TALLER 2: PRONÓSTICO

Definiciones

La variación de precios de las acciones de Amazon en el mercado NASDAQ es una serie de tiempo con gran importancia económica y de comercio electrónico. La solidez y la volatilidad del mercado de acciones de Amazon ofrecen un escenario propicio para analizar y modelar patrones complejos. La base de datos proporciona una variedad de campos, entre ellos Date, Open, High, Low, Close y Volume, que posibilitan un análisis integral y una comprensión profunda de la dinámica del mercado financiero. La temporalidad otorgada por el campo “Date” resulta esencial para identificar patrones estacionales y tendencias durante los doce meses del estudio (de 07/03/2023 a 07/03/2024). En conjunto, esta elección ofrece una base sólida para la aplicación de diversos métodos de pronóstico, facilitando la comprensión del comportamiento futuro de las acciones de Amazon en NASDAQ.

Detallando los campos de la base de datos:

Date (Fecha): Este campo indica la fecha correspondiente a los datos registrados. La temporalidad es esencial en el análisis, ya que permite identificar patrones estacionales, tendencias a largo plazo y eventos específicos que pueden afectar a las acciones.

Open (Precio de apertura): Representa el precio de apertura de las acciones en el mercado al inicio del día de negociación. Este valor es fundamental para entender cómo se inicia la jornada bursátil y puede proporcionar insights sobre la dirección que el mercado podría tomar.

High (Precio máximo): Indica el precio máximo alcanzado durante el día de negociación. Este dato es valioso para evaluar la volatilidad diaria y las posibles tendencias al alsa o a la baja.

Low (Precio mínimo): Representa el precio más bajo alcanzado durante el día de negociación. Al igual que el precio máximo, el precio mínimo proporciona información sobre la volatilidad diaria y posibles patrones a la baja.

Close (Precio de cierre): Es el precio de cierre de las acciones al final del día de negociación. Este valor es esencial para evaluar el rendimiento diario de las acciones y puede utilizarse para identificar patrones de comportamiento a lo largo del tiempo.

Volume (Volumen de transacciones): Indica la cantidad total de acciones negociadas durante el día. El volumen es crucial para comprender la liquidez del mercado y la intensidad del interés de los inversionistas en un período específico. Cambios significativos en el volumen pueden señalar eventos importantes en el mercado.

En el gráfico se observa que:

1. Tendencia: A la alza

2. Ciclo: NO

3. Estacionalidad: NO

4. Irregularidad: SI

Pronosticos de “Open”

Modelos de suavizamiento

Modelo de Promedios Moviles

Forecast

Medida de Precision

##                       RMSE     MAE     MAPE
## Promedios Moviles 7.531029 4.53781 6.763405

Modelo de Suavización Exponencial

Forecast

Medida de Error y Precision

##                       RMSE      MAE     MAPE
## Promedios Moviles 7.459883 4.465301 6.713787

Modelo de Índices Estacionales

Medidas de Error y Precision

##                          RMSE       MAE      MAPE
## Estacional simple   27.609910 18.169616 25.965102
## Estacional parcial   7.167818  4.679931  9.807565
## Estacional completo  7.167558  4.681202  9.821548

Modelo de Índices Regresiones polinómicas (Tendencia lineal y al menos dos polinomios que recojan el ciclo). Incluya análisis de viabilidad.

## 
## Call:
## lm(formula = y ~ t)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -51.692 -18.856  -5.319  14.981  66.118 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -22.22539    3.64634  -6.095  7.4e-09 ***
## t             1.02824    0.03743  27.474  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 23.53 on 166 degrees of freedom
## Multiple R-squared:  0.8197, Adjusted R-squared:  0.8186 
## F-statistic: 754.8 on 1 and 166 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = y ~ t + t2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -60.408 -13.300  -2.941   8.355  67.934 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -6.2866448  5.2736760  -1.192  0.23494    
## t            0.4656980  0.1440776   3.232  0.00148 ** 
## t2           0.0033287  0.0008258   4.031 8.47e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 22.51 on 165 degrees of freedom
## Multiple R-squared:  0.8359, Adjusted R-squared:  0.8339 
## F-statistic: 420.2 on 2 and 165 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = y ~ t + t2 + t3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -52.889 -12.607  -1.956  10.334  53.008 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.926e+01  5.801e+00   5.044 1.20e-06 ***
## t           -2.021e+00  2.964e-01  -6.820 1.66e-10 ***
## t2           4.001e-02  4.069e-03   9.834  < 2e-16 ***
## t3          -1.447e-04  1.583e-05  -9.142 2.28e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.38 on 164 degrees of freedom
## Multiple R-squared:  0.8913, Adjusted R-squared:  0.8893 
## F-statistic: 448.2 on 3 and 164 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = y ~ t + t2 + t3 + t4)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -55.442  -4.693  -0.257   4.522  48.261 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  8.802e-01  6.424e+00   0.137 0.891188    
## t            1.250e+00  5.234e-01   2.388 0.018099 *  
## t2          -4.649e-02  1.254e-02  -3.707 0.000287 ***
## t3           6.499e-04  1.113e-04   5.837 2.79e-08 ***
## t4          -2.351e-06  3.269e-07  -7.192 2.19e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.06 on 163 degrees of freedom
## Multiple R-squared:  0.9175, Adjusted R-squared:  0.9155 
## F-statistic: 453.1 on 4 and 163 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = y ~ t + t2 + t3 + t4 + t5)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.657  -5.172   0.602   5.529  44.602 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -5.226e+00  7.820e+00  -0.668  0.50489   
## t            2.290e+00  9.253e-01   2.475  0.01434 * 
## t2          -8.908e-02  3.368e-02  -2.645  0.00897 **
## t3           1.319e-03  5.035e-04   2.619  0.00965 **
## t4          -6.798e-06  3.281e-06  -2.072  0.03984 * 
## t5           1.053e-08  7.727e-09   1.362  0.17500   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.02 on 162 degrees of freedom
## Multiple R-squared:  0.9184, Adjusted R-squared:  0.9159 
## F-statistic: 364.7 on 5 and 162 DF,  p-value: < 2.2e-16

