Introduction

Background

Forecasting the price of a stock is a necessary work for every stock investors. choosing the right model might be challenging and confusing especially if the historical data has no trend and seasonality. This report describes different timeseries forecasting models applied to a real stock close price dataset. For this project we will start with a general idea of the stock price, incluiding dataset analysis. Followed by a general description and analysis of the dataset, our objective is to apply different forecasting predictive models for “Unilever Indonesia” stock daily close price. The models will be evaluated, analysed and compared to find the best model that has as little error as possible.

Dataset

The dataset is consist of stock price and several basic stock indicators of Unilever Indonesia, an issuer with code UNVR from Indonesia Stock Exchange. the dataset is acquire from IDX Stock API

Library and Setup

library(tidyverse) #data manipulation
library(lubridate) # date manipulation
library(forecast) # time series library
library(TTR) # for Simple moving average function
library(MLmetrics) # calculate error
library(tseries) # adf.test
library(fpp)
library(ggplot2)
library(gridExtra)

Import Data

unvr <- read_csv("data_unvr_2015-2019.csv")
glimpse(unvr)

## Rows: 1,213
## Columns: 25
## $ date                  <dttm> 2015-01-02, 2015-01-05, 2015-01-06, 2015-01-07,~
## $ previous              <dbl> 32300, 32525, 32475, 32475, 33125, 33375, 33225,~
## $ open_price            <dbl> 32300, 32475, 32300, 32500, 33100, 33100, 32875,~
## $ first_trade           <dbl> 32400, 32400, 32200, 32525, 33125, 33125, 32875,~
## $ high                  <dbl> 32625, 32525, 32500, 33700, 33450, 33450, 32900,~
## $ low                   <dbl> 32275, 32275, 32150, 32500, 33100, 32975, 32100,~
## $ close                 <dbl> 32525, 32475, 32475, 33125, 33375, 33225, 32100,~
## $ change                <dbl> 225, -50, 0, 650, 250, -150, -1125, 900, -350, 7~
## $ volume                <dbl> 766900, 1761400, 1400700, 2901400, 2654900, 2078~
## $ value                 <dbl> 24931422500, 57201080000, 45438150000, 960229150~
## $ frequency             <dbl> 987, 1229, 2123, 2549, 1574, 1663, 1518, 1693, 1~
## $ index_individual      <dbl> 166538.7, 166282.6, 166282.6, 169610.9, 170890.9~
## $ offer                 <dbl> 32550, 32500, 32500, 33200, 33375, 33275, 32275,~
## $ offer_volume          <dbl> 255200, 125500, 235800, 15200, 39600, 200, 2000,~
## $ bid                   <dbl> 32525, 32475, 32475, 33125, 33350, 33200, 32100,~
## $ bid_volume            <dbl> 46300, 96700, 103000, 252300, 80000, 200, 133200~
## $ listed_shares         <dbl> 7.63e+09, 7.63e+09, 7.63e+09, 7.63e+09, 7.63e+09~
## $ tradeble_shares       <dbl> 7.63e+09, 7.63e+09, 7.63e+09, 7.63e+09, 7.63e+09~
## $ weight_for_index      <dbl> 7.63e+09, 7.63e+09, 7.63e+09, 7.63e+09, 7.63e+09~
## $ foreign_sell          <dbl> 352000, 1055200, 823400, 1091000, 931700, 107070~
## $ foreign_buy           <dbl> 288100, 265700, 356800, 1754900, 1107300, 970800~
## $ delisting_date        <lgl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, ~
## $ non_regular_volume    <dbl> 683000, 0, 378, 86600, 157407, 5000, 157400, 154~
## $ non_regular_value     <dbl> 22171705000, 0, 12271891, 2851565300, 5254861400~
## $ non_regular_frequency <dbl> 15, 0, 4, 5, 7, 1, 8, 1, 0, 1, 1, 6, 2, 1, 18, 5~

Because we only want to forcast the stock price we will only use 2 variables which is date and close (closing price).

data <- unvr %>% select(date,close)
data

Data Preprocessing

Imputation Missing Date

As we can see from the data that there are some missing date due to there are no market transaction on weekends. to impute the missing date values we will perform padding using pad function from padr package. for the missing values we will use the price of the date before.

library(padr)
library(imputeTS)
data <- data %>% 
  pad() %>% 
  mutate(close = na_locf(close))

# add first jan 2015 as the first row data
data <- rbind(data.frame(date = ymd("2015-01-01"), close = data[1,]$close), data)
data

Time Series Object

After the data is free from missing date. we will create time series object with ts function from stats package.

freq <- 365
data_ts <- ts(data = data$close,
              start = c(2015,01,02),
              frequency = freq)

data_ts %>% autoplot()

Now, we will try to explore whether our timeseries object has trend and seasonal properties (one-seasonal/multiseasonal).

data_dec <- decompose(data_ts)
data_dec %>% autoplot()

note that the data is an additive time series and it has both trend and seasonality. This is a valuable information for the model building later.

Stationary Test

When forecasting or predicting the future, most time series models assume that each point is independent of one another. The best indication of this is when the dataset of past instances is stationary. For data to be stationary, the statistical properties of a system do not change over time.

Now we test whether the data is stationary or not with Augmented Dickey-Fuller Test. we will see the alpha and determine whether to reject or accept the null hypothesis. the null hypothesis is data is not stationary H0 : data is not stationary H1 : data is stationary

adf.test(data_ts)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  data_ts
## Dickey-Fuller = -2.4592, Lag order = 12, p-value = 0.3839
## alternative hypothesis: stationary

Based on Augmented Dickey-Fuller Test (adf.test) result, the p-value is < alpha and therefore the data is already stationary. Therefore, for a model building using SARIMA, we do not need to perform differencing on the data first.

Model Building

Cross Validation

For time series the cross-validation scheme should not be sampled randomly, but split sequentially. We will split the data 80:20. Because the data is consist of five years transactions. we will use the first four years for train dataset and the rest for test dataset.

# the first 4 years data
train <- data_ts %>% head(length(data_ts) - freq)

# the last year data
test <- data_ts %>% tail(freq)

Model Training

For model building, I will compare between two of the widely used time series modeling in business and industry: ETS Holt-Winters and Seasonal Arima. I use ETS Holt-Winters because my data contain both seasonal and trend. I also want to compare it between seasonal Arima (SARIMA) to check whether the seasonal ARIMA can give better forecasting performance.

SARIMA is an acronyms for Seasonal Auto Regresive Integreted Moving Averange. SARIMA is great for predicting a stationary time series data that has seasonality. Stationary time series data has values which move around its mean (no trend nor seasonality). It predicts future values by combining Auto Regresive and Moving Averange method, whereas integrated stands for how many times differencing applied to the data (a way to transform non-stationary into stationary time series data). There is also a special approach for stationary data with seasonality that is using Seasonal ARIMA (SARIMA).

ETS Holt-Winters is a method that predict future values by smoothing on error, trend, and seasonal, where it gives different weights on each data point used for prediction.

# ETS Holt-Winters
data_ets <- stlm(train, method = "ets", lambda = 0) # no log transformation for addivite data

# SARIMA TODO
data_arima <- stlm(train, method = "arima", lambda = 0)

Forecasting

# forecast
ets_forcast <- forecast(data_ets, h = freq)
arima_forcast <- forecast(data_arima, h = freq)

ETS

The table below is the forecasted value plus confidence interval from ETS model.

