Dedicado con mucho amor a mi hermosa novia que me ha apoyado a seguir adelante

MODELO DE AUTORREGRESIÓN (AR)

Introducción

Los modelos de autorregresión, también conocidos como modelos AR, se utilizan para realizar pronósticos sobre variables ex-post (observaciones que conocemos completamente su valor) en determinados momentos del tiempo normalmente ordenados cronológicamente.

Los modelos autorregresivos, como bien dice su nombre, son modelos que se regresan en sí mismos. Es decir, la variable dependiente y la variable explicativa son la misma con la diferencia que la variable dependiente estará en un momento del tiempo posterior (t) al de la variable independiente (t-1). Decimos ordenados cronológicamente porque actualmente nos encontramos en el momento (t) del tiempo. Si avanzamos un período nos trasladamos a (t+1) y si retrocedemos un período nos vamos a (t-1).

En esta ocasión utilizaremos como base el Producto Interno Bruto de Estados Unidos que la FRED ofrece dentro de su página oficial como base de datos que se puede encontrar dentro del siguiente link:

[Gross Domestic Product] (https://fred.stlouisfed.org/series/GDP)

Cargaamos las librerías que se utilizarán dentro de nuestro modelo Autorregresivo:

library(ggplot2)
library(ggfortify)
library(timeSeries)
## Loading required package: timeDate
library(tidyverse)
## -- Attaching packages --------------------------------------------------------------------- tidyverse 1.3.0 --
## v tibble  3.0.1     v dplyr   1.0.2
## v tidyr   1.1.0     v stringr 1.4.0
## v readr   1.3.1     v forcats 0.5.0
## v purrr   0.3.4
## -- Conflicts ------------------------------------------------------------------------ tidyverse_conflicts() --
## x dplyr::filter() masks timeSeries::filter(), stats::filter()
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library(lubridate)
## 
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library(car)
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library(urca)
library(tseries)
## Registered S3 method overwritten by 'quantmod':
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library(astsa)
library(forecast)
## Registered S3 methods overwritten by 'forecast':
##   method                 from     
##   autoplot.Arima         ggfortify
##   autoplot.acf           ggfortify
##   autoplot.ar            ggfortify
##   autoplot.bats          ggfortify
##   autoplot.decomposed.ts ggfortify
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##   autoplot.ts            ggfortify
##   fitted.ar              ggfortify
##   fortify.ts             ggfortify
##   residuals.ar           ggfortify
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library(foreign)
library(vars)
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library(lmtest)
library(mFilter)
library(dynlm)
library(nlme)
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library(quantmod)
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library(NTS)
library(broom)
library(rugarch)
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library(rmgarch)
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library(kableExtra)
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library(knitr)
library(MASS)
library(FinTS)
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library(fGarch)
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library(parallel)
library(openxlsx)
library(readxl)

Cargamos los datos que empiezan desde 1947 de manera Trimestral tal y como se muestran a continuación:

ProductoInternoBruto = read.xlsx("C:\\Users\\Brook\\Desktop\\GDP 1947.xlsx")

Generamos ahora nuestra serie de tiempo de nuestra variable, observamos los datos y visualizamos la gráfica:

PIB = ts(ProductoInternoBruto$GDP, start = c(1947, 1), frequency = 4)
PIB
##           Qtr1      Qtr2      Qtr3      Qtr4
## 1947   243.164   245.968   249.585   259.745
## 1948   265.742   272.567   279.196   280.366
## 1949   275.034   271.351   272.889   270.627
## 1950   280.828   290.383   308.153   319.945
## 1951   336.000   344.090   351.385   356.178
## 1952   359.820   361.030   367.701   380.812
## 1953   387.980   391.749   391.171   385.970
## 1954   385.345   386.121   390.996   399.734
## 1955   413.073   421.532   430.221   437.092
## 1956   439.746   446.010   451.191   460.463
## 1957   469.779   472.025   479.490   474.864
## 1958   467.540   471.978   485.841   499.555
## 1959   510.330   522.653   525.034   528.600
## 1960   542.648   541.080   545.604   540.197
## 1961   545.018   555.545   567.664   580.612
## 1962   594.013   600.366   609.027   612.280
## 1963   621.672   629.752   644.444   653.938
## 1964   669.822   678.674   692.031   697.319
## 1965   717.790   730.191   749.323   771.857
## 1966   795.734   804.981   819.638   833.302
## 1967   844.170   848.983   865.233   881.439
## 1968   909.387   934.344   950.825   968.030
## 1969   993.337  1009.020  1029.956  1038.147
## 1970  1051.200  1067.375  1086.059  1088.608
## 1971  1135.156  1156.271  1177.675  1190.297
## 1972  1230.609  1266.369  1290.566  1328.904
## 1973  1377.490  1413.887  1433.838  1476.289
## 1974  1491.209  1530.056  1560.026  1599.679
## 1975  1616.116  1651.853  1709.820  1761.831
## 1976  1820.487  1852.332  1886.558  1934.273
## 1977  1988.648  2055.909  2118.473  2164.270
## 1978  2202.760  2331.633  2395.053  2476.949
## 1979  2526.610  2591.247  2667.565  2723.883
## 1980  2789.842  2797.352  2856.483  2985.557
## 1981  3124.206  3162.532  3260.609  3280.818
## 1982  3274.302  3331.972  3366.322  3402.561
## 1983  3473.413  3578.848  3689.179  3794.706
## 1984  3908.054  4009.601  4084.250  4148.551
## 1985  4230.168  4294.887  4386.773  4444.094
## 1986  4507.894  4545.340  4607.669  4657.627
## 1987  4722.156  4806.160  4884.555  5007.994
## 1988  5073.372  5190.036  5282.835  5399.509
## 1989  5511.253  5612.463  5695.365  5747.237
## 1990  5872.701  5960.028  6015.116  6004.733
## 1991  6035.178  6126.862  6205.937  6264.540
## 1992  6363.102  6470.763  6566.641  6680.803
## 1993  6729.459  6808.939  6882.098  7013.738
## 1994  7115.652  7246.931  7331.075  7455.288
## 1995  7522.289  7580.997  7683.125  7772.586
## 1996  7868.468  8032.840  8131.408  8259.771
## 1997  8362.655  8518.825  8662.823  8765.907
## 1998  8866.480  8969.699  9121.097  9293.991
## 1999  9417.264  9524.152  9681.856  9899.378
## 2000 10002.857 10247.679 10319.825 10439.025
## 2001 10472.879 10597.822 10596.294 10660.294
## 2002 10788.952 10893.207 10992.051 11071.463
## 2003 11183.507 11312.875 11567.326 11769.275
## 2004 11920.169 12108.987 12303.340 12522.425
## 2005 12761.337 12910.022 13142.873 13332.316
## 2006 13603.933 13749.806 13867.469 14037.228
## 2007 14208.569 14382.363 14535.003 14681.501
## 2008 14651.039 14805.611 14835.187 14559.543
## 2009 14394.547 14352.850 14420.312 14628.021
## 2010 14721.350 14926.098 15079.917 15240.843
## 2011 15285.828 15496.189 15591.850 15796.460
## 2012 16019.758 16152.257 16257.151 16358.863
## 2013 16569.591 16637.926 16848.748 17083.137
## 2014 17104.555 17432.909 17721.657 17849.912
## 2015 18003.399 18223.577 18347.425 18378.803
## 2016 18470.156 18656.207 18821.359 19032.580
## 2017 19237.435 19379.232 19617.288 19937.963
## 2018 20242.215 20552.653 20742.723 20909.853
## 2019 21115.309 21329.877 21540.325 21747.394
## 2020 21561.139 19520.114 21157.635
autoplot(PIB, col = "Blue") + labs(title = "Producto Interno Bruto de Estados Unidos (1947/T1 - 2020/T3)", subtitle = "Fuente: FRED", caption = "En línea: https://fred.stlouisfed.org/series/GDP [23/11/20]") + geom_point(pch = 20) + theme_gray() + ylab("Producto Interno Bruto") + xlab("")

Observamos a primera vista que:

  • Nuestros datos no son estacionarios.

  • Tiene una tendencia creciente

Pasaremos a logaritmo nuestra variable.

Frecuentemente las regresiones simples y/o múltiples incorporan logaritmos en la ecuación con el objetivo de aportar estabilidad en los regresores, reducir las observaciones atípicas y establecer distintas visiones de la estimación, entre otras aplicaciones.