Medidas de Error y Precision

##                RMSE      MAE      MAPE
## Lineal     23.38509 18.51797 0.6582849
## Cuadratico 22.31225 16.15739 0.4041579
## Grado3     18.15969 13.93914 0.4135110
## Grado4     15.82184 10.05132 0.1488743
## Grado5     15.73199 10.61152 0.1976600

##                RMSE      MAE     MAPE
## Lineal     23.38509 18.51797 65.82849
## Cuadratico 22.31225 16.15739 40.41579
## Grado3     18.15969 13.93914 41.35110
## Grado4     15.82184 10.05132 14.88743
## Grado5     15.73199 10.61152 19.76600

Forecast

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       151.5475 151.5475 151.5475 151.5475 151.5475
## 170       152.5757 152.5757 152.5757 152.5757 152.5757
## 171       153.6040 153.6040 153.6040 153.6040 153.6040
## 172       154.6322 154.6322 154.6322 154.6322 154.6322
## 173       155.6605 155.6605 155.6605 155.6605 155.6605
## 174       156.6887 156.6887 156.6887 156.6887 156.6887
## 175       157.7169 157.7169 157.7169 157.7169 157.7169
## 176       158.7452 158.7452 158.7452 158.7452 158.7452
## 177       159.7734 159.7734 159.7734 159.7734 159.7734
## 178       160.8017 160.8017 160.8017 160.8017 160.8017

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       167.4796 167.4710 167.4882 167.4664 167.4927
## 170       169.0604 169.0412 169.0796 169.0310 169.0897
## 171       170.6412 170.6090 170.6733 170.5920 170.6903
## 172       172.2220 172.1749 172.2690 172.1500 172.2939
## 173       173.8028 173.7391 173.8665 173.7054 173.9002
## 174       175.3836 175.3016 175.4655 175.2583 175.5089
## 175       176.9644 176.8628 177.0660 176.8090 177.1198
## 176       178.5452 178.4225 178.6678 178.3576 178.7328
## 177       180.1260 179.9810 180.2709 179.9042 180.3477
## 178       181.7068 181.5383 181.8753 181.4491 181.9645

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       132.0057 131.9513 132.0601 131.9225 132.0889
## 170       131.2084 131.0869 131.3299 131.0225 131.3943
## 171       130.4111 130.2077 130.6144 130.1001 130.7221
## 172       129.6137 129.3161 129.9114 129.1585 130.0690
## 173       128.8164 128.4134 129.2194 128.2000 129.4328
## 174       128.0190 127.5007 128.5374 127.2262 128.8119
## 175       127.2217 126.5787 127.8647 126.2383 128.2051
## 176       126.4244 125.6482 127.2005 125.2373 127.6114
## 177       125.6270 124.7096 126.5444 124.2240 127.0301
## 178       124.8297 123.7634 125.8960 123.1990 126.4604

##     Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## 169      103.79444 103.69251 103.89638 103.63855 103.95034
## 170       99.97986  99.75195 100.20777  99.63130 100.32841
## 171       96.16527  95.78392  96.54663  95.58204  96.74851
## 172       92.35069  91.79244  92.90894  91.49692  93.20446
## 173       88.53611  87.78023  89.29198  87.38010  89.69211
## 174       84.72152  83.74925  85.69379  83.23456  86.20848
## 175       80.90694  79.70098  82.11289  79.06259  82.75128
## 176       77.09235  75.63662  78.54808  74.86600  79.31870
## 177       73.27777  71.55713  74.99840  70.64629  75.90925
## 178       69.46318  67.46334  71.46303  66.40468  72.52168