ets_forcast

##           Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2019.0000       43009.39 42379.92 43648.22 42050.43 43990.22
## 2019.0027       43005.30 42151.13 43876.77 41705.86 44345.22
## 2019.0055       42995.78 41966.11 44050.72 41431.05 44619.60
## 2019.0082       42295.24 41135.77 43487.39 40534.90 44132.02
## 2019.0110       42458.60 41164.74 43793.12 40495.86 44516.48
## 2019.0137       43208.63 41772.35 44694.30 41031.45 45501.33
## 2019.0164       42959.64 41422.34 44553.98 40630.94 45421.79
## 2019.0192       43069.07 41425.96 44777.35 40581.68 45708.92
## 2019.0219       43117.93 41377.25 44931.85 40484.41 45922.77
## 2019.0247       42778.95 40962.23 44676.24 40031.97 45714.43
## 2019.0274       42660.99 40764.30 44645.93 39794.63 45733.81
## 2019.0301       42614.24 40638.63 44685.90 39630.15 45823.04
## 2019.0329       42797.96 40735.90 44964.40 39684.84 46155.30
## 2019.0356       42491.76 40370.11 44724.91 39290.20 45954.19
## 2019.0384       42588.20 40389.94 44906.11 39272.57 46183.77
## 2019.0411       42850.28 40568.72 45260.16 39410.53 46590.25
## 2019.0438       42941.94 40587.83 45432.60 39394.33 46809.03
## 2019.0466       42689.20 40283.70 45238.34 39065.67 46648.83
## 2019.0493       43259.01 40757.19 45914.40 39491.91 47385.44
## 2019.0521       43086.34 40532.29 45801.33 39242.13 47307.14
## 2019.0548       43580.31 40935.69 46395.79 39601.31 47959.11
## 2019.0575       43479.51 40781.38 46356.17 39421.52 47955.23
## 2019.0603       43554.44 40793.33 46502.43 39403.26 48142.94
## 2019.0630       44004.73 41157.49 47048.93 39725.60 48744.78
## 2019.0658       44265.60 41344.82 47392.71 39877.50 49136.57
## 2019.0685       44047.72 41086.12 47222.79 39599.82 48995.20
## 2019.0712       44313.22 41279.36 47570.05 39758.34 49389.92
## 2019.0740       44257.74 41174.40 47571.97 39630.12 49425.73
## 2019.0767       44057.91 40936.46 47417.37 39374.61 49298.26
## 2019.0795       44350.80 41157.17 47792.24 39560.74 49720.85
## 2019.0822       44115.93 40888.96 47597.57 39277.38 49550.53
## 2019.0849       43995.94 40728.49 47525.52 39098.21 49507.19
## 2019.0877       44295.99 40957.49 47906.63 39293.28 49935.64
## 2019.0904       44902.21 41469.37 48619.21 39759.69 50709.86
## 2019.0932       45421.22 41900.27 49238.04 40148.26 51386.71
## 2019.0959       45325.21 41764.11 49189.95 39993.68 51367.49
## 2019.0986       45442.68 41825.34 49372.87 40028.50 51589.16
## 2019.1014       45293.76 41642.10 49265.63 39829.76 51507.33
## 2019.1041       45478.58 41766.31 49520.80 39925.44 51804.10
## 2019.1068       45348.14 41601.57 49432.13 39745.23 51740.91
## 2019.1096       45461.19 41660.82 49608.25 39779.37 51954.58
## 2019.1123       45925.52 42042.01 50167.76 40120.95 52569.88
## 2019.1151       45793.32 41877.35 50075.47 39941.80 52502.09
## 2019.1178       45849.38 41885.49 50188.41 39927.80 52649.19
## 2019.1205       45772.82 41773.01 50155.62 39799.12 52643.14
## 2019.1233       45798.44 41754.34 50234.22 39760.14 52753.75
## 2019.1260       45348.90 41303.35 49790.70 39309.97 52315.54
## 2019.1288       45685.43 41568.88 50209.63 39542.06 52783.24
## 2019.1315       45672.27 41516.41 50244.13 39471.77 52846.78
## 2019.1342       45959.29 41737.01 50608.71 39661.24 53257.45
## 2019.1370       45969.06 41706.02 50667.87 39611.74 53346.68
## 2019.1397       45729.32 41449.26 50451.34 39348.16 53145.32
## 2019.1425       46097.77 41744.08 50905.52 39608.38 53650.36
## 2019.1452       45935.91 41558.90 50773.90 39413.30 53537.96
## 2019.1479       45720.63 41326.10 50582.47 39173.44 53362.08
## 2019.1507       45989.27 41531.04 50926.08 39348.70 53750.51
## 2019.1534       46333.65 41804.25 51353.81 39588.61 54227.90
## 2019.1562       46092.63 41549.56 51132.45 39328.77 54019.76
## 2019.1589       46088.76 41509.18 51173.58 39272.09 54088.63
## 2019.1616       46208.15 41580.08 51351.34 39320.83 54301.83
## 2019.1644       46546.24 41847.75 51772.26 39555.66 54772.24
## 2019.1671       47028.03 42244.30 52353.47 39912.19 55412.53
## 2019.1699       46647.55 41866.54 51974.55 39537.29 55036.50
## 2019.1726       46367.14 41579.40 51706.17 39248.42 54777.02
## 2019.1753       46749.43 41886.77 52176.59 39520.85 55300.15
## 2019.1781       46510.49 41637.73 51953.51 39268.42 55088.18
## 2019.1808       45918.06 41073.12 51334.49 38718.87 54455.82
## 2019.1836       45671.94 40819.21 51101.58 38462.67 54232.48
## 2019.1863       45572.37 40696.80 51032.05 38330.67 54182.23
## 2019.1890       45545.18 40639.38 51043.18 38260.08 54217.43
## 2019.1918       45844.29 40873.19 51419.99 38463.72 54641.07
## 2019.1945       45825.61 40823.72 51440.35 38400.83 54685.96
## 2019.1973       45851.72 40814.40 51510.74 38375.85 54783.94
## 2019.2000       46008.76 40921.75 51728.15 38460.64 55038.24
## 2019.2027       45905.49 40797.76 51652.68 38328.14 54980.85
## 2019.2055       46088.79 40928.65 51899.50 38435.19 55266.45
## 2019.2082       45815.21 40654.10 51631.52 38161.68 55003.69
## 2019.2110       46145.07 40915.21 52043.41 38391.09 55465.14
## 2019.2137       46193.32 40926.60 52137.80 38386.18 55588.30
## 2019.2164       46193.48 40895.57 52177.73 38341.60 55653.32
## 2019.2192       46399.42 41046.79 52450.06 38467.96 55966.22
## 2019.2219       45891.49 40566.90 51914.95 38003.08 55417.32
## 2019.2247       45761.66 40421.88 51806.82 37852.23 55323.80
## 2019.2274       45806.16 40431.12 51895.78 37845.97 55440.62
## 2019.2301       45729.55 40333.67 51847.30 37739.98 55410.51
## 2019.2329       45700.10 40278.09 51852.00 37673.32 55437.10
## 2019.2356       45665.04 40217.78 51850.09 37602.37 55456.50
## 2019.2384       45363.99 39923.63 51545.70 37312.99 55152.15
## 2019.2411       45863.55 40334.14 52150.99 37682.24 55821.13
## 2019.2438       45874.15 40314.48 52200.53 37649.56 55895.40
## 2019.2466       45610.04 40053.76 51937.08 37391.93 55634.34
## 2019.2493       45678.27 40085.19 52051.75 37407.20 55778.15
## 2019.2521       45848.84 40206.46 52283.05 37506.33 56046.97
## 2019.2548       45979.99 40293.14 52469.47 37573.21 56267.74
## 2019.2575       46035.71 40313.78 52569.80 37578.54 56396.20
## 2019.2603       46373.22 40581.10 52992.05 37813.81 56870.12
## 2019.2630       46722.63 40858.59 53428.28 38058.44 57359.28
## 2019.2658       46406.31 40554.05 53103.09 37761.00 57030.94
## 2019.2685       46507.62 40614.76 53255.48 37803.83 57215.34
## 2019.2712       46802.82 40844.72 53630.05 38004.16 57638.54
## 2019.2740       46881.74 40885.87 53756.91 38028.79 57795.62
## 2019.2767       46592.68 40606.37 53461.52 37755.35 57498.56
## 2019.2795       47189.97 41099.32 54183.22 38200.11 58295.48
## 2019.2822       46360.47 40349.91 53266.36 37490.30 57329.31
## 2019.2849       46231.61 40211.02 53153.65 37348.10 57228.13
## 2019.2877       46320.51 40261.69 53291.10 37382.07 57396.23
## 2019.2904       46661.32 40531.22 53718.57 37619.20 57876.80
## 2019.2932       46628.79 40476.42 53716.32 37555.30 57894.46
## 2019.2959       46814.43 40611.05 53965.38 37667.21 58182.99
## 2019.2986       47289.45 40996.49 54548.38 38011.64 58831.78
## 2019.3014       47349.97 41022.42 54653.52 38022.66 58965.35
## 2019.3041       47800.30 41385.93 55208.84 38346.52 59584.78
## 2019.3068       48232.23 41733.14 55743.42 38655.12 60182.15
## 2019.3096       48408.02 41858.53 55982.29 38758.16 60460.46
## 2019.3123       47634.49 41163.50 55122.73 38101.80 59552.16
## 2019.3151       46425.67 40093.53 53757.88 37098.99 58097.08
## 2019.3178       45855.99 39576.62 53131.67 36608.48 57439.48
## 2019.3205       46720.58 40297.54 54167.40 37262.96 58578.63
## 2019.3233       45726.74 39415.70 53048.26 36435.49 57387.31
## 2019.3260       45551.50 39240.24 52877.83 36261.35 57221.78
## 2019.3288       45600.94 39258.51 52968.01 36266.35 