Results of Autocorrelation Analysis
Using Eviews, plot the term spread with a line chart and draw a horizontal line at value zero on the graph. Explain why the term spread typically takes a negative value
Hint: to draw the horizontal line, from a graph object, click the Line/Shadeà Type: Line; Orientation: Horizontal –left axis; Position: Data Value 0.
(2) Compute the sample ACF and the sample PACF for this series for the first 12 lags using Eviews. Comment on the pattern of the correlogram. (3) Consider three models
(i) select the best model using AIC and BIC.
Hint: Sample sizes need to be the same when comparing models with AIC or BIC. That is, use the sample period starting from 1957m04 to 2015m03 for the estimation of all three models.
(ii) Using Eviews, compute the ACF and PACF (the first 12 lags) of the residual of the preferred model, estimated from the sample running from 1957m04 to 2015m03.
(iii) Conduct Ljung-Box Q(5) tests for the residual of the preferred model at the 5% significance level.
Hint: Calculate the Ljung-Box Q-statistics by hand (i.e., use the sample ACF obtained from Eviews and show all your working) and show the decision rules used.
Compare the forecasting performance of the ARMA(1,1) and AR(3) models. The insample estimation period is 957M01 to 2012M12 and the out-of-sample forecast period is 2013M01 to 2015M03.
0 11 11
0 11 2 2
0 11 2 2 33
Model 1: y
Model 2: y
Model 3: y
a a be e
aa a e
aa a a e
- -
- -
-- -
=+ + +
=+ + +
=+ + + +
t t tt
t t tt
t t t tt
(4) What are the similarities and differences between a static and a rolling window one-step-ahead forecast
(5) What are the similarities and differences between a one-step-ahead static and a dynamic forecast
(6) Estimate the ARMA(1,1) and AR(3) models over the period 1957M01 to 2012M12.
(7) Use the estimation results of each model in (6) to provide a dynamic forecast for the rest of the sample period.
(i) What are the RMSEs Which model would you select based on the RMSEs of this forecasting method? Attach the Eviews forecast evaluation results.
(ii) For the preferred model, plot the original time series, your forecasts and prediction intervals for this period (see tutorial week 10 for an example of the graph) and comment on the forecasting results.
(8) Conduct rolling window forecasts based on the ARMA(1,1) and AR(3) models. Specifically, estimate each model over the period 1957m04 to 2012m12; obtain the one-step-ahead forecast and the one-step-ahead forecast error; continue to update the estimate period so as to obtain the 27 one-stepahead forecast and forecast error.
Hint: you could obtain the rolling window one-step-ahead forecasts and forecast errors by revising the Eviews program rolling_forecasting.prg described in Lecture week 9.
(i) What are the RMSEs Which model would you select based on the RMSEs of this forecasting method Attach the programs in an appendix (at the end of the assignment) and highlight the changes you made to the code.
(ii) For the preferred model, plot the original time series, your forecasts and prediction intervals for this period and comment on the forecasting results.
- 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 |
|
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 |
|
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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) |
|
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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
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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 |
||||||
|
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.
