Homework 1
Name:
Exercise 1
Define an AR(1) process with mean 1, standard deviation 3 and autoregressive coefficient 0.8. Plot the
simple and partial autocorrelation functions. Plot a histogram of the distribution of the estimates of the
three parameters of the model, both for the maximum likelihood estimates and the OLS estimates and
compare its center to the true values of the parameters.
Exercise 2
Define an AR(2) process with mean 1, standard deviation 3 and autoregressive coefficients 1.5 and -0.6.
Plot the simple and partial autocorrelation functions. Determine the period of this process. Plot a
histogram of the distribution of the estimates of the four parameters of the model, both for the
maximum likelihood estimates and the OLS estimates, and compare its center to the true values of the
parameters.
Exercise 3
Define an ARMA(1,1) process with mean 1, standard deviation 3, autoregressive coefficient 0.5, and
moving average coefficient 0.8. Plot the simple and partial autocorrelation functions. Plot a histogram of
the distribution of the estimates of the four parameters of the model and compare its center to the true
values of the parameters.
Exercise 4
Define an MA(1) process with mean 0, standard deviation 1 and moving average coefficient 0.8.
Simulate the model one time with sample size 200 and obtain the value of the likelihood function. With
the same simulated errors of the previous simulation, simulate the MA(1) model with moving average
coefficient 1.6 and sample size 200, and obtain the value of the likelihood function. Are they the same?
Why or why not?
Yes, they are the same.
Exercise 5
Download the real GDP data of Ecuador (constant dollars of 2007) and get the t/t-1 quarterly growth
rate of real GDP from the first quarter of 2000 to the fourth quarter of 2019. Write a Matlab code to
determine the optimal model for the growth rate of real GDP.
The model I will select is an ar(2) ma(2).
Exercise 6
Once you have selected the most parsimonious model from the last exercise, estimate that model in
Eviews
Dependent Variable: GDP_G
Method: ARMA Maximum Likelihood (OPG - BHHH)
Date: 06/26/20 Time: 22:04
Sample: 2001Q1 2019Q4
Included observations: 76
Convergence achieved after 44 iterations
Coefficient covariance computed using outer product of gradients
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
AR(1)
AR(2)
MA(1)
MA(2)
SIGMASQ
3.508940
1.013603
-0.363551
0.036054
0.860277
1.377069
0.747175
0.153909
0.136342
0.091288
0.064341
0.220015
4.696278
6.585718
-2.666469
0.394949
13.37068
6.258976
0.0000
0.0000
0.0095
0.6941
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic
Prob(F-statistic)
Inverted AR Roots
Inverted MA Roots
0.835263
0.823496
1.222744
104.6572
-122.2289
70.98415
0.000000
.51+.33i
-.02+.93i
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Hannan-Quinn criter.
Durbin-Watson stat
3.495282
2.910442
3.374444
3.558449
3.447982
1.871818
.51-.33i
-.02-.93i
12
8
4
6
4
0
2
-4
0
-2
-4
2002
2004
2006
2008
Residual
2010
2012
Actual
2014
2016
2018
Fitted
gdp_g = 3.508940 + alpha + 0.036054*beta + 0.860277*gamma
@state alpha = 1.013603*alpha(-1) + - 0.363551*beta(-1) + [var=1.377069^2]
@state beta = alpha(-1)
@state gamma = beta(-1) + [var=1.377069^2]
12
10
8
6
4
2
0
-2
-4
00
02
04
06
08
10
GDP_GF
12
14
GDP_G
16
18
20