a) Examine your Y data
(excluding the hold out period) to determine if it needs to be differenced...

Question

a) Examine your Y data
(excluding the hold out period) to determine if it needs to be differenced to
make it stationary. Show a time series plot of the raw Y data and
autocorrelation functions (ACFs).b) From your time series data plot and AFCs
determine if you have seasonality. If you do, use seasonal differences to
remove it and run the ACFs and PACFs on the non seasonal Y data series.c) Fill out the ARIMA seasonal menu (P,D,Q)
appropriately. If you have no trend as shown by the seasonally
differenced ACFs run the ARIMA model and note the significance of each
coefficient. Make model adjustments accordingly to improve results.Note: You maynotuse an
ARIMA model with non significant coefficients to forecast. If the
coeffcients are not signficant derive another model that has signficant
coefficents and the lowest residual MS value.d) If it requires differencing for trendto
make it stationary do so and run another time series plot and ACFs on the
differenced data. If this requires differencing again do so but run time series
plots and ACFs each time you do.e) Run and show the PACFs on your stationary data
series and identify the appropriate ARIMA model and show the initial ARIMA non
seasonal menu section(p,d,q) filled out appropriately and any seasonal
(P,D,Q) components in the seasonal menu filled out.Explainf) Run the ARIMA model and note the significance
of each coefficient. Make model adjustments accordingly to improve
results shown by the residual MS or MSE.Explaing) Calculate the two error measures that you used
in other model analysis and comment on the acceptability of the size of the
measure.h) Note the LBQ associated P values for the
selected lags. They should each be significant (above .05) to declare the
residuals random. If they are not random select an alternative ARIMA
model form that has random residuals.i) Run an ARIMA forecast for your hold out period
and show a time series plot of the residuals (Y actual and Y forecast) for the
hold out period.j) Calculate the hold out period MSE, RMSE and
MAPE (Refer back to earlier chapters for the error measure formulas) and
compare them to the Fit period ARIMA error measures (from g above). Explain

Robert · Accepted Answer

a) Examine your Y data (excluding the hold out period) to determine if it needs to be differenced to make it stationary. Show a time series plot of the raw Y data and autocorrelation functions 
Date
Y
2011.22008.32005.42003.12000.21997.31994.41992.11989.21986.31984.1
1600
1400
1200
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0
Time Series Plot of Y
The above graph is the time series plot of S&P 500 series. From the graph it can be seen that there is an upward trend visible. Also, the data exhibit a seasonal variation. However, there is no such cyclical variation exist in the data. These conclude that the series is not stationary. 
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282624222018161412108642
1.0
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-0.4
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-0.8
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Autocorrelation Function for Y
(with 5% significance limits for the autocorrelations)
The autocorrelation function of the S&P 500 series is given above. This shows that the data is not stationary as the autocorrelation functions of different orders are dying out very slowly.  
b) From your time series data plot and AFCs determine if you have seasonality. If you do, use seasonal differences to remove it and run the ACFs and PACFs on the non seasonal Y data series.
The time series plot shows that a seasonal variation exist in the data. The seasonal difference of lag 4 has been done and the seasonally differenced series (DesY) is presented below. 
Date
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Y
2011.22008.32005.42003.12000.21997.31994.41992.11989.21986.31984.1
500
250
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-250
-500
Time Series Plot of DesY
The autocorrelation function and partial autocorrelation function of the seasonally differenced series (DesY) are presented below. 
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2624222018161412108642
1.0
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Autocorrelation Function for DesY
(with 5% significance limits for the autocorrelations)
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2624222018161412108642
1.0
0.8
0.6
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0.0
-0.2
-0.4
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-0.8
-1.0
Partial Autocorrelation Function for DesY
(with 5% significance limits for the partial autocorrelations)
c) Fill out the ARIMA seasonal menu (P,D,Q) appropriately. If you have no trend as shown by the seasonally differenced ACFs run the ARIMA model and note the significance of each coefficient. Make model adjustments accordingly to improve results.

a) Examine your Y data (excluding the hold out period) to determine if it needs to be differenced to make it stationary. Show a time series plot of the raw Y data and autocorrelation functions (ACFs)....

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