Este tutorial apresenta uma análise prática de estacionariedade em séries temporais. Aplica-se os testes ADF, KPSS, Phillips-Perron e Zivot-Andrews, além de visualizações como gráficos de série e correlogramas. Cada teste é explicado e seus resultados são interpretados, oferecendo uma introdução e aplicação para análise de estacionariedade.
1 Testes de Estacionariedade em Séries Temporais
Uma série temporal é considerada estacionária quando suas propriedades estatísticas, como a média, variância e autocorrelação, permanecem constantes ao longo do tempo.
1.1 Definição Formal de Estacionariedade
Uma série temporal \({X_t}\) é considerada estritamente estacionária se a distribuição conjunta de \((X_{t_1}, X_{t_2}, \dots, X_{t_k})\) for a mesma que a de \((X_{t_1+h}, X_{t_2+h}, \dots, X_{t_k+h})\) para qualquer deslocamento \(h\) e qualquer conjunto de tempos \(t_1, t_2, \dots, t_k\). Em outras palavras, as propriedades estatísticas da série não dependem do tempo.
Na prática, utilizamos o conceito de estacionariedade fraca (ou estacionariedade de segunda ordem), que requer as seguintes condições:
1. Média constante: \(\mathbb{E}[X_t] = \mu\), para todo \(t\).
2. Variância constante e finita: \(\text{Var}(X_t) = \sigma^2\), para todo \(t\).
3. Autocovariância que depende apenas da defasagem (lag): \(\text{Cov}(X_t, X_{t+h}) = \gamma(h)\), onde \(\gamma(h)\) é uma função que depende apenas da defasagem \(h\), e não do tempo \(t\).
Essas condições garantem que as propriedades estatísticas da série, como a média, a variância e a covariância, não mudam ao longo do tempo.
Testes de estacionariedade são fundamentais para analisar a adequação de modelos temporais, como ARIMA.
Nesta aula, explora-se alguns dos principais testes de estacionariedade e como aplicá-los em R:
Testes ADF (Dickey-Fuller aumentado)
Testes KPSS
Testes PP (Phillips-Perron)
Teste com quebra estrutural: Zivot-Andrews
1.2 Visualização de Séries Temporais
Vamos primeiro visualizar as séries temporais que serão utilizadas nesta aula. Para isso, vamos usar a série AirPassengers, que é fornecida no R e a série de produção industrial do Brasil (com e sem ajuste sazonal)
# Carregar a série AirPassengersdata("AirPassengers")plot(AirPassengers, main="Série AirPassengers", ylab="Passageiros", xlab="Ano")
# usando a série sazonalmente ajustadalibrary(seasonal)passageiros_sa <-final(seas(AirPassengers))# Visualizando a série temporalplot(passageiros_sa, main="Passageiros Aéreos Sazonalmente Ajustada", ylab="Produção", xlab="Ano")
# Leitura dos dados da Produção Industriallibrary(readxl)dados <-read_excel("tabela8888.xlsx")# Preparando a série temporalpimpf <-ts(dados$pimpf, start=c(2002,1), frequency=12)# Visualizando a série temporalplot(pimpf, main="Produção Industrial Brasileira", ylab="Produção", xlab="Ano")
# usando a série sazonalmente ajustadalibrary(seasonal)pimpf_sa <-final(seas(pimpf))# Visualizando a série temporalplot(pimpf_sa, main="Produção Industrial Brasileira Sazonalmente Ajustada", ylab="Produção", xlab="Ano")
Agora vamos examinar o Correlograma (ACF) que ajuda a identificar a estacionariedade observando os padrões de autocorrelação.
Observe os gráficos anteriores. Em séries não estacionárias, a função de autocorrelação tende a decair lentamente, indicando a presença de tendências ou sazonalidades. Na sequência é apresentado a função de autocorrelação para a primeira diferença.
Nesta seção, vamos introduzir os principais testes de estacionariedade e como interpretá-los.