##     Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## 169      109.81937 109.72703 109.91171 109.67815 109.96059
## 170      106.92228 106.71583 107.12874 106.60653 107.23803
## 171      104.02520 103.67973 104.37067 103.49686 104.55355
## 172      101.12812 100.62241 101.63383 100.35470 101.90153
## 173       98.23104  97.54630  98.91577  97.18383  99.27824
## 174       95.33395  94.45319  96.21472  93.98694  96.68097
## 175       92.43687  91.34442  93.52932  90.76610  94.10763
## 176       89.53979  88.22106  90.85851  87.52297  91.55660
## 177       86.64270  85.08401  88.20140  84.25888  89.02652
## 178       83.74562  81.93399  85.55725  80.97497  86.51627

Análisis de correlación y ARIMA

Prueba de estacionariedad

## 
##  Augmented Dickey-Fuller Test
## 
## data:  y
## Dickey-Fuller = -2.5333, Lag order = 5, p-value = 0.3537
## alternative hypothesis: stationary

## Warning in adf.test(diferencia): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diferencia
## Dickey-Fuller = -4.6909, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

## Warning in adf.test(variacion): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  variacion
## Dickey-Fuller = -5.4617, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

Funciones de autocorrelación

## Series: y 
## ARIMA(0,1,0) with drift 
## 
## Coefficients:
##        drift
##       0.8978
## s.e.  0.5804
## 
## sigma^2 = 56.59:  log likelihood = -573.45
## AIC=1150.9   AICc=1150.98   BIC=1157.14

## Series: variacion 
## ARIMA(0,0,0)(1,0,0)[12] with non-zero mean 
## 
## Coefficients:
##          sar1    mean
##       -0.1135  0.0232
## s.e.   0.0796  0.0060
## 
## sigma^2 = 0.007335:  log likelihood = 174.38
## AIC=-342.75   AICc=-342.61   BIC=-333.4

Y ARIMA(0,1,0)[12]

Variacion ARIMA(0,0,0)(1,0,0))[12]

## 
## Call:
## arima(x = y, order = c(0, 1, 0), seasonal = list(order = c(2, 0, 0)), include.mean = FALSE)
## 
## Coefficients:
##          sar1    sar2
##       -0.0734  0.0942
## s.e.   0.0776  0.0955
## 
## sigma^2 estimated as 56.27:  log likelihood = -573.63,  aic = 1151.26
## 
## Training set error measures:
##                     ME     RMSE      MAE      MPE     MAPE      MASE      ACF1
## Training set 0.8539053 7.478998 4.496187 1.560806 6.569043 0.9674197 -0.120093

## 
## z test of coefficients:
## 
##       Estimate Std. Error z value Pr(>|z|)
## sar1 -0.073395   0.077600 -0.9458   0.3442
## sar2  0.094206   0.095461  0.9868   0.3237

## 
## Call:
## arima(x = variacion, order = c(0, 0, 0), seasonal = list(order = c(1, 0, 0)), 
##     include.mean = TRUE)
## 
## Coefficients:
##          sar1  intercept
##       -0.1135     0.0232
## s.e.   0.0796     0.0060
## 
## sigma^2 estimated as 0.007247:  log likelihood = 174.38,  aic = -344.75
## 
## Training set error measures:
##                        ME       RMSE        MAE          MPE     MAPE
## Training set 0.0001043701 0.08512962 0.06654832 0.0006766693 0.274339
##                    MASE        ACF1
## Training set 0.01463976 -0.07403497

## 
## z test of coefficients:
## 
##             Estimate Std. Error z value  Pr(>|z|)    
## sar1      -0.1135432  0.0795811 -1.4268 0.1536488    
## intercept  0.0231699  0.0059616  3.8865 0.0001017 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

## 
##  Jarque Bera Test
## 
## data:  e1
## X-squared = 401.51, df = 2, p-value < 2.2e-16

## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  e1
## D = 0.30722, p-value = 3.375e-14
## alternative hypothesis: two-sided

