Module 14 Discussion

Setup

library(fpp3)
library(fredr)
library(dplyr)
library(tseries)
library(fabletools)
library(ggplot2)
library(readr)
library(tidyverse)
library(tsibble)
library(feasts)
library(scales)
library(patchwork)
library(ggtime)
library(tseries)
library(readxl)
library(lubridate)
library(forecast)
library(tidyquant)
library(tidyverse)
library(quantmod)
library(lubridate)
library(rugarch)
fredr_has_key()

[1] TRUE

Get S&P Data

# Get S&P 500 data
getSymbols("^GSPC", from = "2015-01-01")

[1] "GSPC"

# Compute log returns
returns <- dailyReturn(Cl(GSPC), type = "log")

# Plot time series
plot(returns, main = "S&P 500 Log Returns", col = "blue")

spec <- ugarchspec(
  variance.model = list(model = "sGARCH", garchOrder = c(1,1)),
  mean.model = list(armaOrder = c(1,0)),
  distribution.model = "norm"
)

fit <- ugarchfit(spec = spec, data = returns)

show(fit)

*---------------------------------*
*          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      0.000780    0.000065 11.97504 0.000000
ar1    -0.052760    0.020048 -2.63172 0.008495
omega   0.000004    0.000006  0.71008 0.477658
alpha1  0.170699    0.025323  6.74084 0.000000
beta1   0.796417    0.037286 21.35958 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000780    0.002602  0.29966  0.76444
ar1    -0.052760    0.120379 -0.43828  0.66118
omega   0.000004    0.000128  0.03103  0.97525
alpha1  0.170699    0.669075  0.25513  0.79863
beta1   0.796417    0.907417  0.87767  0.38012

LogLikelihood : 9426.07 

Information Criteria
------------------------------------
                    
Akaike       -6.6229
Bayes        -6.6124
Shibata      -6.6229
Hannan-Quinn -6.6191

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                      1.599  0.2060
Lag[2*(p+q)+(p+q)-1][2]     1.656  0.3558
Lag[4*(p+q)+(p+q)-1][5]     2.919  0.4519
d.o.f=1
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.9051  0.3414
Lag[2*(p+q)+(p+q)-1][5]    3.0341  0.4009
Lag[4*(p+q)+(p+q)-1][9]    5.3953  0.3737
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]  0.001433 0.500 2.000  0.9698
ARCH Lag[5]  4.198468 1.440 1.667  0.1568
ARCH Lag[7]  5.280840 2.315 1.543  0.1976

Nyblom stability test
------------------------------------
Joint Statistic:  5.9954
Individual Statistics:             
mu     0.1007
ar1    0.1919
omega  0.2080
alpha1 0.2495
beta1  0.6348

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           3.5530 3.870e-04 ***
Negative Sign Bias  0.1058 9.158e-01    
Positive Sign Bias  0.3761 7.069e-01    
Joint Effect       25.2587 1.363e-05 ***


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     142.1    7.371e-21
2    30     155.6    2.870e-19
3    40     161.5    7.700e-17
4    50     174.1    6.411e-16


Elapsed time : 0.1984742 

coef(fit)

           mu           ar1         omega        alpha1         beta1 
 7.797799e-04 -5.276042e-02  3.981383e-06  1.706988e-01  7.964171e-01