AR(8) Model for DowJones
Il primo modello che specifichiamo per i rendimenti del DowJones è un modello autoregressivo di ordine 8. La specificazione del modello:
\[y_t = \alpha_0 + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \alpha_3 y_{t-3} + \alpha_4 y_{t-4} + \\ + \alpha_5 y_{t-5} + \alpha_6 y_{t-6} + \alpha_7 y_{t-7} + \alpha_8 y_{t-8} + u_t \]
Le assunzioni sul termine di errore:
- \(E(u_t) = 0\)
- \(E(u_t u_{t-k} ) = 0\)
- \(E(u_t^2 ) = \sigma^2 \, costant\)
lag1 = Lag(r_DJ, 1)
lag2 = Lag(r_DJ, 2)
lag3 = Lag(r_DJ, 3)
lag4 = Lag(r_DJ, 4)
lag5 = Lag(r_DJ, 5)
lag6 = Lag(r_DJ, 6)
lag7 = Lag(r_DJ, 7)
lag8 = Lag(r_DJ, 8)
df = cbind(r_DJ, lag1, lag2, lag3, lag4, lag5, lag6, lag7, lag8) %>% na.omit()
ar8 = lm(r_DJ ~ Lag.1 + Lag.2 + Lag.3 + Lag.4 + Lag.5 + Lag.6 + Lag.7 + Lag.8, data = df)
broom::tidy(ar8) %>% mutate_if(is.numeric, round, 2) %>% knitr::kable(caption = "Modello AR(8) Dowjones") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1), font_size = 14) %>%
row_spec(c(1), bold = T, color = "black", background = "yellow") %>% column_spec(c(1,
2, 3, 4, 5), background = "white", color = "black")
Modello AR(8) Dowjones
|
term
|
estimate
|
std.error
|
statistic
|
p.value
|
|
(Intercept)
|
0.03
|
0.01
|
3.22
|
0.00
|
|
Lag.1
|
-0.02
|
0.01
|
-2.21
|
0.03
|
|
Lag.2
|
-0.01
|
0.01
|
-1.07
|
0.28
|
|
Lag.3
|
0.00
|
0.01
|
-0.52
|
0.60
|
|
Lag.4
|
-0.02
|
0.01
|
-2.77
|
0.01
|
|
Lag.5
|
0.00
|
0.01
|
0.52
|
0.60
|
|
Lag.6
|
-0.04
|
0.01
|
-4.34
|
0.00
|
|
Lag.7
|
0.01
|
0.01
|
1.65
|
0.10
|
|
Lag.8
|
-0.02
|
0.01
|
-1.77
|
0.08
|
broom::glance(ar8)[, -c(11, 12)] %>% knitr::kable(caption = "Modello AR(8) Dowjones") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1), font_size = 14) %>%
row_spec(c(1), bold = T, color = "black", background = "yellow") %>% column_spec(c(1:10),
background = "white", color = "black")
Modello AR(8) Dowjones
|
r.squared
|
adj.r.squared
|
sigma
|
statistic
|
p.value
|
df
|
logLik
|
AIC
|
BIC
|
deviance
|
|
0.0029278
|
0.0023275
|
1.070122
|
4.877377
|
5e-06
|
8
|
-19764.3
|
39548.6
|
39623.55
|
15216.9
|
Test sulla significatività di tutti i coefficenti
Usiamo gli errori corretti per l’eteroschedasicità.
car::linearHypothesis(ar8, c("Lag.1=0", "Lag.2=0", "Lag.3=0", "Lag.4=0", "Lag.5=0",
"Lag.6=0", "Lag.7=0", "Lag.8=0"), white.adjust = "hc0") %>% broom::tidy() %>%
knitr::kable(caption = "Test F sui Coefficenti AR(8)") %>% kableExtra::kable_classic() %>%
kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>% row_spec(c(2), bold = T,
color = "black", background = "yellow") %>% column_spec(c(1, 2, 4), background = "white",
color = "black")
Test F sui Coefficenti AR(8)
|
res.df
|
df
|
statistic
|
p.value
|
|
13296
|
NA
|
NA
|
NA
|
|
13288
|
8
|
0.734854
|
0.6607994
|
Come possiamo vedere usando gli errori corretti abbiamo un non rifiuto dell’ipotesi nulla, di conseguenza i coefficenti sono uguali a zero e dobbiamo ricercare un modello migliore.
