Analysis Of Autocorrelation And ARIMA Model Selection

Calculating the ACF and the PFC of the data set

Date: 10/25/18 Time: 22:48
Sample: 1957M01 2015M03
Included observations: 699


Autocorrelation Save Time On Research and Writing Hire a Pro to Write You a 100% Plagiarism-Free Paper. Get My Paper	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

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.

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
Date: 10/26/18 Time: 07:50
Sample: 1957M01 2012M12
Included observations: 672
Forecast length: 0


Number of estimated ARMA models: 4
Number of non-converged estimations: 0
Selected ARMA model: (1,1)(0,0)
AIC value: 0.677963363957

Model Selection Criteria Table
Dependent Variable: SPREAD
Date: 10/26/18 Time: 07:50
Sample: 1957M01 2012M12
Included observations: 672


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
Date: 10/26/18 Time: 07:59
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

Model Selection Criteria Table
Dependent Variable: SPREAD
Date: 10/26/18 Time: 07:59
Sample: 1957M01 2015M03
Included observations: 696


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
Date: 10/26/18 Time: 08:02
Sample: 1957M01 2015M03
Included observations: 699
Evaluation sample: 1957M01 2015M03
Training sample: 1957M01 2012M12
Number of forecasts: 2


Combination tests
Null hypothesis: Forecast i includes all information contained in others


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
MSE ranks	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000

Dependent Variable: SPREAD
Method: ARMA Maximum Likelihood (OPG – BHHH)
Date: 10/26/18 Time: 08:51
Sample: 2013M01 2014M12
Included observations: 24
Failure to improve objective (non-zero gradients) after 106 iterations
Coefficient covariance computed using outer product of gradients


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
Method: Least Squares
Date: 10/26/18 Time: 09:15
Sample (adjusted): 1957M01 2014M12
Included observations: 696 after adjustments


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
Dependent Variable: SPREAD
Date: 10/26/18 Time: 09:15
Sample: 1957M01 2015M03
Included observations: 696


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
Sample: 2013M01 2015M03
Included observations: 27
Evaluation sample: 2013M01 2015M03
Training sample: 1957M01 2012M12
Number of forecasts: 3


Combination tests
Null hypothesis: Forecast i includes all information contained in others


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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