57338.16
## 2019.3315       45594.84 39229.07 52993.60 36227.32 57384.59
## 2019.3342       45884.72 39454.23 53363.28 36423.41 57803.69
## 2019.3370       45829.89 39382.99 53332.13 36345.87 57788.66
## 2019.3397       46553.70 39980.62 54207.44 36885.51 58756.05
## 2019.3425       46759.08 40132.64 54479.64 37013.88 59070.06
## 2019.3452       46400.63 39800.93 54094.69 36696.20 58671.44
## 2019.3479       47237.59 40494.45 55103.58 37323.72 59784.75
## 2019.3507       47143.94 40389.95 55027.33 37215.59 59720.98
## 2019.3534       47686.41 40830.31 55693.77 37609.44 60463.38
## 2019.3562       47495.42 40642.58 55503.73 37424.73 60276.04
## 2019.3589       47229.67 40391.23 55225.90 37181.61 59993.15
## 2019.3616       47308.15 40434.45 55350.34 37209.77 60147.13
## 2019.3644       47328.64 40428.18 55406.91 37192.40 60227.36
## 2019.3671       47661.53 40688.67 55829.33 37420.44 60705.36
## 2019.3699       47470.37 40501.82 55637.89 37237.08 60515.91
## 2019.3726       46560.95 39702.80 54603.77 36491.22 59409.42
## 2019.3753       46546.52 39667.48 54618.50 36447.58 59443.68
## 2019.3781       46652.00 39734.42 54773.91 36497.92 59631.05
## 2019.3808       46820.26 39854.78 55003.11 36597.33 59898.83
## 2019.3836       46642.78 39680.94 54826.04 36426.63 59724.13
## 2019.3863       46832.05 39819.20 55079.99 36542.49 60018.92
## 2019.3890       46475.40 39493.46 54691.67 36232.63 59613.75
## 2019.3918       46967.07 39888.62 55301.63 36584.17 60296.71
## 2019.3945       47181.25 40047.86 55585.24 36719.23 60624.10
## 2019.3973       47403.62 40213.94 55878.71 36860.51 60962.35
## 2019.4000       47141.27 39968.93 55600.67 36625.03 60677.06
## 2019.4027       47312.72 40091.84 55834.14 36726.78 60949.89
## 2019.4055       47059.56 39855.08 55566.38 36499.12 60675.52
## 2019.4082       46951.48 39741.45 55469.59 36384.34 60587.66
## 2019.4110       46350.44 39210.97 54789.86 35888.14 59862.77
## 2019.4137       45938.85 38841.31 54333.33 35539.42 59381.33
## 2019.4164       46054.17 38917.39 54499.71 35598.65 59580.54
## 2019.4192       46249.08 39060.66 54760.41 35719.32 59882.93
## 2019.4219       46025.46 38850.53 54525.45 35516.89 59643.25
## 2019.4247       45861.41 38690.96 54360.73 35360.80 59480.24
## 2019.4274       46325.84 39061.54 54941.08 35689.21 60132.54
## 2019.4301       46070.83 38825.48 54668.26 35463.37 59851.10
## 2019.4329       46163.59 38882.65 54807.91 35505.43 60021.15
## 2019.4356       45671.84 38447.76 54253.29 35098.31 59430.71
## 2019.4384       45873.29 38596.62 54521.83 35224.20 59741.85
## 2019.4411       45654.19 38391.73 54290.47 35027.29 59505.18
## 2019.4438       45770.54 38469.05 54457.86 35087.92 59705.50
## 2019.4466       45671.82 38365.68 54369.31 34983.79 59625.20
## 2019.4493       45854.16 38498.44 54615.32 35095.00 59911.80
## 2019.4521       45944.95 38554.28 54752.38 35136.07 60078.96
## 2019.4548       46046.63 38619.25 54902.47 35185.47 60260.46
## 2019.4575       45902.05 38477.78 54758.85 35046.82 60119.53
## 2019.4603       46017.28 38554.16 54925.06 35106.66 60318.74
## 2019.4630       46000.98 38520.39 54934.29 35066.21 60345.56
## 2019.4658       45941.60 38450.64 54891.95 34993.07 60315.67
## 2019.4685       45791.00 38304.70 54740.42 34850.68 60165.70
## 2019.4712       46033.78 38487.87 55059.15 35007.73 60532.61
## 2019.4740       45648.59 38146.12 54626.63 34687.40 60073.50
## 2019.4767       45832.56 38280.15 54875.02 34799.80 60363.11
## 2019.4795       45721.83 38168.07 54770.55 34688.49 60264.55
## 2019.4822       45781.56 38198.37 54870.16 34706.62 60390.52
## 2019.4849       45646.31 38066.10 54736.00 34577.10 60259.13
## 2019.4877       45491.90 37918.03 54578.60 34433.32 60102.04
## 2019.4904       45580.02 37972.20 54712.08 34473.25 60265.22
## 2019.4932       46355.27 38598.52 55670.82 35032.47 61337.70
## 2019.4959       46479.05 38682.06 55847.66 35098.92 61548.98
## 2019.4986       46209.40 38438.29 55551.60 34868.43 61239.01
## 2019.5014       46214.19 38422.99 55585.26 34845.29 61292.40
## 2019.5041       46183.00 38377.83 55575.55 34795.12 61297.93
## 2019.5068       46509.35 38629.74 55996.22 35014.26 61778.24
## 2019.5096       46187.41 38343.25 55636.30 34745.42 61397.34
## 2019.5123       46092.15 38245.17 55549.13 34647.44 61317.25
## 2019.5151       45972.45 38126.97 55432.32 34531.30 61204.36
## 2019.5178       45838.46 37997.07 55298.06 34404.66 61072.09
## 2019.5205       46037.36 38143.15 55565.38 34527.93 61383.32
## 2019.5233       45862.48 37979.60 55381.51 34370.93 61196.12
## 2019.5260       46492.23 38482.24 56169.48 34816.78 62082.92
## 2019.5288       46385.00 38374.72 56067.34 34710.52 61986.06
## 2019.5315       46520.01 38467.66 56257.95 34785.61 62212.85
## 2019.5342       46419.92 38366.23 56164.21 34684.96 62125.18
## 2019.5370       46466.97 38386.50 56248.42 34694.37 62234.30
## 2019.5397       46137.52 38095.90 55876.64 34422.90 61838.79
## 2019.5425       45959.73 37930.79 55688.19 34264.96 61645.99
## 2019.5452       45724.72 37718.67 55430.11 34064.66 61375.93
## 2019.5479       45453.53 37476.96 55127.82 33837.76 61056.74
## 2019.5507       45578.14 37561.71 55305.44 33905.68 61268.99
## 2019.5534       45365.80 37368.85 55074.09 33723.07 61028.13
## 2019.5562       45290.30 37288.88 55008.65 33642.41 60970.99
## 2019.5589       45333.85 37307.00 55087.73 33650.28 61074.02
## 2019.5616       45492.58 37419.87 55306.84 33743.61 61332.33
## 2019.5644       45452.33 37369.08 55284.05 33689.37 61322.43
## 2019.5671       45346.84 37264.76 55181.80 33586.93 61224.30
## 2019.5699       45939.22 37733.78 55928.98 34001.19 62068.76
## 2019.5726       46316.47 38025.78 56414.76 34255.78 62623.45
## 2019.5753       45362.04 37224.75 55278.14 33525.85 61376.96
## 2019.5781       45619.54 37418.56 55617.93 33692.06 61769.52
## 2019.5808       45514.94 37315.35 55516.29 33590.84 61671.88
## 2019.5836       45831.17 37557.13 55928.03 33800.15 62144.57
## 2019.5863       45721.40 37449.79 55819.99 33695.27 62039.77
## 2019.5890       45464.42 37222.05 55531.96 33482.16 61734.78
## 2019.5918       45659.98 37364.89 55796.60 33602.42 62044.16
## 2019.5945       45426.93 37157.04 55537.42 33407.34 61771.03
## 2019.5973       45413.96 37129.35 55547.11 33374.32 61796.85
## 2019.6000       44938.16 36723.49 54990.38 33001.49 61192.35
## 2019.6027       44716.26 36525.42 54743.89 32815.53 60932.83
## 2019.6055       44982.72 36726.29 55095.28 32988.02 61338.78
## 2019.6082       44844.82 36597.01 54951.42 32863.97 61193.38
## 2019.6110       44449.87 36258.21 54492.24 32551.89 60696.66
## 2019.6137       44616.13 36377.32 54720.90 32650.98 60965.99
## 2019.6164       44267.62 36076.82 54318.04 32373.51 60531.67
## 2019.6192       43979.57 35825.88 53988.98 32140.63 60179.35
## 2019.6219       43774.46 35642.72 53761.42 31968.69 59940.00
## 2019.6247       44698.97 36379.12 54921.55 32621.41 61248.05
## 2019.6274       44654.53 36326.64 54891.58 32566.61 61229.17
## 2019.6301       44574.69 36245.46 54817.99 32486.13 61161.58
## 2019.6329       44463.57 36138.95 54705.78 32383.01 61050.83
## 2019.6356       44368.88 36045.91 54613.62 32292.01 60962.37
## 2019.6384       44151.07 35853.00 54369.70 32111.62 60704.40
## 2019.6411       44400.90 36039.87 54701.65 32271.41 61089.38
## 2019.6438       44436.61 36052.87 54769.90 32275.48 61179.95
## 2019.6466       44862.78 36382.54 55319.62 32562.99 61808.48
## 2019.6493       44975.93 36458.22 55483.63 32623.10 62006.20
## 2019.6521       45667.99 37002.92 56362.19 33102.78 63002.72
## 2019.6548       45705.28 37016.86 56433.00 33107.56 63096.55
## 2019.6575       45804.38 37080.87 56580.16 33157.10 63275.78
## 2019.6603       45821.70 37078.65 56626.32 33147.44 63342.07
## 2019.6630       46071.55 37264.56 56959.96 33305.94 63730.01