1.3.1 Teste Dickey-Fuller Aumentado (ADF)
O Teste Dickey-Fuller Aumentado (ADF) é um dos testes mais comuns para verificar a estacionariedade de uma série temporal. A equação básica para o teste ADF é:
Onde: - \(\Delta X_t\) é a primeira diferença da série \(X_t\). - \(\alpha\) é uma constante (ou termo de drift). - \(\beta\) é o coeficiente do termo de tendência. - \(\gamma\) é o coeficiente que estamos interessados em testar. - \(\delta_i\) são os coeficientes das defasagens das primeiras diferenças para lidar com autocorrelação. - \(\epsilon_t\) é o erro branco.
O teste ADF verifica a presença de uma raiz unitária, ou seja, se a série é não estacionária.
Hipótese Nula (\(H_0\)): A série tem uma raiz unitária (não estacionária).\(H_0: \gamma = 0\)
Hipótese Alternativa (\(H_1\)): A série é estacionária.\(H_1: \gamma < 0\)
Se rejeitamos a hipótese nula \(H_0\), concluímos que a série é estacionária.
Se não rejeitamos \(H_0\), a série apresenta uma raiz unitária, ou seja, é não estacionária.
O teste ADF é uma extensão do teste Dickey-Fuller original, que inclui defasagens das diferenças para capturar possíveis autocorrelações na série residual e melhorar a robustez do teste.
# Instalar e carregar pacotes necessárioslibrary(urca)# Aplicar o teste ADFadf_test_1a <-ur.df(passageiros_sa, type ="none")adf_test_1b <-ur.df(passageiros_sa, type ="drift")adf_test_1c <-ur.df(passageiros_sa, type ="trend")summary(adf_test_1a)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-17.2858 -3.0838 0.3265 3.5808 18.1680
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 0.010016 0.001708 5.865 3.08e-08 ***
z.diff.lag -0.172151 0.084423 -2.039 0.0433 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.475 on 140 degrees of freedom
Multiple R-squared: 0.1991, Adjusted R-squared: 0.1877
F-statistic: 17.4 on 2 and 140 DF, p-value: 1.781e-07
Value of test-statistic is: 5.8655
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62
summary(adf_test_1b)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-17.8816 -3.3550 -0.0715 3.6738 18.7181
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.499921 1.261133 1.189 0.236
z.lag.1 0.005396 0.004242 1.272 0.205
z.diff.lag -0.175820 0.084354 -2.084 0.039 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.467 on 139 degrees of freedom
Multiple R-squared: 0.03677, Adjusted R-squared: 0.02291
F-statistic: 2.653 on 2 and 139 DF, p-value: 0.07401
Value of test-statistic is: 1.2722 17.9601
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.46 -2.88 -2.57
phi1 6.52 4.63 3.81
summary(adf_test_1c)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression trend
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-16.7016 -3.5280 0.0256 3.1695 19.0653
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.02286 3.05782 2.951 0.00373 **
z.lag.1 -0.08378 0.03343 -2.507 0.01335 *
tt 0.23896 0.08887 2.689 0.00805 **
z.diff.lag -0.13901 0.08365 -1.662 0.09882 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.348 on 138 degrees of freedom
Multiple R-squared: 0.08472, Adjusted R-squared: 0.06482
F-statistic: 4.258 on 3 and 138 DF, p-value: 0.006543
Value of test-statistic is: -2.5066 14.9199 4.4604
Critical values for test statistics:
1pct 5pct 10pct
tau3 -3.99 -3.43 -3.13
phi2 6.22 4.75 4.07
phi3 8.43 6.49 5.47
adf_test_2a <-ur.df(pimpf_sa, type ="none")adf_test_2b <-ur.df(pimpf_sa, type ="drift")adf_test_2c <-ur.df(pimpf_sa, type ="trend")summary(adf_test_2a)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-20.4050 -0.4450 0.1030 0.5846 10.9919
