Homework Assignment 3

This homework assignment must be completed individually. To ensure a comprehensive understanding of the material, students will be selected at random for a brief oral examination to discuss their solutions and the underlying methodology.

Problem 1: Modeling trend

• USA GROSS DOMESTIC PRODUCT GROWTH

• Data from 1947.Q2 to 2025.Q4

• Number of observations: 315 quarterly measurements

• U.S. Bureau of Economic Analysis, Gross Domestic Product [GDP], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GDP, March 12, 2026.

Task:

1. Fit AR(1) and AR(2) models from the Bayesian viewpoint for thedata from 1947.Q2 to 2023.Q4.

2. Use the period of 2024.Q1 to 2025.Q4 (8 observations) for out-of-sample forecasting and model comparison.

3. Provide thorough details describing your analysis.

gdp = scan("https://hedibert.org/wp-content/uploads/2026/03/us-gdp-growth.txt")
par(mfrow=c(1,1))
ts.plot(gdp,main="USA GDP Growth")
abline(h=0,lty=2)

Problem 2: Modeling seasonality

Produção física industrial, Bens de consumo semiduráveis e não duráveis (Índice de base fixa sem ajuste sazonal (base: média de 2022 = 100)), janeiro 2002 - janeiro 2026 - 289 monthly observations

Source: https://www.ibge.gov.br/estatisticas/economicas/industria/9294-pesquisa-industrial-mensal-producao-fisica-brasil.html?=&t=series-historicas

Consider the standard seasonal ARIMA (SARIMA) models.

Task:

1. Use the data from January 2002 to December 2023 for fit.

2. Forecast the period between January 2024 to December 2025 (24 observations).

3. Compare the performance of the three best SARIMA models both in-sample and out-of-sample.

4. Provide thorough details describing your analysis.

pim = scan("https://hedibert.org/wp-content/uploads/2026/03/pim.txt")
par(mfrow=c(1,1))
ts.plot(pim,main="Bens de consumo semiduráveis e não duráveis")

Problem 3: Modeling time-varying volatility

• NYSE - Nasdaq Real Time Price - USD

• Petróleo Brasileiro S.A. - Petrobras (PBR)

• https://finance.yahoo.com/quote/PBR/

• Data from August 10th 2000 to March 11th 2026

• Number of observed prices: 6433

• Number of observed returns: 6432

Task:

1. Fit the GARCH-family models we discussed in class. Which model you would suggest is the best and why?

2. Fit the GARCH(1,1) from a Bayesian viewpoint as discussed in class.

3. Provide thorough details describing your analysis.

#install.packages("quantmod")
library("quantmod")

## Loading required package: xts

## Loading required package: zoo

## 
## Attaching package: 'zoo'

## The following objects are masked from 'package:base':
## 
##     as.Date, as.Date.numeric

## Loading required package: TTR

## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo

getSymbols("PBR",src="yahoo",from="2000-01-01",to=Sys.Date())

## [1] "PBR"

price = as.vector(PBR$PBR.Adjusted)
n     = length(price)
ret   = 100*(1-price[2:n]/price[1:(n-1)])
n1    = length(ret)

par(mfrow=c(2,2))
ts.plot(price,xlab="Day",ylab="PBR price")
abline(h=12,lty=2,col=2)
plot((n-200):n,price[(n-200):n],type="l",xlab="Day",ylab="PBR price")
abline(h=12,lty=2,col=2)

ts.plot(ret,xlab="Day",ylab="PBR return")
abline(h=0,lty=2,col=2)
plot((n1-200):n1,ret[(n1-200):n1],type="l",xlab="Day",ylab="PBR return")
abline(h=0,lty=2,col=2)