##       LAG ACF          PACF        Q        Prob>Q    
##  [1,] 1   -0.120093    -0.120093   2.422953 0.1195693 
##  [2,] 2   0.09045526   0.07714554  3.797555 0.1497516 
##  [3,] 3   -0.05362659  -0.03504799 4.280691 0.2327063 
##  [4,] 4   0.191453     0.1786584   10.43861 0.03365326
##  [5,] 5   -0.002447229 0.04556238  10.43961 0.06369523
##  [6,] 6   -0.03844743  -0.06608772 10.68795 0.09851287
##  [7,] 7   0.08064446   0.08406986  11.78054 0.1080107 
##  [8,] 8   0.069691     0.06556443  12.59649 0.1265079 
##  [9,] 9   -0.10868     -0.1272507  14.5808  0.1031132 
## [10,] 10  0.052432     0.0487763   15.04265 0.1305146 
## [11,] 11  0.03742829   0.04566283  15.278   0.1701205 
## [12,] 12  -0.01984782  -0.06826786 15.34418 0.2231517 
## [13,] 13  -0.1085887   -0.07533005 17.32515 0.1848631 
## [14,] 14  -0.1213529   -0.1546213  19.79921 0.1366001 
## [15,] 15  0.1298583    0.08144402  22.63222 0.0922668 
## [16,] 16  -0.02867146  0.04820018  22.77033 0.1200135 
## [17,] 17  -0.02432262  -0.01485338 22.86971 0.1535405 
## [18,] 18  -0.1894344   -0.1815113  28.89846 0.0496343 
## [19,] 19  0.03168712   -0.02949049 29.06714 0.06493636
## [20,] 20  -0.1298071   -0.1066101  31.89792 0.04439825
## [21,] 21  -0.07886025  -0.09025656 32.9427  0.04685634
## [22,] 22  -0.1857199   -0.169655   38.73734 0.01512156
## [23,] 23  0.07324599   0.01908887  39.63865 0.01686532
## [24,] 24  0.002716485  0.1229511   39.63989 0.02336987
## [25,] 25  -0.03392937  0.01626699  39.83329 0.03033485
## [26,] 26  0.01330276   0.0406361   39.86302 0.04023576
## [27,] 27  0.007492705  -0.03193983 39.87246 0.05266144
## [28,] 28  -0.07410606  -0.0749309  40.79506 0.05607284
## [29,] 29  -0.0313958   0.02309138  40.96066 0.06941261
## [30,] 30  -0.04531833  -0.08095605 41.30569 0.08195102
## [31,] 31  0.08781893   -0.01046226 42.60133 0.08016259
## [32,] 32  -0.05353967  -0.01302918 43.0829  0.09129018
## [33,] 33  0.09088014   0.1258474   44.47045 0.08768107
## [34,] 34  -0.04505514  -0.08070367 44.81148 0.1016323 
## [35,] 35  0.1035602    0.05367885  46.61324 0.09068508
## [36,] 36  0.03760238   0.05583877  46.85078 0.106401

##       LAG ACF        PACF        Q        Prob>Q      
##  [1,] 1   0.1091626  0.1091626   2.001966 0.1570954   
##  [2,] 2   0.1868105  0.1770033   7.864856 0.01959603  
##  [3,] 3   0.421707   0.4028743   37.74144 3.206135e-08
##  [4,] 4   0.1103598  0.04023814  39.78755 4.789159e-08
##  [5,] 5   0.1285179  -0.01029772 42.56239 4.531999e-08
##  [6,] 6   0.0889221  -0.1291428  43.89079 7.770015e-08
##  [7,] 7   0.1083686  0.03911166  45.86374 9.292231e-08
##  [8,] 8   0.1118617  0.07889212  47.96593 1.002881e-07
##  [9,] 9   0.1289675  0.1481862   50.76021 7.748387e-08
## [10,] 10  0.09680475 0.02424318  52.33456 9.893796e-08
## [11,] 11  0.09397949 -0.01699665 53.81836 1.273915e-07
## [12,] 12  0.1415919  0.01332037  57.18647 7.314498e-08

Pronostico

## $pred
##           Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
## 2024 156.7630 157.2198 153.0924 151.9235 149.6759 152.1764 150.8986 150.6225
##           Sep      Oct      Nov      Dec
## 2024 149.2288 147.6857 146.1929 147.4832
## 
## $se
##            Jan       Feb       Mar       Apr       May       Jun       Jul
## 2024  7.501357 10.608521 12.992731 15.002714 16.773544 18.374497 19.846725
##            Aug       Sep       Oct       Nov       Dec
## 2024 21.217041 22.504070 23.721373 24.879186 25.985462

## $pred
##              Jan         Feb         Mar         Apr         May         Jun
## 2024 0.035390921 0.015603970 0.022859481 0.008771944 0.016270578 0.023431262
##              Jul         Aug         Sep         Oct         Nov         Dec
## 2024 0.020776080 0.035717230 0.019841659 0.015595742 0.021492306 0.022556413
## 
## $se
##             Jan        Feb        Mar        Apr        May        Jun
## 2024 0.08512962 0.08512962 0.08512962 0.08512962 0.08512962 0.08512962
##             Jul        Aug        Sep        Oct        Nov        Dec
## 2024 0.08512962 0.08512962 0.08512962 0.08512962 0.08512962 0.08512962

## [1] 161.3864

## [1] 163.9047

Pronosticos de “Close”

Modelos de suavizamiento

Modelo de Promedios Moviles

Forecast

Medidas de Error y Precision

##                       RMSE      MAE     MAPE
## Promedios Moviles 7.634659 4.619943 6.766519

Modelo de Suavización Exponencial

Forecast

Medidas de Error y Precision

##                       RMSE      MAE     MAPE
## Promedios Moviles 7.558425 4.479443 6.627999

Modelo de Índices Estacionales

Medidas de Error y Precision

##                         RMSE       MAE     MAPE
## Estacional simple   29.08732 19.402450 27.47779
## Estacional parcial  10.15539  6.447852 12.05354
## Estacional completo 10.15467  6.448947 12.06704

Modelo de Regresiones polinómicas (Tendencia lineal y al menos dos polinomios que recojan el ciclo). Incluya análisis de viabilidad.