logpib = log(PIB)
logpib
##          Qtr1     Qtr2     Qtr3     Qtr4
## 1947 5.493736 5.505201 5.519800 5.559700
## 1948 5.582526 5.607884 5.631914 5.636096
## 1949 5.616895 5.603413 5.609065 5.600741
## 1950 5.637742 5.671201 5.730596 5.768149
## 1951 5.817111 5.840903 5.861882 5.875431
## 1952 5.885604 5.888961 5.907270 5.942306
## 1953 5.960954 5.970621 5.969145 5.955760
## 1954 5.954139 5.956151 5.968697 5.990799
## 1955 6.023624 6.043896 6.064299 6.080144
## 1956 6.086197 6.100341 6.111891 6.132233
## 1957 6.152262 6.157032 6.172723 6.163028
## 1958 6.147485 6.156932 6.185881 6.213718
## 1959 6.235058 6.258918 6.263463 6.270232
## 1960 6.296461 6.293567 6.301893 6.291934
## 1961 6.300819 6.319950 6.341530 6.364083
## 1962 6.386901 6.397539 6.411863 6.417190
## 1963 6.432413 6.445326 6.468388 6.483013
## 1964 6.507012 6.520141 6.539631 6.547243
## 1965 6.576177 6.593306 6.619170 6.648799
## 1966 6.679265 6.690819 6.708863 6.725396
## 1967 6.738354 6.744039 6.762999 6.781556
## 1968 6.812771 6.839845 6.857330 6.875263
## 1969 6.901070 6.916735 6.937271 6.945193
## 1970 6.957688 6.972958 6.990311 6.992655
## 1971 7.034525 7.052955 7.071297 7.081958
## 1972 7.115264 7.143909 7.162836 7.192110
## 1973 7.228018 7.254098 7.268110 7.297287
## 1974 7.307342 7.333060 7.352458 7.377558
## 1975 7.387781 7.409653 7.444143 7.474109
## 1976 7.506859 7.524201 7.542509 7.567487
## 1977 7.595210 7.628473 7.658451 7.679838
## 1978 7.697466 7.754324 7.781161 7.814783
## 1979 7.834634 7.859895 7.888921 7.909814
## 1980 7.933740 7.936429 7.957346 8.001542
## 1981 8.046935 8.059128 8.089669 8.095848
## 1982 8.093860 8.111320 8.121576 8.132284
## 1983 8.152893 8.182796 8.213159 8.241362
## 1984 8.270795 8.296447 8.314893 8.330514
## 1985 8.349997 8.365181 8.386349 8.399331
## 1986 8.413585 8.421858 8.435477 8.446261
## 1987 8.460021 8.477654 8.493833 8.518791
## 1988 8.531761 8.554496 8.572218 8.594063
## 1989 8.614547 8.632745 8.647408 8.656474
## 1990 8.678070 8.692830 8.702031 8.700303
## 1991 8.705361 8.720438 8.733262 8.742660
## 1992 8.758271 8.775049 8.789758 8.806993
## 1993 8.814250 8.825992 8.836679 8.855626
## 1994 8.870052 8.888333 8.899877 8.916679
## 1995 8.925626 8.933400 8.946782 8.958358
## 1996 8.970619 8.991293 9.003489 9.019152
## 1997 9.031531 9.050034 9.066796 9.078625
## 1998 9.090033 9.101607 9.118345 9.137123
## 1999 9.150300 9.161586 9.178009 9.200227
## 2000 9.210626 9.234807 9.241822 9.253306
## 2001 9.256544 9.268404 9.268260 9.274281
## 2002 9.286278 9.295895 9.304928 9.312126
## 2003 9.322195 9.333697 9.355940 9.373248
## 2004 9.385987 9.401703 9.417626 9.435276
## 2005 9.454175 9.465759 9.483635 9.497946
## 2006 9.518114 9.528780 9.537301 9.549468
## 2007 9.561601 9.573758 9.584315 9.594344
## 2008 9.592267 9.602762 9.604757 9.586002
## 2009 9.574605 9.571704 9.576393 9.590694
## 2010 9.597054 9.610867 9.621119 9.631734
## 2011 9.634681 9.648349 9.654504 9.667541
## 2012 9.681578 9.689815 9.696288 9.702525
## 2013 9.715324 9.719440 9.732032 9.745847
## 2014 9.747100 9.766115 9.782543 9.789754
## 2015 9.798316 9.810471 9.817245 9.818953
## 2016 9.823912 9.833934 9.842748 9.853908
## 2017 9.864613 9.871957 9.884166 9.900381
## 2018 9.915526 9.930745 9.939951 9.947976
## 2019 9.957754 9.967864 9.977682 9.987249
## 2020 9.978648 9.879201 9.959756
autoplot(logpib, col = "blue") + labs(title = "Logaritmo del Producto Interno Bruto de Estados Unidos (1947/T1 - 2020/T3)", subtitle = "Fuente: FRED", caption = "En línea: https://fred.stlouisfed.org/series/GDP [23/11/20]") + geom_point(pch = 20) + theme_gray() + ylab("Logaritmo") + xlab("")

Lo que toca ahora es estacionalizar la serie a través de obtener el primer diferencial:

dlogpib = diff(logpib)
dlogpib
##              Qtr1         Qtr2         Qtr3         Qtr4
## 1947               0.011465333  0.014598092  0.039900842
## 1948  0.022825533  0.025358542  0.024029589  0.004181848
## 1949 -0.019201166 -0.013481541  0.005651933 -0.008323630
## 1950  0.037000893  0.033458359  0.059395672  0.037552693
## 1951  0.048962054  0.023792091  0.020979237  0.013548117
## 1952  0.010173301  0.003357151  0.018309051  0.035035707
## 1953  0.018647977  0.009667537 -0.001476524 -0.013385158
## 1954 -0.001620609  0.002011755  0.012546538  0.022101996
## 1955  0.032825007  0.020271361  0.020403336  0.015844669
## 1956  0.006053588  0.014144086  0.011549380  0.020341752
## 1957  0.020029866  0.004769579  0.015691089 -0.009694592
## 1958 -0.015543541  0.009447468  0.028949036  0.027836292
## 1959  0.021339873  0.023860189  0.004545258  0.006768980
## 1960  0.026228857 -0.002893717  0.008326296 -0.009959550
## 1961  0.008884934  0.019130792  0.021580080  0.022553025
## 1962  0.022818485  0.010638265  0.014323132  0.005327093
## 1963  0.015222928  0.012913469  0.023061838  0.014624616
## 1964  0.023999460  0.013128889  0.019489858  0.007612230
## 1965  0.028934065  0.017129097  0.025863988  0.029629167
## 1966  0.030465659  0.011553716  0.018044105  0.016533342
## 1967  0.012957775  0.005685266  0.018959672  0.018556964
## 1968  0.031214947  0.027073933  0.017485350  0.017933050
## 1969  0.025806904  0.015664860  0.020536520  0.007921310
## 1970  0.012494976  0.015269994  0.017353185  0.002344268
## 1971  0.041870271  0.018430085  0.018341984  0.010660700
## 1972  0.033306313  0.028644582  0.018927131  0.029273661
## 1973  0.035908460  0.026079647  0.014012116  0.029176741
## 1974  0.010055694  0.025717136  0.019398152  0.025100496
## 1975  0.010222756  0.021871948  0.034490414  0.029965507
## 1976  0.032750439  0.017341338  0.018308619  0.024977541
## 1977  0.027723466  0.033263075  0.029977460  0.021387576
## 1978  0.017627997  0.056857761  0.026836480  0.033622201
## 1979  0.019850922  0.025260744  0.029026844  0.020892365
## 1980  0.023926528  0.002688292  0.020917892  0.044195181
## 1981  0.045393842  0.012192802  0.030541015  0.006178794
## 1982 -0.001988065  0.017459604  0.010256433  0.010707630
## 1983  0.020609301  0.029903277  0.030362979  0.028202997
## 1984  0.029432614  0.025652183  0.018446378  0.015621004
## 1985  0.019482592  0.015183537  0.021168632  0.012982147
## 1986  0.014254058  0.008272450  0.013619557  0.010784002
## 1987  0.013759384  0.017632954  0.016179758  0.024957250
## 1988  0.012970250  0.022734948  0.017722252  0.021845138
## 1989  0.020483979  0.018197659  0.014663026  0.009066531
## 1990  0.021595447  0.014760515  0.009200455 -0.001727643
## 1991  0.005057357  0.015077362  0.012823704  0.009398747
## 1992  0.015610832  0.016778036  0.014708408  0.017235751
## 1993  0.007256564  0.011741554  0.010687240  0.018947250
## 1994  0.014426068  0.018281204  0.011544094  0.016801417
## 1995  0.008946901  0.007774241  0.013381644  0.011576562
## 1996  0.012260453  0.020674759  0.012195955  0.015662769
## 1997  0.012379097  0.018502460  0.016762230  0.011829342
## 1998  0.011407881  0.011574245  0.016737963  0.018777980
## 1999  0.013176538  0.011286288  0.016422730  0.022218308
## 2000  0.010398825  0.024180489  0.007015562  0.011484385
## 2001  0.003237776  0.011859545 -0.000144191  0.006021681
## 2002  0.011996649  0.009616736  0.009032991  0.007198522
## 2003  0.010069207  0.011501354  0.022242942  0.017307923
## 2004  0.012739517  0.015716065  0.015922866  0.017650267
## 2005  0.018899016  0.011583856  0.017875726  0.014311228
## 2006  0.020168080  0.010665773  0.008521023  0.012167206
## 2007  0.012132290  0.012157431  0.010557076  0.010028526
## 2008 -0.002077011  0.010494976  0.001995628 -0.018755204
## 2009 -0.011397201 -0.002900926  0.004689239  0.014301168
## 2010  0.006359885  0.013812403  0.010252634  0.010615005
## 2011  0.002947261  0.013667998  0.006154219  0.013037522
## 2012  0.014036971  0.008236957  0.006473081  0.006236956
## 2013  0.012799318  0.004115640  0.012591563  0.013815485
## 2014  0.001252966  0.019014939  0.016427709  0.007211127
## 2015  0.008561995  0.012155622  0.006773042  0.001708752
## 2016  0.004958251  0.010022665  0.008813436  0.011159906
## 2017  0.010705873  0.007343857  0.012209241  0.016214385
## 2018  0.015144672  0.015219757  0.009205454  0.008024997
## 2019  0.009777839  0.010110444  0.009817994  0.009567174
## 2020 -0.008601360 -0.099447153  0.080555412
autoplot(dlogpib, col = "blue") + labs(title = "Primer diferencial del Logaritmo del PIB de Estadis Unidos (1947/T1 - 2020/T3)", subtitle = "Fuente: FRED", caption = "En línea: https://fred.stlouisfed.org/series/GDP [23/11/20]") + geom_point(pch = 20) + theme_gray() + ylab("Logaritmo") + xlab("")

Lo que ahora debemos de comprobar es que la serie de tiempo sea estacionaria, lo que significa es que el tiempo no depende de los datos obsenidos. Para eso lo comprobaremos a través de la prueba de Dickey - Fuller Aumentada y la prueba Phillips - Perron.

Dickey - Fuller Aumentada:

adf.test(dlogpib)
## Warning in adf.test(dlogpib): p-value smaller than printed p-value
## 
##  Augmented Dickey-Fuller Test
## 
## data:  dlogpib
## Dickey-Fuller = -5.3898, Lag order = 6, p-value = 0.01
## alternative hypothesis: stationary

Phillips - Perron:

pp.test(dlogpib)
## Warning in pp.test(dlogpib): p-value smaller than printed p-value
## 
##  Phillips-Perron Unit Root Test
## 
## data:  dlogpib
## Dickey-Fuller Z(alpha) = -251.69, Truncation lag parameter = 5, p-value
## = 0.01
## alternative hypothesis: stationary

Observamos que el p-value en ambos casos es menor que 0.01, por lo que aseguramos entonces que nuestra serie de tiempo es Estacionaria. Podemos proceder a correr el modelo Autorregresivo (AR).

