Ekonometrika - Week 4
Kefas Ronaldo Immanuel Labiro
February 23, 2021
The impact of how a dollar spent on an organization’s marketing efforts on its sales is something that all organizations should consider. A fiscally prudent organization should be using its relatively scarce resources wisely. Thus, all organizations need to ask themselves, “Is the money I’m spending worth the return on sales?” Furthermore, organizations can delve deeper by asking, “For every dollar spent on marketing, how much are we getting in return on sales?” One can answer these questions using a simple linear regression model. As always, we will use a fabricated example to examine a store’s marketing efforts and their impact on sales. This will also be a more comprehensive primer on the simple linear regression model, the model that the majority of econometrics students are first exposed to.
marketing <- readRDS("marketing.rds")
marketing## youtube facebook newspaper sales
## 1 276.12 45.36 83.04 26.52
## 2 53.40 47.16 54.12 12.48
## 3 20.64 55.08 83.16 11.16
## 4 181.80 49.56 70.20 22.20
## 5 216.96 12.96 70.08 15.48
## 6 10.44 58.68 90.00 8.64
## 7 69.00 39.36 28.20 14.16
## 8 144.24 23.52 13.92 15.84
## 9 10.32 2.52 1.20 5.76
## 10 239.76 3.12 25.44 12.72
## 11 79.32 6.96 29.04 10.32
## 12 257.64 28.80 4.80 20.88
## 13 28.56 42.12 79.08 11.04
## 14 117.00 9.12 8.64 11.64
## 15 244.92 39.48 55.20 22.80
## 16 234.48 57.24 63.48 26.88
## 17 81.36 43.92 136.80 15.00
## 18 337.68 47.52 66.96 29.28
## 19 83.04 24.60 21.96 13.56
## 20 176.76 28.68 22.92 17.52
## 21 262.08 33.24 64.08 21.60
## 22 284.88 6.12 28.20 15.00
## 23 15.84 19.08 59.52 6.72
## 24 273.96 20.28 31.44 18.60
## 25 74.76 15.12 21.96 11.64
## 26 315.48 4.20 23.40 14.40
## 27 171.48 35.16 15.12 18.00
## 28 288.12 20.04 27.48 19.08
## 29 298.56 32.52 27.48 22.68
## 30 84.72 19.20 48.96 12.60
## 31 351.48 33.96 51.84 25.68
## 32 135.48 20.88 46.32 14.28
## 33 116.64 1.80 36.00 11.52
## 34 318.72 24.00 0.36 20.88
## 35 114.84 1.68 8.88 11.40
## 36 348.84 4.92 10.20 15.36
## 37 320.28 52.56 6.00 30.48
## 38 89.64 59.28 54.84 17.64
## 39 51.72 32.04 42.12 12.12
## 40 273.60 45.24 38.40 25.80
## 41 243.00 26.76 37.92 19.92
## 42 212.40 40.08 46.44 20.52
## 43 352.32 33.24 2.16 24.84
## 44 248.28 10.08 31.68 15.48
## 45 30.12 30.84 51.96 10.20
## 46 210.12 27.00 37.80 17.88
## 47 107.64 11.88 42.84 12.72
## 48 287.88 49.80 22.20 27.84
## 49 272.64 18.96 59.88 17.76
## 50 80.28 14.04 44.16 11.64
## 51 239.76 3.72 41.52 13.68
## 52 120.48 11.52 4.32 12.84
## 53 259.68 50.04 47.52 27.12
## 54 219.12 55.44 70.44 25.44
## 55 315.24 34.56 19.08 24.24
## 56 238.68 59.28 72.00 28.44
## 57 8.76 33.72 49.68 6.60
## 58 163.44 23.04 19.92 15.84
## 59 252.96 59.52 45.24 28.56
## 60 252.84 35.40 11.16 22.08
## 61 64.20 2.40 25.68 9.72
## 62 313.56 51.24 65.64 29.04
## 63 287.16 18.60 32.76 18.84
## 64 123.24 35.52 10.08 16.80
## 65 157.32 51.36 34.68 21.60
## 66 82.80 11.16 1.08 11.16
## 67 37.80 29.52 2.64 11.40
## 68 167.16 17.40 12.24 16.08
## 69 284.88 33.00 13.20 22.68
## 70 260.16 52.68 32.64 26.76
## 71 238.92 36.72 46.44 21.96
## 72 131.76 17.16 38.04 14.88
## 73 32.16 39.60 23.16 10.56
## 74 155.28 6.84 37.56 13.20
## 75 256.08 29.52 15.72 20.40
## 76 20.28 52.44 107.28 10.44
## 77 33.00 1.92 24.84 8.28
## 78 144.60 34.20 17.04 17.04
## 79 6.48 35.88 11.28 6.36
## 80 139.20 9.24 27.72 13.20
## 81 91.68 32.04 26.76 14.16
## 82 287.76 4.92 44.28 14.76
## 83 90.36 24.36 39.00 13.56
## 84 82.08 53.40 42.72 16.32
## 85 256.20 51.60 40.56 26.04
## 86 