## 
## Call:
## lm(formula = x ~ t)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -53.551 -18.592  -4.912  15.585  64.014 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -22.04258    3.61153  -6.103  7.1e-09 ***
## t             1.03632    0.03707  27.957  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 23.3 on 166 degrees of freedom
## Multiple R-squared:  0.8248, Adjusted R-squared:  0.8238 
## F-statistic: 781.6 on 1 and 166 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = x ~ t + t2)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -61.735 -13.179  -2.739   8.680  66.093 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -6.2318064  5.2225512  -1.193 0.234485    
## t            0.4782960  0.1426809   3.352 0.000994 ***
## t2           0.0033019  0.0008178   4.038 8.25e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 22.3 on 165 degrees of freedom
## Multiple R-squared:  0.8406, Adjusted R-squared:  0.8386 
## F-statistic:   435 on 2 and 165 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = x ~ t + t2 + t3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -55.922 -11.769  -1.708   9.803  51.496 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.818e+01  5.810e+00   4.850 2.85e-06 ***
## t           -1.929e+00  2.968e-01  -6.500 9.24e-10 ***
## t2           3.881e-02  4.075e-03   9.525  < 2e-16 ***
## t3          -1.401e-04  1.585e-05  -8.837 1.46e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.41 on 164 degrees of freedom
## Multiple R-squared:  0.892,  Adjusted R-squared:   0.89 
## F-statistic: 451.5 on 3 and 164 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = x ~ t + t2 + t3 + t4)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -59.184  -5.212  -0.416   4.388  63.172 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.744e+00  6.695e+00   0.559  0.57674    
## t            8.870e-01  5.455e-01   1.626  0.10583    
## t2          -3.566e-02  1.307e-02  -2.729  0.00706 ** 
## t3           5.441e-04  1.160e-04   4.689 5.78e-06 ***
## t4          -2.024e-06  3.406e-07  -5.942 1.65e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.74 on 163 degrees of freedom
## Multiple R-squared:  0.9112, Adjusted R-squared:  0.909 
## F-statistic: 418.3 on 4 and 163 DF,  p-value: < 2.2e-16

## 
## Call:
## lm(formula = x ~ t + t2 + t3 + t4 + t5)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -55.550  -5.963   0.682   5.571  53.707 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -7.572e+00  8.047e+00  -0.941 0.348147    
## t            2.816e+00  9.522e-01   2.957 0.003571 ** 
## t2          -1.146e-01  3.466e-02  -3.307 0.001163 ** 
## t3           1.784e-03  5.182e-04   3.443 0.000733 ***
## t4          -1.027e-05  3.376e-06  -3.041 0.002755 ** 
## t5           1.951e-08  7.952e-09   2.453 0.015219 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 16.49 on 162 degrees of freedom
## Multiple R-squared:  0.9144, Adjusted R-squared:  0.9118 
## F-statistic: 346.1 on 5 and 162 DF,  p-value: < 2.2e-16

Medidas de Error del Pronóstico

##                RMSE      MAE      MAPE
## Lineal     23.16188 18.46320 0.6428859
## Cuadratico 22.09595 16.08703 0.3981964
## Grado3     18.18644 13.85413 0.3884788
## Grado4     16.48817 10.22447 0.1400106
## Grado5     16.19019 11.07866 0.2201290

##                RMSE      MAE     MAPE
## Lineal     23.16188 18.46320 64.28859
## Cuadratico 22.09595 16.08703 39.81964
## Grado3     18.18644 13.85413 38.84788
## Grado4     16.48817 10.22447 14.00106
## Grado5     16.19019 11.07866 22.01290

Forecast

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       153.0961 153.0961 153.0961 153.0961 153.0961
## 170       154.1324 154.1324 154.1324 154.1324 154.1324
## 171       155.1687 155.1687 155.1687 155.1687 155.1687
## 172       156.2050 156.2050 156.2050 156.2050 156.2050
## 173       157.2414 157.2414 157.2414 157.2414 157.2414
## 174       158.2777 158.2777 158.2777 158.2777 158.2777
## 175       159.3140 159.3140 159.3140 159.3140 159.3140
## 176       160.3503 160.3503 160.3503 160.3503 160.3503
## 177       161.3866 161.3866 161.3866 161.3866 161.3866
## 178       162.4230 162.4230 162.4230 162.4230 162.4230

