Caso de Negocios 1: Costos en aerolineas

Contexto

La base de datos es de la Universidad de NY y contiene 90 observaciones que incluyen los costos de 6 aerolineas estadounidenses durante 15 años, de 1970 a 1984

Las variables son: * I = Aerolinea * T = Año * Q = Millas voladas por los pasajeros * C = Costo toal en $1,000 * PF = Precio del combustible * LF = Factor de carga (Utilización promedio de la capacidad de la flota)

Fuente: [Tabla F7.1] https://pages.stern.nyu.edu/~wgreene/Text/tables/tablelist5.htm

Instalar paquetes y llamar librerías

#install.packages("plm")
library(plm)
#install.packages("tidyverse")
library(tidyverse)
#install.packages("forecast")
library(forecast)
#install.packages("lavaan")
library(lavaan)
#install.packages("lavaanPlot")
library(lavaanPlot)
#install.packages("DataExplorer")
library(DataExplorer)
#install.packages("ggplot2")
library(ggplot2)
#install.packages("gplots")
library(gplots)

Importar la base de datos

df <- read.csv("C:\\Users\\Cristina\\Desktop\\Cost Data for U.S. Airlines.csv")

Análisis descriptivo

summary(df)

##        I             T            C                 Q          
##  Min.   :1.0   Min.   : 1   Min.   :  68978   Min.   :0.03768  
##  1st Qu.:2.0   1st Qu.: 4   1st Qu.: 292046   1st Qu.:0.14213  
##  Median :3.5   Median : 8   Median : 637001   Median :0.30503  
##  Mean   :3.5   Mean   : 8   Mean   :1122524   Mean   :0.54499  
##  3rd Qu.:5.0   3rd Qu.:12   3rd Qu.:1345968   3rd Qu.:0.94528  
##  Max.   :6.0   Max.   :15   Max.   :4748320   Max.   :1.93646  
##        PF                LF        
##  Min.   : 103795   Min.   :0.4321  
##  1st Qu.: 129848   1st Qu.:0.5288  
##  Median : 357434   Median :0.5661  
##  Mean   : 471683   Mean   :0.5605  
##  3rd Qu.: 849840   3rd Qu.:0.5947  
##  Max.   :1015610   Max.   :0.6763

str(df)

## 'data.frame':    90 obs. of  6 variables:
##  $ I : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ T : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ C : int  1140640 1215690 1309570 1511530 1676730 1823740 2022890 2314760 2639160 3247620 ...
##  $ Q : num  0.953 0.987 1.092 1.176 1.16 ...
##  $ PF: int  106650 110307 110574 121974 196606 265609 263451 316411 384110 569251 ...
##  $ LF: num  0.534 0.532 0.548 0.541 0.591 ...

head(df)

##   I T       C        Q     PF       LF
## 1 1 1 1140640 0.952757 106650 0.534487
## 2 1 2 1215690 0.986757 110307 0.532328
## 3 1 3 1309570 1.091980 110574 0.547736
## 4 1 4 1511530 1.175780 121974 0.540846
## 5 1 5 1676730 1.160170 196606 0.591167
## 6 1 6 1823740 1.173760 265609 0.575417

df$I <- as.factor(df$I) #cambiar aerolinea de número a factor
df$Y <- df$T + 1969
summary(df)

##  I            T            C                 Q                 PF         
##  1:15   Min.   : 1   Min.   :  68978   Min.   :0.03768   Min.   : 103795  
##  2:15   1st Qu.: 4   1st Qu.: 292046   1st Qu.:0.14213   1st Qu.: 129848  
##  3:15   Median : 8   Median : 637001   Median :0.30503   Median : 357434  
##  4:15   Mean   : 8   Mean   :1122524   Mean   :0.54499   Mean   : 471683  
##  5:15   3rd Qu.:12   3rd Qu.:1345968   3rd Qu.:0.94528   3rd Qu.: 849840  
##  6:15   Max.   :15   Max.   :4748320   Max.   :1.93646   Max.   :1015610  
##        LF               Y       
##  Min.   :0.4321   Min.   :1970  
##  1st Qu.:0.5288   1st Qu.:1973  
##  Median :0.5661   Median :1977  
##  Mean   :0.5605   Mean   :1977  
##  3rd Qu.:0.5947   3rd Qu.:1981  
##  Max.   :0.6763   Max.   :1984

str(df)