## 2019.6658       45913.49 37120.53 56789.28 33169.56 63553.70
## 2019.6685       45715.06 36944.04 56568.45 33004.25 63321.14
## 2019.6712       45808.75 37003.68 56708.98 33049.94 63493.03
## 2019.6740       45861.22 37030.03 56798.54 33065.89 63607.89
## 2019.6767       46054.74 37170.21 57062.86 33183.48 63918.53
## 2019.6795       45199.64 36464.34 56027.54 32545.88 62773.15
## 2019.6822       45122.96 36386.81 55956.57 32469.28 62707.93
## 2019.6849       45880.07 36981.45 56919.91 32992.39 63802.00
## 2019.6877       45960.66 37030.53 57044.35 33028.68 63956.00
## 2019.6904       45640.75 36757.05 56671.54 32777.33 63552.42
## 2019.6932       46167.38 37165.29 57349.93 33133.88 64327.72
## 2019.6959       46096.06 37092.06 57285.75 33061.13 64270.23
## 2019.6986       46283.76 37227.26 57543.49 33174.17 64573.94
## 2019.7014       46895.56 37703.33 58328.88 33590.85 65470.00
## 2019.7041       46586.18 37438.73 57968.64 33347.63 65080.25
## 2019.7068       46482.48 37339.59 57864.07 33251.88 64977.39
## 2019.7096       46600.89 37418.91 58035.97 33315.07 65184.99
## 2019.7123       46304.22 37165.04 57690.81 33081.67 64811.76
## 2019.7151       46172.18 37043.48 57550.48 32966.12 64668.52
## 2019.7178       46375.64 37191.10 57828.35 33090.14 64995.19
## 2019.7205       46510.90 37283.95 58021.32 33165.40 65226.53
## 2019.7233       46542.95 37294.05 58085.59 33167.04 65313.24
## 2019.7260       45985.26 36831.80 57413.53 32748.71 64571.83
## 2019.7288       46176.18 36969.32 57675.93 32863.73 64881.24
## 2019.7315       46008.21 36819.53 57490.02 32723.37 64686.35
## 2019.7342       45631.79 36503.13 57043.34 32435.05 64197.85
## 2019.7370       45384.45 36290.23 56757.64 32238.81 63890.33
## 2019.7397       45956.37 36732.36 57496.65 32624.44 64736.38
## 2019.7425       46166.26 36884.90 57783.09 32752.76 65073.10
## 2019.7452       46169.61 36872.39 57811.10 32734.51 65118.84
## 2019.7479       46053.70 36764.70 57689.68 32631.80 64996.21
## 2019.7507       45625.53 36407.94 57176.79 32308.13 64432.37
## 2019.7534       45778.21 36514.81 57391.63 32395.94 64688.50
## 2019.7562       45850.31 36557.37 57505.53 32426.67 64830.91
## 2019.7589       45563.90 36314.18 57169.64 32204.00 64466.17
## 2019.7616       45298.96 36088.32 56860.40 31996.80 64131.28
## 2019.7644       45256.56 36039.88 56830.28 31946.97 64111.12
## 2019.7671       45192.48 35974.24 56772.86 31881.93 64060.11
## 2019.7699       45541.65 36237.50 57234.70 32108.36 64595.09
## 2019.7726       45553.66 36232.39 57272.96 32096.95 64652.11
## 2019.7753       45600.30 36254.83 57354.75 32109.97 64758.29
## 2019.7781       45494.56 36156.19 57244.84 32015.78 64647.99
## 2019.7808       45343.99 36022.03 57078.35 31890.18 64473.70
## 2019.7836       45279.16 35956.07 57019.64 31825.02 64421.08
## 2019.7863       45271.02 35935.20 57032.26 31799.79 64449.02
## 2019.7890       44971.21 35682.92 56677.25 31569.86 64061.42
## 2019.7918       45528.75 36110.87 57402.84 31941.72 64895.28
## 2019.7945       45491.14 36066.65 57378.32 31895.87 64881.26
## 2019.7973       45474.39 36039.02 57380.05 31864.71 64896.89
## 2019.8000       45518.89 36059.94 57459.04 31876.50 64999.91
## 2019.8027       45473.50 36009.68 57424.53 31825.38 64974.52
## 2019.8055       45332.98 35884.19 57269.77 31707.82 64813.01
## 2019.8082       45267.17 35817.92 57209.26 31642.64 64758.09
## 2019.8110       45174.80 35730.72 57115.07 31559.00 64664.99
## 2019.8137       44806.36 35425.33 56671.58 31282.74 64176.27
## 2019.8164       45331.82 35826.67 57358.76 31630.56 64967.98
## 2019.8192       45323.65 35806.15 57370.97 31605.86 64995.33
## 2019.8219       45111.33 35624.43 57124.61 31438.93 64729.67
## 2019.8247       44913.72 35454.49 56896.66 31282.47 64484.74
## 2019.8274       44805.83 35355.50 56782.18 31188.67 64368.32
## 2019.8301       44965.42 35467.58 57006.68 31281.08 64636.16
## 2019.8329       44701.36 35245.56 56694.00 31078.85 64294.90
## 2019.8356       44373.43 34973.39 56299.99 30832.51 63861.22
## 2019.8384       44370.08 34957.17 56317.61 30811.87 63894.35
## 2019.8411       44560.18 35093.33 56580.84 30925.53 64206.17
## 2019.8438       44454.74 34996.73 56468.81 30834.09 64092.17
## 2019.8466       44610.67 35105.91 56688.79 30923.96 64355.02
## 2019.8493       44416.27 34939.45 56463.54 30771.03 64112.41
## 2019.8521       44234.09 34782.73 56253.62 30626.77 63887.08
## 2019.8548       44317.30 34834.76 56381.13 30666.33 64044.92
## 2019.8575       43393.16 34095.26 55226.63 30009.22 62746.26
## 2019.8603       43145.98 33888.04 54933.10 29820.79 62425.43
## 2019.8630       42329.52 33234.05 53914.24 29239.36 61280.02
## 2019.8658       42430.78 33300.82 54063.87 29292.17 61462.53
## 2019.8685       42618.75 33435.58 54324.11 29404.77 61770.86
## 2019.8712       42575.75 33389.12 54289.98 29357.99 61744.51
## 2019.8740       42898.14 33629.15 54721.88 29563.08 62248.25
## 2019.8767       43127.14 33795.83 55034.90 29703.64 62616.91
## 2019.8795       43099.49 33761.35 55020.48 29667.38 62613.08
## 2019.8822       43072.72 33727.61 55007.13 29631.79 62610.44
## 2019.8849       43143.88 33770.56 55118.85 29663.58 62750.15
## 2019.8877       43090.94 33716.39 55072.00 29610.09 62709.35
## 2019.8904       42901.07 33555.18 54850.00 29462.62 62469.03
## 2019.8932       42991.86 33613.54 54986.78 29507.99 62637.29
## 2019.8959       43307.55 33847.64 55411.37 29707.59 63133.50
## 2019.8986       43515.53 33997.43 55698.37 29833.13 63473.11
## 2019.9014       43379.89 33878.77 55545.56 29723.10 63311.53
## 2019.9041       43689.58 34107.87 55963.03 29918.18 63800.00
## 2019.9068       43873.34 34238.53 56219.40 30026.85 64104.94
## 2019.9096       43470.54 33911.54 55724.04 29734.22 63552.64
## 2019.9123       43522.22 33939.22 55811.07 29752.62 63664.45
## 2019.9151       43459.62 33877.80 55751.51 29692.93 63609.03
## 2019.9178       43694.87 34048.54 56074.12 29836.71 63989.68
## 2019.9205       43814.78 34129.32 56248.85 29901.64 64201.68
## 2019.9233       44131.80 34363.54 56676.82 30100.94 64702.83
## 2019.9260       44190.01 34396.15 56772.55 30123.61 64824.80
## 2019.9288       44161.47 34361.25 56756.82 30087.16 64819.51
## 2019.9315       44078.55 34284.10 56671.14 30013.75 64734.29
## 2019.9342       43903.06 34135.03 56466.28 29877.43 64512.85
## 2019.9370       44082.03 34261.59 56717.31 29982.37 64812.25
## 2019.9397       44147.60 34299.97 56822.53 30010.13 64945.10
## 2019.9425       43967.86 34147.81 56611.92 29871.21 64716.94
## 2019.9452       43914.70 34094.04 56564.16 29818.40 64674.87
## 2019.9479       44031.08 34171.91 56734.79 29880.72 64882.51
## 2019.9507       44044.56 34169.90 56772.86 29873.20 64938.59
## 2019.9534       43875.28 34026.18 56575.27 29741.81 64725.05
## 2019.9562       44223.66 34283.88 57045.24 29961.29 65275.30
## 2019.9589       44557.72 34530.31 57497.04 30170.85 65804.93
## 2019.9616       44016.58 34098.58 56819.36 29787.90 65041.81
## 2019.9644       44343.86 34339.67 57262.57 29992.77 65561.73
## 2019.9671       44866.49 34731.83 57958.42 30329.48 66371.15
## 2019.9699       44560.11 34482.20 57583.43 30105.73 65954.34
## 2019.9726       45049.08 34848.01 58236.31 30419.30 66714.86
## 2019.9753       44952.78 34760.99 58132.75 30337.56 66608.93
## 2019.9781       44923.57 34725.91 58115.87 30301.17 66602.27
## 2019.9808       44924.17 34713.91 58137.53 30284.94 66639.75
## 2019.9836       44928.27 34704.63 58163.70 30271.09 66682.41
## 2019.9863       44637.86 34467.95 57808.44 30058.96 66287.69
## 2019.9890       45119.80 34827.63 58453.48 30366.88 67040.04
## 2019.9918       45229.81 34900.08 58616.93 30424.29 67240.21
## 2019.9945       45721.87 35267.18 59275.77 30738.50 68008.81
## 2019.9973       45376.96 34988.67 58849.57 30490.01 67532.55