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       168.9002 168.8917 168.9087 168.8872 168.9133
## 170       170.4847 170.4656 170.5037 170.4556 170.5138
## 171       172.0691 172.0373 172.1010 172.0204 172.1178
## 172       173.6536 173.6069 173.7002 173.5822 173.7249
## 173       175.2380 175.1749 175.3012 175.1414 175.3346
## 174       176.8225 176.7412 176.9037 176.6982 176.9467
## 175       178.4069 178.3062 178.5076 178.2528 178.5610
## 176       179.9913 179.8697 180.1130 179.8054 180.1773
## 177       181.5758 181.4320 181.7195 181.3560 181.7956
## 178       183.1602 182.9932 183.3273 182.9047 183.4157

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       134.5609 134.5077 134.6140 134.4796 134.6421
## 170       133.8432 133.7244 133.9620 133.6615 134.0249
## 171       133.1256 132.9268 133.3244 132.8215 133.4296
## 172       132.4080 132.1169 132.6990 131.9629 132.8530
## 173       131.6903 131.2963 132.0844 131.0877 132.2930
## 174       130.9727 130.4658 131.4796 130.1975 131.7479
## 175       130.2550 129.6264 130.8837 129.2935 131.2166
## 176       129.5374 128.7785 130.2963 128.3768 130.6981
## 177       128.8198 127.9228 129.7168 127.4479 130.1916
## 178       128.1021 127.0596 129.1447 126.5077 129.6966

##     Point Forecast     Lo 80     Hi 80     Lo 95     Hi 95
## 169      110.27312 110.18414 110.36210 110.13704 110.40921
## 170      106.95859 106.75996 107.15723 106.65481 107.26238
## 171      103.64406 103.31191 103.97621 103.13609 104.15204
## 172      100.32953  99.84350 100.81556  99.58622 101.07284
## 173       97.01500  96.35707  97.67292  96.00879  98.02121
## 174       93.70047  92.85433  94.54661  92.40641  94.99453
## 175       90.38594  89.33656  91.43531  88.78105  91.99082
## 176       87.07140  85.80480  88.33801  85.13429  89.00851
## 177       83.75687  82.25989  85.25386  81.46743  86.04631
## 178       80.44234  78.70254  82.18214  77.78155  83.10313

##     Point Forecast    Lo 80    Hi 80    Lo 95    Hi 95
## 169       121.4439 121.3507 121.5370 121.3014 121.5863
## 170       119.8287 119.6353 120.0220 119.5330 120.1244
## 171       118.2135 117.8942 118.5329 117.7251 118.7019
## 172       116.5983 116.1327 117.0640 115.8862 117.3105
## 173       114.9831 114.3537 115.6126 114.0204 115.9459
## 174       113.3680 112.5589 114.1771 112.1305 114.6054
## 175       111.7528 110.7496 112.7560 110.2186 113.2870
## 176       110.1376 108.9269 111.3483 108.2860 111.9892
## 177       108.5224 107.0917 109.9532 106.3342 110.7106
## 178       106.9073 105.2445 108.5700 104.3642 109.4503

Análisis de correlación y ARIMA

Prueba de estacionariedad

## 
##  Augmented Dickey-Fuller Test
## 
## data:  x
## Dickey-Fuller = -2.5408, Lag order = 5, p-value = 0.3506
## alternative hypothesis: stationary

## Warning in adf.test(diferencia): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diferencia
## Dickey-Fuller = -4.1806, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

## Warning in adf.test(variacion): p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  variacion
## Dickey-Fuller = -5.2076, Lag order = 5, p-value = 0.01
## alternative hypothesis: stationary

Funciones de autocorrelación

## Series: x 
## ARIMA(0,1,0) with drift 
## 
## Coefficients:
##        drift
##       1.0178
## s.e.  0.5873
## 
## sigma^2 = 57.95:  log likelihood = -575.43
## AIC=1154.87   AICc=1154.94   BIC=1161.1

## Series: variacion 
## ARIMA(0,0,1)(1,0,0)[12] with non-zero mean 
## 
## Coefficients:
##           ma1     sar1    mean
##       -0.0957  -0.1210  0.0231
## s.e.   0.0755   0.0796  0.0054
## 
## sigma^2 = 0.007547:  log likelihood = 172.49
## AIC=-336.98   AICc=-336.73   BIC=-324.5

Y ARIMA(0,1,0)[12]

Variacion ARIMA(0,0,0)(1,0,0))[12]

## 
## Call:
## arima(x = x, order = c(0, 1, 0), seasonal = list(order = c(2, 0, 0)), include.mean = FALSE)
## 
## Coefficients:
##          sar1    sar2
##       -0.0474  0.0810
## s.e.   0.0789  0.0961
## 
## sigma^2 estimated as 58.19:  log likelihood = -576.38,  aic = 1156.76
## 
## Training set error measures:
##                     ME     RMSE      MAE      MPE     MAPE      MASE
## Training set 0.9656816 7.605551 4.592177 1.522331 6.712723 0.9880734
##                     ACF1
## Training set -0.09782822

