# BUA Autocovariance Worksheet

1.
Consider a moving average process of order 2, MA(2), model as follows

𝑌𝑡 =𝜖𝑡 −𝜃1𝜖𝑡−1 −𝜃2𝜖𝑡−2where 𝐸(𝜖𝑡)=0,𝑉(𝜖𝑡)=𝜎2 and 𝜖𝑡𝑠 are uncorrelated.- Show how you can obtain 𝑉(𝑌𝑡)- Show how you can obtain the autocovariance and autocorrelation functions at lagsk=1,2.2. Consider an ARMA model of order (2,1) as follows𝑌𝑡 =𝐶+𝜙1𝑌𝑡−1 +𝜙2𝑌𝑡−2 +𝜖𝑡 −𝜃1𝜖𝑡−1where 𝐸(𝜖𝑡)=0,𝑉(𝜖𝑡)=𝜎2 and 𝜖𝑡𝑠 are uncorrelated.- Show how you can obtain 𝐸(𝑌𝑡) and 𝑉(𝑌𝑡).3. Write a simulation sampler for a second order MA process (as in from question 1above) where 𝜃1 =0.8,𝜃2 =−0.85,𝜎2 =2. Please use a seed of 1 for your sampler andsimulate a total of 100 observations. Obtain the time series, acf and pacf plots of thesimulated series. Estimate an MA(2) model using the simulated series and investigate ifthe residuals are white noise.4. JJ_Earnings.txt contains the quarterly earnings of Johnson and Johnson between 1960-1980.Consider the modeling of the J&J earnings as a SARIMA process:1. Obtain the time series plot of the series. Assess if you need to transform the seriesfor variance stabilization purposes. Obtain the ACF and PACF plots to determinea suitable SARIMA model for the series.2. Determine the SARIMA model that provides the best fit to the data. Estimate yourbest fit model and investigate if the parameters are significant.3. Test if the residuals from your SARIMA model are white noise.4. Plot the in-sample fit against the actual data.5. Exclude the last eight observations (test sample) of the data and use the rest asyour training sample to re-estimate your best fit SARIMA model. In addition, alsoestimate a multiple linear regression model with seasonal dummy variables

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(indicators) as we did in Lecture Set 2 using the training set. Obtain 8 step-aheadpredictions for both models. Plot your predictions against actual data. Comparethe predictive performance of the SARIMA model against the seasonal regressionmodel using MAPE, MAE, and MSE estimates.

DS 809 – Assignment 4
For statistical inference purposes you can use an 𝜶 (significance) level of 0.05. For
when needed.
1. Consider a moving average process of order 2, MA(2), model as follows
𝑌𝑡 = 𝜖𝑡 − 𝜃1 𝜖𝑡−1 − 𝜃2 𝜖𝑡−2
where 𝐸(𝜖𝑡 ) = 0, 𝑉(𝜖𝑡 ) = 𝜎 2 and 𝜖𝑡 𝑠 are uncorrelated.

Show how you can obtain 𝑉(𝑌𝑡 )
Show how you can obtain the autocovariance and autocorrelation functions at lags
k=1,2.
2. Consider an ARMA model of order (2,1) as follows
𝑌𝑡 = 𝐶 + 𝜙1 𝑌𝑡−1 + 𝜙2 𝑌𝑡−2 + 𝜖𝑡 − 𝜃1 𝜖𝑡−1
where 𝐸(𝜖𝑡 ) = 0, 𝑉(𝜖𝑡 ) = 𝜎 2 and 𝜖𝑡 𝑠 are uncorrelated.

Show how you can obtain 𝐸(𝑌𝑡 ) and 𝑉(𝑌𝑡 ).
3. Write a simulation sampler for a second order MA process (as in from question 1
above) where 𝜃1 = 0.8, 𝜃2 = −0.85, 𝜎 2 = 2. Please use a seed of 1 for your sampler and
simulate a total of 100 observations. Obtain the time series, acf and pacf plots of the
simulated series. Estimate an MA(2) model using the simulated series and investigate if
the residuals are white noise.
4. JJ_Earnings.txt contains the quarterly earnings of Johnson and Johnson between 19601980.
Consider the modeling of the J&J earnings as a SARIMA process:
1. Obtain the time series plot of the series. Assess if you need to transform the series
for variance stabilization purposes. Obtain the ACF and PACF plots to determine
a suitable SARIMA model for the series.
2. Determine the SARIMA model that provides the best fit to the data. Estimate your
best fit model and investigate if the parameters are significant.
3. Test if the residuals from your SARIMA model are white noise.
4. Plot the in-sample fit against the actual data.
5. Exclude the last eight observations (test sample) of the data and use the rest as
estimate a multiple linear regression model with seasonal dummy variables
(indicators) as we did in Lecture Set 2 using the training set. Obtain 8 step-ahead
predictions for both models. Plot your predictions against actual data. Compare
the predictive performance of the SARIMA model against the seasonal regression
model using MAPE, MAE, and MSE estimates.

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