Test autocorrelazione dei residui fino al 8 ordine
Construction of a Breush Godfrey
u_hat_ar8 = residuals(ar8)
lag1_u_hat_ar8 = Lag(u_hat_ar8, 1)
lag2_u_hat_ar8 = Lag(u_hat_ar8, 2)
lag3_u_hat_ar8 = Lag(u_hat_ar8, 3)
lag4_u_hat_ar8 = Lag(u_hat_ar8, 4)
lag5_u_hat_ar8 = Lag(u_hat_ar8, 5)
lag6_u_hat_ar8 = Lag(u_hat_ar8, 6)
lag7_u_hat_ar8 = Lag(u_hat_ar8, 7)
lag8_u_hat_ar8 = Lag(u_hat_ar8, 8)
df_u_hat_ar8 = cbind(u_hat_ar8, lag1_u_hat_ar8, lag2_u_hat_ar8, lag3_u_hat_ar8, lag4_u_hat_ar8,
lag5_u_hat_ar8, lag6_u_hat_ar8, lag7_u_hat_ar8, lag8_u_hat_ar8) %>% na.omit() %>%
as_tibble()
mod1_u_hat_ar8 = lm(u_hat_ar8 ~ Lag.1 + Lag.2, data = df_u_hat_ar8)
mod2_u_hat_ar8 = lm(u_hat_ar8 ~ Lag.1 + Lag.2 + Lag.3 + Lag.4 + Lag.5, data = df_u_hat_ar8)
mod3_u_hat_ar8 = lm(u_hat_ar8 ~ Lag.1 + Lag.2 + Lag.3 + Lag.4 + Lag.5 + Lag.6 + Lag.7 +
Lag.8, data = df_u_hat_ar8)
car::linearHypothesis(mod1_u_hat_ar8, c("Lag.1=0", "Lag.2=0")) %>% broom::tidy() %>%
knitr::kable(caption = "Test F residui Breush Godfrey AR(8) 2 Lag") %>% kableExtra::kable_classic() %>%
kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>% row_spec(c(2), bold = T,
color = "black", background = "yellow") %>% column_spec(c(1, 2, 3, 4), background = "white",
color = "black")
Test F residui Breush Godfrey AR(8) 2 Lag
|
res.df
|
rss
|
df
|
sumsq
|
statistic
|
p.value
|
|
13288
|
15213.37
|
NA
|
NA
|
NA
|
NA
|
|
13286
|
15213.37
|
2
|
0.0013394
|
0.0005849
|
0.9994153
|
car::linearHypothesis(mod2_u_hat_ar8, c("Lag.1=0", "Lag.2=0", "Lag.3=0", "Lag.4=0",
"Lag.5=0")) %>% knitr::kable(caption = "Test F residui Breush Godfrey AR(8) 5 Lag") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>%
row_spec(c(2), bold = T, color = "black", background = "yellow") %>% column_spec(c(1,
2, 3, 4), background = "white", color = "black")
Test F residui Breush Godfrey AR(8) 5 Lag
|
Res.Df
|
RSS
|
Df
|
Sum of Sq
|
F
|
Pr(>F)
|
|
13288
|
15213.37
|
NA
|
NA
|
NA
|
NA
|
|
13283
|
15213.36
|
5
|
0.0116396
|
0.0020325
|
0.9999994
|
car::linearHypothesis(mod3_u_hat_ar8, c("Lag.1=0", "Lag.2=0", "Lag.3=0", "Lag.4=0",
"Lag.5=0", "Lag.6=0", "Lag.7=0", "Lag.8=0")) %>% knitr::kable(caption = "Test F residui Breush Godfrey AR(8) 8 Lag") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>%
row_spec(c(2), bold = T, color = "black", background = "yellow") %>% column_spec(c(1,
2, 3, 4), background = "white", color = "black")
Test F residui Breush Godfrey AR(8) 8 Lag
|
Res.Df
|
RSS
|
Df
|
Sum of Sq
|
F
|
Pr(>F)
|
|
13288
|
15213.37
|
NA
|
NA
|
NA
|
NA
|
|
13280
|
15213.32
|
8
|
0.0450503
|
0.0049157
|
1
|
Come possiamo vedere da tutti e tre i test sembra che i residui non siano serialmente autocorrelati, tuttavia dovremmo ripetere il test considerando i quadrati dei residui.