Dados da produção industrial

pim = c(87.71970,83.11683,89.70269,92.75267,92.94413,88.98536,99.79610,103.66860,100.76334,
        113.73182,106.20027,93.57646,85.55091,81.94634,85.45267,85.47378,90.30211,86.89062,
        94.21479,95.72981,101.47236,108.49665,102.60542,94.05932,86.17560,80.64803,94.59889,
        88.78858,92.59840,93.62235,98.10789,102.84493,106.63144,108.98440,111.96525,103.76027,
        95.14461,85.97299,97.74678,98.33256,102.05317,102.31380,103.07006,111.84126,108.10040,
        111.59283,112.72675,107.06415,95.54773,92.28469,105.40895,96.35514,106.24948,102.93799,
        106.24289,113.39767,111.53816,118.10057,117.17701,106.23216,97.48499,93.15808,105.86009,
        99.47235,110.29380,108.79420,110.99726,119.46788,114.32566,125.88792,122.62162,108.83540,
        104.64416,95.20297,103.05501,106.24276,111.32517,110.22067,116.24257,116.59774,121.77258,
        127.36630,119.11797,104.38322,93.18346,90.01224,103.50557,99.62742,107.44469,104.51804,
        112.16857,115.58437,116.32600,125.25375,120.06276,110.08595,99.39619,99.13293,115.64975,
        107.79226,112.76277,111.40264,117.82544,120.87247,123.10825,126.45879,124.16332,111.12167,
        101.92356,104.48581,115.04015,106.58498,117.73065,110.31223,117.67677,124.33195,120.67810,
        123.90867,122.85466,110.04925,100.87854,102.04774,113.51115,104.30036,113.92819,107.97763,
        116.87830,126.53944,120.10647,131.65242,126.78199,108.20757,106.96031,99.35423,107.59216,
        114.63436,117.84018,114.15400,122.02782,127.57635,124.44110,132.56219,126.37443,107.40969,
        106.10194,106.38608,110.55684,110.37854,119.35443,110.49589,123.67214,124.10183,127.37815,
        132.71877,122.01103,106.16318,100.77764,96.08610,107.83123,100.34119,106.81061,108.13828,
        112.69541,115.49168,118.09805,122.38543,115.61430,100.98842,94.05887,93.95102,104.18159,
        104.07643,105.11821,107.05618,106.29700,113.94301,113.99022,113.81435,110.10009,97.73616,
        96.55549,93.07249,105.32007,94.06159,107.36209,105.61873,110.82996,118.21215,113.51182,
        119.20024,113.07747,98.54489,98.65891,92.19956,103.13301,103.18175,97.32194,107.67353,
        112.38267,117.40335,111.37505,119.29543,112.19098,96.54790,96.89664,95.44862,97.95312,
        102.39148,109.23645,102.55246,114.11676,117.11991,113.68491,123.80491,113.47167,
        97.62655,96.33999,93.72499,90.81747,75.87591,88.05621,97.33476,108.61583,109.68134,
        115.95653,119.66066,111.66040,99.48688,95.33225,92.10168,96.02629,89.87943,100.74080,
        99.40791,106.65119,108.61420,109.51141,107.71345,104.70169,91.96436,85.21055,85.61565,
        92.07931,90.44198,104.13901,103.38563,110.70190,113.75400,108.75261,108.39278,104.22589,
        93.30069,89.55306,86.24938,96.94244,90.69546,105.81213,103.23152,109.08393,117.84967,
        111.83009,112.84863,107.99431,92.31876,92.45704,90.48019,93.42381,100.62153,107.39104,
        108.75780,115.32769,118.13866,114.67011,117.51183,105.50950,90.56210,92.73243,90.50286,
        96.46420,94.97920,105.23906,99.90380,111.30356,112.73548,113.77899,115.67371,105.03481,94.74008,93.46062)

