Código
library(showtext)
library(tidyverse)
library(GetTDData)
library(knitr)
library(rbcb)
library(ggthemes)
library(urca)
library(fBasics)
library(forecast)
library(FinTS)
first_year <- 2020
last_year <- 2024ARIMA para LTN
UNIVERSIDADE FEDERAL DA PARAÍBA
O que é LTN
As Letras do Tesouro Nacional (LTNs) são títulos públicos emitidos pelo governo federal do Brasil através do Tesouro Nacional. Elas são instrumentos financeiros utilizados para captar recursos para o financiamento da dívida pública e para o financiamento de diversas atividades do governo.
LTN são títulos pre fixados, o que significa que têm uma taxa de retorno definida no momento da compra.
Pacotes Usados
Baixando Dados
Os dados foram requeridos através do pacote GetTDData, que usa a base de dados do próprio Tesouro Nacional. A série foi se inicia em 01/01/2020 até 16/06/2024, um pouco mais de 4 anos.
Código
df_td <- td_get("LTN",
first_year,
last_year)Visualização de Dados Requeridos
Código
kable(head(df_td))| ref_date | yield_bid | price_bid | asset_code | matur_date |
|---|---|---|---|---|
| 2020-01-02 | 0.0451 | 957.18 | LTN 010121 | 2021-01-01 |
| 2020-01-03 | 0.0453 | 957.16 | LTN 010121 | 2021-01-01 |
| 2020-01-06 | 0.0448 | 957.78 | LTN 010121 | 2021-01-01 |
| 2020-01-07 | 0.0448 | 957.95 | LTN 010121 | 2021-01-01 |
| 2020-01-08 | 0.0445 | 958.38 | LTN 010121 | 2021-01-01 |
| 2020-01-09 | 0.0440 | 959.00 | LTN 010121 | 2021-01-01 |
Código
showtext_auto()
font_add_google("ubuntu", "ubuntu")
ggplot(data = df_td,
aes(x = as.Date(ref_date),
y = yield_bid,
color = asset_code))+
geom_line(size = 1)+ scale_x_date() + labs(title = "", x = "Data")+
theme_calc()+
theme(legend.text = element_text(family = "ubuntu", face = "bold",
color = "#11122233", size = 8),
legend.key.size = unit(0.5, "cm"),
axis.text.y = element_text(family = "ubuntu", color = "#22233344",
size = 8, hjust = 1.25),
plot.title = element_text(family = "ubuntu", face = "bold", size = 20),
plot.subtitle = element_text(family = "ubuntu", face = "italic", size = 10),
legend.title = element_text(size = 8),
plot.margin = margin(t = 20, r = 8, b = 7, l = 5),
panel.grid = element_blank())+
labs(x = "ano",
y = "Retorno",
title = "Evolução dos Retornos da LTN",
subtitle = "(2020-2024)",
caption = "Fonte: Tesouro Direto")
Isolando LTN Mensal
A LTN mensal foi criada isolando as últimas observações de cada mês.
Código
# Isolando apenas o que será usado
LTN <- data.frame(data = as.Date(df_td$ref_date), retorno = df_td$yield_bid*100)
# Separando a Ultima Observação de cada mês
LTN_m <- LTN |>
group_by(ano_mes = format(data, "%Y-%m")) |>
slice_tail(n = 1)
# Data Frame Final com Gráfico
ltn <- LTN_m[1:2]
ggplot(data = ltn,
aes(x = data, y = retorno))+
geom_line()
Trazendo IPCA e calculando a Diferença
O IPCA foi requerido através da função rbcb, em seguida os valores mensais do IPCA foi subtraido dos valores mensais da LTN para, por fim, se chegar ao valor da Diferença entre eles.
Código
IPCA <- get_series(13522, start_date = "2020-01-01", end_date = "2024-06-01")
desvio <- data.frame(data = ltn$data, retorno = ltn$retorno - IPCA$`13522`)
desvio |>
ggplot(aes(x = data, y = retorno))+
geom_line()
Código
desvio_ts <- ts(desvio$retorno, frequency = 12, start = c(2020,01,01))Transformando a Série em Ìndice
O índice será calculado dividindo-se todos os valores por um valor de referência, que nesse caso será o primeiro valor da série no periodo 2020/01/01 que será a base 100. O objetivo é tornar a série estacionaria.
