"   "95% CI: 0.705936892864646"   
[301] "95% CI: -0.630470156714925"   "95% CI: NaN"                 
[303] "95% CI: -0.206463581565571"   "95% CI: NaN"                 
[305] "95% CI: -0.15797719039076"    "95% CI: NaN"                 
[307] "95% CI: 0.701146955463913"    "95% CI: NaN"                 
[309] "95% CI: 0.0151348898872841"   "95% CI: NaN"                 
[311] "95% CI: -0.111757542121077"   "95% CI: NaN"                 
[313] "95% CI: NaN"                  "95% CI: 0.05686462644086"    
[315] "95% CI: -0.0151106312348738"  "95% CI: NaN"                 
[317] "95% CI: 0.0761703814614659"   "95% CI: NaN"                 
[319] "95% CI: NaN"                  "95% CI: 0.170172603968121"   
[321] "95% CI: NaN"                  "95% CI: 0.169799639811064"   
[323] "95% CI: NaN"                  "95% CI: -0.00433847912670843"

acf(homes.ar$res[-1])

Error in na.fail.default(as.ts(x)) : missing values in object

ts.plot(homes.ts)

DP <- diff(homes.ts)
plot(as.ts(DP))

acf(DP)

mean(DP) + c(-2, 2) * sd(DP)/sqrt(length(DP))

[1] -3.565491  4.569294

boxplot(homes.ts~cycle(homes.ts),xlab="Date", ylab = "Home Sales (1000's)" ,main ="Existing Home Sales Boxplot from 1999 to 2020")

#Test stationarity of the time series (ADF)
adf.test(homes.ts) 

Warning in adf.test(homes.ts) : p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  homes.ts
Dickey-Fuller = -4.2119, Lag order = 6, p-value = 0.01
alternative hypothesis: stationary

#Test stationarity of the time series (Autocorrelation)

autoplot(acf(homes.ts,plot=FALSE))+ labs(title="Correlogram of Existing Home Sales from 1999 to 2020") 

# Review random time series for any missing values
decomposehomes$random 

           Jan       Feb       Mar       Apr       May       Jun       Jul
1999        NA        NA        NA        NA        NA        NA 1.1032138
2000 0.7471242 0.9647163 1.0861664 0.8958302 0.9721120 1.0445071 0.9562007
2001 0.8959457 0.7994850 1.1391593 0.9683893 0.9705731 1.0303365 1.0521361
2002 1.6262728 1.3400270 1.0502201 1.0852613 1.0217421 0.8509614 0.9684505
2003 1.7534877 1.2839186 0.9490118 0.9656290 0.9166626 0.8123944 0.9521887
2004 1.4442928 1.4816236 1.2023740 1.1029251 0.8671025 0.7759704 0.8619054
2005 1.7919323 1.5852357 1.2420813 1.0279263 0.8434099 0.7520451 0.8447768
2006 1.6669975 1.6224832 1.2762148 1.0070171 0.8981061 0.8111830 0.8741685
2007 1.3275135 1.3018466 1.0083741 0.9750104 1.0345445 1.0610047 1.0888602
2008 0.2606451 0.6648082 0.7157827 0.9353234 1.0903689 1.1357747 1.2788053
2009 0.1121819 0.3435429 0.5572081 0.8234928 0.8231217 1.0757494 1.2348246
2010 0.1474391 0.4288815 1.0574149 1.6052556 1.4723900 1.4924202 0.5071150
2011 0.5977799 0.5220470 1.0816703 1.0884550 0.9479613 1.1842651 0.8776538
2012 0.6673561 0.9362595 0.9248319 0.9667644 1.1269514 1.0438293 0.9189143
2013 0.9663403 0.9074717 0.8878940 1.0947429 1.1400443 0.9189579 1.1392809
2014 0.6540340 0.5667614 0.7057765 0.9507975 1.0670037 1.0174648 1.0989700
2015 0.5822650 0.7278423 0.9177545 0.9791645 0.9634593 1.1020116 1.1708390
2016 0.9101947 0.9731486 0.9753523 1.0284439 0.9979752 1.0216221 0.9131125
2017 1.3281877 0.9338656 1.1801175 0.8482081 1.0859714 0.9986222 0.8904037
2018 1.1307373 1.0553368 1.0310295 0.9710272 1.0513212 1.0386968 1.0022745
2019 0.6579942 0.9448600 0.8371957 0.9970403 1.1517246 0.8951613 1.0726252
2020 1.5379973 1.4225577 0.9810897 0.4900149 0.3641739 0.7437414        NA
           Aug       Sep       Oct       Nov       Dec
1999 1.0609259 0.9532918 0.8688172 1.0064462 0.9206730
2000 1.0951518 1.0343578 0.9800087 1.1004968 0.8045493
2001 1.1205794 0.7797275 0.9409636 0.9529452 0.8740892
2002 0.9342772 0.9170661 1.0507936 1.0553706 1.1488710
2003 0.9164478 1.2058768 1.1309633 0.9675022 1.1473590
2004 0.8145749 1.0738350 1.0618611 1.3147930 1.2493292
2005 0.8223871 1.1340074 1.0882428 1.3058569 1.1412458
2006 0.8878485 1.0431984 1.0737694 1.2930181 1.1478331
2007 1.1275575 0.6495928 0.7722568 0.8081698 0.6556508
2008 1.1528771 1.1943650 0.9991209 0.3448566 0.6415508
2009 0.9749603 1.1378990 1.3569016 1.4434377 0.7636518
2010 0.6266981 0.6737898 0.4967009 0.6756946 1.2417400
2011 1.1770663 1.0205276 0.7485791 0.9452040 0.9379527
2012 1.1759497 0.7619739 0.9599266 1.1603551 0.8975498
2013 1.0889929 0.9965762 1.0161778 0.8253968 0.9561006
2014 0.9968449 1.0919241 1.1553238 0.7230650 1.0901564
2015 0.9067078 1.1624320 0.9920142 0.6158981 1.1199111
2016 0.9738097 1.1273928 0.9287702 1.0447404 1.0574169
2017 0.9358159 1.0276552 1.0263462 1.1166599 1.0074414
2018 1.0434286 0.8306383 1.0447795 1.0603488 0.7601642
2019 0.9738180 0.9905919 1.1144020 1.0464635 1.2434834
2020        NA        NA        NA        NA        NA