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 0.0001715 0.0012137 0.141 0.888
z.diff.lag 0.0310667 0.0613735 0.506 0.613
Residual standard error: 2.11 on 266 degrees of freedom
Multiple R-squared: 0.001053, Adjusted R-squared: -0.006458
F-statistic: 0.1401 on 2 and 266 DF, p-value: 0.8693
Value of test-statistic is: 0.1413
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62
summary(adf_test_2b)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-20.6722 -0.4803 0.0800 0.5933 10.6867
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.80375 1.57959 2.408 0.0167 *
z.lag.1 -0.03553 0.01487 -2.389 0.0176 *
z.diff.lag 0.04578 0.06113 0.749 0.4546
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.091 on 265 degrees of freedom
Multiple R-squared: 0.02201, Adjusted R-squared: 0.01463
F-statistic: 2.982 on 2 and 265 DF, p-value: 0.05239
Value of test-statistic is: -2.3886 2.9095
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.44 -2.87 -2.57
phi1 6.47 4.61 3.79
summary(adf_test_2c)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression trend
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-20.5434 -0.4582 0.0626 0.5826 10.7417
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.235914 1.630949 2.597 0.00993 **
z.lag.1 -0.037352 0.014970 -2.495 0.01320 *
tt -0.001765 0.001664 -1.061 0.28988
z.diff.lag 0.042923 0.061178 0.702 0.48355
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.09 on 264 degrees of freedom
Multiple R-squared: 0.02616, Adjusted R-squared: 0.01509
F-statistic: 2.364 on 3 and 264 DF, p-value: 0.07154
Value of test-statistic is: -2.4952 2.3155 3.4165
Critical values for test statistics:
1pct 5pct 10pct
tau3 -3.98 -3.42 -3.13
phi2 6.15 4.71 4.05
phi3 8.34 6.30 5.36
Qual foram as conclusões encontradas?
Vamos testar agora a estacionariedade da primeira diferença:
# Instalar e carregar pacotes necessárioslibrary(urca)# Aplicar o teste ADFadf_test_diff_1a <-ur.df(diff(passageiros_sa), type ="none")adf_test_diff_1b <-ur.df(diff(passageiros_sa), type ="drift")adf_test_diff_1c <-ur.df(diff(passageiros_sa), type ="trend")summary(adf_test_diff_1a)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-19.068 -1.323 1.944 5.421 24.290
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 -0.79216 0.11541 -6.864 2.04e-10 ***
z.diff.lag -0.16821 0.08362 -2.012 0.0462 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 6.045 on 139 degrees of freedom
Multiple R-squared: 0.4908, Adjusted R-squared: 0.4834
F-statistic: 66.98 on 2 and 139 DF, p-value: < 2.2e-16
Value of test-statistic is: -6.8638
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62
summary(adf_test_diff_1b)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-18.6473 -3.3138 -0.1185 3.4084 19.4456
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.10562 0.57392 5.411 2.69e-07 ***
z.lag.1 -1.20643 0.13011 -9.272 3.33e-16 ***
z.diff.lag 0.03944 0.08534 0.462 0.645
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.51 on 138 degrees of freedom
Multiple R-squared: 0.5799, Adjusted R-squared: 0.5738
F-statistic: 95.24 on 2 and 138 DF, p-value: < 2.2e-16
Value of test-statistic is: -9.2723 42.9893
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.46 -2.88 -2.57
phi1 6.52 4.63 3.81
summary(adf_test_diff_1c)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression trend
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-17.5737 -3.1138 0.0527 3.7282 17.9902
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.85789 0.95427 1.947 0.0536 .