Código
desvio_indice <- (desvio$retorno / desvio$retorno[1])*100
desvio_indice <- ts(desvio_indice, frequency = 12, start = c(2020,01,01))
kable(head(desvio_indice))| x |
|---|
| 100.0000 |
| 121.3930 |
| 196.5174 |
| 233.3333 |
| 231.3433 |
| 209.9502 |
Testando a Série em Ìndice
Código
ggtsdisplay(desvio_indice)
Código
shapiro.test(desvio_indice)
Shapiro-Wilk normality test
data: desvio_indice
W = 0.92253, p-value = 0.001865Código
jarqueberaTest(desvio_indice)
Title:
Jarque - Bera Normalality Test
Test Results:
STATISTIC:
X-squared: 4.3523
P VALUE:
Asymptotic p Value: 0.1135 Código
ddicky2 <- ur.df(desvio_indice, lags = 0, type="drift")
summary(ddicky2)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1)
Residuals:
Min 1Q Median 3Q Max
-69.153 -25.522 -0.712 21.692 85.664
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.39774 8.76320 1.072 0.289
z.lag.1 -0.01938 0.03696 -0.524 0.602
Residual standard error: 36.01 on 51 degrees of freedom
Multiple R-squared: 0.005364, Adjusted R-squared: -0.01414
F-statistic: 0.275 on 1 and 51 DF, p-value: 0.6022
Value of test-statistic is: -0.5244 0.7793
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.51 -2.89 -2.58
phi1 6.70 4.71 3.86Código
#library(lmtest)
#length(residuos);length(residuos)*0.15
#gqtest(IS2, fraction=36, alternative = "greater")
#bptest(IS2)
#library(whitestrap)
#white_test(IS2)
#dwtest(IS2)
#bgtest(IS2, order=4)
#library(FinTS)
#ArchTest(residuos, lags = 4)
#### Teste de Estacionariedade do Resíduo #######
#Box.test(residuos, lag=12, type="Box-Pierce")Logarítimo do Desvio Calculado
Código
# serie temporal
log_desvio <- log(desvio_indice + sqrt(min(desvio_indice)**2)+0.001)Testando a Série em log(Ìndice)
Código
ggtsdisplay(log_desvio)
Código
shapiro.test(log_desvio)
Shapiro-Wilk normality test
data: log_desvio
W = 0.56817, p-value = 2.339e-11Código
jarqueberaTest(log_desvio)
Title:
Jarque - Bera Normalality Test
Test Results:
STATISTIC:
X-squared: 1325.4869
P VALUE:
Asymptotic p Value: < 2.2e-16 Código
ddicky2 <- ur.df(log_desvio, lags = 0, type="drift")
summary(ddicky2)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1)
Residuals:
Min 1Q Median 3Q Max
-10.5109 0.0531 0.3827 0.5418 3.7258
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.2835 0.6138 3.720 0.000497 ***
z.lag.1 -0.4765 0.1200 -3.972 0.000224 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.683 on 51 degrees of freedom
Multiple R-squared: 0.2363, Adjusted R-squared: 0.2213
F-statistic: 15.78 on 1 and 51 DF, p-value: 0.0002243
Value of test-statistic is: -3.9721 7.8947
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.51 -2.89 -2.58
phi1 6.70 4.71 3.86Número de diferenças para ter uma Série Estacionaria
Código
# Numero de diferenças necessarias
ndiffs(desvio_indice)[1] 1Testando a Série em diff(Ìndice)
Código
indice_dif <- diff(desvio_indice,1)
ggtsdisplay(indice_dif)
Código
shapiro.test(indice_dif)
Shapiro-Wilk normality test
data: indice_dif
W = 0.97584, p-value = 0.3551Código
jarqueberaTest(indice_dif)
Title:
Jarque - Bera Normalality Test
Test Results:
STATISTIC:
X-squared: 1.2166
P VALUE:
Asymptotic p Value: 0.5443 Código
ddicky2 <- ur.df(indice_dif, lags = 0, type="drift")
summary(ddicky2)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1)
Residuals:
Min 1Q Median 3Q Max
-74.077 -17.401 -4.024 20.525 85.862
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.7549 4.9234 0.763 0.449
z.lag.1 -0.7312 0.1360 -5.377 2e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 35.05 on 50 degrees of freedom
Multiple R-squared: 0.3663, Adjusted R-squared: 0.3537
F-statistic: 28.91 on 1 and 50 DF, p-value: 2.001e-06
Value of test-statistic is: -5.3766 14.4582
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.51 -2.89 -2.58
phi1 6.70 4.71 3.86Série Selecionada diff(ìndice)
Código
arima.1<- arima(indice_dif, order= c(1,1,0), method = c("CSS-ML"))
arima.1
Call:
arima(x = indice_dif, order = c(1, 1, 0), method = c("CSS-ML"))
Coefficients:
ar1
-0.4719
s.e. 0.1222
sigma^2 estimated as 1446: log likelihood = -263.1, aic = 530.21Código
previsao <- predict(arima.1, n.ahead = 12)
previsao$pred
Jan Feb Mar Apr May Jun
2024
2025 0.09418747 0.07829393 0.08579339 0.08225473 0.08392447 0.08313659
Jul Aug Sep Oct Nov Dec
2024 1.06175308 -0.37825739 0.30121972 -0.01939541 0.13188866 0.06050441
2025
$se
Jan Feb Mar Apr May Jun Jul Aug
2024 38.02790 43.00578
2025 72.76105 77.19086 81.40954 85.40678 89.23115 92.89548
Sep Oct Nov Dec
2024 51.62034 57.16339 62.98240 67.96811
2025 Código
autoplot(previsao$pred)
Testando os Residuos do Modelo
Código
residuos <- arima.1$residuals
Box.test(residuos, lag = 1)
Box-Pierce test
data: residuos
X-squared = 0.21638, df = 1, p-value = 0.6418Código
shapiro.test(residuos)
Shapiro-Wilk normality test
data: residuos
W = 0.98963, p-value = 0.9249Código
jarqueberaTest(residuos)
Title:
Jarque - Bera Normalality Test
Test Results:
STATISTIC:
X-squared: 0.3554
P VALUE:
Asymptotic p Value: 0.8372 Código
ArchTest(residuos, lags = 4)
ARCH LM-test; Null hypothesis: no ARCH effects
data: residuos
Chi-squared = 0.75406, df = 4, p-value = 0.9445Código
ddicky2 <- ur.df(residuos, lags = 0, type="drift")
summary(ddicky2)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1)
Residuals:
Min 1Q Median 3Q Max
-88.94 -29.59 -4.30 26.51 77.05
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.7426 5.3660 -0.138 0.89
z.lag.1 -1.0644 0.1416 -7.514 9.44e-10 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 38.69 on 50 degrees of freedom
Multiple R-squared: 0.5304, Adjusted R-squared: 0.521
F-statistic: 56.46 on 1 and 50 DF, p-value: 9.44e-10
Value of test-statistic is: -7.5143 28.236
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.51 -2.89 -2.58
phi1 6.70 4.71 3.86