Dados da PIB dos EUA

gdp = c(1.153131220,1.470516490,4.070757457,2.308802864,2.568280513,2.432062575,0.419060445 -1.901799790
 ,-1.339107165,0.566793563 -0.828908457,3.769394776,3.402438503,6.119504241,3.826670518,5.018049977
 ,2.407738095,2.120084862,1.364030906,1.022522447,0.336279251,1.847768883,3.565668845,1.882293625
 ,0.971441827 -0.147543453 -1.329597542 -0.161929684,0.201377986,1.262557592,2.234805471,3.336969084
 ,2.047822056,2.061290721,1.597086149,0.607194824,1.424458665,1.161633147,2.055005530,2.023181016
 ,0.478097148,1.581484032 -0.964775074 -1.542336332,0.949223596,2.937213175,2.822734187,2.156919658
 ,2.414712049,0.455560381,0.679194109,2.657586076 -0.288953428,0.836105567 -0.991011796,0.892452198
 ,1.931495841,2.181461448,2.280926745,2.308081817,1.069505213,1.442620002,0.534130671,1.533938721
 ,1.299720753,2.332981872,1.473207913,2.428976447,1.321545127,1.968102506,0.764127619,2.935672196
 ,1.727664080,2.620136375,3.007247876,3.093448657,1.162071748,1.820788317,1.667077417,1.304209038
 ,0.570145824,1.914054816,1.873021487,3.170724236,2.744376157,1.763911365,1.809481240,2.614278483
 ,1.578819675,2.074884541,0.795276691,1.257336389,1.538717656,1.750462583,0.234701798,4.275919339
 ,1.860096762,1.851123136,1.071772773,3.386717769,2.905878309,1.910738497,2.970634590,3.656095549
,2.642269635,1.411074577,2.960655248,1.010642225,2.605067432,1.958751837,2.541816611,1.027518646
,2.211289289,3.509210565,3.041899147,3.329263703,1.749257204,1.847724922,2.529209280,2.811133692
,3.382247638,3.043130800,2.161792952,1.778428754,5.850523888,2.719982090,3.419381534,2.004926222
,2.558249987,2.945222898,2.111213785,2.421506357,0.269190872,2.113820499,4.518633578,4.643991054
,1.226743691,3.101217632,0.619792192 -0.198609005,1.761291414,1.030921028,1.076516150,2.082313881
,3.035486998,3.082863536,2.860446728,2.987003473,2.598403195,1.861756319,1.574364938,1.967361616
,1.529939236,2.139427650,1.306678052,1.435613198,0.830676143,1.371272556,1.084235869,1.385447997
,1.778933182,1.631135876,2.527128879,1.305472810,2.299535693,1.788022280,2.208549008,2.069521506
,1.836424494,1.477105506,0.910775692,2.183031603,1.486998912,0.924290960 -0.172615125,0.507016715
,1.519159833,1.290628057,0.944305429,1.573331801,1.691957790,1.481710889,1.738514409,0.728295685
,1.181075626,1.074455212,1.912788804,1.453062547,1.844932833,1.161098402,1.694335415,0.898704383
,0.780453928,1.347157900,1.164382982,1.233592012,2.088996232,1.227062907,1.578607297,1.245603540
,1.867469123,1.690350488,1.189958516,1.147319952,1.164148569,1.687882726,1.895539539,1.266312825
,1.216870693,1.683943562,2.204513729,1.030386451,2.454875083,0.687421202,1.139534016,0.330469969
,1.229858252 -0.009246155,0.589213834,1.154124140,0.964065470,0.887075590,0.704595031,1.018819171
,1.240696255,2.244393635,1.777218662,1.284488569,1.588198446,1.589159910,1.803343874,1.916403759
,1.216938353,1.702328066,1.381472614,2.063582935,1.134364181,0.848981315,1.221122598,1.254248709
,1.311448909,1.125080388,1.036389642 -0.057899874,1.082260149,0.223992128 -1.951741859 -1.213749064
 -0.344164211,0.470376816,1.400572030,0.773729257,1.460133387,1.077516158,1.108647187,0.274170099
,1.342485738,0.579403979,1.243500634,1.430010707,0.860736066,0.693682990,0.618142385,1.387114421
,0.483524064,1.345897619,1.404879532,0.033265436,1.865187154,1.630960810,0.605760609,0.845518826
,1.197191313,0.666539605,0.182108907,0.492516004,1.002751117,0.966972433,1.041357960,0.999011021
,0.822397869,1.306428643,1.749352993,1.454627539,1.241401688,1.058349601,0.572809013,0.926160397
,1.356306486,1.491886742,0.994816498 -0.829695890 -8.242965297,8.748975551,1.763339911,2.687230952
,3.285688846,2.375442405,3.465965574,1.760111391,2.419550908,1.836768248,1.648712743,1.665754345
,1.152281277,1.978895429,1.246225892,0.997156630,1.528774344,1.250967337,1.062352838,0.727341748
,1.476647132,2.008474195,1.260668402)