Walmart
Erick Santillan Suarez
1 de diciembre de 2019
Walmart (WMT)
Walmart (WMT) Walmart (oficialmente, Wal-Mart, Inc.) es una corporación multinacional de tiendas de origen estadounidense, que opera cadenas de grandes almacenes de descuento y clubes de almacenes. Fue fundada por Sam Walton en 1962, incorporada el 31 de octubre de 1969, y ha cotizado en la Bolsa de Nueva York desde 1972. Su sede principal está ubicada en Bentonville, Arkansas.
Comportamiento del precio de cierre de Walmart: 01 de enero de 2013 al 22 de noviembre de 2019
En el siguiente grafico se presenta el comportamiento de Walmart a partir del 1 de enero de 2013 al 22 de noviembre de 2019. La tendencia al inicio de la serie (Enero 2013) es lateral hasta finales de 2014; en este periodo puede apreciarse volatilidad a la baja y despues a finales de 2015 a la alza. En el periodo comprendido en el 2016-2019 se es notable una tendencia a la alza. A inicios del 2018 Walmart presenta una fuerte subida en su precio a $109.55, siguiendo la tendencia a la alza se puede observar que el precio sigue subiendo con variaciones y bajas en el corto plazo, llegando a su maximo en las fechas de fin de año del 2019.
Rendimientos WMT: 01 de enero de 2013 al 22 de noviembre de 2019
Walmart presenta rendimientos sin gran varicacion pero principalmente positivos; tiene dos notables perdidas, la primera el 24 de agosoto del 2015 con -.03 en sus rendimientos, mientras la segunda perdida notable se da el 05 de febrero del 2018 con -.04.
En contraste tiene distintos rendimientos en forma positiva; 19 mayo del 2016 con redimientos de .09 y el 16 de noviembre del 2017 con .1 en rendimientos.
Figura 2. Rendimientos de Walmart (WMT)
Autocorrelacion de rendimientos de Walmart (WMT)
Walmart no presenta ningun tipo de autoccorelacion, esto es visible en los graficos continuos donde ningun valor sale de la banda de autocorrelacion.
Figura 3. Autocorrelacion de rendimientos de Walmart (WMT)
Prueba ARCH en precios de cierre de Walmart (WMT)
Los datos presentados en el precio de cierre de Walmart son homocedasticos; la prueba ARCH nos permite verificarlo ya que el p-value es mayor que 0.05, es decir, se puede tomar la hipotesis nula donde los residuales son homocedasticos.
Figura 3. Autocorrelacion de rendimientos de Walmart (WMT)
## Series: WMT_P
## ARIMA(0,1,1)
##
## Coefficients:
## ma1
## -0.0471
## s.e. 0.0246
##
## sigma^2 estimated as 0.8929: log likelihood=-2363.05
## AIC=4730.09 AICc=4730.1 BIC=4741.01##
## Box-Ljung test
##
## data: fit1$residuals^2
## X-squared = 23.624, df = 30, p-value = 0.7888Modelos ARCH y GARCH en rendimientos de Walmart (WMT)
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(0,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.0e+00 5.4227 0
## beta1 0.99899 1.4e-05 73961.4505 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.000000 0.90208 0.36701
## beta1 0.99899 0.001067 936.10086 0.00000
##
## LogLikelihood : 5310.892
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1198
## Bayes -6.1135
## Shibata -6.1198
## Hannan-Quinn -6.1174
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.341 0.1260
## Lag[2*(p+q)+(p+q)-1][2] 3.162 0.1268
## Lag[4*(p+q)+(p+q)-1][5] 3.989 0.2554
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.097 0.07841
## Lag[2*(p+q)+(p+q)-1][2] 3.139 0.12869
## Lag[4*(p+q)+(p+q)-1][5] 3.325 0.35092
## d.o.f=1
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[2] 0.08247 0.500 2.000 0.7740
## ARCH Lag[4] 0.25869 1.397 1.611 0.9428
## ARCH Lag[6] 0.45000 2.222 1.500 0.9774
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 291.0841
## Individual Statistics:
## omega 9.5013
## beta1 0.1341
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.61 0.749 1.07
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5106 0.609708
