ARIMA.RPF

ARIMA.RPF demonstrates a wide variety of procedures and instructions for handling ARIMA (Box-Jenkins) models, including the instructions BOXJENK, UFORECAST and THEIL and the procedures @REGCORRS, @UFOREERRORS, @BJIDENT, @BJFORE, @BJAUTOFIT, @DMARIANO and @GNEWBOLD.

Full Program

open data quarterly.xls
calendar(q) 1960:1
data(format=xls,org=columns) / tbill r10
graph(header="T-Bill and 10-year Bond Rates",key=upleft) 2
# tbill
# r10
*
* Compute spread and first difference of spread:
*
set spread = r10 - tbill
diff spread / dspread

spgraph(vfields=2,footer="FIGURE 2.5 Time Path of Interest Rate Spread")
graph(header="Panel (a): The interest rate spread")
# spread
graph(header="Panel (b): First difference of the spread")
# dspread
spgraph(done)
*
* Compute and graph autocorrelations of the spread itself
*
corr(results=corrs,partial=pcorrs,number=12) spread
graph(footer="FIGURE 2.6 ACF and PACF of the Spread",key=below,$
   style=bar,nodates,min=-1.0,max=1.0,number=0) 2
# corrs
# pcorrs
*
* Compute and graph autocorrelations for the first difference
*
corr(results=dcorrs,partial=dpcorrs,number=12) dspread
graph(header="Correlations of the first difference of spread",key=below,$
   style=bar,nodates,min=-1.0,max=1.0,number=0) 2
# dcorrs
# dpcorrs
*
* Estimating various candidate models. To ensure comparability, they are
* all run over the period from 1961:4 to the end of data. 1961:4 is the
* earliest period for which an AR(7) model (the longest AR considered)
* can be run using conditional least squares.
*
boxjenk(constant,ar=7) spread 1961:4 *
correlate(results=rescorrs,number=12,span=4,qstats,dfc=%narma) %resids
graph(header="AR(7)",style=bar,nodates,min=-1.0,max=1.0,number=1)
# rescorrs 2 *
*
@regcorrs(dfc=%narma,number=20,qstats,report,$
   method=burg,title="AR(7) model diagnostics")
*
* AR(2)
*
boxjenk(constant,ar=2) spread 1961:4 2008:1
@regcorrs(dfc=%narma,number=20,qstats,report,$
   method=burg,title="AR(2) model diagnostics")
*
* AR({1,2,7})
*
boxjenk(constant,ar=||1,2,7||) spread 1961:4 2008:1
@regcorrs(dfc=%narma,number=20,qstats,report,$
   method=burg,title="AR({1,2,7}) model diagnostics")
*
* ARMA(1,1)
*
boxjenk(constant,ar=1,ma=1) spread 1961:4 2008:1
@regcorrs(dfc=%narma,number=20,qstats,report,$
   method=burg,title="ARMA(1,1) model diagnostics")
*
* ARMA(2,(1,7))
*
boxjenk(constant,ar=2,ma=||1,7||) spread 1961:4 2008:1
@regcorrs(dfc=%narma,number=20,qstats,report,$
   method=burg,title="ARMA(2,(1,7)) model diagnostics")
*
* Estimate two candidate models and compare forecasting performance:
*
* Use all available data through 1995:3 to estimate each model (use *
* for "start" parameter to let RATS determine start date)
*
boxjenk(constant,define=ar7eq,ar=7)                spread * 1995:3
boxjenk(constant,define=ar2ma17eq,ar=2,ma=||1,7||) spread * 1995:3
*
* Compute and print one-step forecast (and corresponding actual value)
* for 1995:4.
*
uforecast(equation=ar7eq,static,print) onestep_ar7 1995:4 1995:4
uforecast(equation=ar2ma17eq,static,print) onestep_ar2ma17 1995:4 1995:4
*
* Now, compute and graph dynamic forecasts for 1995:4 through 2008:1.
*
uforecast(equation=ar7eq,print) fore_ar7 1995:4 2008:1
uforecast(equation=ar2ma17eq,print) fore_ar2ma17 1995:4 2008:1
graph(header="Forecasts vs Actuals",key=upleft) 3
# spread 1995:1 *
# fore_ar7
# fore_ar2ma17 / 5
*
* Next, compute one-step forecasts for all 50 periods, re-estimating at
* each period.
*
do time=1995:4,2008:1