z.lag.1 -1.24870 0.13191 -9.467 <2e-16 ***
tt 0.01885 0.01156 1.631 0.1052
z.diff.lag 0.06099 0.08585 0.710 0.4786
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.477 on 137 degrees of freedom
Multiple R-squared: 0.5879, Adjusted R-squared: 0.5789
F-statistic: 65.15 on 3 and 137 DF, p-value: < 2.2e-16
Value of test-statistic is: -9.4666 29.8912 44.8353
Critical values for test statistics:
1pct 5pct 10pct
tau3 -3.99 -3.43 -3.13
phi2 6.22 4.75 4.07
phi3 8.43 6.49 5.47
adf_test_diff_2a <-ur.df(diff(pimpf_sa), type ="none")adf_test_diff_2b <-ur.df(diff(pimpf_sa), type ="drift")adf_test_diff_2c <-ur.df(diff(pimpf_sa), type ="trend")summary(adf_test_diff_2a)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression none
Call:
lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-20.3731 -0.4334 0.1249 0.6326 11.0360
Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1 -1.03223 0.08538 -12.089 <2e-16 ***
z.diff.lag 0.06535 0.06137 1.065 0.288
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.109 on 265 degrees of freedom
Multiple R-squared: 0.4861, Adjusted R-squared: 0.4823
F-statistic: 125.4 on 2 and 265 DF, p-value: < 2.2e-16
Value of test-statistic is: -12.0893
Critical values for test statistics:
1pct 5pct 10pct
tau1 -2.58 -1.95 -1.62
summary(adf_test_diff_2b)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression drift
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-20.4172 -0.4769 0.0812 0.5893 10.9886
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.04356 0.12932 0.337 0.737
z.lag.1 -1.03299 0.08556 -12.074 <2e-16 ***
z.diff.lag 0.06575 0.06148 1.070 0.286
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.112 on 264 degrees of freedom
Multiple R-squared: 0.4864, Adjusted R-squared: 0.4825
F-statistic: 125 on 2 and 264 DF, p-value: < 2.2e-16
Value of test-statistic is: -12.0737 72.8874
Critical values for test statistics:
1pct 5pct 10pct
tau2 -3.44 -2.87 -2.57
phi1 6.47 4.61 3.79
summary(adf_test_diff_2c)
###############################################
# Augmented Dickey-Fuller Test Unit Root Test #
###############################################
Test regression trend
Call:
lm(formula = z.diff ~ z.lag.1 + 1 + tt + z.diff.lag)
Residuals:
Min 1Q Median 3Q Max
-20.3090 -0.4160 0.0847 0.5210 11.0427
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.226406 0.261722 0.865 0.388
z.lag.1 -1.038220 0.085861 -12.092 <2e-16 ***
tt -0.001353 0.001683 -0.804 0.422
z.diff.lag 0.068364 0.061607 1.110 0.268
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.114 on 263 degrees of freedom
Multiple R-squared: 0.4876, Adjusted R-squared: 0.4818
F-statistic: 83.43 on 3 and 263 DF, p-value: < 2.2e-16
Value of test-statistic is: -12.0918 48.7418 73.1125
Critical values for test statistics:
1pct 5pct 10pct
tau3 -3.98 -3.42 -3.13
phi2 6.15 4.71 4.05
phi3 8.34 6.30 5.36
1.3.2 Teste KPSS
Diferente do ADF, o teste KPSS tem como hipótese nula que a série é estacionária. Isso complementa os resultados do ADF.
# Teste KPSSlibrary(urca)kpss_test_1a <-ur.kpss(passageiros_sa,type="mu")kpss_test_1b <-ur.kpss(passageiros_sa,type="tau")summary(kpss_test_1a)
#######################
# KPSS Unit Root Test #
#######################
Test is of type: mu with 4 lags.
Value of test-statistic is: 2.9471
Critical value for a significance level of:
10pct 5pct 2.5pct 1pct
critical values 0.347 0.463 0.574 0.739
summary(kpss_test_1b)
#######################
# KPSS Unit Root Test #
#######################
Test is of type: tau with 4 lags.