231.84 22.08 78.84 18.24
## 87 91.56 33.00 19.20 14.40
## 88 132.84 48.72 75.84 19.20
## 89 105.96 30.60 88.08 15.48
## 90 131.76 57.36 61.68 20.04
## 91 161.16 5.88 11.16 13.44
## 92 34.32 1.80 39.60 8.76
## 93 261.24 40.20 70.80 23.28
## 94 301.08 43.80 86.76 26.64
## 95 128.88 16.80 13.08 13.80
## 96 195.96 37.92 63.48 20.28
## 97 237.12 4.20 7.08 14.04
## 98 221.88 25.20 26.40 18.60
## 99 347.64 50.76 61.44 30.48
## 100 162.24 50.04 55.08 20.64
## 101 266.88 5.16 59.76 14.04
## 102 355.68 43.56 121.08 28.56
## 103 336.24 12.12 25.68 17.76
## 104 225.48 20.64 21.48 17.64
## 105 285.84 41.16 6.36 24.84
## 106 165.48 55.68 70.80 23.04
## 107 30.00 13.20 35.64 8.64
## 108 108.48 0.36 27.84 10.44
## 109 15.72 0.48 30.72 6.36
## 110 306.48 32.28 6.60 23.76
## 111 270.96 9.84 67.80 16.08
## 112 290.04 45.60 27.84 26.16
## 113 210.84 18.48 2.88 16.92
## 114 251.52 24.72 12.84 19.08
## 115 93.84 56.16 41.40 17.52
## 116 90.12 42.00 63.24 15.12
## 117 167.04 17.16 30.72 14.64
## 118 91.68 0.96 17.76 11.28
## 119 150.84 44.28 95.04 19.08
## 120 23.28 19.20 26.76 7.92
## 121 169.56 32.16 55.44 18.60
## 122 22.56 26.04 60.48 8.40
## 123 268.80 2.88 18.72 13.92
## 124 147.72 41.52 14.88 18.24
## 125 275.40 38.76 89.04 23.64
## 126 104.64 14.16 31.08 12.72
## 127 9.36 46.68 60.72 7.92
## 128 96.24 0.00 11.04 10.56
## 129 264.36 58.80 3.84 29.64
## 130 71.52 14.40 51.72 11.64
## 131 0.84 47.52 10.44 1.92
## 132 318.24 3.48 51.60 15.24
## 133 10.08 32.64 2.52 6.84
## 134 263.76 40.20 54.12 23.52
## 135 44.28 46.32 78.72 12.96
## 136 57.96 56.40 10.20 13.92
## 137 30.72 46.80 11.16 11.40
## 138 328.44 34.68 71.64 24.96
## 139 51.60 31.08 24.60 11.52
## 140 221.88 52.68 2.04 24.84
## 141 88.08 20.40 15.48 13.08
## 142 232.44 42.48 90.72 23.04
## 143 264.60 39.84 45.48 24.12
## 144 125.52 6.84 41.28 12.48
## 145 115.44 17.76 46.68 13.68
## 146 168.36 2.28 10.80 12.36
## 147 288.12 8.76 10.44 15.84
## 148 291.84 58.80 53.16 30.48
## 149 45.60 48.36 14.28 13.08
## 150 53.64 30.96 24.72 12.12
## 151 336.84 16.68 44.40 19.32
## 152 145.20 10.08 58.44 13.92
## 153 237.12 27.96 17.04 19.92
## 154 205.56 47.64 45.24 22.80
## 155 225.36 25.32 11.40 18.72
## 156 4.92 13.92 6.84 3.84
## 157 112.68 52.20 60.60 18.36
## 158 179.76 1.56 29.16 12.12
## 159 14.04 44.28 54.24 8.76
## 160 158.04 22.08 41.52 15.48
## 161 207.00 21.72 36.84 17.28
## 162 102.84 42.96 59.16 15.96
## 163 226.08 21.72 30.72 17.88
## 164 196.20 44.16 8.88 21.60
## 165 140.64 17.64 6.48 14.28
## 166 281.40 4.08 101.76 14.28
## 167 21.48 45.12 25.92 9.60
## 168 248.16 6.24 23.28 14.64
## 169 258.48 28.32 69.12 20.52
## 170 341.16 12.72 7.68 18.00
## 171 60.00 13.92 22.08 10.08
## 172 197.40 25.08 56.88 17.40
## 173 23.52 24.12 20.40 9.12
## 174 202.08 8.52 15.36 14.04
## 175 266.88 4.08 15.72 13.80
## 176 332.28 58.68 50.16 32.40
## 177 298.08 36.24 24.36 24.24
## 178 204.24 9.36 42.24 14.04
## 179 332.04 2.76 28.44 14.16
## 180 198.72 12.00 21.12 15.12
## 181 187.92 3.12 9.96 12.60
## 182 262.20 6.48 32.88 14.64
## 183 67.44 6.84 35.64 10.44
## 184 345.12 51.60 86.16 31.44
## 185 304.56 25.56 36.00 21.12
## 186 246.00 54.12 23.52 27.12
## 187 167.40 2.52 31.92 12.36
## 188 229.32 34.44 21.84 20.76
## 189 343.20 16.68 4.44 19.08
## 190 22.44 14.52 28.08 8.04
## 191 47.40 49.32 6.96 12.96
## 192 90.60 12.96 7.20 11.88
## 193 20.64 4.92 37.92 7.08
## 194 200.16 50.40 4.32 23.52
## 195 179.64 42.72 7.20 20.76
## 196 45.84 4.44 16.56 9.12
## 197 113.04 5.88 9.72 11.64
## 198 212.40 11.16 7.68 15.36
## 199 340.32 50.40 79.44 30.60
## 200 278.52 10.32 10.44 16.081 Is there a relationship between advertising budget and sales?
Hipotesisnya adalah \[ \begin{align} \tag{1} H_{0}&:\beta_{youtube}=\beta_{facebook}= \beta_{newspaper}=0 \\ H_{a}&:\beta_{youtube}≠\beta_{facebook}≠\beta_{newspaper}≠0 \end{align} \]