   boxjenk(noprint,constant,define=ar7eq,ar=7) spread * time-1
   boxjenk(noprint,constant,define=ar2ma17eq,ar=2,ma=||1,7||) spread * time-1
   uforecast(equation=ar7eq,static) forecast_ar7 time time
   uforecast(equation=ar2ma17eq,static) forecast_ar2ma17 time time
end do
*
* Graph the forecast with the actuals.
*
graph(header="Forecasts vs Actuals",key=upleft) 3
# spread 1995:1 *
# forecast_ar7
# forecast_ar2ma17eq / 5
*
* Compute means, variances:
*
set e1 = forecast_ar7 - spread
set e2 = forecast_ar2ma17 - spread
stats e1
stats e2
*
* Using UForeErrors procedure:
*
@uforeerrors spread forecast_ar7
@uforeerrors spread forecast_ar2ma17
*
* Check for bias:
*
linreg spread
# constant forecast_ar7
test(all)
# 0.0 1.0
linreg spread
# constant forecast_ar2ma17
test(all)
# 0.0 1.0
*
* Granger-Newbold test
* Computing it directly:
*
set x = e1 + e2
set z = e1 - e2
cmom(corr)
# x z
compute corrcoef = %cmom(2,1)
compute gnstat = corrcoef/sqrt((1-corrcoef^2)/(%nobs-1))
cdf(noprint) ttest gnstat %nobs-1
display "G-N statistic = " *.### gnstat "Signif. Level = " #.##### %signif/2
display
*
* Using the GNewbold procedure:
*
@gnewbold spread forecast_ar7 forecast_ar2ma17
*
* Diebold-Mariano statistic on 4th power
*
set d = e1^4 - e2^4
statistics(noprint) d
compute dmstat = sqrt((%nobs-1)/%variance)*%mean
cdf(noprint) ttest dmstat %nobs-1
display "D-M statistic on 4th Power= " *.### dmstat "Signif. Level = " #.##### %signif/2
display
*
* RATS also includes a procedure for doing D-M tests based on MSE or MAE:
*
@dmariano(criterion=mse) spread forecast_ar7 forecast_ar2ma17
@dmariano(criterion=mae) spread forecast_ar7 forecast_ar2ma17
*
* Using THEIL to generate forecast performance statistics for dynamic
* forecasts up to 8 steps ahead.
*
boxjenk(noprint,constant,ar=7,define=ar7eq) spread * 1995:3
theil(setup,steps=8,to=2008:1)
# ar7eq
do time=1995:4,2008:1
   theil time
   boxjenk(noprint,constant,ar=7,define=ar7eq) spread * time
end do time
theil(dump,title="Theil U results for AR(7) Model")
*
boxjenk(noprint,constant,define=ar2ma17eq,ar=2,ma=||1,7||) spread * 1995:3
theil(setup,steps=8,to=2008:1)
# ar7eq
do time=1995:4,2008:1
   theil time
   boxjenk(noprint,constant,define=ar2ma17eq,ar=2,ma=||1,7||) spread * time
end do time
theil(dump,title="Theil U results for ARMA(2,(1,7)) Model")
*
* Using the procedures
*
* We'll read in two additional series used to demonstrate additional
* features of some of the procedures:
*
open data quarterly.xls
calendar(q) 1960:1
data(format=xls,org=columns) / tbill r10 ppinsa m1nsa
set spread = r10 - tbill
*
* Compare log, square root, and no transformation:
*
@bjtrans ppinsa
@bjtrans m1nsa
*
* Enders suggests the possibility of transforming a version of SPREAD
* adjusted to have positive values for all entries. Here's how you could
* do that:
*
set spreadshift = spread + 2
@bjtrans spreadshift
*
* BJIDENT plots for SPREAD:
*
@bjident(diffs=1) spread
*
* BJIDENT plots for the log of M1NSA (via the TRANS=LOG option):
*
@bjident(diffs=1,sdiffs=1,trans=log,number=24,report,qstat) m1nsa
*
* Using @BJDIFF procedure. This recommends 1 regular and 1 seasonal, but
* without the constant.
*
@bjdiff(diffs=1,sdiffs=1,trans=log) m1nsa
*
* Using BJFORE, first on the AR(7) model for SPREAD:
*
@bjfore(ars=7,constant) spread 1995:4 2008:1 fore
graph(header="Forecasts vs Actuals") 2
# fore
# spread 1993:1 *
@regcorrs(dfc=%narma,method=burg)
*
* And on an AR(1)SMA(1) model for M1NSA:
*