## Negative Sign Bias 3.0495 0.002327 ***
## Positive Sign Bias 1.7749 0.076085 *
## Joint Effect 15.0779 0.001751 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 213.0 8.694e-35
## 2 30 220.6 2.314e-31
## 3 40 230.1 6.165e-29
## 4 50 256.5 6.730e-30
##
##
## Elapsed time : 0.235878##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(0,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.0e+00 16.408 0
## beta1 0.50274 8.1e-05 6203.435 0
## beta2 0.49614 8.0e-05 6163.518 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.000000 10.954 0
## beta1 0.50274 0.000181 2782.418 0
## beta2 0.49614 0.000178 2784.524 0
##
## LogLikelihood : 5311.244
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1190
## Bayes -6.1096
## Shibata -6.1190
## Hannan-Quinn -6.1155
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.316 0.1281
## Lag[2*(p+q)+(p+q)-1][2] 3.140 0.1286
## Lag[4*(p+q)+(p+q)-1][5] 3.969 0.2580
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.089 0.07881
## Lag[2*(p+q)+(p+q)-1][5] 3.312 0.35295
## Lag[4*(p+q)+(p+q)-1][9] 3.613 0.65494
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.1129 0.500 2.000 0.7368
## ARCH Lag[5] 0.3179 1.440 1.667 0.9355
## ARCH Lag[7] 0.5317 2.315 1.543 0.9754
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 558.4174
## Individual Statistics:
## omega 17.7566
## beta1 0.1177
## beta2 0.1177
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5089 0.610885
## Negative Sign Bias 3.0512 0.002314 ***
## Positive Sign Bias 1.7707 0.076788 *
## Joint Effect 15.0777 0.001751 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 218.3 7.626e-36
## 2 30 226.0 2.120e-32
## 3 40 237.4 2.913e-30
## 4 50 264.8 2.227e-31
##
##
## Elapsed time : 0.152426##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(0,3)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.000000 5.4451e-01 0.58609
## beta1 0.34208 0.000027 1.2458e+04 0.00000
## beta2 0.33356 0.000118 2.8159e+03 0.00000
## beta3 0.32332 0.000133 2.4372e+03 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.000023 0.007353 0.99413
## beta1 0.34208 0.009751 35.080276 0.00000
## beta2 0.33356 0.004616 72.269226 0.00000
## beta3 0.32332 0.000937 344.951910 0.00000
##
## LogLikelihood : 5311.382
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1180
## Bayes -6.1054
## Shibata -6.1180
## Hannan-Quinn -6.1134
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.311 0.1285
## Lag[2*(p+q)+(p+q)-1][2] 3.137 0.1288
## Lag[4*(p+q)+(p+q)-1][5] 3.967 0.2582
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.093 0.07865
## Lag[2*(p+q)+(p+q)-1][8] 3.556 0.58405
## Lag[4*(p+q)+(p+q)-1][14] 4.329 0.84712
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.1822 0.500 2.000 0.6695
## ARCH Lag[6] 0.4750 1.461 1.711 0.8990
## ARCH Lag[8] 0.6045 2.368 1.583 0.9728
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: -209.339
## Individual Statistics:
## omega 29.0458
## beta1 0.1167
## beta2 0.1167
## beta3 0.1167
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5083 0.611320
## Negative Sign Bias 3.0524 0.002305 ***
## Positive Sign Bias 1.7701 0.076888 *
## Joint Effect 15.0841 0.001746 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 215.5 2.761e-35
## 2 30 224.9 3.453e-32
## 3 40 236.1 4.930e-30
## 4 50 262.1 6.792e-31
##
##
## Elapsed time : 0.1748919##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(0,4)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.000001 0.2605 0.79448
## beta1 0.26928 0.000208 1295.9232 0.00000
## beta2 0.25654 0.000232 1105.7570 0.00000
## beta3 0.24350 0.000242 1004.3246 0.00000
## beta4 0.22963 0.000261 878.6199 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.00000 0.000094 0.001889 0.99849
## beta1 0.26928 0.016085 16.741658 0.00000
## beta2 0.25654 0.011962 21.446832 0.00000
## beta3 0.24350 0.011994 20.302723 0.00000
## beta4 0.22963 0.009352 24.554374 0.00000
##
## LogLikelihood : 5311.497
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1170
## Bayes -6.1013
## Shibata -6.1170
## Hannan-Quinn -6.1112
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.293 0.1299
## Lag[2*(p+q)+(p+q)-1][2] 3.123 0.1300
## Lag[4*(p+q)+(p+q)-1][5] 3.954 0.2599
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.088 0.07889
## Lag[2*(p+q)+(p+q)-1][11] 3.942 0.74152
## Lag[4*(p+q)+(p+q)-1][19] 4.884 0.94532
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.1033 0.500 2.000 0.7479