Value of test-statistic is: 0.4692
Critical value for a significance level of:
10pct 5pct 2.5pct 1pct
critical values 0.119 0.146 0.176 0.216
#######################
# KPSS Unit Root Test #
#######################
Test is of type: mu with 5 lags.
Value of test-statistic is: 0.9226
Critical value for a significance level of:
10pct 5pct 2.5pct 1pct
critical values 0.347 0.463 0.574 0.739
summary(kpss_test_2b)
#######################
# KPSS Unit Root Test #
#######################
Test is of type: tau with 5 lags.
Value of test-statistic is: 0.8245
Critical value for a significance level of:
10pct 5pct 2.5pct 1pct
critical values 0.119 0.146 0.176 0.216
No teste KPSS, se o valor-p for menor que 0,05, rejeitamos a hipótese nula de estacionariedade. Assim, uma série não estacionária será identificada com valores baixos de p.
1.3.3 Teste Phillips-Perron (PP)
O teste PP é uma variação do teste Dickey-Fuller e corrige para heterocedasticidade na série. Ele é útil em séries com variância não constante.
# Teste Phillips-Perronpp_test_1a <-ur.pp(passageiros_sa, model ="constant")pp_test_1b <-ur.pp(passageiros_sa, model ="trend")summary(pp_test_1a)
##################################
# Phillips-Perron Unit Root Test #
##################################
Test regression with intercept
Call:
lm(formula = y ~ y.l1)
Residuals:
Min 1Q Median 3Q Max
-20.529 -3.194 -0.116 3.714 20.291
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.408422 1.257980 1.12 0.265
y.l1 1.004102 0.004201 238.99 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.512 on 141 degrees of freedom
Multiple R-squared: 0.9975, Adjusted R-squared: 0.9975
F-statistic: 5.712e+04 on 1 and 141 DF, p-value: < 2.2e-16
Value of test-statistic, type: Z-alpha is: 0.665
aux. Z statistics
Z-tau-mu 1.3381
summary(pp_test_1b)
##################################
# Phillips-Perron Unit Root Test #
##################################
Test regression with intercept and trend
Call:
lm(formula = y ~ y.l1 + trend)
Residuals:
Min 1Q Median 3Q Max
-18.7575 -3.2299 0.2098 3.2261 20.3375
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 27.41522 8.99673 3.047 0.00276 **
y.l1 0.91030 0.03241 28.090 < 2e-16 ***
trend 0.25130 0.08612 2.918 0.00411 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.371 on 140 degrees of freedom
Multiple R-squared: 0.9977, Adjusted R-squared: 0.9976
F-statistic: 3.008e+04 on 2 and 140 DF, p-value: < 2.2e-16
Value of test-statistic, type: Z-alpha is: -9.5524
aux. Z statistics
Z-tau-mu 4.4969
Z-tau-beta 2.6323
pp_test_2a <-ur.pp(pimpf_sa, model ="constant")pp_test_2b <-ur.pp(pimpf_sa, model ="trend")summary(pp_test_2a)
##################################
# Phillips-Perron Unit Root Test #
##################################
Test regression with intercept
Call:
lm(formula = y ~ y.l1)
Residuals:
Min 1Q Median 3Q Max
-20.7417 -0.4638 0.0917 0.5916 10.2023
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.68430 1.55865 2.364 0.0188 *
y.l1 0.96562 0.01468 65.772 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.085 on 267 degrees of freedom
Multiple R-squared: 0.9419, Adjusted R-squared: 0.9416
F-statistic: 4326 on 1 and 267 DF, p-value: < 2.2e-16
Value of test-statistic, type: Z-alpha is: -8.5195
aux. Z statistics
Z-tau-mu 2.2893
summary(pp_test_2b)
##################################
# Phillips-Perron Unit Root Test #
##################################
Test regression with intercept and trend
Call:
lm(formula = y ~ y.l1 + trend)
Residuals:
Min 1Q Median 3Q Max
-20.6043 -0.4249 0.0752 0.5878 10.2920
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.868141 1.567270 2.468 0.0142 *
y.l1 0.963887 0.014762 65.294 <2e-16 ***
trend -0.001790 0.001646 -1.087 0.2780
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.084 on 266 degrees of freedom
Multiple R-squared: 0.9421, Adjusted R-squared: 0.9417
F-statistic: 2165 on 2 and 266 DF, p-value: < 2.2e-16
Value of test-statistic, type: Z-alpha is: -8.8466
aux. Z statistics
Z-tau-mu 2.8467
Z-tau-beta -1.1426
1.4 Testes de Estacionariedade com Quebra Estrutural (Zivot-Andrews)
Quando há mudanças estruturais em uma série, como crises econômicas ou mudanças de regime, os testes tradicionais de estacionariedade podem não ser adequados. O teste de Zivot-Andrews lida com essas quebras estruturais.