@bjfore(ars=1,smas=1,trans=log,diffs=1,sdiffs=1,constant) m1nsa 2008:2+1 2008:2+12 m1fore m1resids
graph(header="M1NSA Out of Sample Forecasts and Actuals") 2
# m1nsa 2000:1 *
# m1fore
*
* Do an "autofit" allowing up to 7 AR and 7 MA, using the Schwarz criterion
*
@bjautofit(constant,pmax=7,qmax=7,crit=sbc) spread

boxjenk(constant,define=ar2ma1eq,ar=%%autop,ma=%%autoq) spread
@regcorrs(dfc=%narma,method=burg)

do time=1995:4,2008:1
   boxjenk(noprint,constant,define=ar2ma1eq,ar=2,ma=1) spread * time-1
   uforecast(equation=ar2ma1eq,static) forecast_ar2ma1 time time
end do
@uforeerrors spread forecast_ar2ma1

Output

(This also generates a large number of graphs).

Correlations of Series SPREAD

Quarterly Data From 1960:01 To 2008:01

Autocorrelations

1 2 3 4 5 6 7 8 9 10

0.88976 0.74070 0.61628 0.48089 0.35679 0.22163 0.12109 0.06359 -0.01413 -0.10563

11 12

-0.18184 -0.21220

Partial Autocorrelations

1 2 3 4 5 6 7 8 9 10

0.88976 -0.24468 0.06984 -0.18353 0.01813 -0.20758 0.14985 -0.00209 -0.16063 -0.12736

11 12

-0.02063 0.12637

Correlations of Series DSPREAD

Quarterly Data From 1960:02 To 2008:01

Autocorrelations

1 2 3 4 5 6 7 8 9 10

0.17688 -0.11064 0.05038 -0.05136 0.04976 -0.15675 -0.19497 0.09099 0.06202 -0.06936

11 12

-0.20771 -0.05892

Partial Autocorrelations

1 2 3 4 5 6 7 8 9 10

0.17688 -0.14651 0.10415 -0.10592 0.11183 -0.24115 -0.07135 0.08017 0.02941 -0.08264

11 12

-0.20944 0.02459

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 3 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 2008:01

Usable Observations 186

Degrees of Freedom 178

Centered R^2 0.8248732

R-Bar^2 0.8179862

Uncentered R^2 0.9226102

Mean of Dependent Variable 1.3747320591

Std Error of Dependent Variable 1.2265941859

Standard Error of Estimate 0.5233027517

Sum of Squared Residuals 48.744547045

Regression F(7,178) 119.7724

Significance Level of F 0.0000000

Log Likelihood -139.3813

Durbin-Watson Statistic 1.9946

Q(36-7) 28.7150

Significance Level of Q 0.4799810

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.389213608 0.253444027 5.48134 0.00000014

2. AR{1} 1.176820330 0.074292627 15.84034 0.00000000

3. AR{2} -0.465765535 0.111809622 -4.16570 0.00004833

4. AR{3} 0.386060435 0.114616262 3.36829 0.00092711

5. AR{4} -0.338622292 0.115528650 -2.93107 0.00382132

6. AR{5} 0.318757035 0.114738661 2.77811 0.00605399

7. AR{6} -0.379106197 0.111797717 -3.39100 0.00085788

8. AR{7} 0.150404878 0.074428800 2.02079 0.04480018

Correlations of Series %RESIDS

Quarterly Data From 1961:04 To 2008:01

Autocorrelations

1 2 3 4 5 6 7 8 9 10

0.00040 0.02388 -0.01403 -0.02093 -0.04870 0.02612 -0.05768 0.12420 0.08036 0.01155

