- Calculating the ACF and the PFC of the data set
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Date: 10/25/18 Time: 22:48 |
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Sample: 1957M01 2015M03 |
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Included observations: 699 |
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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 |
Model selection
First Model
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Automatic ARIMA Forecasting |
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Selected dependent variable: SPREAD |
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Date: 10/26/18 Time: 09:28 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Forecast length: 0 |
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Number of estimated ARMA models: 2 |
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Number of non-converged estimations: 0 |
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Selected ARMA model: (1,0)(0,0) |
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AIC value: 0.75597960146 |
Second Model
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Automatic ARIMA Forecasting |
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Selected dependent variable: SPREAD |
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Date: 10/26/18 Time: 09:29 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Forecast length: 0 |
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Number of estimated ARMA models: 3 |
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Number of non-converged estimations: 0 |
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Selected ARMA model: (2,0)(0,0) |
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AIC value: 0.683346733664 |
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Third Model
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Automatic ARIMA Forecasting |
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Selected dependent variable: SPREAD |
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Date: 10/26/18 Time: 09:30 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Forecast length: 0 |
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Number of estimated ARMA models: 4 |
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Number of non-converged estimations: 0 |
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Selected ARMA model: (3,0)(0,0) |
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AIC value: 0.660012693068 |
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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
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Date: 10/26/18 Time: 09:31 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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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 |
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.|*** | |
.|. | |
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.
- 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.
- 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.
- Forecasting
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Automatic ARIMA Forecasting |
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Selected dependent variable: SPREAD |
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Date: 10/26/18 Time: 07:50 |
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Sample: 1957M01 2012M12 |
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Included observations: 672 |
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Forecast length: 0 |
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Number of estimated ARMA models: 4 |
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Number of non-converged estimations: 0 |
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Selected ARMA model: (1,1)(0,0) |
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AIC value: 0.677963363957 |
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Model Selection Criteria Table |
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Dependent Variable: SPREAD |
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Date: 10/26/18 Time: 07:50 |
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Sample: 1957M01 2012M12 |
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Included observations: 672 |
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Model |
LogL |
AIC* |
BIC |
HQ |
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(1,1)(0,0) |
-223.795690 |
0.677963 |
0.704810 |
0.688361 |
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(1,0)(0,0) |
-260.215708 |
0.783380 |
0.803515 |
0.791178 |
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(0,1)(0,0) |
-699.965715 |
2.092160 |
2.112295 |
2.099958 |
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(0,0)(0,0) |
-1091.533293 |
3.254563 |
3.267987 |
3.259762 |
The results from the AR (1,1) models is shown in the table above
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Automatic ARIMA Forecasting |
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Selected dependent variable: SPREAD |
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Date: 10/26/18 Time: 07:59 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Forecast length: 0 |
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Number of estimated ARMA models: 4 |
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Number of non-converged estimations: 0 |
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Selected ARMA model: (3,0)(0,0) |
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AIC value: 0.660012693068 |
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Model Selection Criteria Table |
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Dependent Variable: SPREAD |
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Date: 10/26/18 Time: 07:59 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Model |
LogL |
AIC* |
BIC |
HQ |
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(3,0)(0,0) |
-224.684417 |
0.660013 |
0.692666 |
0.672638 |
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(2,0)(0,0) |
-233.804663 |
0.683347 |
0.709469 |
0.693447 |
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(1,0)(0,0) |
-260.080901 |
0.755980 |
0.775572 |
0.763555 |
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(0,0)(0,0) |
-1125.854148 |
3.240960 |
3.254022 |
3.246011 |
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Forecast Evaluation |
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Date: 10/26/18 Time: 08:02 |
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Sample: 1957M01 2015M03 |
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Included observations: 699 |
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Evaluation sample: 1957M01 2015M03 |
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Training sample: 1957M01 2012M12 |
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Number of forecasts: 2 |
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Combination tests |
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Null hypothesis: Forecast i includes all information contained in others |
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Forecast |
F-stat |
F-prob |
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SPREAD |
NA |
NA |
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Evaluation statistics |