ets_plot <- autoplot(ets_forcast, series = "ETS", fcol = "red") +
  autolayer(data_ts, series = "Actual", color = "black") + 
  labs(subtitle = "Unilever Stock Price, from Jan 2015 - Dec 2018",
       y = "Closing Price") +
  theme_minimal()

ets_plot

ARIMA

arima_forcast

##           Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 2019.0000       43031.78 42404.20 43668.64 42075.69 44009.59
## 2019.0027       43102.49 42251.30 43970.83 41807.53 44437.56
## 2019.0055       43124.80 42090.01 44185.02 41552.32 44756.78
## 2019.0082       42446.45 41293.06 43632.06 40695.23 44273.04
## 2019.0110       42629.32 41361.76 43935.73 40706.08 44643.43
## 2019.0137       43397.30 42012.99 44827.23 41298.15 45603.16
## 2019.0164       43158.74 41700.47 44668.00 40948.57 45488.20
## 2019.0192       43277.63 41742.14 44869.60 40951.48 45735.91
## 2019.0219       43333.68 41729.33 44999.72 40904.22 45907.44
## 2019.0247       42998.35 41344.96 44717.85 40495.61 45655.76
## 2019.0274       42883.91 41177.53 44661.01 40301.89 45631.37
## 2019.0301       42840.12 41081.27 44674.29 40179.61 45676.81
## 2019.0329       43027.31 41208.90 44925.96 40277.61 45964.73
## 2019.0356       42721.38 40866.57 44660.39 39917.51 45722.21
## 2019.0384       42819.84 40913.15 44815.40 39938.41 45909.16
## 2019.0411       43084.51 41119.74 45143.16 40116.18 46272.47
## 2019.0438       43177.57 41163.58 45290.10 40135.75 46449.93
## 2019.0466       42924.14 40878.51 45072.13 39835.39 46252.38
## 2019.0493       43497.63 41381.78 45721.67 40303.70 46944.67
## 2019.0521       43324.44 41175.31 45585.73 40081.14 46830.18
## 2019.0548       43821.47 41606.48 46154.38 40479.62 47439.21
## 2019.0575       43720.37 41470.24 46092.58 40326.36 47400.02
## 2019.0603       43795.90 41502.40 46216.15 40337.32 47551.04
## 2019.0630       44248.84 41892.49 46737.74 40696.34 48111.46
## 2019.0658       44511.29 42102.32 47058.09 40880.32 48464.75
## 2019.0685       44292.29 41857.41 46868.81 40623.12 48292.86
## 2019.0712       44559.34 42072.43 47193.24 40812.63 48650.00
## 2019.0740       44503.61 41983.13 47175.40 40707.18 48654.09
## 2019.0767       44302.71 41757.68 47002.86 40470.14 48498.23
## 2019.0795       44597.26 41999.69 47355.49 40686.43 48884.01
## 2019.0822       44361.11 41742.39 47144.12 40419.29 48687.35
## 2019.0849       44240.48 41594.57 47054.69 40258.59 48616.20
## 2019.0877       44542.22 41844.22 47414.17 40482.78 49008.71
## 2019.0904       45151.81 42382.85 48101.68 40986.46 49740.47
## 2019.0932       45673.72 42838.78 48696.27 41409.99 50376.46
## 2019.0959       45577.18 42714.78 48631.41 41273.01 50330.22
## 2019.0986       45695.31 42792.36 48795.19 41331.05 50520.40
## 2019.1014       45545.57 42619.53 48672.49 41147.46 50413.77
## 2019.1041       45731.42 42761.11 48908.05 41267.64 50678.04
## 2019.1068       45600.26 42606.61 48804.25 41102.27 50590.49
## 2019.1096       45713.94 42681.26 48962.10 41158.17 50773.99
## 2019.1123       46180.85 43085.67 49498.38 41532.07 51349.99
## 2019.1151       46047.92 42930.56 49391.63 41366.70 51258.88
## 2019.1178       46104.29 42952.34 49487.54 41371.99 51377.90
## 2019.1205       46027.30 42850.22 49439.95 41258.13 51347.77
## 2019.1233       46053.06 42844.10 49502.37 41236.91 51431.71
## 2019.1260       45601.02 42394.07 49050.58 40788.73 50981.07
## 2019.1288       45939.43 42679.26 49448.63 41048.15 51413.54
## 2019.1315       45926.19 42637.85 49468.14 40993.52 51452.41
## 2019.1342       46214.82 42876.80 49812.70 41208.50 51829.34
## 2019.1370       46224.64 42857.18 49856.70 41175.03 51893.53
## 2019.1397       45983.56 42605.35 49629.64 40918.69 51675.36
## 2019.1425       46354.06 42920.35 50062.47 41206.85 52144.22
## 2019.1452       46191.30 42741.73 49919.28 41021.17 52013.05
## 2019.1479       45974.83 42513.88 49717.52 40788.51 51820.59
## 2019.1507       46244.96 42736.22 50041.78 40987.89 52176.30
## 2019.1534       46591.26 43028.82 50448.63 41254.60 52618.25
## 2019.1562       46348.90 42777.95 50217.94 41000.35 52395.16
## 2019.1589       46345.00 42747.53 50245.21 40957.61 52441.03
## 2019.1616       46465.06 42831.61 50406.73 41024.65 52626.94
## 2019.1644       46805.03 43118.36 50806.90 41285.80 53062.08
## 2019.1671       47289.49 43537.98 51364.27 41674.06 53661.59
## 2019.1699       46906.90 43159.47 50979.72 41298.46 53277.00
## 2019.1726       46624.93 42874.12 50703.88 41012.29 53005.67
## 2019.1753       47009.34 43201.69 51152.59 41312.53 53491.73
## 2019.1781       46769.08 42955.31 50921.46 41063.97 53266.82
## 2019.1808       46173.35 42383.09 50302.57 40504.26 52635.90
## 2019.1836       45925.86 42131.17 50062.35 40250.99 52400.82
## 2019.1863       45825.74 42014.80 49982.35 40127.43 52333.25
## 2019.1890       45798.40 41965.41 49981.48 40067.97 52348.39
## 2019.1918       46099.17 42216.71 50338.69 40295.62 52738.58
## 2019.1945       46080.39 42175.38 50346.96 40243.99 52763.21
## 2019.1973       46106.64 42175.44 50404.28 40231.95 52839.17
## 2019.2000       46264.56 42296.01 50605.48 40334.90 53065.95
## 2019.2027       46160.71 42177.40 50520.22 40209.85 52992.28
## 2019.2055       46345.03 42322.21 50750.24 40336.00 53249.27
## 2019.2082       46069.93 42047.68 50476.95 40062.59 52978.06
## 2019.2110       46401.62 42327.09 50868.38 40317.06 53404.46
## 2019.2137       46450.14 42348.16 50949.46 40325.43 53505.09
## 2019.2164       46450.31 42325.26 50977.38 40292.02 53549.84
## 2019.2192       46657.39 42490.96 51232.37 40438.17 53833.11
## 2019.2219       46146.64 42003.20 50698.80 39962.60 53287.63
## 2019.2247       46016.08 41861.97 50582.42 39816.94 53180.36
## 2019.2274       46060.83 41880.39 50658.55 39823.25 53275.41
## 2019.2301       45983.80 41788.24 50600.59 39724.50 53229.36
## 2019.2329       45954.19 41739.37 50594.62 39666.99 53237.90
## 2019.2356       45918.92 41685.52 50582.25 39604.85 53239.63
## 2019.2384       45616.21 41389.16 50274.95 39312.45 52930.77
## 2019.2411       46118.54 41823.30 50854.91 39713.92 53556.04
## 2019.2438       46129.20 41811.42 50892.86 39691.82 53610.61
## 2019.2466       45863.62 41549.41 50625.78 39432.40 53343.73
## 2019.2493       45932.23 41590.37 50727.36 39460.63 53465.19
## 2019.2521       46103.75 41724.52 50942.61 39577.28 53706.47
## 2019.2548       46235.63 41822.77 51114.10 39659.88 53901.66
## 2019.2575       46291.66 41852.44 51201.74 39677.48 54008.43
## 2019.2603       46631.05 42138.23 51602.89 39937.85 54445.96
## 2019.2630       46982.40 42434.64 52017.55 40208.20 54897.91
## 2019.2658       46664.32 42126.50 51690.94 39905.79 54567.49
## 2019.2685       46766.19 42197.70 51829.29 39962.81 54727.80
## 2019.2712       47063.04 42444.75 52183.83 40186.36 55116.45
## 2019.2740       47142.40 42495.60 52297.31 40224.12 55250.57
## 2019.2767       46851.73 42213.09 52000.09 39946.45 54950.68
## 2019.2795       47452.34 42733.59 52692.14 40428.67 55696.24
## 2019.2822       46618.22 41962.24 51790.81 39688.82 54757.45
## 2019.2849       46488.65 41825.59 51671.59 39549.55 54645.25
## 2019.2877       46578.04 41886.06 51795.62 39596.73 54790.23
## 2019.2904       46920.75 42174.23 52201.47 39859.15 55233.41
## 2019.2932       46888.04 42124.93 52189.71 39802.60 55234.78
## 2019.2959       47074.71 42272.76 52422.13 39932.34 55494.56
## 2019.2986       47552.37 42681.72 52978.85 40308.67 56097.81
## 2019.3014       47613.23 42716.42 53071.37 40331.49 56209.65
## 2019.3041       48066.06 43102.68 53600.99 40686.19 56784.53
## 2019.3068       48500.39 43472.07 54110.33 41024.83 57338.16
## 2019.3096       48677.16 43610.43 54332.54 41145.38 57587.65
## 2019.3123       47899.32 42893.90 53488.85 40459.54 56707.16
## 2019.3151       46683.79 41786.31 52155.26 39405.29 55306.69
## 2019.3178       46110.94 41254.81 51538.69 38894.71 54666.01
## 2019.3205       46980.34 42013.63 52534.20 39600.63 55735.29
## 