11 12

-0.18448 0.04826

Ljung-Box Q-Statistics

Lags Statistic Signif Lvl

4 0.230

8 4.501 0.033868

12 13.072 0.022714

Lag Corr Partial LB Q Q Signif

1 0.000 0.000 0.000031

2 0.024 0.024 0.108430

3 -0.014 -0.014 0.146041

4 -0.021 -0.022 0.230212

5 -0.049 -0.048 0.688489

6 0.026 0.027 0.820999

7 -0.058 -0.056 1.470848

8 0.124 0.122 4.501357 0.0339

9 0.080 0.082 5.777170 0.0557

10 0.012 0.003 5.803670 0.1216

11 -0.184 -0.191 12.603911 0.0134

12 0.048 0.051 13.071935 0.0227

13 -0.033 -0.005 13.286264 0.0387

14 -0.054 -0.063 13.877188 0.0534

15 -0.061 -0.061 14.628409 0.0668

16 -0.045 -0.063 15.045535 0.0897

17 -0.111 -0.124 17.614942 0.0618

18 0.016 -0.020 17.669738 0.0896

19 0.030 0.088 17.862936 0.1199

20 0.069 0.089 18.863287 0.1274

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 3 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 2008:01

Usable Observations 186

Degrees of Freedom 183

Centered R^2 0.8052464

R-Bar^2 0.8031180

Uncentered R^2 0.9139370

Mean of Dependent Variable 1.3747320591

Std Error of Dependent Variable 1.2265941859

Standard Error of Estimate 0.5442568905

Sum of Squared Residuals 54.207448001

Regression F(2,183) 378.3245

Significance Level of F 0.0000000

Log Likelihood -149.2602

Durbin-Watson Statistic 1.9583

Q(36-2) 49.5634

Significance Level of Q 0.0412570

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.376715393 0.292984914 4.69893 0.00000513

2. AR{1} 1.107804742 0.071986352 15.38909 0.00000000

3. AR{2} -0.244015722 0.072036345 -3.38740 0.00086380

Lag Corr Partial LB Q Q Signif

1 0.018 0.018 0.058249

2 -0.102 -0.102 2.031976

3 0.156 0.161 6.658597 0.0099

4 -0.034 -0.055 6.874352 0.0322

5 0.140 0.184 10.670502 0.0136

6 -0.094 -0.155 12.375405 0.0148

7 -0.164 -0.102 17.651420 0.0034

8 0.136 0.071 21.285935 0.0016

9 0.056 0.075 21.909475 0.0026

10 -0.040 -0.012 22.230520 0.0045

11 -0.190 -0.212 29.461984 0.0005

12 -0.022 0.017 29.558816 0.0010

13 0.026 -0.073 29.692901 0.0018

14 -0.113 -0.057 32.268361 0.0013

15 -0.071 -0.045 33.303399 0.0015

16 -0.049 -0.001 33.796549 0.0022

17 -0.094 -0.152 35.625921 0.0020

18 0.031 -0.006 35.831591 0.0031

19 0.012 0.071 35.863803 0.0048

20 0.020 0.081 35.951353 0.0072

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 3 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 2008:01

Usable Observations 186

Degrees of Freedom 182

Centered R^2 0.8068368

R-Bar^2 0.8036528

Uncentered R^2 0.9146398

Mean of Dependent Variable 1.3747320591

Std Error of Dependent Variable 1.2265941859

Standard Error of Estimate 0.5435171169

Sum of Squared Residuals 53.764775863

Regression F(3,182) 253.4029

Significance Level of F 0.0000000

Log Likelihood -148.4976

Durbin-Watson Statistic 1.9415

Q(36-3) 48.2082

Significance Level of Q 0.0424680

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.385762764 0.238885717 5.80094 0.00000003

2. AR{1} 1.088017061 0.073683472 14.76609 0.00000000

3. AR{2} -0.210106352 0.077087421 -2.72556 0.00704617

4. AR{7} -0.044798883 0.036596480 -1.22413 0.22248511

Lag Corr Partial LB Q Q Signif

1 0.026 0.026 0.128872

2 -0.122 -0.123 2.964685

3 0.121 0.130 5.760102

4 -0.050 -0.077 6.246283 0.0124

5 0.119 0.163 8.972965 0.0113

6 -0.114 -0.175 11.505607 0.0093

7 -0.137 -0.063 15.191381 0.0043

8 0.177 0.116 21.320325 0.0007

9 0.096 0.115 23.146308 0.0007

10 -0.005 0.015 23.150444 0.0016

11 -0.157 -0.175 28.055688 0.0005

12 0.011 0.046 28.081331 0.0009

13 0.053 -0.052 28.643715 0.0014

14 -0.098 -0.039 30.612805 0.0013

15 -0.063 -0.039 31.416859 0.0017

16 -0.038 -0.007 31.708075 0.0027

17 -0.080 -0.156 33.019707 0.0029

18 0.036 0.006 33.291988 0.0043

19 0.021 0.080 33.383616 0.0066

20 0.030 0.086 33.578167 0.0095

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 11 Iterations. Final criterion was 0.0000049 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 2008:01