Empezando por un modelo de autorregresivo a un rezago (t-1) para empezar a comprobar los datos que se pueden obtener a través del resumen estadístico.

AR1 = dynlm(dlogpib ~ L(dlogpib))
summary(AR1)
## 
## Time series regression with "ts" data:
## Start = 1947(3), End = 2020(3)
## 
## Call:
## dynlm(formula = dlogpib ~ L(dlogpib))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.108553 -0.005110 -0.000309  0.005458  0.094954 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.011331   0.001166   9.714  < 2e-16 ***
## L(dlogpib)  0.258727   0.059274   4.365 1.77e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01296 on 291 degrees of freedom
## Multiple R-squared:  0.06145,    Adjusted R-squared:  0.05822 
## F-statistic: 19.05 on 1 and 291 DF,  p-value: 1.769e-05

Observamos que:

*Los estimadores \((\beta_0, \beta_1)\) son significativas, teniendo sus respectivos valores de 0.011331 y de 0.258727, con un valor significativo t de Student junto con una probabilidad de p-value menor que 0.05. Por último, podemos notar una \(R^{2}\) demasiado baja porque debemos de recordar que estamos aplicando una regresión a una serie de tiempo Estacionaria a través de la misma variable dependiente como independiente.

Podemos generar distintas pruebas de dicho modelo y ver si existe algún otro modelo que se llegue a ajustar a diferentes rezagos. En este caso lo haremos a 2 rezagos para ver el comportamiento de dicha serie de tiempo:

AR2 = dynlm(dlogpib ~ L(dlogpib, 1:2))
summary(AR2)
## 
## Time series regression with "ts" data:
## Start = 1947(4), End = 2020(3)
## 
## Call:
## dynlm(formula = dlogpib ~ L(dlogpib, 1:2))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.109615 -0.005058 -0.000474  0.004838  0.088695 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      0.008906   0.001355   6.575 2.27e-10 ***
## L(dlogpib, 1:2)1 0.148450   0.066843   2.221 0.027134 *  
## L(dlogpib, 1:2)2 0.265341   0.078472   3.381 0.000821 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01276 on 289 degrees of freedom
## Multiple R-squared:  0.09716,    Adjusted R-squared:  0.09092 
## F-statistic: 15.55 on 2 and 289 DF,  p-value: 3.85e-07

Podemos generar otro a 3 rezagos:

AR3 = dynlm(dlogpib ~ L(dlogpib, 1:3))
summary(AR3)
## 
## Time series regression with "ts" data:
## Start = 1948(1), End = 2020(3)
## 
## Call:
## dynlm(formula = dlogpib ~ L(dlogpib, 1:3))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.109563 -0.005157 -0.000549  0.004829  0.089502 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       0.009116   0.001467   6.212 1.83e-09 ***
## L(dlogpib, 1:3)1  0.152136   0.067113   2.267 0.024143 *  
## L(dlogpib, 1:3)2  0.289120   0.085460   3.383 0.000816 ***
## L(dlogpib, 1:3)3 -0.046548   0.079960  -0.582 0.560932    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0127 on 287 degrees of freedom
## Multiple R-squared:  0.1005, Adjusted R-squared:  0.09106 
## F-statistic: 10.68 on 3 and 287 DF,  p-value: 1.116e-06

Por último podemos generar otro modelo a 4 rezagos:

AR4 = dynlm(dlogpib ~ L(dlogpib, 1:4))
summary(AR4)
## 
## Time series regression with "ts" data:
## Start = 1948(2), End = 2020(3)
## 
## Call:
## dynlm(formula = dlogpib ~ L(dlogpib, 1:4))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.109771 -0.005211 -0.000496  0.004841  0.088849 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)       0.009406   0.001575   5.972 6.97e-09 ***
## L(dlogpib, 1:4)1  0.145196   0.068373   2.124 0.034566 *  
## L(dlogpib, 1:4)2  0.299812   0.087645   3.421 0.000716 ***
## L(dlogpib, 1:4)3 -0.027310   0.087399  -0.312 0.754911    
## L(dlogpib, 1:4)4 -0.042828   0.081340  -0.527 0.598930    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01274 on 285 degrees of freedom
## Multiple R-squared:  0.1006, Adjusted R-squared:  0.08798 
## F-statistic:  7.97 on 4 and 285 DF,  p-value: 4.205e-06

Podemos resumir que los mejores modelos de la serie de tiempo del Producto Interno Bruto de Estados Unidos se explica a través de la Autorregresión a 1 o 2 rezagos.