## ARCH Lag[7] 0.3964 1.473 1.746 0.9244
## ARCH Lag[9] 0.5312 2.402 1.619 0.9817
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: -897.4691
## Individual Statistics:
## omega 33.0124
## beta1 0.1066
## beta2 0.1066
## beta3 0.1066
## beta4 0.1066
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5071 0.612148
## Negative Sign Bias 3.0535 0.002296 ***
## Positive Sign Bias 1.7672 0.077368 *
## Joint Effect 15.0836 0.001747 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 223.0 8.506e-37
## 2 30 230.9 2.426e-33
## 3 40 243.6 2.042e-31
## 4 50 273.0 7.559e-33
##
##
## Elapsed time : 0.175245##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000000 1.5052 0.13227
## alpha1 0.012676 0.000149 85.0981 0.00000
## beta1 0.983873 0.000607 1620.9678 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000005 0.099279 0.92092
## alpha1 0.012676 0.005171 2.451146 0.01424
## beta1 0.983873 0.003086 318.767677 0.00000
##
## LogLikelihood : 5339.682
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1518
## Bayes -6.1424
## Shibata -6.1518
## Hannan-Quinn -6.1483
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.149 0.2838
## Lag[2*(p+q)+(p+q)-1][2] 1.531 0.3539
## Lag[4*(p+q)+(p+q)-1][5] 1.879 0.6472
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.7678 0.3809
## Lag[2*(p+q)+(p+q)-1][5] 0.8954 0.8829
## Lag[4*(p+q)+(p+q)-1][9] 1.1469 0.9793
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 7.051e-05 0.500 2.000 0.9933
## ARCH Lag[5] 2.748e-01 1.440 1.667 0.9469
## ARCH Lag[7] 3.582e-01 2.315 1.543 0.9894
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 310.2195
## Individual Statistics:
## omega 20.05725
## alpha1 0.07359
## beta1 0.13755
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.6251 0.53201
## Negative Sign Bias 2.2895 0.02217 **
## Positive Sign Bias 0.9677 0.33332
## Joint Effect 9.2911 0.02566 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 167.4 9.171e-26
## 2 30 180.7 7.362e-24
## 3 40 191.5 5.154e-22
## 4 50 195.9 1.872e-19
##
##
## Elapsed time : 0.4067609##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000000 4.2146 2.5e-05
## alpha1 0.021332 0.002547 8.3743 0.0e+00
## beta1 0.370766 0.000553 670.3547 0.0e+00
## beta2 0.602417 0.000916 657.4715 0.0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000001 0.60201 0.54717
## alpha1 0.021332 0.029683 0.71865 0.47236
## beta1 0.370766 0.015249 24.31332 0.00000
## beta2 0.602417 0.013436 44.83607 0.00000
##
## LogLikelihood : 5341.543
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1528
## Bayes -6.1402
## Shibata -6.1528
## Hannan-Quinn -6.1481
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.172 0.2790
## Lag[2*(p+q)+(p+q)-1][2] 1.570 0.3451
## Lag[4*(p+q)+(p+q)-1][5] 1.928 0.6356
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4613 0.4970
## Lag[2*(p+q)+(p+q)-1][8] 0.7626 0.9876
## Lag[4*(p+q)+(p+q)-1][14] 1.3500 0.9985
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.02316 0.500 2.000 0.8791
## ARCH Lag[6] 0.40282 1.461 1.711 0.9186
## ARCH Lag[8] 0.54595 2.368 1.583 0.9781
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 359.5354
## Individual Statistics:
## omega 10.8775
## alpha1 0.0723
## beta1 0.1331
## beta2 0.1337
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.6435 0.52001
## Negative Sign Bias 2.1407 0.03244 **
## Positive Sign Bias 0.8411 0.40043
## Joint Effect 8.4150 0.03817 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 164.5 3.384e-25
## 2 30 184.4 1.539e-24
## 3 40 194.1 1.844e-22
## 4 50 197.6 9.670e-20
##
##
## Elapsed time : 0.215028##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(2,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000000 1.4679e+00 0.14212
## alpha1 0.012560 0.011637 1.0793e+00 0.28046
## alpha2 0.000000 0.011681 2.0000e-06 1.00000
## beta1 0.984141 0.000600 1.6397e+03 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000005 0.096055 0.92348
## alpha1 0.012560 0.020245 0.620398 0.53499
## alpha2 0.000000 0.018683 0.000002 1.00000
## beta1 0.984141 0.003258 302.115104 0.00000
##
## LogLikelihood : 5339.548
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1505
## Bayes -6.1379
## Shibata -6.1505
## Hannan-Quinn -6.1458
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.142 0.2852