u2_hat_ar8 = residuals(ar8)^2
lag1_u2_hat_ar8 = Lag(u2_hat_ar8, 1)
lag2_u2_hat_ar8 = Lag(u2_hat_ar8, 2)
lag3_u2_hat_ar8 = Lag(u2_hat_ar8, 3)
lag4_u2_hat_ar8 = Lag(u2_hat_ar8, 4)
lag5_u2_hat_ar8 = Lag(u2_hat_ar8, 5)
lag6_u2_hat_ar8 = Lag(u2_hat_ar8, 6)
lag7_u2_hat_ar8 = Lag(u2_hat_ar8, 7)
lag8_u2_hat_ar8 = Lag(u2_hat_ar8, 8)
df_u2_hat_ar8 = cbind(u2_hat_ar8, lag1_u2_hat_ar8, lag2_u2_hat_ar8, lag3_u2_hat_ar8,
lag4_u2_hat_ar8, lag5_u2_hat_ar8, lag6_u2_hat_ar8, lag7_u2_hat_ar8, lag8_u2_hat_ar8) %>%
na.omit() %>% as_tibble()
mod1_u2_hat_ar8 = lm(u2_hat_ar8 ~ Lag.1 + Lag.2, data = df_u2_hat_ar8)
mod2_u2_hat_ar8 = lm(u2_hat_ar8 ~ Lag.1 + Lag.2 + Lag.3 + Lag.4 + Lag.5, data = df_u2_hat_ar8)
mod3_u2_hat_ar8 = lm(u2_hat_ar8 ~ Lag.1 + Lag.2 + Lag.3 + Lag.4 + Lag.5 + Lag.6 +
Lag.7 + Lag.8, data = df_u2_hat_ar8)
car::linearHypothesis(mod1_u2_hat_ar8, c("Lag.1=0", "Lag.2=0")) %>% broom::tidy() %>%
knitr::kable(caption = "Test F quadrati Breush Godfrey AR(8) 2 Lag") %>% kableExtra::kable_classic() %>%
kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>% row_spec(c(2), bold = T,
color = "black", background = "yellow") %>% column_spec(c(1, 2, 3, 4), background = "white",
color = "black")
Test F quadrati Breush Godfrey AR(8) 2 Lag
|
res.df
|
rss
|
df
|
sumsq
|
statistic
|
p.value
|
|
13288
|
646971.8
|
NA
|
NA
|
NA
|
NA
|
|
13286
|
607223.9
|
2
|
39747.88
|
434.8399
|
0
|
car::linearHypothesis(mod2_u2_hat_ar8, c("Lag.1=0", "Lag.2=0", "Lag.3=0", "Lag.4=0",
"Lag.5=0")) %>% knitr::kable(caption = "Test F quadrati Breush Godfrey AR(8) 5 Lag") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>%
row_spec(c(2), bold = T, color = "black", background = "yellow") %>% column_spec(c(1,
2, 3, 4), background = "white", color = "black")
Test F quadrati Breush Godfrey AR(8) 5 Lag
|
Res.Df
|
RSS
|
Df
|
Sum of Sq
|
F
|
Pr(>F)
|
|
13288
|
646971.8
|
NA
|
NA
|
NA
|
NA
|
|
13283
|
596242.4
|
5
|
50729.39
|
226.0284
|
0
|
car::linearHypothesis(mod3_u2_hat_ar8, c("Lag.1=0", "Lag.2=0", "Lag.3=0", "Lag.4=0",
"Lag.5=0", "Lag.6=0", "Lag.7=0", "Lag.8=0")) %>% knitr::kable(caption = "Test F quadrati Breush Godfrey AR(8) 8 Lag") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>%
row_spec(c(2), bold = T, color = "black", background = "yellow") %>% column_spec(c(1,
2, 3, 4), background = "white", color = "black")
Test F quadrati Breush Godfrey AR(8) 8 Lag
|
Res.Df
|
RSS
|
Df
|
Sum of Sq
|
F
|
Pr(>F)
|
|
13288
|
646971.8
|
NA
|
NA
|
NA
|
NA
|
|
13280
|
593759.5
|
8
|
53212.24
|
148.7678
|
0
|
Come possiamo vedere dai test i quadrati dei residui risultano correlati, il che rappresenta un ulteriore problema di mispecificazione del modello.
