ECON-4200_Assignment_3_Winter_2022 TRENT UNIVERSITY DEPARTMENT OF ECONOMICS ECON-4200H: TOPICS IN ECONOMETRICS ASSIGNMENT 3 DUE: APRIL 12, 2022 Question 1: The posted dataset contains information on...

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ECON-4200_Assignment_3_Winter_2022
TRENT UNIVERSITY
DEPARTMENT OF ECONOMICS
ECON-4200H: TOPICS IN ECONOMETRICS
ASSIGNMENT 3
DUE: APRIL 12, 2022
Question 1:
The posted dataset contains information on the daily returns of the DOW Index for the period
March 2010 to March 2020.
(a) Test for the presence of ARCH effects in the daily returns (variable r). What do you
conclude?
(b) Plot the daily returns. What do you observe?
(c) Estimate an ARCH(1) model on the daily return series. Interpret the results.
(d) Estimate a GARCH(1,1) model on the daily return series. Interpret the results.
(e) Estimate a T-GARCH(1,1) model on the daily return series. Interpret the results.
(f) Of the models estimated in parts (c), (d), and (e), which provides the best fit?
Answered 2 days AfterApr 06, 2022

Solution

Manoj answered on Apr 09 2022
11 Votes
Question 1.
a.
R- Output:
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: dt$
## Chi-squared = 259.23, df = 1, p-value < 2.2e-16
The Null and alternative hypotheses given by,
Null Hypothesis: H0: There is no presence of ARCH effect
Alternative Hypothesis: H1: There is presence of ARCH effect
Because the p-value is less than aplha = 0.05, we conclude that there is presence of GARCH effect.
.
From above plot we can conclude that there is no trend, seasonality or any cyclical variation. There is no any trend present in the given data.
c.
R Output:
summary(arch.fit)
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 0), data = dt$r, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 0)
## ## [data = dt$r]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1
## 0.056573 0.607614 0.316754
##
## Std. E
ors:
## based on Hessian
##
## E
or Analysis:
## Estimate Std. E
or t value Pr(>|t|)
## mu 0.05657 0.01651 3.427 0.000609 ***
## omega 0.60761 0.02286 26.585< 2e-16 ***
## alpha1 0.31675 0.03783 8.373< 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## -3258.849 normalized: -1.29525
##
## Description:
## Sat Apr 09 10:31:52 2022 by user: manoj
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 3671.878 0
## Shapiro-Wilk Test R W 0.9375725 0
## Ljung-Box Test R Q(10) 22.53408 0.01260335
## Ljung-Box Test R Q(15) 29.88333 0.01234921
## Ljung-Box Test R Q(20) 35.12458 0.01945074
## Ljung-Box Test R^2 Q(10) 404.4623 0
## Ljung-Box Test R^2 Q(15) 450.4108 0
## Ljung-Box Test R^2 Q(20) 474.5421 0
## LM Arch Test R TR^2 313.9569 0
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## 2.592885 2.599837 2.592882 2.595408
From Jarque Bera test and Ljung-Box tests p-values are less than α = 0.05, residuals are not normally distributed.
Value of AIC = 2.592885
d.
R Output:
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. E
or t value...
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