2019.3233       45980.97 41101.37 51439.87 38731.52 54587.30
## 2019.3260       45804.75 40925.48 51265.75 38556.61 54415.46
## 2019.3288       45854.47 40951.58 51344.35 38572.06 54511.79
## 2019.3315       45848.34 40927.87 51360.36 38540.64 54541.65
## 2019.3342       46139.83 41169.81 51709.83 38759.36 54925.67
## 2019.3370       46084.70 41102.44 51670.88 38686.89 54897.14
## 2019.3397       46812.53 41733.21 52510.05 39271.43 55801.71
## 2019.3425       47019.05 41898.94 52764.85 39418.23 56085.50
## 2019.3452       46658.61 41559.59 52383.25 39089.93 55692.76
## 2019.3479       47500.22 42290.80 53351.33 39768.53 56735.08
## 2019.3507       47406.05 42188.67 53268.66 39663.37 56660.18
## 2019.3534       47951.54 42655.68 53904.89 40093.26 57350.03
## 2019.3562       47759.48 42466.55 53712.11 39906.40 57157.95
## 2019.3589       47492.26 42210.83 53434.50 39657.09 56875.44
## 2019.3616       47571.17 42262.91 53546.16 39697.03 57007.19
## 2019.3644       47591.78 42263.21 53592.17 39688.37 57069.04
## 2019.3671       47926.51 42542.42 53992.01 39941.59 57507.74
## 2019.3699       47734.29 42353.88 53798.21 39755.68 57314.14
## 2019.3726       46819.82 41524.99 52789.80 38968.95 56252.38
## 2019.3753       46805.30 41494.68 52795.59 38931.86 56271.05
## 2019.3781       46911.38 41571.33 52937.39 38995.13 56434.67
## 2019.3808       47080.58 41703.87 53150.47 39110.83 56674.35
## 2019.3836       46902.10 41528.52 52971.00 38937.82 56495.39
## 2019.3863       47092.43 41679.78 53207.98 39071.07 56760.58
## 2019.3890       46733.80 41345.30 52824.57 38749.06 56363.89
## 2019.3918       47228.19 41765.51 53405.36 39134.36 56996.00
## 2019.3945       47443.57 41938.78 53670.90 39288.19 57291.83
## 2019.3973       47667.17 42119.23 53945.89 39448.70 57597.83
## 2019.4000       47403.36 41869.07 53669.19 39205.95 57314.74
## 2019.4027       47575.77 42004.29 53886.25 39324.12 57558.92
## 2019.4055       47321.21 41762.64 53619.61 39089.52 57286.37
## 2019.4082       47212.52 41649.93 53518.04 38975.69 57190.06
## 2019.4110       46608.14 41100.22 52854.18 38453.10 56492.67
## 2019.4137       46194.26 40718.94 52405.83 38088.29 56025.34
## 2019.4164       46310.22 40804.85 52558.37 38160.59 56200.30
## 2019.4192       46506.22 40961.23 52801.83 38298.76 56472.53
## 2019.4219       46281.35 40747.00 52567.38 38090.45 56233.60
## 2019.4247       46116.39 40585.70 52400.75 37931.72 56067.09
## 2019.4274       46583.40 40980.54 52952.28 38292.74 56669.05
## 2019.4301       46326.98 40738.93 52681.52 38059.06 56391.01
## 2019.4329       46420.25 40804.95 52808.28 38112.82 56538.44
## 2019.4356       45925.77 40354.51 52266.18 37684.30 55969.63
## 2019.4384       46128.33 40516.71 52517.18 37827.96 56250.01
## 2019.4411       45908.02 40307.53 52286.66 37624.93 56014.62
## 2019.4438       46025.01 40394.60 52440.22 37698.47 56190.65
## 2019.4466       45925.75 40291.92 52347.33 37594.95 56102.59
## 2019.4493       46109.10 40437.21 52576.57 37722.83 56359.76
## 2019.4521       46200.40 40501.71 52700.90 37775.32 56504.52
## 2019.4548       46302.64 40575.81 52837.75 37836.77 56662.74
## 2019.4575       46157.26 40432.97 52691.96 37695.95 56517.81
## 2019.4603       46273.12 40519.04 52844.34 37768.58 56692.68
## 2019.4630       46256.74 40489.32 52845.68 37733.30 56705.51
## 2019.4658       46197.03 40421.75 52797.45 37662.78 56665.10
## 2019.4685       46045.59 40274.04 52644.23 37517.66 56511.95
## 2019.4712       46289.72 40472.34 52943.27 37694.87 56844.29
## 2019.4740       45902.39 40118.63 52519.97 37358.01 56401.00
## 2019.4767       46087.38 40265.24 52751.38 37487.11 56660.73
## 2019.4795       45976.04 40152.97 52643.58 37375.20 56556.11
## 2019.4822       46036.09 40190.46 52731.96 37402.72 56662.23
## 2019.4849       45900.10 40056.86 52595.71 37271.07 56526.92
## 2019.4877       45744.82 39906.58 52437.19 37123.96 56367.60
## 2019.4904       45833.43 39969.12 52558.16 37174.88 56508.68
## 2019.4932       46613.00 40633.98 53471.79 37785.89 57502.19
## 2019.4959       46737.47 40727.53 53634.26 37865.52 57688.12
## 2019.4986       46466.31 40476.41 53342.62 37624.76 57385.56
## 2019.5014       46471.13 40465.83 53367.65 37607.65 57423.59
## 2019.5041       46439.76 40423.79 53351.05 37561.33 57416.80
## 2019.5068       46767.93 40694.66 53747.59 37805.75 57854.68
## 2019.5096       46444.20 40398.32 53394.88 37523.25 57486.05
## 2019.5123       46348.41 40300.43 53304.02 37425.17 57399.21
## 2019.5151       46228.05 40181.29 53184.77 37307.40 57281.73
## 2019.5178       46093.31 40049.77 53048.83 37178.22 57146.19
## 2019.5205       46293.32 40209.13 53298.13 37319.06 57425.65
## 2019.5233       46117.47 40042.06 53114.67 37156.97 57238.82
## 2019.5260       46750.71 40577.40 53863.21 37646.62 58056.45
## 2019.5288       46642.89 40469.41 53758.12 37539.36 57954.10
## 2019.5315       46778.66 40572.80 53933.74 37628.18 58154.35
## 2019.5342       46678.00 40471.16 53836.76 37526.89 58060.67
## 2019.5370       46725.32 40497.88 53910.37 37544.63 58150.93
## 2019.5397       46394.03 40196.57 53547.01 37258.35 57769.77
## 2019.5425       46215.26 40027.60 53359.43 37094.83 57578.11
## 2019.5452       45978.94 39808.96 53105.20 36885.36 57314.42
## 2019.5479       45706.24 39559.01 52808.71 36646.98 57004.98
## 2019.5507       45831.54 39653.62 52971.97 36727.84 57191.78
## 2019.5534       45618.02 39455.14 52743.54 36537.27 56955.65
## 2019.5562       45542.10 39375.80 52674.05 36457.09 56891.08
## 2019.5589       45585.90 39400.01 52742.98 36472.81 56975.98
## 2019.5616       45745.51 39524.30 52945.95 36581.17 57205.70
## 2019.5644       45705.03 39475.71 52917.35 36529.53 57185.24
## 2019.5671       45598.96 39370.55 52812.71 36425.58 57082.56
## 2019.5699       46194.63 39871.16 53520.98 36882.05 57858.60
## 2019.5726       46573.98 40184.82 53978.98 37165.45 58364.30
## 2019.5753       45614.24 39343.30 52884.71 36380.58 57191.48
## 2019.5781       45873.18 39553.15 53203.05 36568.03 57546.12
## 2019.5808       45768.00 39449.04 53099.12 36465.21 57444.05
## 2019.5836       46085.98 39709.65 53486.19 36699.51 57873.18
## 2019.5863       45975.60 39601.13 53376.15 36592.66 57764.47
## 2019.5890       45717.19 39365.25 53094.08 36368.20 57469.49
## 2019.5918       45913.84 39521.25 53340.43 36505.81 57746.44
## 2019.5945       45679.49 39306.32 53086.03 36300.81 57481.26
## 2019.5973       45666.45 39281.91 53088.68 36271.83 57494.34
## 2019.6000       45188.01 38857.35 52550.07 35873.44 56921.11
## 2019.6027       44964.87 38652.55 52308.04 35678.07 56668.97
## 2019.6055       45232.81 38869.93 52637.28 35872.38 57035.73
## 2019.6082       45094.14 38737.88 52493.37 35744.22 56889.80
## 2019.6110       44697.00 38383.98 52048.33 35411.45 56417.40
## 2019.6137       44864.19 38514.80 52260.32 35525.91 56657.11
## 2019.6164       44513.74 38201.32 51869.23 35230.60 56242.96
## 2019.6192       44224.08 37940.23 51548.70 34983.70 55905.16
## 2019.6219       44017.83 37750.86 51325.18 34803.03 55672.44
## 2019.6247       44947.49 38535.50 52426.36 35520.23 56876.78
## 2019.6274       44902.80 38484.58 52391.40 35467.14 56848.70
## 2019.6301       44822.52 38403.22 52314.83 35386.04 56775.45
## 2019.6329       44710.78 38295.00 52201.44 35280.22 56662.17
## 2019.6356       44615.56 38201.00 52107.22 35187.56 56569.65
## 2019.6384       44396.54 38001.12 51868.27 34997.43 56319.92
## 2019.6411       44647.76 38203.77 52178.69 35178.03 56666.70
## 2019.6438       44683.67 38222.13 52237.55 35188.91 56740.33
## 2019.6466       45112.20 38576.24 52755.55 35508.85 57312.78
## 2019.6493       45225.99 38661.08 52905.65 35580.87 57485.66
## 2019.6521       45921.90 39243.36 53737.01 36110.61 58398.91
## 2019.6548       45959.39 39262.80 53798.14 36122.37 58475.28
## 2019.6575       46059.05 39335.34 53932.06 36182.97 58630.78
## 2019.6603       46076.45 39337.63 