## 'data.frame':    90 obs. of  7 variables:
##  $ I : Factor w/ 6 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ T : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ C : int  1140640 1215690 1309570 1511530 1676730 1823740 2022890 2314760 2639160 3247620 ...
##  $ Q : num  0.953 0.987 1.092 1.176 1.16 ...
##  $ PF: int  106650 110307 110574 121974 196606 265609 263451 316411 384110 569251 ...
##  $ LF: num  0.534 0.532 0.548 0.541 0.591 ...
##  $ Y : num  1970 1971 1972 1973 1974 ...

head(df)

##   I T       C        Q     PF       LF    Y
## 1 1 1 1140640 0.952757 106650 0.534487 1970
## 2 1 2 1215690 0.986757 110307 0.532328 1971
## 3 1 3 1309570 1.091980 110574 0.547736 1972
## 4 1 4 1511530 1.175780 121974 0.540846 1973
## 5 1 5 1676730 1.160170 196606 0.591167 1974
## 6 1 6 1823740 1.173760 265609 0.575417 1975

#create_report(df)
plot_missing(df)

plot_histogram(df)

plot_correlation(df)

ggplot(df, aes(x=Y, y=Q, color=I, group=I)) + geom_line() + labs(title= "Costo por Aerolinea (en miles)", x= "Año", y="Costo (USD)", color="Aerolinea") + theme_minimal()

ggplot(df, aes(x=Y, y=C, color=I, group=I)) + geom_line() + labs(title= "Millas voladas por pasajero", x= "Año", y="Índice normalizado", color="Aerolinea") + theme_minimal()

ggplot(df, aes(x=Y, y=PF, color=I, group=I)) + geom_line() + labs(title= "Precio de Combustible", x= "Año", y="Costo (USD)", color="Aerolinea") + theme_minimal()

ggplot(df, aes(x=Y, y=LF, color=I, group=I)) + geom_line() + labs(title= "Factor de Carga", x= "Año", y="porcentaje", color="Aerolinea") + theme_minimal()

Tema 1. Datos de Panel

Heterogeneidad

plotmeans(C~I, main= "Heterogeneidad entre aerolineas", xlab= "Aerolinea", ylab = "Costo en miles de USD", data= df)

## Warning in arrows(x, li, x, pmax(y - gap, li), col = barcol, lwd = lwd, :
## zero-length arrow is of indeterminate angle and so skipped

## Warning in arrows(x, ui, x, pmin(y + gap, ui), col = barcol, lwd = lwd, :
## zero-length arrow is of indeterminate angle and so skipped

Como el valor promedio (círculo) y el rango intercuartil (líneas azules) varían entre individuos, se observa presencia de heterogeneidad

Creación de datos panel

df_panel <- pdata.frame(df, index= c("I", "Y"))
df_panel <- df_panel%>% select(-c("I", "T", "Y"))

Modelo 1. Regresión Agrupada (Pooled)

#El modelo re regresión agrupada (Pooled) es una técnica de estimación de datos panel donde se asume que no hay efectos individuales especídicos para cada unidad (Ej. Aerolineas) ni variaciones en el tiempo, Ignora heterogeneidades
pooled <- plm(C ~ Q + PF + LF, data= df_panel, model= "pooling")
summary(pooled)

## Pooling Model
## 
## Call:
## plm(formula = C ~ Q + PF + LF, data = df_panel, model = "pooling")
## 
## Balanced Panel: n = 6, T = 15, N = 90
## 
## Residuals:
##    Min. 1st Qu.  Median 3rd Qu.    Max. 
## -520654 -250270   37333  208690  849700 
## 
## Coefficients:
##                Estimate  Std. Error t-value  Pr(>|t|)    
## (Intercept)  1.1586e+06  3.6059e+05  3.2129   0.00185 ** 
## Q            2.0261e+06  6.1807e+04 32.7813 < 2.2e-16 ***
## PF           1.2253e+00  1.0372e-01 11.8138 < 2.2e-16 ***
## LF          -3.0658e+06  6.9633e+05 -4.4027 3.058e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    1.2647e+14
## Residual Sum of Squares: 6.8177e+12
## R-Squared:      0.94609
## Adj. R-Squared: 0.94421
## F-statistic: 503.118 on 3 and 86 DF, p-value: < 2.22e-16

#Prueba de Breusch-Pagan (BP): Para verficar si el modelo pooled es adecuado 
#P- value <0.05 Avanzamos para usar un modeloo de efectos fijos o aleatorios
# p-value >0.05 Podemos usar el mddelo pooled 
plmtest(pooled, type="bp")