Usable Observations 186

Degrees of Freedom 183

Centered R^2 0.8084206

R-Bar^2 0.8063268

Uncentered R^2 0.9153397

Mean of Dependent Variable 1.3747320591

Std Error of Dependent Variable 1.2265941859

Standard Error of Estimate 0.5398034133

Sum of Squared Residuals 53.323953677

Log Likelihood -147.7320

Durbin-Watson Statistic 2.0655

Q(36-2) 50.0686

Significance Level of Q 0.0372206

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.3764740935 0.2951469258 4.66369 0.00000598

2. AR{1} 0.8168202206 0.0468118958 17.44899 0.00000000

3. MA{1} 0.3684977270 0.0755692740 4.87629 0.00000234

Lag Corr Partial LB Q Q Signif

1 -0.036 -0.036 0.243884

2 -0.019 -0.020 0.310737

3 0.167 0.166 5.648186 0.0175

4 -0.070 -0.061 6.597796 0.0369

5 0.148 0.155 10.830199 0.0127

6 -0.090 -0.121 12.420552 0.0145

7 -0.153 -0.133 16.998880 0.0045

8 0.133 0.075 20.481212 0.0023

9 0.013 0.067 20.516603 0.0046

10 -0.018 0.003 20.578636 0.0084

11 -0.186 -0.226 27.461877 0.0012

12 -0.024 -0.001 27.579145 0.0021

13 0.009 -0.052 27.595705 0.0037

14 -0.115 -0.060 30.283598 0.0025

15 -0.070 -0.059 31.272548 0.0031

16 -0.046 0.007 31.706023 0.0044

17 -0.099 -0.141 33.743214 0.0037

18 0.035 -0.019 33.999298 0.0054

19 0.011 0.085 34.023768 0.0083

20 0.016 0.075 34.079455 0.0123

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 19 Iterations. Final criterion was 0.0000097 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 2008:01

Usable Observations 186

Degrees of Freedom 181

Centered R^2 0.8233157

R-Bar^2 0.8194110

Uncentered R^2 0.9219219

Mean of Dependent Variable 1.3747320591

Std Error of Dependent Variable 1.2265941859

Standard Error of Estimate 0.5212505160

Sum of Squared Residuals 49.178080183

Log Likelihood -140.2048

Durbin-Watson Statistic 1.9349

Q(36-4) 29.5105

Significance Level of Q 0.5931429

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.399969290 0.327016177 4.28104 0.00003013

2. AR{1} 0.312137887 0.103267109 3.02263 0.00286945

3. AR{2} 0.488178387 0.101238541 4.82206 0.00000300

4. MA{1} 0.862435712 0.062507359 13.79735 0.00000000

5. MA{7} -0.143794824 0.038119159 -3.77225 0.00021900

Lag Corr Partial LB Q Q Signif

1 0.029 0.029 0.164134

2 0.073 0.072 1.179351

3 0.020 0.016 1.254742

4 0.023 0.017 1.354938

5 0.019 0.015 1.421503 0.2332

6 -0.051 -0.055 1.921974 0.3825

7 -0.073 -0.074 2.972068 0.3960

8 0.026 0.037 3.107256 0.5400

9 0.087 0.099 4.610502 0.4652

10 -0.093 -0.100 6.345342 0.3856

11 -0.160 -0.171 11.442026 0.1205

12 -0.063 -0.047 12.230953 0.1412

13 -0.008 0.015 12.243919 0.1999

14 -0.111 -0.104 14.752291 0.1414

15 -0.081 -0.060 16.095155 0.1376

16 -0.063 -0.035 16.916471 0.1528

17 -0.117 -0.148 19.746639 0.1017

18 0.024 -0.000 19.870797 0.1343

19 0.010 0.064 19.892655 0.1761

20 0.068 0.096 20.862370 0.1839

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 3 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 1995:03

