Financial Econometrics: ACF, PACF, Model Selection, Forecasting

Financial Econometrics
1) Plot
2) Calculating the ACF and the PFC of the data set
Date: 10/25/18 Time: 22:48
Sample: 1957M01 2015M03
Included observations: 699
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.|******* .|******* 1 0.957 0.957 643.11 0.000
.|******| **|. | 2 0.894 -0.267 1204.6 0.000
.|******| .|* | 3 0.841 0.158 1702.0 0.000
.|******| *|. | 4 0.788 -0.111 2139.5 0.000
.|***** | .|. | 5 0.739 0.071 2525.0 0.000
.|***** | .|. | 6 0.697 0.002 2868.0 0.000
.|***** | .|* | 7 0.664 0.092 3179.9 0.000
.|***** | .|. | 8 0.635 -0.025 3465.6 0.000

.|**** | *|. | 9 0.599 -0.092 3720.0 0.000
.|**** | *|. | 10 0.550 -0.133 3935.2 0.000
.|**** | .|. | 11 0.502 0.033 4114.6 0.000
.|*** | .|. | 12 0.458 -0.012 4264.3 0.000
3) Model selection
First Model
Automatic ARIMA
Forecasting
Selected dependent
variable: SPREAD
Date: 10/26/18 Time:
09:28
Sample: 1957M01
2015M03
Included observations: 696
Forecast length: 0
Number of estimated
ARMA models: 2
Number of non-converged
estimations: 0
Selected ARMA model:
(1,0)(0,0)
AIC value: 0.75597960146
Second Model
Automatic ARIMA Forecasting
Selected dependent variable: SPREAD
Date: 10/26/18 Time: 09:29
Sample: 1957M01 2015M03
Included observations: 696
Forecast length: 0
Number of estimated ARMA models: 3
Number of non-converged estimations: 0
Selected ARMA model: (2,0)(0,0)
AIC value: 0.683346733664

Third Model
Automatic ARIMA Forecasting
Selected dependent variable: SPREAD
Date: 10/26/18 Time: 09:30
Sample: 1957M01 2015M03
Included observations: 696
Forecast length: 0
Number of estimated ARMA models: 4
Number of non-converged estimations: 0
Selected ARMA model: (3,0)(0,0)
AIC value: 0.660012693068
On the basis of the results from the above table, the AIC is lowest for the third model with three
lags. So the best model is the third model.
Plotting the ACF and PCF for the Preferred Model
Date: 10/26/18 Time: 09:31
Sample: 1957M01 2015M03
Included observations: 696
Autocorrelation Partial Correlation AC PAC Q-Stat Prob
.|******* .|******* 1 0.957 0.957 640.22 0.000
.|******| **|. | 2 0.894 -0.265 1199.3 0.000
.|******| .|* | 3 0.840 0.155 1694.5 0.000
.|******| *|. | 4 0.788 -0.106 2130.3 0.000
.|***** | .|. | 5 0.739 0.063 2514.1 0.000
.|***** | .|. | 6 0.696 0.005 2855.3 0.000
.|***** | .|* | 7 0.663 0.085 3165.0 0.000
.|***** | .|. | 8 0.633 -0.017 3448.4 0.000
.|**** | *|. | 9 0.597 -0.099 3700.3 0.000
.|**** | *|. | 10 0.549 -0.122 3913.4 0.000
.|**** | .|. | 11 0.501 0.038 4091.7 0.000
.|*** | .|. | 12 0.459 -0.006 4241.5 0.000

To calculate the Ljung-Box for the residuals we have to use the chi square test. The Q statistics is
used to test following null hypothesis:
Null hypothesis:
There is no autocorrelation up to order k:
On the basis of the results from the ACF and PACF, all the p values are less than 0.05. So, the
null hypothesis can be rejected. So There is autocorrelation in the order 5 as mentioned.
4) The similarity between the static forecasting and the rolling window forecasting is that in
both the model the previous data is used to forecast the future values. Based on the
historical data the forecasting is done. However the major differences arise on the basis
of the values is taken into consideration for forecasting. In case of the static forecasting
only a fixed period data is used for forecasting. On the other hand, in case of the rolling
window first the rolling window is selected and then to forecast the future values, the
window in the previous period is used. For example, in rolling window, a sample rolling
window a data for one quarter can be taken and based on that the value for next quarter
can be forecasted. The rolling window keeps changing.
5) In case of the one step ahead forecasting and the dynamic forecasting also, the
forecasting process is same, i.e taking the previous value to forecast the values in future.
However in case of the one step ahead static forecasting only the actual values are used
for forecasting. On the other hand for the dynamic forecasting can take into consideration
the previously forecasted values for further forecasting.
6) Forecasting
Automatic ARIMA Forecasting
Selected dependent variable: SPREAD
Date: 10/26/18 Time: 07:50
Sample: 1957M01 2012M12
Included observations: 672
Forecast length: 0
Number of estimated ARMA models: 4