## 
##  Lagrange Multiplier Test - (Breusch-Pagan)
## 
## data:  C ~ Q + PF + LF
## chisq = 0.61309, df = 1, p-value = 0.4336
## alternative hypothesis: significant effects

Como el p-value es mayor a 0.05, podemos utilizar el modelo pooled

Modelo 2. Efectos fijos (Within)

within <-  plm(C ~ Q + PF + LF, data= df_panel, model= "within")
summary(within)

## Oneway (individual) effect Within Model
## 
## Call:
## plm(formula = C ~ Q + PF + LF, data = df_panel, model = "within")
## 
## Balanced Panel: n = 6, T = 15, N = 90
## 
## Residuals:
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -551783 -159259    1796       0  137226  499296 
## 
## Coefficients:
##       Estimate  Std. Error t-value  Pr(>|t|)    
## Q   3.3190e+06  1.7135e+05 19.3694 < 2.2e-16 ***
## PF  7.7307e-01  9.7319e-02  7.9437 9.698e-12 ***
## LF -3.7974e+06  6.1377e+05 -6.1869 2.375e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    5.0776e+13
## Residual Sum of Squares: 3.5865e+12
## R-Squared:      0.92937
## Adj. R-Squared: 0.92239
## F-statistic: 355.254 on 3 and 81 DF, p-value: < 2.22e-16

Modelo 3. Efectos Aleatorios (Within) Método Walhus

walhus <- plm(C ~ Q + PF + LF, data= df_panel, model= "random", random.method =  "walhus")
summary(walhus)

## Oneway (individual) effect Random Effect Model 
##    (Wallace-Hussain's transformation)
## 
## Call:
## plm(formula = C ~ Q + PF + LF, data = df_panel, model = "random", 
##     random.method = "walhus")
## 
## Balanced Panel: n = 6, T = 15, N = 90
## 
## Effects:
##                     var   std.dev share
## idiosyncratic 7.339e+10 2.709e+05 0.969
## individual    2.363e+09 4.861e+04 0.031
## theta: 0.1788
## 
## Residuals:
##    Min. 1st Qu.  Median 3rd Qu.    Max. 
## -524180 -243611   39332  199517  824905 
## 
## Coefficients:
##                Estimate  Std. Error z-value  Pr(>|z|)    
## (Intercept)  1.1267e+06  3.6994e+05  3.0455  0.002323 ** 
## Q            2.0647e+06  7.1927e+04 28.7051 < 2.2e-16 ***
## PF           1.2075e+00  1.0358e-01 11.6578 < 2.2e-16 ***
## LF          -3.0314e+06  7.1431e+05 -4.2438 2.198e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    1.0182e+14
## Residual Sum of Squares: 6.5784e+12
## R-Squared:      0.93539
## Adj. R-Squared: 0.93314
## Chisq: 1245.09 on 3 DF, p-value: < 2.22e-16

Modelo 3. Efectos Aleatorios (Within) Método Amemiya

amemiya <- plm(C ~ Q + PF + LF, data= df_panel, model= "random", random.method =  "amemiya")
summary(amemiya)

## Oneway (individual) effect Random Effect Model 
##    (Amemiya's transformation)
## 
## Call:
## plm(formula = C ~ Q + PF + LF, data = df_panel, model = "random", 
##     random.method = "amemiya")
## 
## Balanced Panel: n = 6, T = 15, N = 90
## 
## Effects:
##                     var   std.dev share
## idiosyncratic 4.270e+10 2.066e+05 0.084
## individual    4.640e+11 6.812e+05 0.916
## theta: 0.9219
## 
## Residuals:
##    Min. 1st Qu.  Median 3rd Qu.    Max. 
## -603585 -144415   22641  158005  485417 
## 
## Coefficients:
##                Estimate  Std. Error z-value  Pr(>|z|)    
## (Intercept)  1.0746e+06  4.2105e+05  2.5522    0.0107 *  
## Q            3.2090e+06  1.6482e+05 19.4695 < 2.2e-16 ***
## PF           8.1014e-01  9.6147e-02  8.4260 < 2.2e-16 ***
## LF          -3.7168e+06  6.1330e+05 -6.0603 1.359e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    5.1238e+13
## Residual Sum of Squares: 3.8227e+12
## R-Squared:      0.92539
## Adj. R-Squared: 0.92279
## Chisq: 1066.71 on 3 DF, p-value: < 2.22e-16