53969.68 36178.96 58681.61
## 2019.6630       46327.70 39539.52 54281.27 36358.50 59030.37
## 2019.6658       46168.76 39391.33 54112.26 36216.13 58856.48
## 2019.6685       45969.23 39208.64 53895.52 36042.10 58630.59
## 2019.6712       46063.43 39276.54 54023.09 36098.46 58779.23
## 2019.6740       46116.20 39309.09 54102.08 36122.33 58875.04
## 2019.6767       46310.79 39462.50 54347.53 36257.24 59152.03
## 2019.6795       45450.94 38717.60 53355.27 35566.91 58081.74
## 2019.6822       45373.83 38639.76 53281.51 35489.50 58011.09
## 2019.6849       46135.15 39275.76 54192.51 36067.65 59012.77
## 2019.6877       46216.19 39332.43 54304.72 36113.70 59144.76
## 2019.6904       45894.51 39046.45 53943.60 35845.19 58761.18
## 2019.6932       46424.06 39484.66 54583.05 36241.49 59467.56
## 2019.6959       46352.34 39411.39 54515.70 36168.27 59403.98
## 2019.6986       46541.09 39559.57 54754.71 36298.29 59674.24
## 2019.7014       47156.29 40070.05 55495.69 36760.65 60491.73
## 2019.7041       46845.19 39793.38 55146.66 36500.84 60121.13
## 2019.7068       46740.91 39692.53 55040.90 36402.38 60015.65
## 2019.7096       46859.98 39781.37 55198.14 36477.90 60196.94
## 2019.7123       46561.66 39515.95 54863.64 36228.61 59841.89
## 2019.7151       46428.88 39391.15 54724.00 36108.32 59699.30
## 2019.7178       46633.48 39552.60 54982.01 36250.43 59990.51
## 2019.7205       46769.49 39655.81 55159.26 36339.13 60193.65
## 2019.7233       46801.72 39671.02 55214.15 36347.18 60263.30
## 2019.7260       46240.92 39183.70 54569.19 35894.91 59568.98
## 2019.7288       46432.91 39334.41 54812.44 36027.16 59844.17
## 2019.7315       46264.01 39179.42 54629.66 35879.42 59654.20
## 2019.7342       45885.50 38847.08 54199.15 35569.37 59193.60
## 2019.7370       45636.77 38624.81 53921.69 35360.18 58900.01
## 2019.7397       46211.88 39099.73 54617.70 35789.23 59669.84
## 2019.7425       46422.94 39266.46 54883.71 35936.10 59970.03
## 2019.7452       46426.30 39257.48 54904.23 35922.16 60002.01
## 2019.7479       46309.75 39147.15 54782.86 35815.50 59878.90
## 2019.7507       45879.20 38771.56 54289.82 35466.24 59349.44
## 2019.7534       46032.73 38889.65 54487.82 35568.62 59575.33
## 2019.7562       46105.23 38939.25 54589.95 35608.34 59696.45
## 2019.7589       45817.22 38684.46 54265.14 35369.76 59350.64
## 2019.7616       45550.82 38448.07 53965.70 35148.08 59032.44
## 2019.7644       45508.18 38400.65 53931.22 35099.20 59004.02
## 2019.7671       45443.74 38334.89 53870.86 35033.59 58947.24
## 2019.7699       45794.86 38619.63 54303.18 35288.27 59429.63
## 2019.7726       45806.93 38618.38 54333.57 35281.60 59472.20
## 2019.7753       45853.82 38646.50 54405.26 35301.77 59559.99
## 2019.7781       45747.50 38545.53 54295.13 35204.04 59448.70
## 2019.7808       45596.10 38406.65 54131.36 35071.73 59278.63
## 2019.7836       45530.90 38340.46 54069.85 35005.84 59220.48
## 2019.7863       45522.72 38322.33 54076.00 34983.85 59236.42
## 2019.7890       45221.24 38057.38 53733.61 34736.61 58870.48
## 2019.7918       45781.88 38517.94 54415.69 35151.53 59626.99
## 2019.7945       45744.06 38474.89 54386.62 35106.83 59604.35
## 2019.7973       45727.22 38449.52 54382.44 35078.26 59608.96
## 2019.8000       45771.97 38475.95 54451.49 35096.97 59693.84
## 2019.8027       45726.32 38426.42 54412.99 35046.40 59660.81
## 2019.8055       45585.02 38296.57 54260.58 34922.61 59502.83
## 2019.8082       45518.85 38229.91 54197.50 34856.48 59442.76
## 2019.8110       45425.97 38140.88 54102.54 34769.98 59347.69
## 2019.8137       45055.47 37818.89 53676.77 34471.19 58889.63
## 2019.8164       45583.85 38251.38 54321.89 34860.08 59606.50
## 2019.8192       45575.64 38233.50 54327.73 34838.48 59621.98
## 2019.8219       45362.13 38043.46 54088.75 34660.05 59368.73
## 2019.8247       45163.43 37865.96 53867.25 34493.10 59134.59
## 2019.8274       45054.94 37764.19 53753.24 34395.18 59018.36
## 2019.8301       45215.42 37887.88 53960.11 34502.62 59254.46
## 2019.8329       44949.89 37654.64 53658.54 34285.04 58932.19
## 2019.8356       44620.14 37367.76 53280.07 34018.71 58525.35
## 2019.8384       44616.77 37354.32 53291.20 34001.35 58546.39
## 2019.8411       44807.93 37503.71 53534.71 34132.21 58822.76
## 2019.8438       44701.89 37404.36 53423.17 34036.68 58709.00
## 2019.8466       44858.69 37524.94 53625.73 34141.29 58940.43
## 2019.8493       44663.21 37350.87 53407.13 33977.83 58708.95
## 2019.8521       44480.02 37187.18 53203.08 33823.87 58493.38
## 2019.8548       44563.69 37246.64 53318.17 33872.90 58628.65
## 2019.8575       43634.42 36459.69 52221.03 33152.30 57430.78
## 2019.8603       43385.86 36241.83 51938.14 32949.30 57128.16
## 2019.8630       42564.87 35546.05 50969.59 32311.94 56071.15
## 2019.8658       42666.69 35621.11 51105.82 32375.38 56229.34
## 2019.8685       42855.70 35768.92 51346.57 32504.91 56502.58
## 2019.8712       42812.47 35722.86 51309.08 32458.26 56469.67
## 2019.8740       43136.64 35983.33 51711.99 32690.10 56921.50
## 2019.8767       43366.92 36165.36 52002.51 32850.63 57249.71
## 2019.8795       43339.11 36132.14 51983.61 32815.63 57237.32
## 2019.8822       43312.19 36099.68 51965.72 32781.34 57226.03
## 2019.8849       43383.75 36149.31 52065.99 32821.60 57344.85
## 2019.8877       43330.52 36094.98 52016.49 32767.47 57298.71
## 2019.8904       43139.59 35926.01 51801.58 32609.31 57070.32
## 2019.8932       43230.89 35992.12 51925.53 32664.56 57215.22
## 2019.8959       43548.33 36246.43 52321.21 32890.57 57659.61
## 2019.8986       43757.47 36410.50 52586.93 33034.63 57960.87
## 2019.9014       43621.08 36287.05 52437.40 32917.85 57804.46
## 2019.9041       43932.49 36536.09 52826.22 33138.96 58241.53
## 2019.9068       44117.26 36679.72 53062.92 33264.42 58510.96
## 2019.9096       43712.23 36333.04 52590.12 32945.25 57998.00
## 2019.9123       43764.20 36366.31 52667.00 32970.66 58091.18
## 2019.9151       43701.24 36304.11 52605.57 32909.53 58031.79
## 2019.9178       43937.80 36490.71 52904.72 33073.91 58370.20
## 2019.9205       44058.38 36580.91 53064.32 33150.90 58554.70
## 2019.9233       44377.17 36835.60 53462.76 33376.92 59002.84
## 2019.9260       44435.69 36874.20 53547.77 33407.10 59105.13
## 2019.9288       44407.00 36840.42 53527.66 33371.72 59091.39
## 2019.9315       44323.62 36761.32 53441.58 33295.32 59004.80
## 2019.9342       44147.15 36605.09 53243.16 33149.08 58794.11
## 2019.9370       44327.11 36744.42 53474.60 33270.51 59058.10
## 2019.9397       44393.05 36789.19 53568.55 33306.31 59170.27
## 2019.9425       44212.31 36629.57 53364.78 33157.09 58953.57
## 2019.9452       44158.86 36575.47 53314.54 33103.42 58906.44
## 2019.9479       44275.88 36662.59 53470.14 33177.57 59086.73
## 2019.9507       44289.44 36664.01 53500.81 33174.16 59128.99
## 2019.9534       44119.21 36513.35 53309.41 33033.17 58925.78
## 2019.9562       44469.54 36793.46 53747.03 33281.89 59417.89
## 2019.9589       44805.45 37061.53 54167.45 33519.65 59891.10
## 2019.9616       44261.30 36601.70 53523.82 33099.10 59187.79
## 2019.9644       44590.40 36864.06 53936.11 33331.67 59652.09
## 2019.9671       45115.94 37288.65 54586.26 33710.85 60379.62
## 2019.9699       44807.86 37024.22 54227.86 33467.10 59991.58
## 2019.9726       45299.54 37420.59 54837.40 33820.66 60674.40
## 2019.9753       45202.71 37330.75 54734.63 33734.74 60569.15
## 2019.9781       45173.33 37296.66 54713.48 33699.23 60554.19
## 2019.9808       45173.93 37287.34 54728.62 33686.12 60579.39
## 2019.9836       45178.06 37280.94 54748.00 33675.65 60609.28
## 2019.9863       44886.04 37030.24 54408.41 33444.55 60241.70
## 2019.9890       45370.66 37420.23 55010.25 33792.09 60916.53
## 2019.9918       45481.28 37501.65 55158.81 33860.91 61089.50
## 2019.9945       45976.07 37899.72 55773.47 34215.61 61778.80
## 2019.9973       45629.24 37604.00 55367.19 33943.94 61337.25