Usable Observations 136

Degrees of Freedom 128

Centered R^2 0.8205491

R-Bar^2 0.8107354

Uncentered R^2 0.9189417

Mean of Dependent Variable 1.3878197059

Std Error of Dependent Variable 1.2643087248

Standard Error of Estimate 0.5500318407

Sum of Squared Residuals 38.724483302

Regression F(7,128) 83.6124

Significance Level of F 0.0000000

Log Likelihood -107.5552

Durbin-Watson Statistic 2.0038

Q(34-7) 22.3708

Significance Level of Q 0.7183058

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.351487121 0.301909009 4.47647 0.00001659

2. AR{1} 1.159915901 0.087275081 13.29034 0.00000000

3. AR{2} -0.494951526 0.128316005 -3.85729 0.00018076

4. AR{3} 0.446059748 0.131067297 3.40329 0.00088925

5. AR{4} -0.351758180 0.133567471 -2.63356 0.00949082

6. AR{5} 0.391453999 0.131261596 2.98224 0.00342675

7. AR{6} -0.466884011 0.128607847 -3.63029 0.00040798

8. AR{7} 0.159779881 0.088100655 1.81361 0.07208107

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 17 Iterations. Final criterion was 0.0000070 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1960:03 To 1995:03

Usable Observations 141

Degrees of Freedom 136

Centered R^2 0.8140898

R-Bar^2 0.8086219

Uncentered R^2 0.9179234

Mean of Dependent Variable 1.3917741135

Std Error of Dependent Variable 1.2418091815

Standard Error of Estimate 0.5432516282

Sum of Squared Residuals 40.136637084

Log Likelihood -111.4892

Durbin-Watson Statistic 2.0268

Q(35-4) 27.9503

Significance Level of Q 0.6237293

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.437927566 0.354494834 4.05627 0.00008359