Tendencia = seq_along(dlogpib)
Tendencia = ts(Tendencia, start = c(1974, 01), frequency = 4)
Tendencia
##      Qtr1 Qtr2 Qtr3 Qtr4
## 1974    1    2    3    4
## 1975    5    6    7    8
## 1976    9   10   11   12
## 1977   13   14   15   16
## 1978   17   18   19   20
## 1979   21   22   23   24
## 1980   25   26   27   28
## 1981   29   30   31   32
## 1982   33   34   35   36
## 1983   37   38   39   40
## 1984   41   42   43   44
## 1985   45   46   47   48
## 1986   49   50   51   52
## 1987   53   54   55   56
## 1988   57   58   59   60
## 1989   61   62   63   64
## 1990   65   66   67   68
## 1991   69   70   71   72
## 1992   73   74   75   76
## 1993   77   78   79   80
## 1994   81   82   83   84
## 1995   85   86   87   88
## 1996   89   90   91   92
## 1997   93   94   95   96
## 1998   97   98   99  100
## 1999  101  102  103  104
## 2000  105  106  107  108
## 2001  109  110  111  112
## 2002  113  114  115  116
## 2003  117  118  119  120
## 2004  121  122  123  124
## 2005  125  126  127  128
## 2006  129  130  131  132
## 2007  133  134  135  136
## 2008  137  138  139  140
## 2009  141  142  143  144
## 2010  145  146  147  148
## 2011  149  150  151  152
## 2012  153  154  155  156
## 2013  157  158  159  160
## 2014  161  162  163  164
## 2015  165  166  167  168
## 2016  169  170  171  172
## 2017  173  174  175  176
## 2018  177  178  179  180
## 2019  181  182  183  184
## 2020  185  186  187  188
## 2021  189  190  191  192
## 2022  193  194  195  196
## 2023  197  198  199  200
## 2024  201  202  203  204
## 2025  205  206  207  208
## 2026  209  210  211  212
## 2027  213  214  215  216
## 2028  217  218  219  220
## 2029  221  222  223  224
## 2030  225  226  227  228
## 2031  229  230  231  232
## 2032  233  234  235  236
## 2033  237  238  239  240
## 2034  241  242  243  244
## 2035  245  246  247  248
## 2036  249  250  251  252
## 2037  253  254  255  256
## 2038  257  258  259  260
## 2039  261  262  263  264
## 2040  265  266  267  268
## 2041  269  270  271  272
## 2042  273  274  275  276
## 2043  277  278  279  280
## 2044  281  282  283  284
## 2045  285  286  287  288
## 2046  289  290  291  292
## 2047  293  294
autoplot(Tendencia, col = "blue") + labs(title = "Tendencia del Primer Diferencial del Logaritmo del Producto Interno Bruto de Estados Unidos", subtitle = "(1947/T1 - 2020/T3)") + ylab("") + xlab("")

Podemos agregar ahora la tendencia a nuestro modelo autorregresivo:

AR5 = dynlm(dlogpib ~ Tendencia + L(dlogpib, 1))
summary(AR5)
## 
## Time series regression with "ts" data:
## Start = 1974(1), End = 2020(3)
## 
## Call:
## dynlm(formula = dlogpib ~ Tendencia + L(dlogpib, 1))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.106017 -0.003945  0.000459  0.004959  0.061081 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    0.0273791  0.0027140  10.088  < 2e-16 ***
## Tendencia     -0.0001185  0.0000190  -6.238 2.97e-09 ***
## L(dlogpib, 1) -0.1433563  0.0824139  -1.739   0.0836 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01212 on 184 degrees of freedom
## Multiple R-squared:  0.1841, Adjusted R-squared:  0.1752 
## F-statistic: 20.76 on 2 and 184 DF,  p-value: 7.425e-09

Observamos que: Al incluir la tendencia a nuestro modelo Autorregresivo, a penas es valido el primer rezago del primer diferencial del logaritmo del PIB de Estados Unidos. Por lo que podemos decir que nuestro segundo modelo, AR2 fue el mejor que explicaba en sí el modelo de Autorregresión. Resultado por el cual lo tomaremos y obtaremos por manejarlo.

Podemos ahora utilizar el modelo AR2 para estimar nuestra serie de tiempo:

Pronostico = predict(AR2)
Pronostico
##      1947 Q4      1948 Q1      1948 Q2      1948 Q3      1948 Q4      1949 Q1 
##  0.014115492  0.018702947  0.022881973  0.018727215  0.019202043  0.015903011 
##      1949 Q2      1949 Q3      1949 Q4      1950 Q1      1950 Q2      1950 Q3 
##  0.007165366  0.001809969  0.006167995  0.009170216  0.012190369  0.023690945 
##      1950 Q4      1951 Q1      1951 Q2      1951 Q3      1951 Q4      1952 Q1 
##  0.026601371  0.030241010  0.026138891  0.025429772  0.018333573  0.016484054 
##      1952 Q2      1952 Q3      1952 Q4      1953 Q1      1953 Q2      1953 Q3 
##  0.014011278  0.012103939  0.012514950  0.018965386  0.020970892  0.015289400 
##      1953 Q4      1954 Q1      1954 Q2      1954 Q3      1954 Q4      1955 Q1 
##  0.011252177  0.006527356  0.005113957  0.008774803  0.011302513  0.015516338 
##      1955 Q2      1955 Q3      1955 Q4      1956 Q1      1956 Q2      1956 Q3 
##  0.019643631  0.020625294  0.017313885  0.016672167  0.014009074  0.012612135 
##      1956 Q4      1957 Q1      1957 Q2      1957 Q3      1957 Q4      1958 Q1 
##  0.014373692  0.014990442  0.017277121  0.014928968  0.012501087  0.011630498 
##      1958 Q2      1958 Q3      1958 Q4      1959 Q1      1959 Q2      1959 Q3 
##  0.004026350  0.006184310  0.015710473  0.020719856  0.019460203  0.018110577 
##      1959 Q4      1960 Q1      1960 Q2      1960 Q3      1960 Q4      1961 Q1 
##  0.015912010  0.011117075  0.014595948  0.015436196  0.009374392  0.009636982 
##      1961 Q2      1961 Q3      1961 Q4      1962 Q1      1962 Q2      1962 Q3 
##  0.007582465  0.014103686  0.017185934  0.017980265  0.018277835  0.016540113 
##      1962 Q4      1963 Q1      1963 Q2      1963 Q3      1963 Q4      1964 Q1 
##  0.013855218  0.013497499  0.012579520  0.014862453  0.015756189  0.017196460 
##      1964 Q2      1964 Q3      1964 Q4      1965 Q1      1965 Q2      1965 Q3 
##  0.016349417  0.017223209  0.015283086  0.015207674  0.015221286  0.019126396 
##      1965 Q4      1966 Q1      1966 Q2      1966 Q3      1966 Q4      1967 Q1 
##  0.017290749  0.020167419  0.021290654  0.018705123  0.014650505  0.016148400 
##      1967 Q2      1967 Q3      1967 Q4      1968 Q1      1968 Q2      1968 Q3 
##  0.015216737  0.013188385  0.013229280  0.016691745  0.018463974  0.021207923 
##      1968 Q4      1969 Q1      1969 Q2      1969 Q3      1969 Q4      1970 Q1 
##  0.018685712  0.016207926  0.017495597  0.018079264  0.016111362  0.015531280 
##      1970 Q2      1970 Q3      1970 Q4      1971 Q1      1971 Q2      1971 Q3 
##  0.012862907  0.014488442  0.015534020  0.013858695  0.015743864  0.022752037 
##      1971 Q4      1972 Q1      1972 Q2      1972 Q3      1972 Q4      1973 Q1 
##  0.016519310  0.015355643  0.016679233  0.021996013  0.019316503  0.018274010 
##      1973 Q2      1973 Q3      1973 Q4      1974 Q1      1974 Q2      1974 Q3 
##  0.022004310  0.022305704  0.017906284  0.016955465  0.018140738  0.015392083 
##      1974 Q4      1975 Q1      1975 Q2      1975 Q3      1975 Q4      1976 Q1 
##  0.018609654  0.017779482  0.017083942  0.014865592  0.019829821  0.022506296 
##      1976 Q2      1976 Q3      1976 Q4      1977 Q1      1977 Q2      1977 Q3 
##  0.021719076  0.020170544  0.016225468  0.017472132  0.019649306  0.021200271 
##      1977 Q4      1978 Q1      1978 Q2      1978 Q3      1978 Q4      1979 Q1 
##  0.022182406  0.020035425  0.017198063  0.022024169  0.027976770  0.021018229 
##      1979 Q2      1979 Q3      1979 Q4      1980 Q1      1980 Q2      1980 Q3 
##  0.020774408  0.017923410  0.019917938  0.019709673  0.018001683  0.015653945 
##      1980 Q4      1981 Q1      1981 Q2      1981 Q3      1981 Q4      1982 Q1 
##  0.012724758  0.021017347  0.027371715  0.022761058  0.016675253  0.017927208 
##      1982 Q2      1982 Q3      1982 Q4      1983 Q1      1983 Q2      1983 Q3 
##  0.010250531  0.010970543  0.015061498  0.013217179  0.014806808  0.018813825 
##      1983 Q4      1984 Q1      1984 Q2      1984 Q3      1984 Q4      1985 Q1 
##  0.021348144  0.021149471  0.020758876  0.020523938  0.018451128  0.016119702 
##      1985 Q2      1985 Q3      1985 Q4      1986 Q1      1986 Q2      1986 Q3 
##  0.015943269  0.016329711  0.016077484  0.016450289  0.014466893  0.013916411 
##      1986 Q4      1987 Q1      1987 Q2      1987 Q3      1987 Q4      1988 Q1 
##  0.013123024  0.014120892  0.013810199  0.015174724  0.015986815  0.016904245 
##      1988 Q2      1988 Q3      1988 Q4      1989 Q1      1989 Q2      1989 Q3 
##  0.017453800  0.015722728  0.017569568  0.016851538  0.017743445  0.017042868 
##      1989 Q4      1990 Q1      1990 Q2      1990 Q3      1990 Q4      1991 Q1 
##  0.015911495  