O teste de Zivot-Andrews permite uma quebra estrutural no processo da série temporal, o que pode melhorar a identificação de estacionariedade em séries com eventos marcantes.
O Teste Zivot-Andrews testa a hipótese nula de que a série tem uma raiz unitária com uma possível quebra estrutural em algum ponto do tempo. Ele é útil quando há eventos que podem causar mudanças abruptas no comportamento da série (como crises econômicas, mudanças políticas, etc.).
Hipóteses: Hipótese Nula (\(H_0\)): A série tem uma raiz unitária com uma quebra estrutural. Hipótese Alternativa (\(H_1\)): A série é estacionária com uma quebra estrutural.
# Teste de Zivot-Andrewsza_test_1a <-ur.za(passageiros_sa, model ="intercept")za_test_1b <-ur.za(passageiros_sa, model ="trend")za_test_1c <-ur.za(passageiros_sa, model ="both")summary(za_test_1a)
################################
# Zivot-Andrews Unit Root Test #
################################
Call:
lm(formula = testmat)
Residuals:
Min 1Q Median 3Q Max
-19.1081 -2.7904 0.4833 2.8499 19.1576
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 13.34275 3.21420 4.151 5.74e-05 ***
y.l1 0.85844 0.03649 23.526 < 2e-16 ***
trend 0.43985 0.10699 4.111 6.70e-05 ***
du -5.38306 1.89114 -2.846 0.00509 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.239 on 139 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9978, Adjusted R-squared: 0.9978
F-statistic: 2.108e+04 on 3 and 139 DF, p-value: < 2.2e-16
Teststatistic: -3.8796
Critical values: 0.01= -5.34 0.05= -4.8 0.1= -4.58
Potential break point at position: 52
summary(za_test_1b)
################################
# Zivot-Andrews Unit Root Test #
################################
Call:
lm(formula = testmat)
Residuals:
Min 1Q Median 3Q Max
-17.3688 -2.8397 0.2805 2.9601 18.4119
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 23.04645 5.53154 4.166 5.41e-05 ***
y.l1 0.79936 0.04934 16.201 < 2e-16 ***
trend 0.41739 0.10129 4.121 6.46e-05 ***
dt 0.20232 0.06916 2.926 0.00402 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.231 on 139 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9978, Adjusted R-squared: 0.9978
F-statistic: 2.114e+04 on 3 and 139 DF, p-value: < 2.2e-16
Teststatistic: -4.0665
Critical values: 0.01= -4.93 0.05= -4.42 0.1= -4.11
Potential break point at position: 61
summary(za_test_1c)
################################
# Zivot-Andrews Unit Root Test #
################################
Call:
lm(formula = testmat)
Residuals:
Min 1Q Median 3Q Max
-17.8053 -2.8958 0.5532 2.8325 18.0546
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 23.23639 5.32722 4.362 2.51e-05 ***
y.l1 0.78590 0.04835 16.255 < 2e-16 ***
trend 0.50736 0.11297 4.491 1.48e-05 ***
du -4.57457 1.91270 -2.392 0.0181 *
dt 0.16514 0.06886 2.398 0.0178 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.184 on 138 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9979, Adjusted R-squared: 0.9978
F-statistic: 1.615e+04 on 4 and 138 DF, p-value: < 2.2e-16
Teststatistic: -4.4281
Critical values: 0.01= -5.57 0.05= -5.08 0.1= -4.82
Potential break point at position: 53