2. AR{1} 0.401851212 0.123272973 3.25985 0.00140855

3. AR{2} 0.386997961 0.117754950 3.28647 0.00129088

4. MA{1} 0.791255153 0.080435453 9.83714 0.00000000

5. MA{7} -0.151967684 0.050831942 -2.98961 0.00331627

Entry SPREAD

1995:04 0.806674241

0.623330000

Entry SPREAD

1995:04 1.075241479

0.623330000

Entry SPREAD

1995:04 0.806674241

1996:01 0.693679473

1996:02 0.612664189

1996:03 0.609518579

1996:04 0.816869540

1997:01 0.921044416

1997:02 0.977391972

1997:03 1.083993148

1997:04 1.171822150

1998:01 1.279116387

1998:02 1.331295579

1998:03 1.346943158

1998:04 1.388298667

1999:01 1.407670032

1999:02 1.416323037

1999:03 1.414079870

1999:04 1.400195951

2000:01 1.399468113

2000:02 1.392226338

2000:03 1.379733579

2000:04 1.371563587

2001:01 1.362291112

2001:02 1.358393258

2001:03 1.354498251

2001:04 1.349021837

2002:01 1.347597894

2002:02 1.346479094

2002:03 1.346311397

2002:04 1.346775417

2003:01 1.346450429

2003:02 1.347539646

2003:03 1.348581705

2003:04 1.349172296

2004:01 1.350022912

2004:02 1.350428265

2004:03 1.351026564

2004:04 1.351539044

2005:01 1.351637653

2005:02 1.351846409

2005:03 1.351913784

2005:04 1.351933191

2006:01 1.351966828

2006:02 1.351847790

2006:03 1.351795586

2006:04 1.351746796

2007:01 1.351660606

2007:02 1.351618241

2007:03 1.351549158

2007:04 1.351509230

2008:01 1.351494784

Entry SPREAD

1995:04 1.075241479

1996:01 1.038280695

1996:02 1.140234323

1996:03 1.177748017

1996:04 1.311052030

1997:01 1.368644822

1997:02 1.351422600

1997:03 1.376353160

1997:04 1.379706571

1998:01 1.390702219

1998:02 1.396418597

1998:03 1.402971023

1998:04 1.407816351

1999:01 1.412299227

1999:02 1.415975808

1999:03 1.419188111

1999:04 1.421901808

2000:01 1.424235465

2000:02 1.426223443

2000:03 1.427925435

2000:04 1.429378726

2001:01 1.430621400

2001:02 1.431683190

2001:03 1.432590785

2001:04 1.433366413

2002:01 1.434029338

2002:02 1.434595902

2002:03 1.435080126

2002:04 1.435493972

2003:01 1.435847670

2003:02 1.436149961

2003:03 1.436408318

2003:04 1.436629125

2004:01 1.436817840

2004:02 1.436979127

2004:03 1.437116973

2004:04 1.437234785

2005:01 1.437335473

2005:02 1.437421528

2005:03 1.437495075

2005:04 1.437557933

2006:01 1.437611656

2006:02 1.437657570

2006:03 1.437696811

2006:04 1.437730349

2007:01 1.437759013

2007:02 1.437783510

2007:03 1.437804447

2007:04 1.437822341

2008:01 1.437837635

Statistics on Series E1

Quarterly Data From 1995:04 To 2008:01

Observations 50

Sample Mean -0.017824 Variance 0.212819

Standard Error 0.461323 SE of Sample Mean 0.065241

t-Statistic (Mean=0) -0.273200 Signif Level (Mean=0) 0.785847

Skewness -0.637663 Signif Level (Sk=0) 0.074187

Kurtosis (excess) 0.366175 Signif Level (Ku=0) 0.622627

Jarque-Bera 3.667793 Signif Level (JB=0) 0.159790

Statistics on Series E2

Quarterly Data From 1995:04 To 2008:01

Observations 50

Sample Mean 0.002352 Variance 0.201455

Standard Error 0.448837 SE of Sample Mean 0.063475

t-Statistic (Mean=0) 0.037052 Signif Level (Mean=0) 0.970594

Skewness -0.647366 Signif Level (Sk=0) 0.069890

Kurtosis (excess) 0.208134 Signif Level (Ku=0) 0.779687

Jarque-Bera 3.582610 Signif Level (JB=0) 0.166742

Forecast Analysis for SPREAD

From 1995:04 to 2008:01

Mean Error 0.01782385

Mean Absolute Error 0.35514485

Root Mean Square Error 0.45703438

Mean Square Error 0.208880

Theil's U 0.986644

Forecast Analysis for SPREAD

From 1995:04 to 2008:01

Mean Error -0.0023519

Mean Absolute Error 0.3547898

Root Mean Square Error 0.4443326

Mean Square Error 0.197431

Theil's U 0.959223

Linear Regression - Estimation by Least Squares

Dependent Variable SPREAD

Quarterly Data From 1995:04 To 2008:01

Usable Observations 50

Degrees of Freedom 48

Centered R^2 0.8331962

R-Bar^2 0.8297212

Uncentered R^2 0.9315142

Mean of Dependent Variable 1.3391336600

Std Error of Dependent Variable 1.1290023426

Standard Error of Estimate 0.4658811966

Sum of Squared Residuals 10.418173890

Regression F(1,48) 239.7633

Significance Level of F 0.0000000

Log Likelihood -31.7351

Durbin-Watson Statistic 1.8091

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant 0.0358565878 0.1068882860 0.33546 0.73874103

2. FORECAST_AR7 0.9863523737 0.0637001922 15.48429 0.00000000

F(2,48)= 0.05954 with Significance Level 0.94226394

Linear Regression - Estimation by Least Squares

Dependent Variable SPREAD

Quarterly Data From 1995:04 To 2008:01

Usable Observations 50

Degrees of Freedom 48

Centered R^2 0.8420871

R-Bar^2 0.8387973

Uncentered R^2 0.9351646

Mean of Dependent Variable 1.3391336600

Std Error of Dependent Variable 1.1290023426

Standard Error of Estimate 0.4532951392

Sum of Squared Residuals 9.8628711968

Regression F(1,48) 255.9650

Significance Level of F 0.0000000

Log Likelihood -30.3658

Durbin-Watson Statistic 1.8278

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.019547910 0.106402598 -0.18372 0.85500979

2. FORECAST_AR2MA17 1.012818664 0.063305489 15.99891 0.00000000

F(2,48)= 0.02117 with Significance Level 0.97905786

G-N statistic = 0.826 Signif. Level = 0.20642

Granger-Newbold Forecast Comparison Test

Forecasts of SPREAD over 1995:04 to 2008:01

Forecast Test Stat P(GN>x)