0.014142808  0.014517751  0.016827540  0.014188557  0.011090963 
##      1991 Q2      1991 Q3      1991 Q4      1992 Q1      1992 Q2      1992 Q3 
##  0.009198524  0.012486339  0.014810503  0.013704078  0.013717483  0.015539077 
##      1992 Q4      1993 Q1      1993 Q2      1993 Q3      1993 Q4      1994 Q1 
##  0.015541547  0.015367575  0.014556768  0.012574677  0.013608216  0.014554666 
##      1994 Q2      1994 Q3      1994 Q4      1995 Q1      1995 Q2      1995 Q3 
##  0.016075216  0.015447856  0.015470656  0.014463474  0.014692453  0.012434244 
##      1995 Q4      1996 Q1      1996 Q2      1996 Q3      1996 Q4      1997 Q1 
##  0.012955510  0.014175420  0.013797981  0.015228554  0.016202534  0.014467407 
##      1997 Q2      1997 Q3      1997 Q4      1998 Q1      1998 Q2      1998 Q3 
##  0.014899834  0.014937556  0.016303999  0.015109955  0.013738490  0.013651356 
##      1998 Q4      1999 Q1      1999 Q2      1999 Q3      1999 Q4      2000 Q1 
##  0.014462055  0.016135044  0.015844808  0.014077906  0.014338852  0.016562118 
##      2000 Q2      2000 Q3      2000 Q4      2001 Q1      2001 Q2      2001 Q3 
##  0.016345316  0.015255014  0.016363717  0.012472552  0.012434102  0.011525842 
##      2001 Q4      2002 Q1      2002 Q2      2002 Q3      2002 Q4      2003 Q1 
##  0.012031593  0.009761833  0.012284880  0.013516987  0.012798841  0.012371621 
##      2003 Q2      2003 Q3      2003 Q4      2004 Q1      2004 Q2      2004 Q3 
##  0.012311015  0.013285329  0.015259931  0.017377511  0.015389865  0.014619549 
##      2004 Q4      2005 Q1      2005 Q2      2005 Q3      2005 Q4      2006 Q1 
##  0.015440049  0.015751356  0.016395084  0.015640489  0.014633507  0.015773848 
##      2006 Q2      2006 Q3      2006 Q4      2007 Q1      2007 Q2      2007 Q3 
##  0.015697492  0.015840935  0.013001191  0.012973378  0.013935678  0.013930145 
##      2007 Q4      2008 Q1      2008 Q2      2008 Q3      2008 Q4      2009 Q1 
##  0.013699243  0.013196139  0.011258820  0.009913039  0.011987174  0.006651475 
##      2009 Q2      2009 Q3      2009 Q4      2010 Q1      2010 Q2      2010 Q3 
##  0.002237722  0.005451380  0.008832556  0.012273436  0.013644990  0.012644170 
##      2010 Q4      2011 Q1      2011 Q2      2011 Q3      2011 Q4      2012 Q1 
##  0.014093181  0.013202421  0.012160293  0.011717222  0.013446452  0.012474566 
##      2012 Q2      2012 Q3      2012 Q4      2013 Q1      2013 Q2      2013 Q3 
##  0.014449359  0.013853540  0.012052708  0.011549626  0.012461158  0.012913328 
##      2013 Q4      2014 Q1      2014 Q2      2014 Q3      2014 Q4      2015 Q1 
##  0.011867444  0.014298148  0.012757993  0.012061412  0.016390321  0.014335616 
##      2015 Q2      2015 Q3      2015 Q4      2016 Q1      2016 Q2      2016 Q3 
##  0.012090614  0.012982530  0.013137021  0.010957004  0.010095629  0.011709670 
##      2016 Q4      2017 Q1      2017 Q2      2017 Q3      2017 Q4      2018 Q1 
##  0.012873957  0.012901433  0.013456647  0.012837081  0.012667268  0.014552820 
##      2018 Q2      2018 Q3      2018 Q4      2019 Q1      2019 Q2      2019 Q3 
##  0.015456751  0.015184058  0.014311155  0.012540073  0.012487059  0.013001536 
##      2019 Q4      2020 Q1      2020 Q2      2020 Q3 
##  0.013046375  0.012931542  0.010167862 -0.008139101
Pronostico = ts(Pronostico, start = c(1947, 1), frequency = 4)
autoplot(Pronostico, col = "Red", ts.linetype = "dashed") + labs(title = "Estimación del primer diferencial del logaritmo del PIB de Estados UNidos (1947/T1 - 2020/T3)", subtitle = "Fuente: FRED", caption = "https://fred.stlouisfed.org/series/GDP") + ylab("") + xlab("")
## Warning: Ignoring unknown parameters: ts.linetype

Podemos ahora graficar ambas series de tiempo para ver cómo es el comportamiento entre la variable observada y la variable estimada del Primer Diferencial del Producto Interno Bruto de Estados Unidos.

autoplot(dlogpib, series = "dlogpib") + autolayer(Pronostico, series = "dlogpib Estimado") + labs(title = "Primer diferencial del logaritmo del PIB (Observado y Estimado)", subtitle = "Fuente: Calculos propios a partir de datos de FRED") + ylab("") + xlab("")

Por último podemos definir si existe o no autocorrelación dentro del modelo dentro de la Prueba Durbin - Watson:

dwtest(AR2)
## 
##  Durbin-Watson test
## 
## data:  AR2
## DW = 1.8058, p-value = 0.04463
## alternative hypothesis: true autocorrelation is greater than 0

Observamos que nuestro valor del Durbin-Watson es de 1.8058, por lo cual entra dentro de la zona de no rechazar \(H_0\), lo que significa que no existe problema de autocorrelación positiva o negativa dentro del modelo AR(2). Siendo un excelente modelo para Estimar la variable del PIB de Estados Unidos.

Bibliografía