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Forecast |
RMSE |
MAE |
MAPE |
SMAPE |
Theil U1 |
Theil U2 |
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SPREAD |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
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MSE ranks |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
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Dependent Variable: SPREAD |
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Method: ARMA Maximum Likelihood (OPG – BHHH) |
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Date: 10/26/18 Time: 08:51 |
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Sample: 2013M01 2014M12 |
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Included observations: 24 |
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Failure to improve objective (non-zero gradients) after 106 iterations |
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Coefficient covariance computed using outer product of gradients |
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Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
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C |
-2.370419 |
0.141478 |
-16.75472 |
0.0000 |
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AR(1) |
0.233121 |
0.562167 |
0.414683 |
0.6851 |
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AR(2) |
1.137205 |
0.640238 |
1.776223 |
0.0991 |
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AR(3) |
0.190303 |
0.734960 |
0.258930 |
0.7997 |
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AR(4) |
-0.830724 |
0.628845 |
-1.321031 |
0.2093 |
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MA(1) |
0.572083 |
3.782041 |
0.151263 |
0.8821 |
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MA(1) |
0.577825 |
11.81595 |
0.048902 |
0.9617 |
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MA(2) |
-0.424238 |
8.292924 |
-0.051157 |
0.9600 |
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MA(3) |
-0.985814 |
17.48258 |
-0.056388 |
0.9559 |
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MA(4) |
-0.167773 |
3.705652 |
-0.045275 |
0.9646 |
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SIGMASQ |
0.013356 |
0.285408 |
0.046795 |
0.9634 |
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R-squared |
0.871241 |
Mean dependent var |
-2.410000 |
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Adjusted R-squared |
0.772195 |
S.D. dependent var |
0.328991 |
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S.E. of regression |
0.157024 |
Akaike info criterion |
-0.250678 |
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Sum squared resid |
0.320534 |
Schwarz criterion |
0.289263 |
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Log likelihood |
14.00814 |
Hannan-Quinn criter. |
-0.107432 |
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F-statistic |
8.796349 |
Durbin-Watson stat |
2.187936 |
Dynamic forecasting
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Automatic ARIMA Forecasting |
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Selected dependent variable: SPREAD |
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Date: 10/26/18 Time: 09:15 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Forecast length: 0 |
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Number of estimated ARMA models: 4 |
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Number of non-converged estimations: 0 |
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Selected ARMA model: (0,0)(0,0) |
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(0,0)(0,0) |
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Dependent Variable: SPREAD |
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Method: Least Squares |
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Date: 10/26/18 Time: 09:15 |
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Sample (adjusted): 1957M01 2014M12 |
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Included observations: 696 after adjustments |
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Variable |
Coefficient |
Std. Error |
t-Statistic |
Prob. |
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C |
-1.519009 |
0.046269 |
-32.83010 |
0.0000 |
|
R-squared |
0.000000 |
Mean dependent var |
-1.519009 |
|
|
Adjusted R-squared |
0.000000 |
S.D. dependent var |
1.220654 |
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S.E. of regression |
1.220654 |
Akaike info criterion |
3.238087 |
|
|
Sum squared resid |
1035.548 |
Schwarz criterion |
3.244617 |
|
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Log likelihood |
-1125.854 |
Hannan-Quinn criter. |
3.240612 |
|
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Durbin-Watson stat |
0.084520 |
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Model Selection Criteria Table |
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Dependent Variable: SPREAD |
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Date: 10/26/18 Time: 09:15 |
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Sample: 1957M01 2015M03 |
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Included observations: 696 |
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Model |
LogL |
AIC |
BIC |
HQ |
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(0,0)(0,0) |
-1125.854148 |
3.240960 |
3.254022 |
3.246011 |
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(0,1)(0,0) |
-719.672654 |
2.076646 |
2.096238 |
2.084221 |
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(1,0)(0,0) |
-260.080901 |
0.755980 |
0.775572 |
0.763555 |
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(1,1)(0,0) |
-222.593793 |
0.651132 |
0.677254 |
0.661232 |
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Forecast Evaluation |
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Date: 10/26/18 Time: 09:18 |
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Sample: 2013M01 2015M03 |
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Included observations: 27 |
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Evaluation sample: 2013M01 2015M03 |
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Training sample: 1957M01 2012M12 |
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Number of forecasts: 3 |
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Combination tests |
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Null hypothesis: Forecast i includes all information contained in others |
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Forecast |
F-stat |
F-prob |
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|
SPREAD |
NA |
NA |
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Evaluation statistics |
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|
Forecast |
RMSE |
MAE |
MAPE |
SMAPE |
Theil U1 |
Theil U2 |
|
|
SPREAD |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
|
|
Mean square error |
NA |
NA |
NA |
NA |
NA |
NA |
|
|
MSE ranks |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
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On the basis of results from the forecasting it can be said that the spread is going to decline for some time and then increase after 2014. In terms of the forecasting accuracy, the original series do not show any trend, however the results from forecasting is continuously declining after 1957.
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