Modello AR(1)-GARCH(1,1)
garchSpec1 <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,
1)), mean.model = list(armaOrder = c(1, 0)), distribution.model = "norm")
garch11 <- ugarchfit(spec = garchSpec1, data = na.omit(r_DJ))
garch11
*---------------------------------*
* GARCH Model Fit *
*---------------------------------*
Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model : ARFIMA(1,0,0)
Distribution : norm
Optimal Parameters
------------------------------------
Estimate Std. Error t value Pr(>|t|)
mu 0.051834 0.007094 7.3072 0.000000
ar1 0.036227 0.009405 3.8520 0.000117
omega 0.016037 0.001730 9.2677 0.000000
alpha1 0.086025 0.004654 18.4827 0.000000
beta1 0.899858 0.005326 168.9472 0.000000
Robust Standard Errors:
Estimate Std. Error t value Pr(>|t|)
mu 0.051834 0.007351 7.0509 0.000000
ar1 0.036227 0.009880 3.6666 0.000246
omega 0.016037 0.003393 4.7264 0.000002
alpha1 0.086025 0.014422 5.9647 0.000000
beta1 0.899858 0.014312 62.8731 0.000000
LogLikelihood : -17311.27
Information Criteria
------------------------------------
Akaike 2.6030
Bayes 2.6058
Shibata 2.6030
Hannan-Quinn 2.6039
Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
statistic p-value
Lag[1] 2.419 0.11985
Lag[2*(p+q)+(p+q)-1][2] 2.514 0.08863
Lag[4*(p+q)+(p+q)-1][5] 3.608 0.29490
d.o.f=1
H0 : No serial correlation
Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
statistic p-value
Lag[1] 0.2171 0.6413
Lag[2*(p+q)+(p+q)-1][5] 1.3313 0.7813
Lag[4*(p+q)+(p+q)-1][9] 2.2795 0.8691
d.o.f=2
Weighted ARCH LM Tests
------------------------------------
Statistic Shape Scale P-Value
ARCH Lag[3] 1.273 0.500 2.000 0.2593
ARCH Lag[5] 1.315 1.440 1.667 0.6424
ARCH Lag[7] 1.685 2.315 1.543 0.7835
Nyblom stability test
------------------------------------
Joint Statistic: 9.6128
Individual Statistics:
mu 0.3536
ar1 8.0362
omega 0.3397
alpha1 0.1090
beta1 0.4880
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 0.9773 3.285e-01
Negative Sign Bias 2.6815 7.339e-03 ***
Positive Sign Bias 2.4800 1.315e-02 **
Joint Effect 34.1055 1.882e-07 ***
Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
group statistic p-value(g-1)
1 20 583.1 1.825e-111
2 30 561.6 5.810e-100
3 40 650.9 5.308e-112
4 50 656.1 3.867e-107
Elapsed time : 1.49227
# residui modello garch
u_hat_garch = residuals(garch11)
# varianza condizionata stimata
h = garch11@fit$var
# residui standardizzati
std_uhat = u_hat_garch/sqrt(h)
Test per l’autocorrelazione dei redesidui
# regressors
hh = 1/h
uu = Lag(u_hat_garch, 1)^2/h
colnames(uu) = "uu"
# epsilon^2 - 1
eps2 = (std_uhat)^2 - 1
lag1_eps2 = Lag(eps2, 1)
lag2_eps2 = Lag(eps2, 2)
lag3_eps2 = Lag(eps2, 3)
lag4_eps2 = Lag(eps2, 4)
lag5_eps2 = Lag(eps2, 5)
df_eps = cbind(eps2, hh, uu, lag1_eps2, lag2_eps2, lag3_eps2, lag4_eps2, lag5_eps2) %>%
na.omit() %>% as_tibble()
mod_garch_eps = lm(eps2 ~ hh + uu + Lag.1 + Lag.2 + Lag.3 + Lag.4 + Lag.5 - 1, data = df_eps)
car::linearHypothesis(mod_garch_eps, c("Lag.1=0", "Lag.2=0", "Lag.3=0", "Lag.4=0",
"Lag.5=0"), white.adjust = "hc0") %>% knitr::kable(caption = "Test F autocorrelazione residui garch(1,1) 5 Lag") %>%
kableExtra::kable_classic() %>% kable_styling() %>% row_spec(c(1, 2), font_size = 14) %>%
row_spec(c(2), bold = T, color = "black", background = "yellow") %>% column_spec(c(1,
2, 3, 4), background = "white", color = "black")
Test F autocorrelazione residui garch(1,1) 5 Lag
|
Res.Df
|
Df
|
F
|
Pr(>F)
|
|
13298
|
NA
|
NA
|
NA
|
|
13293
|
5
|
0.8945529
|
0.483569
|