## 
## z test of coefficients:
## 
##       Estimate Std. Error z value Pr(>|z|)
## sar1 -0.047416   0.078937 -0.6007   0.5481
## sar2  0.080974   0.096110  0.8425   0.3995

## 
## Call:
## arima(x = variacion, order = c(0, 0, 0), seasonal = list(order = c(1, 0, 0)), 
##     include.mean = TRUE)
## 
## Coefficients:
##          sar1  intercept
##       -0.1106     0.0231
## s.e.   0.0790     0.0061
## 
## sigma^2 estimated as 0.007484:  log likelihood = 171.69,  aic = -339.39
## 
## Training set error measures:
##                        ME       RMSE        MAE        MPE      MAPE       MASE
## Training set 3.592011e-05 0.08651062 0.06739527 0.01753155 0.2754462 0.01482607
##                    ACF1
## Training set -0.0989939

## 
## z test of coefficients:
## 
##             Estimate Std. Error z value  Pr(>|z|)    
## sar1      -0.1106227  0.0789694 -1.4008 0.1612647    
## intercept  0.0231331  0.0060742  3.8084 0.0001399 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

## 
##  Jarque Bera Test
## 
## data:  e1
## X-squared = 276.99, df = 2, p-value < 2.2e-16

## 
##  Asymptotic one-sample Kolmogorov-Smirnov test
## 
## data:  e1
## D = 0.30908, p-value = 2.298e-14
## alternative hypothesis: two-sided

##       LAG ACF          PACF         Q        Prob>Q      
##  [1,] 1   -0.09782822  -0.09782822  1.60782  0.2047985   
##  [2,] 2   0.06714566   0.05813164   2.365255 0.3064724   
##  [3,] 3   -0.03146753  -0.01985013  2.53161  0.4696056   
##  [4,] 4   0.2121678    0.2063277    10.09416 0.03887109  
##  [5,] 5   -0.01454395  0.02705103   10.1297  0.07164267  
##  [6,] 6   -0.01568631  -0.04020743  10.17103 0.1176317   
##  [7,] 7   0.04724294   0.05115198   10.54599 0.1596851   
##  [8,] 8   0.09030008   0.06223372   11.91588 0.1549966   
##  [9,] 9   -0.06609216  -0.06529125  12.64974 0.1791063   
## [10,] 10  0.04533951   0.04166983   12.99509 0.2239466   
## [11,] 11  0.04293488   0.04250746   13.30478 0.2738727   
## [12,] 12  -0.01441575  -0.05032234  13.33969 0.3448414   
## [13,] 13  -0.09258242  -0.07911965  14.77971 0.3213065   
## [14,] 14  -0.1608306   -0.1991738   19.12527 0.1602175   
## [15,] 15  0.1207465    0.07439507   21.57467 0.1194512   
## [16,] 16  -0.03262203  0.02114535   21.75345 0.1513321   
## [17,] 17  -0.05606137  -0.03968578  22.28146 0.1741358   
## [18,] 18  -0.2425069   -0.2161618   32.16147 0.02103875  
## [19,] 19  0.1321248    0.07151344   35.09424 0.01360624  
## [20,] 20  -0.1376276   -0.1009409   38.27639 0.008191558 
## [21,] 21  -0.1065937   -0.1161985   40.18524 0.0070603   
## [22,] 22  -0.2816848   -0.240117    53.51542 0.0001923974
## [23,] 23  0.1242781    0.04549858   56.11019 0.0001366093
## [24,] 24  -0.004566349 0.1319401    56.1137  0.0002211399
## [25,] 25  -0.07941663  -0.02821375  57.17327 0.0002533883
## [26,] 26  -0.04057913  -0.007035011 57.44991 0.0003656489
## [27,] 27  0.06913049   0.01457088   58.25279 0.0004448573
## [28,] 28  -0.07663442  -0.03250783  59.23942 0.0005089198
## [29,] 29  -0.06351981  -0.01386108  59.91726 0.0006324622
## [30,] 30  -0.03069475  -0.05702953  60.07555 0.0009014831
## [31,] 31  0.08738386   0.01100403   61.35839 0.0009308754
## [32,] 32  -0.04450052  0.01078253   61.69108 0.001240703 
## [33,] 33  0.06221671   0.1608678    62.34139 0.00150415  
## [34,] 34  0.008419007  -0.1043724   62.3533  0.002135613 
## [35,] 35  0.09143919   -0.01644719  63.75797 0.002102946 
## [36,] 36  0.001543542  -0.03164463  63.75837 0.002940266