# Autoplot the random time series from 7:146 which exclude the NA values

autoplot(acf(decomposehomes$random[7:146],plot=FALSE))+ labs(title="Correlogram of Home Sales Random Component from 1999 to 2020") 

#Linear Model
autoplot(homes.ts) + geom_smooth(method="lm")+ labs(x ="Date", y = "Home Sales (1000's)", title="Existing Home Sales from 1999 to 2020") 

`geom_smooth()` using formula 'y ~ x'

#ARIMA Model
arimahomes <- auto.arima(homes.ts)
arimahomes

Series: homes.ts 
ARIMA(1,0,0)(1,1,0)[12] with drift 

Coefficients:
         ar1     sar1   drift
      0.7427  -0.1052  0.1289
s.e.  0.0440   0.0699  0.3405

sigma^2 estimated as 348.5:  log likelihood=-1094.08
AIC=2196.15   AICc=2196.32   BIC=2210.27

ggtsdiag(arimahomes)

forecasthomes <- forecast(arimahomes, level = c(95), h = 36)
autoplot(forecasthomes)

## Seasonal ARIMA

AIC (arima(log(homes.ts), order = c(1,1,0),
           seas = list(order = c(1,0,0), 12)))

[1] 164.038

AIC (arima(log(homes.ts), order = c(0,1,1),
           seas = list(order = c(0,0,1), 12)))

[1] 359.4662

get.best.arima <- function(x.ts, maxord = c(1,1,1,1,1,1)) {
  best.aic <- 1e8
  n <- length(x.ts)
  for (p in 0:maxord[1]) for(d in 0:maxord[2]) for(q in 0:maxord[3]){
    
    for (P in 0:maxord[4]) for(D in 0:maxord[5]) for(Q in 0:maxord[6]){
      fit <- arima(x.ts, order = c(p,d,q),
                   seas = list(order = c(P,D,Q),
                               frequency(x.ts)), method = "CSS")
      fit.aic <- -2 * fit$loglik + (log(n) + 1) * length(fit$coef)
      if (fit.aic < best.aic){
        best.aic <- fit.aic
        best.fit <- fit
        best.model <- c(p,d,q,P,D,Q)
      }
    }
  }
  return(list(best.aic, best.fit, best.model))
}
best.arima.homes <- get.best.arima( log(homes.ts),
                                   maxord = c(2,2,2,2,2,2))

Warning in arima(x.ts, order = c(p, d, q), seas = list(order = c(P, D, Q),  :
  possible convergence problem: optim gave code = 1
Warning in arima(x.ts, order = c(p, d, q), seas = list(order = c(P, D, Q),  :
  possible convergence problem: optim gave code = 1
Warning in arima(x.ts, order = c(p, d, q), seas = list(order = c(P, D, Q),  :
  possible convergence problem: optim gave code = 1
Warning in arima(x.ts, order = c(p, d, q), seas = list(order = c(P, D, Q),  :
  possible convergence problem: optim gave code = 1
Warning in arima(x.ts, order = c(p, d, q), seas = list(order = c(P, D, Q),  :
  possible convergence problem: optim gave code = 1
Warning in arima(x.ts, order = c(p, d, q), seas = list(order = c(P, D, Q),  :
  possible convergence problem: optim gave code = 1

best.fit.homes <- best.arima.homes[[2]]
acf(resid(best.fit.homes))

best.arima.homes[[3]]

[1] 2 0 0 0 1 2

ts.plot( cbind( window(homes.ts,start = 1999),
                exp(predict(best.fit.homes,12)$pred) ), lty = 1:2)