## Lag[2*(p+q)+(p+q)-1][2] 1.524 0.3556
## Lag[4*(p+q)+(p+q)-1][5] 1.872 0.6490
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.776 0.3784
## Lag[2*(p+q)+(p+q)-1][8] 1.078 0.9683
## Lag[4*(p+q)+(p+q)-1][14] 1.643 0.9962
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.01877 0.500 2.000 0.8910
## ARCH Lag[6] 0.38914 1.461 1.711 0.9223
## ARCH Lag[8] 0.51139 2.368 1.583 0.9809
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 321.6276
## Individual Statistics:
## omega 20.84740
## alpha1 0.07273
## alpha2 0.07150
## beta1 0.13544
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.6223 0.53383
## Negative Sign Bias 2.2908 0.02210 **
## Positive Sign Bias 0.9686 0.33290
## Joint Effect 9.2863 0.02572 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 167.5 8.708e-26
## 2 30 180.9 6.940e-24
## 3 40 192.6 3.355e-22
## 4 50 197.9 8.471e-20
##
##
## Elapsed time : 0.184042##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(2,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000000 3.756223 0.000172
## alpha1 0.021230 0.016680 1.272734 0.203112
## alpha2 0.000000 0.016304 0.000003 0.999998
## beta1 0.380249 0.000689 551.974404 0.000000
## beta2 0.593019 0.000978 606.501868 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000001 0.000002 0.534837 0.59276
## alpha1 0.021230 0.035690 0.594827 0.55196
## alpha2 0.000000 0.049385 0.000001 1.00000
## beta1 0.380249 0.014063 27.039636 0.00000
## beta2 0.593019 0.012419 47.749688 0.00000
##
## LogLikelihood : 5341.546
##
## Information Criteria
## ------------------------------------
##
## Akaike -6.1516
## Bayes -6.1359
## Shibata -6.1517
## Hannan-Quinn -6.1458
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.174 0.2786
## Lag[2*(p+q)+(p+q)-1][2] 1.572 0.3448
## Lag[4*(p+q)+(p+q)-1][5] 1.929 0.6353
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4644 0.4956
## Lag[2*(p+q)+(p+q)-1][11] 1.0424 0.9959
## Lag[4*(p+q)+(p+q)-1][19] 1.9386 0.9996
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.4430 0.500 2.000 0.5057
## ARCH Lag[7] 0.5060 1.473 1.746 0.8956
## ARCH Lag[9] 0.7819 2.402 1.619 0.9583
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 410.7689
## Individual Statistics:
## omega 10.87171
## alpha1 0.07258
## alpha2 0.07158
## beta1 0.13428
## beta2 0.13489
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.644 0.51963
## Negative Sign Bias 2.142 0.03233 **
## Positive Sign Bias 0.843 0.39932
## Joint Effect 8.426 0.03799 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 164.3 3.714e-25
## 2 30 185.3 1.032e-24
## 3 40 194.1 1.810e-22
## 4 50 198.0 8.287e-20
##
##
## Elapsed time : 0.1948791Los distintos modelos ARCH y GARCH de los rendimientos de WMT proporcionan los siguientes datos:
| Modelo | Coeficiente | AIC | – |
|---|---|---|---|
| ARCH(1) | Ω=0.0(0) β=.99(0) | -6.12 | – |
| – | – | – | – |
| ARCH(2) | Ω=0.0(0) β_1=.50(0) β_2=.49(0) | -6.11 | – |
| – | – | – | – |
| ARCH(3) | Ω=0.0(.58) β_1=.34(0) β_2=.33(0) β_3=.32(0) | -6.11 | – |
| – | – | – | – |
| ARCH(4) | Ω=0.0(.79) β_1=.26(0) β_2=.25(0) β_3=.24(0) β_4=.22(0) | -6.11 | – |
| – | – | – | – |
| GARCH(1,2) | Ω=0.0(.0) α_1=.02(0) β_1=.37(0) β_2=.59(0) | -6.15 | – |
| – | – | – | – |
| GARCH(2,1) | Ω=0.0(.13) α_1=.01(.26) α_2=0.0(.99) β_1=.98(0) | -6.15 | – |
| – | – | – | – |
| GARCH(2,2) | Ω=0.0(.0) α_1=.02(.20) α_2=0.0(.99) β_1=.38(0) β_2=.59(0) | -6.15 | – |
| – | – | – | – |
Los coeficientes resultan significativos a partir del modelo ARCH(3), los dos anteriores no tienen un valor favorable, sin embargo en distintos modelos solo hay un coeficiente significativo.
Prueba ARCH en precios de cierre de Walmart (WMT)
Los datos presentados en el precio de cierre de Walmart son homocedasticos; la prueba ARCH nos permite verificarlo ya que el p-value es mayor que 0.05, es decir, se puede tomar la hipotesis nula donde los residuales son homocedasticos.
Figura 3. Autocorrelacion de rendimientos de Walmart (WMT)
La volatilidad afecta de una manera no tan significativa al activo financiero, los modelos que mejor pueden comprender este comportamiento son ARCH(3) y ARCH(4).