##       LAG ACF        PACF         Q        Prob>Q      
##  [1,] 1   0.0934871  0.0934871    1.468293 0.2256152   
##  [2,] 2   0.1736912  0.1664058    6.536626 0.0380706   
##  [3,] 3   0.4462138  0.432783     39.98656 1.072523e-08
##  [4,] 4   0.09382764 0.03534397   41.46557 2.152692e-08
##  [5,] 5   0.1467193  0.01540374   45.08203 1.39622e-08 
##  [6,] 6   0.1059687  -0.1325918   46.96856 1.898346e-08
##  [7,] 7   0.08643278 0.01295307   48.22362 3.219572e-08
##  [8,] 8   0.1113217  0.05084256   50.30557 3.56974e-08 
##  [9,] 9   0.173718   0.2104301    55.37546 1.032876e-08
## [10,] 10  0.09638582 0.05343487   56.93622 1.370205e-08
## [11,] 11  0.09528879 -0.009076707 58.46165 1.787822e-08
## [12,] 12  0.1711324  -0.01213572  63.38175 5.421102e-09

Pronostico

## $pred
##           Jan      Feb      Mar      Apr      May      Jun      Jul      Aug
## 2024 177.0944 173.8570 172.8098 171.2125 173.3823 172.5146 171.9160 170.7778
##           Sep      Oct      Nov      Dec
## 2024 169.6837 168.3909 169.7853 168.0424
## 
## $se
##            Jan       Feb       Mar       Apr       May       Jun       Jul
## 2024  7.628288 10.788029 13.212583 15.256576 17.057371 18.685414 20.182553
##            Aug       Sep       Oct       Nov       Dec
## 2024 21.576057 22.884865 24.122765 25.300170 26.425165

## $pred
##             Jan        Feb        Mar        Apr        May        Jun
## 2024 0.01505600 0.02337878 0.00981993 0.01671974 0.02287479 0.02210896
##             Jul        Aug        Sep        Oct        Nov        Dec
## 2024 0.03442106 0.02049690 0.01488668 0.02126236 0.02331862 0.01032469
## 
## $se
##             Jan        Feb        Mar        Apr        May        Jun
## 2024 0.08651062 0.08651062 0.08651062 0.08651062 0.08651062 0.08651062
##             Jul        Aug        Sep        Oct        Nov        Dec
## 2024 0.08651062 0.08651062 0.08651062 0.08651062 0.08651062 0.08651062

## [1] 179.4213

## [1] 183.616

Mejor Pronostico y Analisis

##                              RMSE       MAE      MAPE
## Promedios moviles        7.531029  4.537810  6.763405
## Suavizacion exponencial  7.459883  4.465301  6.713787
## Estacional simple       27.609910 18.169616 25.965102
## Estacional parcial       7.167818  4.679931  9.807565
## Estacional completo      7.167558  4.681202  9.821548
## Grado 4 Polinomial      15.821837 10.051325 14.887430

##                              RMSE       MAE      MAPE
## Promedios moviles        7.634659  4.619943  6.766519
## Suavizacion exponencial  7.558425  4.479443  6.627999
## Estacional simple       29.087324 19.402450 27.477792
## Estacional parcial      10.155390  6.447852 12.053542
## Estacional completo     10.154671  6.448947 12.067041
## Grado 4 Polinomial      16.488171 10.224475 14.001064

Los resultados obtenidos de la tabla revelan que, para la variable “OPEN”, el método de “Estacional Parcial” exhibe un rendimiento destacado con un RMSE de 7.167818, MAE de 4.679931 y MAPE de 9.807565. Estos valores inferiores indican una mayor precisión en las predicciones. El enfoque de “Estacional Parcial” se destaca al considerar el componente estacional de los datos, lo que resulta beneficioso para capturar patrones cíclicos y variaciones a lo largo del tiempo. Por otro lado, para la variable “CLOSE”, la “Suavización Exponencial” emerge como la técnica más efectiva, evidenciada por su RMSE de 7.558425, MAE de 4.479443 y MAPE de 6.627999. Este método es especialmente apto para modelar series temporales con patrones de tendencia y variabilidad, proporcionando una respuesta rápida a cambios en los datos.

En conclusión, para llevar a cabo el pronóstico de los precios de apertura y cierre de las acciones de Amazon en NASDAQ, se sugiere emplear el método de “Estacional Parcial” para la variable “OPEN” y la “Suavización Exponencial” para la variable “CLOSE”. Estos métodos han demostrado un desempeño significativamente superior en comparación con enfoques estocásticos, ya que capturan de manera efectiva las componentes estacionales y las tendencias inherentes en las series temporales de precios. La elección de estos métodos se basa en su capacidad para modelar patrones específicos y variaciones estacionales, optimizando así la precisión de los pronósticos. En contraste, los métodos estocásticos, al depender de la aleatoriedad y no tener en cuenta las estructuras temporales subyacentes, resultan menos efectivos para este conjunto de datos, lo que respalda la preferencia por enfoques más especializados y deterministas.