arima_p <- autoplot(arima_forcast, series = "ARIMA", fcol = "red") +
  autolayer(data_ts, series = "Actual", color = "black") + 
  labs(subtitle = "Unilever Stock Price, from Jan 2015 - Dec 2018",
       y = "Closing Price") +
  theme_minimal()

arima_p

Evaluation

For the evaluation of prediction accuracy we use root mean squared error (RMSE). that measure the standard deviation of the residuals (prediction errors).

data.frame(ETS = RMSE(ets_forcast$mean, test), 
           ARIMA = RMSE(arima_forcast$mean, test))

From the analysis above, we can conclude that we have successfully forecast the stock price and found ARIMA as the best model with the lowest error (RMSE ~2632.69, although not very different from the ETS model).

Assumption Test

The assumptions in the time series are tested to measure whether the residuals obtained from the modeling results are good enough to describe and capture information in the data. Why we use residual data? Because by using residual data, we can get information from the actual data as well as from the prediction results using the model. A good forecasting method produces the following residual values¹:

Uncorrelated residuals. If there are correlated residuals, it means that there is still information left that should be used to calculate forecast results.
Residuals have a mean of 0.

To test those assumption, there are two assumption for a time series analysis:

Normality: Shapiro.test

H0 : residuals are normally distributed H1 : residuals are not normally distributed

For normality test, we want to make sure that the data is normally distributed (p-value > 0.05)

Autocorrelations: Box.test - Ljng-Box

H0 : No autocorrelations in the forecast errors H1 : there is an autocorrelations in the forecast errors

For Autocorrelations test, we want to make sure that the forecast errors has no autocorrelation (p-value > 0.05)

Normality: Shapiro.test

# p-value < 0.05 = reject H0
shapiro.test(arima_forcast$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  arima_forcast$residuals
## W = 0.96849, p-value < 2.2e-16

hist(arima_forcast$residuals, breaks = 20)

plot(arima_forcast$residuals)

Autocorrelation: Box.test - Ljng-Box

Box.test(arima_forcast$residuals, type = "Ljung-Box")

## 
##  Box-Ljung test
## 
## data:  arima_forcast$residuals
## X-squared = 0.0065312, df = 1, p-value = 0.9356

Conclusion

Based on the assumption check, there is no autocorrelation on our forecast residuals (p-value > 0.05). Still, our forecast’s residuals are not distributed normally, therefore the residuals may not be appeared around its mean as seen in the histogram. But, if we inspect the distribution of residuals through a line plot, it is actually resembles the error plot from our time series object decomposition.

In a time series, such errors might emerge from various unpredictable events and is actually quite unavoidable. One strategy to overcome it is to analyze what kinds of unpredictable events that might occur and occurs frequently. This can be done by time series analysis using seasonality adjustment. From that insight, investors could a better buy/hold/sell strategy to deal with the uncertainty in the future.

Refference

Residuals Diagnostic ↩︎

Stock Price Time Series Forecasting - case (UNVR)

Agung Atidhira

October 1, 2021