- 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
|
Automatic ARIMA Forecasting |
|
|
Selected dependent variable: SPREAD |
|
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Date: 10/26/18 Time: 07:50 |
|
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Sample: 1957M01 2012M12 |
|
|
Included observations: 672 |
|
|
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 |
|
|
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 |
|
(1,1)(0,0) |
-223.795690 |
0.677963 |
0.704810 |
0.688361 |
|
(1,0)(0,0) |
-260.215708 |
0.783380 |
0.803515 |
0.791178 |
|
(0,1)(0,0) |
-699.965715 |
2.092160 |
2.112295 |
2.099958 |
|
(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
|
Automatic ARIMA Forecasting |
|
|
Selected dependent variable: SPREAD |
|
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Date: 10/26/18 Time: 07:59 |
|
|
Sample: 1957M01 2015M03 |
|
|
Included observations: 696 |
|
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Forecast length: 0 |
|
|
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 |
|
(3,0)(0,0) |
-224.684417 |
0.660013 |
0.692666 |
0.672638 |
|
(2,0)(0,0) |
-233.804663 |
0.683347 |
0.709469 |
0.693447 |
|
(1,0)(0,0) |
-260.080901 |
0.755980 |
0.775572 |
0.763555 |
|
(0,0)(0,0) |
-1125.854148 |
3.240960 |
3.254022 |
3.246011 |
|
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 |
||||
|
SPREAD |
NA |
NA |
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|
Evaluation statistics |
||||||
|
Forecast |
RMSE |
MAE |
MAPE |
SMAPE |
Theil U1 |
Theil U2 |
|
SPREAD |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
0.000000 |
|
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. |
|
C |
-2.370419 |
0.141478 |
-16.75472 |
0.0000 |
|
AR(1) |
0.233121 |
0.562167 |
0.414683 |
0.6851 |
|
AR(2) |
1.137205 |
0.640238 |
1.776223 |
0.0991 |
|
AR(3) |
0.190303 |
0.734960 |
0.258930 |
0.7997 |
|
AR(4) |
-0.830724 |
0.628845 |
-1.321031 |
0.2093 |
|
MA(1) |
0.572083 |
3.782041 |
0.151263 |
0.8821 |
|
MA(1) |
0.577825 |
11.81595 |
0.048902 |
0.9617 |
|
MA(2) |
-0.424238 |
8.292924 |
-0.051157 |
0.9600 |
|
MA(3) |
-0.985814 |
17.48258 |
-0.056388 |
0.9559 |
|
MA(4) |
-0.167773 |
3.705652 |
-0.045275 |
0.9646 |
|
SIGMASQ |
0.013356 |
0.285408 |
0.046795 |
0.9634 |
|
R-squared |
0.871241 |
Mean dependent var |
-2.410000 |
|
|
Adjusted R-squared |
0.772195 |
S.D. dependent var |
0.328991 |
|
|
S.E. of regression |
0.157024 |
Akaike info criterion |
-0.250678 |
|
|
Sum squared resid |
0.320534 |
Schwarz criterion |
0.289263 |
|
|
Log likelihood |
14.00814 |
Hannan-Quinn criter. |
-0.107432 |
|
|
F-statistic |
8.796349 |
Durbin-Watson stat |
2.187936 |
|
Dynamic forecasting
|
Automatic ARIMA Forecasting |
|
Selected dependent variable: SPREAD |
|
Date: 10/26/18 Time: 09:15 |
|
Sample: 1957M01 2015M03 |
|
Included observations: 696 |
|
Forecast length: 0 |
|
Number of estimated ARMA models: 4 |
|
Number of non-converged estimations: 0 |
|
Selected ARMA model: (0,0)(0,0) |
|
(0,0)(0,0) |
|
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. |
|
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 |
|
|
S.E. of regression |
1.220654 |
Akaike info criterion |
3.238087 |
|
|
Sum squared resid |
1035.548 |
Schwarz criterion |
3.244617 |
|
|
Log likelihood |
-1125.854 |
Hannan-Quinn criter. |
3.240612 |
|
|
Durbin-Watson stat |
0.084520 |
|||
|
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 |
|
(0,0)(0,0) |
-1125.854148 |
3.240960 |
3.254022 |
3.246011 |
|
(0,1)(0,0) |
-719.672654 |
2.076646 |
2.096238 |
2.084221 |
|
(1,0)(0,0) |
-260.080901 |
0.755980 |
0.775572 |
0.763555 |
|
(1,1)(0,0) |
-222.593793 |
0.651132 |
0.677254 |
0.661232 |
|
Forecast Evaluation |
|||||||
|
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 |
|||||
|
SPREAD |
NA |
NA |
|||||
|
Evaluation statistics |
|||||||
|
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 |
|
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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