za_test_2a <-ur.za(pimpf_sa, model ="intercept")za_test_2b <-ur.za(pimpf_sa, model ="trend")za_test_2c <-ur.za(pimpf_sa, model ="both")summary(za_test_2a)
################################
# Zivot-Andrews Unit Root Test #
################################
Call:
lm(formula = testmat)
Residuals:
Min 1Q Median 3Q Max
-20.6559 -0.4236 0.0713 0.5759 10.2646
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.488694 2.111782 4.020 7.6e-05 ***
y.l1 0.916657 0.020955 43.743 < 2e-16 ***
trend 0.009718 0.004021 2.417 0.01633 *
du -2.211884 0.707404 -3.127 0.00196 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.051 on 265 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9442, Adjusted R-squared: 0.9436
F-statistic: 1494 on 3 and 265 DF, p-value: < 2.2e-16
Teststatistic: -3.9772
Critical values: 0.01= -5.34 0.05= -4.8 0.1= -4.58
Potential break point at position: 155
summary(za_test_2b)
################################
# Zivot-Andrews Unit Root Test #
################################
Call:
lm(formula = testmat)
Residuals:
Min 1Q Median 3Q Max
-20.7282 -0.3928 0.0851 0.5972 9.7563
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.89629 2.13825 3.693 0.000269 ***
y.l1 0.91257 0.02428 37.593 < 2e-16 ***
trend 0.03515 0.01406 2.501 0.013006 *
dt -0.04435 0.01676 -2.646 0.008641 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.061 on 265 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9436, Adjusted R-squared: 0.943
F-statistic: 1478 on 3 and 265 DF, p-value: < 2.2e-16
Teststatistic: -3.6014
Critical values: 0.01= -4.93 0.05= -4.42 0.1= -4.11
Potential break point at position: 68
summary(za_test_2c)
################################
# Zivot-Andrews Unit Root Test #
################################
Call:
lm(formula = testmat)
Residuals:
Min 1Q Median 3Q Max
-20.7446 -0.4404 0.0682 0.6156 9.7599
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.159091 2.624501 4.252 2.94e-05 ***
y.l1 0.884506 0.027812 31.803 < 2e-16 ***
trend 0.021625 0.007035 3.074 0.002334 **
du -2.494599 0.679851 -3.669 0.000294 ***
dt -0.023726 0.010282 -2.307 0.021806 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.039 on 264 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.945, Adjusted R-squared: 0.9442
F-statistic: 1135 on 4 and 264 DF, p-value: < 2.2e-16
Teststatistic: -4.1526
Critical values: 0.01= -5.57 0.05= -5.08 0.1= -4.82
Potential break point at position: 146
O teste Zivot-Andrews considera uma quebra estrutural na série. Se o valor-p for menor que 0,05, rejeitamos a hipótese nula de raiz unitária com quebra, indicando que a série é estacionária, possivelmente com uma quebra.
1.5 Considerações finais
Nesta aula, aprendemos sobre os principais testes de estacionariedade aplicados em séries temporais. Vimos que:
- O teste ADF busca raízes unitárias diretamente.
- O teste KPSS avalia a estacionariedade diretamente, com uma hipótese nula oposta ao ADF.
- O teste PP é uma versão robusta do ADF em relação à heterocedasticidade.
- O teste Zivot-Andrews é utilizado quando há suspeitas de quebra estrutural.
Cada teste tem suas peculiaridades, e é importante avaliar os diferentes resultados para fazer um diagnóstico robusto da série temporal.