Modelo 3. Efectos Aleatorios (Within) Método Nerlove

nerlove <- plm(C ~ Q + PF + LF, data= df_panel, model= "random", random.method =  "nerlove")
summary(nerlove)

## Oneway (individual) effect Random Effect Model 
##    (Nerlove's transformation)
## 
## Call:
## plm(formula = C ~ Q + PF + LF, data = df_panel, model = "random", 
##     random.method = "nerlove")
## 
## Balanced Panel: n = 6, T = 15, N = 90
## 
## Effects:
##                     var   std.dev share
## idiosyncratic 3.985e+10 1.996e+05 0.066
## individual    5.602e+11 7.485e+05 0.934
## theta: 0.9313
## 
## Residuals:
##    Min. 1st Qu.  Median 3rd Qu.    Max. 
## -601947 -145039   18713  154903  483623 
## 
## Coefficients:
##                Estimate  Std. Error z-value  Pr(>|z|)    
## (Intercept)  1.0752e+06  4.4535e+05  2.4142   0.01577 *  
## Q            3.2323e+06  1.6521e+05 19.5652 < 2.2e-16 ***
## PF           8.0229e-01  9.5804e-02  8.3743 < 2.2e-16 ***
## LF          -3.7338e+06  6.0963e+05 -6.1247 9.084e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Total Sum of Squares:    5.1133e+13
## Residual Sum of Squares: 3.7726e+12
## R-Squared:      0.92622
## Adj. R-Squared: 0.92365
## Chisq: 1079.63 on 3 DF, p-value: < 2.22e-16

Comparando las R^2 ajustadas de los 3 modelos, el mejor método en el modelo de ejectos aleatorios de Walhus

Efectos Fijos vs. Efectos Aleatorios

phtest(within, walhus)

## 
##  Hausman Test
## 
## data:  C ~ Q + PF + LF
## chisq = 65.039, df = 3, p-value = 4.919e-14
## alternative hypothesis: one model is inconsistent

Tema 2. Series de Tiempo

Generar el Modelo

df_a1 <- df[df$I == "1" , ]
ts_a1 <- ts(df_a1$C, start= 1970, frequency=1)


df_a2 <- df[df$I == "2" , ]
ts_a2 <- ts(df_a2$C, start= 1970, frequency=1)


df_a3 <- df[df$I == "3" , ]
ts_a3 <- ts(df_a3$C, start= 1970, frequency=1)



df_a4 <- df[df$I == "4" , ]
ts_a4 <- ts(df_a4$C, start= 1970, frequency=1)


df_a5 <- df[df$I == "5" , ]
ts_a5 <- ts(df_a5$C, start= 1970, frequency=1)



df_a6 <- df[df$I == "6" , ]
ts_a6 <- ts(df_a6$C, start= 1970, frequency=1)

Generar el modelo ARIMA

arima_a1 <- auto.arima(ts_a1)
summary(arima_a1)

## Series: ts_a1 
## ARIMA(0,1,0) with drift 
## 
## Coefficients:
##           drift
##       257691.43
## s.e.   44508.78
## 
## sigma^2 = 2.987e+10:  log likelihood = -188.19
## AIC=380.37   AICc=381.46   BIC=381.65
## 
## Training set error measures:
##                    ME   RMSE      MAE       MPE     MAPE     MASE      ACF1
## Training set 58.86321 160892 129527.1 -1.742419 5.395122 0.502644 0.4084903

arima_a2 <- auto.arima(ts_a2)
summary(arima_a2)

## Series: ts_a2 
## ARIMA(0,2,0) 
## 
## sigma^2 = 1.392e+10:  log likelihood = -170.26
## AIC=342.53   AICc=342.89   BIC=343.09
## 
## Training set error measures:
##                    ME     RMSE      MAE      MPE     MAPE      MASE      ACF1
## Training set 11689.89 109830.2 79466.33 1.387268 3.747652 0.3056315 0.3172172

arima_a3 <- auto.arima(ts_a3)
summary(arima_a3)

## Series: ts_a3 
## ARIMA(0,1,0) with drift 
## 
## Coefficients:
##          drift
##       63155.14
## s.e.  13344.11
## 
## sigma^2 = 2.685e+09:  log likelihood = -171.32
## AIC=346.64   AICc=347.74   BIC=347.92
## 
## Training set error measures:
##                    ME     RMSE      MAE        MPE     MAPE     MASE       ACF1
## Training set 14.87618 48235.79 38474.72 -0.9277567 5.324145 0.538349 0.09130379

arima_a4 <- auto.arima(ts_a4)
summary(arima_a4)