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlinmod <- lm(sales~., data=marketing)
slinmod <- linmod %>% summary()
anova(linmod)$'Pr(>F)'[1]## [1] 1.809337e-84Jika dilihat, maka p-valuenya = 1.560676e^-95. Hasil tersebut lebih kecil dari tingkat signifikansi, yaitu 0.05. Hal itu menunjukan bahwa kita tolak H0, sehingga terdapat hubungan antara youtube, facebook, newspaper, dan sales.
2 How strong is the relationship between advertising budget and sales?
Kekuatan hubungan tersebut dapat diketahui dengan R^2
\[ \begin{align} \tag{2} R^2= {SSR\over SST}=1-{{SSE\over SST}} \end{align} \]
residual.se <- summary(linmod)$sigma
residual.se## [1] 2.022612mean1 <- mean(marketing$sales)
residual.se/mean1## [1] 0.1202004r_kuadrat <- summary(linmod)$r.sq
r_kuadrat## [1] 0.8972106Nilai R^2 yaitu 0 atau 1. Jika nilai R^2 semakin mendekati 1, maka hubungan antarvariabelnya semakin kuat.
Dari perhitungan kita di atas, hasil dari R^2 (r_kuadrat) adalah 0.8972106, yang berarti mendekati 1, maka relationship antara advertising budget (youtube, facebook, newspaper) dengan sales adalah sangat kuat
3 Which media contribute to sales?
Untuk mencarinya, dapat menggunakan p-value dari setiap variabel responnya
coef1 <- summary(linmod)$coefficients
coef1## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.526667243 0.374289884 9.4222884 1.267295e-17
## youtube 0.045764645 0.001394897 32.8086244 1.509960e-81
## facebook 0.188530017 0.008611234 21.8934961 1.505339e-54
## newspaper -0.001037493 0.005871010 -0.1767146 8.599151e-01Berdasarkan hasil di atas, didapatkan bahwa hasil p-value youtube dan facebook adalah lebih rendah dari 0.05, sedangkan p-value dari newspaper tinggi. Oleh karena itu, media yang berkontribusi dengan sales adalah youtube dan facebook
4 How accurately can we estimate the effect of each medium on sales?
Untuk mengetahui keakuratan media dalam mengestimasi sales, kita dapat melihat dari koefisien determination atau \(R^2\)
slinmod$adj.r.squared## [1] 0.8956373Hasilnya menunjukkan bahwa koefisien determinationnya adalah \(R^2 = 0.9242541\), yang berarti efek dari advertising budget dapat mengestimasi sales hingga 92.42%.
5 How accurately can we predict future sales?
library(dplyr)
library(magrittr)
b0 <- coef(linmod)[[1]]
b1 <- coef(linmod)[[2]]
x1 <- marketing$youtube
b2 <- coef(linmod)[[3]]
x2 <- marketing$facebook
b3 <- coef(linmod)[[4]]
x3 <- marketing$newspaper
marketing$sales_pred <- (b0 +b1*x1 +b2*x2 +b3*x3) %>% round(2)
marketing$error <- abs(marketing$sales-marketing$sales_pred) %>% round(2)
marketing$'%error' <- round(marketing$error/marketing$sales*100,2)
total_percentage_error <- sum(marketing$`%error`)
MAPE <- total_percentage_error/nrow(marketing)
paste(total_percentage_error/nrow(marketing),"%")## [1] "13.87845 %"paste("The accuracy is ", 100-MAPE,"%")## [1] "The accuracy is 86.12155 %"Jadi, tingkat akurasi untuk memprediksi future sales adalah 86.12%
6 Is there synergy (interaction) among the advertising media?
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recodevif(linmod)## youtube facebook newspaper
## 1.004611 1.144952 1.145187Hasil di atas menunjukkan bahwa semua nilai VIF nya adala kurang dari 5, yang berarti tidak ada sinergi atau interaksi di antara advertising media.