8.8 Zoo of Models
111
σ
2
i = c +
n
j=1
a j u
2
i− j +
m
k=1
b k σ
2
i−k ,
(8.48)
where n and m are the orders of the moving average and autoregressive part, respectively.
Most of the time the volatility is reasonably constant, but occasionally it increases
significantly. One reason for such an increase is political unrest, but others are speculative bubbles and subsequent crashes of a market, the topic of the next chapter.
Exercises
1. Remove the trend and seasonality from the data in file ex8_1.dat. What period
do you use? Plot the residuals and then sort them into a histogram with 30 bins.
Hint: MATLAB has a built-in function histogram().
2. You suspect that the data in file ex8_2.dat come from an MA(q)–process.
Determine the order q of the process.
3. You suspect that the data in file ex8_3.dat come from an AR( p)–process.
Determine the order p of the process and discuss which coefficients φ j are significant.
4. Analyze the time series of the CO 2 –concentration measured on Mauna Loa from
2009 until 2019, available in the file ex8_2009a.dat from the book’s web
site. In your analysis
a. remove the trend and the seasonality from the data;
b. perform a PACF on the residuals (up to order 4: φ 44 ) in order to find the relevant
coefficients for an AR( p) model;
c. determine the model coefficients and display the model together with the data
in order to verify that your fitting makes sense.
5. Derive the coefficients π j for j = 1, 2, and 3 that appear in (8.28) from the θ i
and φ k that appear in (8.27).
6. Use an EWMA filter with m = 1, 3, 10, and 30 to remove the noise from the
time series in file ex8_6.dat, available from the book’s web page. Plot both
the raw time series and the de-noised copies. Discuss what you observe as you
increase m.
References
1. NIST/SEMATECH e-Handbook of Statistical Methods (2016). http://www.itl.nist.gov/div898/
handbook/
2. http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4411.htm
111
σ
2
i = c +
n
j=1
a j u
2
i− j +
m
k=1
b k σ
2
i−k ,
(8.48)
where n and m are the orders of the moving average and autoregressive part, respectively.
Most of the time the volatility is reasonably constant, but occasionally it increases
significantly. One reason for such an increase is political unrest, but others are speculative bubbles and subsequent crashes of a market, the topic of the next chapter.
Exercises
1. Remove the trend and seasonality from the data in file ex8_1.dat. What period
do you use? Plot the residuals and then sort them into a histogram with 30 bins.
Hint: MATLAB has a built-in function histogram().
2. You suspect that the data in file ex8_2.dat come from an MA(q)–process.
Determine the order q of the process.
3. You suspect that the data in file ex8_3.dat come from an AR( p)–process.
Determine the order p of the process and discuss which coefficients φ j are significant.
4. Analyze the time series of the CO 2 –concentration measured on Mauna Loa from
2009 until 2019, available in the file ex8_2009a.dat from the book’s web
site. In your analysis
a. remove the trend and the seasonality from the data;
b. perform a PACF on the residuals (up to order 4: φ 44 ) in order to find the relevant
coefficients for an AR( p) model;
c. determine the model coefficients and display the model together with the data
in order to verify that your fitting makes sense.
5. Derive the coefficients π j for j = 1, 2, and 3 that appear in (8.28) from the θ i
and φ k that appear in (8.27).
6. Use an EWMA filter with m = 1, 3, 10, and 30 to remove the noise from the
time series in file ex8_6.dat, available from the book’s web page. Plot both
the raw time series and the de-noised copies. Discuss what you observe as you
increase m.
References
1. NIST/SEMATECH e-Handbook of Statistical Methods (2016). http://www.itl.nist.gov/div898/
handbook/
2. http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4411.htm