## Series: ts_a4 
## ARIMA(0,2,0) 
## 
## sigma^2 = 1.469e+09:  log likelihood = -155.65
## AIC=313.3   AICc=313.66   BIC=313.86
## 
## Training set error measures:
##                    ME    RMSE      MAE      MPE     MAPE      MASE      ACF1
## Training set 7232.074 35684.5 27472.98 1.761789 5.046326 0.2977402 0.1925091

arima_a5 <- auto.arima(ts_a5)
summary(arima_a5)

## Series: ts_a5 
## ARIMA(1,2,0) 
## 
## Coefficients:
##           ar1
##       -0.4543
## s.e.   0.2354
## 
## sigma^2 = 775697764:  log likelihood = -151.09
## AIC=306.18   AICc=307.38   BIC=307.31
## 
## Training set error measures:
##                   ME     RMSE      MAE      MPE     MAPE      MASE        ACF1
## Training set 3061.06 24911.01 14171.99 2.393894 4.771228 0.3823654 0.008627682

arima_a6 <- auto.arima(ts_a6)
summary(arima_a6)

## Series: ts_a6 
## ARIMA(1,2,0) 
## 
## Coefficients:
##          ar1
##       0.5824
## s.e.  0.2281
## 
## sigma^2 = 386182350:  log likelihood = -146.65
## AIC=297.3   AICc=298.5   BIC=298.43
## 
## Training set error measures:
##                    ME     RMSE      MAE      MPE     MAPE      MASE       ACF1
## Training set 6829.403 17576.86 10190.16 2.076518 3.550582 0.1516841 -0.2989742

Generar el Pronóstico

pronostico_a6 <- forecast(arima_a6, level=95, h=5)
pronostico_a6

##      Point Forecast   Lo 95   Hi 95
## 1985        1234478 1195962 1272994
## 1986        1471026 1364365 1577687
## 1987        1714311 1510670 1917953
## 1988        1961521 1635113 2287929
## 1989        2211016 1738872 2683160

plot(pronostico_a6, main= "Pronóstico de Costo Total (en miles)", xlab="Año", ylab="Dólares")

# Tema 3. Modelo de Ecuaciones Estructurales ## Estructurar el Modelo

modelo <- '
          # Regresiones
          C ~ Q + PF + LF + I + Y
          Q ~ PF + I
          PF ~ Y 
          LF ~ I
          # Variables Latentes
          # Varianzas y Covarianzas
          C ~~ C
          Q ~~ Q
          PF ~~ PF
          LF ~~ LF 
          # Intercepto
          '

Generar el Análisis Factorial Confirmatorio (CFA)

df_escalada <- df
df_escalada$I <- as.numeric(df_escalada$I)
df_escalada <- scale(df_escalada)
cfa <- cfa(modelo, df_escalada)
summary(cfa)

## lavaan 0.6-19 ended normally after 3 iterations
## 
##   Estimator                                         ML
##   Optimization method                           NLMINB
##   Number of model parameters                        13
## 
##   Number of observations                            90
## 
## Model Test User Model:
##                                                       
##   Test statistic                                63.804
##   Degrees of freedom                                 5
##   P-value (Chi-square)                           0.000
## 
## Parameter Estimates:
## 
##   Standard errors                             Standard
##   Information                                 Expected
##   Information saturated (h1) model          Structured
## 
## Regressions:
##                    Estimate  Std.Err  z-value  P(>|z|)
##   C ~                                                 
##     Q                 1.000    0.053   18.826    0.000
##     PF                0.194    0.065    3.000    0.003
##     LF               -0.154    0.025   -6.248    0.000
##     I                 0.105    0.052    1.999    0.046
##     Y                 0.140    0.063    2.211    0.027
##   Q ~                                                 
##     PF                0.239    0.046    5.213    0.000
##     I                -0.871    0.046  -18.985    0.000
##   PF ~                                                
##     Y                 0.931    0.038   24.233    0.000
##   LF ~                                                
##     I                -0.340    0.099   -3.429    0.001
## 
## Variances:
##                    Estimate  Std.Err  z-value  P(>|z|)
##    .C                 0.048    0.007    6.708    0.000
##    .Q                 0.187    0.028    6.708    0.000
##    .PF                0.131    0.020    6.708    0.000
##    .LF                0.875    0.130    6.708    0.000

lavaanPlot(cfa)