FORECAST_AR7 0.8260 0.20642

FORECAST_AR2MA17 -0.8260 0.79358

D-M statistic on 4th Power= 1.616 Signif. Level = 0.05627

Diebold-Mariano Forecast Comparison Test

Forecasts of SPREAD over 1995:04 to 2008:01

Forecast MSE Test Stat P(DM>x)

FORECAST_AR7 0.20888042 0.8781 0.18995

FORECAST_AR2MA17 0.19743145 -0.8781 0.81005

Diebold-Mariano Forecast Comparison Test

Forecasts of SPREAD over 1995:04 to 2008:01

Forecast MAE Test Stat P(DM>x)

FORECAST_AR7 0.35514485 0.0241 0.49039

FORECAST_AR2MA17 0.35478985 -0.0241 0.50961

Theil U results for AR(7) Model

Step Mean Error Mean Abs Err RMS Error Theil U N Obs

Forecast Statistics for Series SPREAD

1 0.0178238 0.3551449 0.4570344 0.9866 50

2 0.0333591 0.6057377 0.7382896 0.9734 49

3 0.0310810 0.7696073 0.9264497 0.9260 48

4 0.0130217 0.8798374 1.0775190 0.8907 47

5 -0.0122001 0.9659965 1.1656551 0.8572 46

6 -0.0207625 1.0294925 1.2124810 0.8334 45

7 -0.0297088 1.0544159 1.2244027 0.8053 44

8 -0.0391965 1.0672524 1.2316369 0.7728 43

Theil U results for ARMA(2,(1,7)) Model

Step Mean Error Mean Abs Err RMS Error Theil U N Obs

Forecast Statistics for Series SPREAD

1 0.0131677 0.3449180 0.4389769 0.9477 50

2 0.0221656 0.5791501 0.7038170 0.9279 49

3 0.0155860 0.7343455 0.8825325 0.8821 48

4 -0.0035745 0.8302931 1.0263826 0.8485 47

5 -0.0270086 0.9029884 1.1094199 0.8158 46

6 -0.0347677 0.9691531 1.1482820 0.7892 45

7 -0.0416368 0.9942000 1.1574798 0.7613 44

8 -0.0474994 1.0144249 1.1664960 0.7320 43

0 Regular 0 Seasonal of M1NSA 1 Regular 0 Seasonal of M1NSA

Lag Corr Corr Partial Corr Partial Corr Partial LB Q Q Signif

1 0.99969 0.94101 0.94101 0.15440 0.15440 0.36687 0.36687 25.84440 0.00000

2 0.99910 0.83880 -0.40790 0.25143 0.23315 0.12653 -0.00932 28.93507 0.00000

3 0.99818 0.72162 -0.06114 0.04203 -0.02572 -0.02851 -0.08314 29.09280 0.00000

4 0.99702 0.60778 0.01553 0.64499 0.63344 -0.34384 -0.35553 52.16275 0.00000

5 0.99538 0.53442 0.31157 0.00947 -0.28322 0.01390 0.34094 52.20065 0.00000

6 0.99351 0.45915 -0.38657 0.16012 -0.00366 0.10594 0.05960 54.41438 0.00000

7 0.99135 0.37101 -0.13983 -0.05211 -0.01732 0.00278 -0.16198 54.41591 0.00000

8 0.98898 0.28189 0.08394 0.53209 0.23854 -0.02156 -0.21798 54.50865 0.00000

9 0.98618 0.19483 0.16371 -0.14875 -0.34373 -0.12089 0.14985 57.43963 0.00000

10 0.98320 0.12162 -0.20572 -0.00668 -0.11621 -0.21899 -0.12333 67.11119 0.00000

11 0.98000 0.07395 0.06113 -0.14961 0.08017 -0.09055 -0.06703 68.77419 0.00000

12 0.97663 0.03694 0.01519 0.43333 0.13218 -0.11082 -0.20407 71.27868 0.00000

13 0.97286 0.01300 0.16394 -0.22426 -0.19475 -0.09586 0.08012 73.16356 0.00000

14 0.96895 0.00054 -0.11740 -0.02112 0.07868 -0.01058 -0.04582 73.18667 0.00000

15 0.96482 -0.01015 0.00668 -0.18435 0.01934 -0.04573 -0.00007 73.62050 0.00000

16 0.96053 -0.01493 -0.03954 0.41380 0.13952 -0.09350 -0.27723 75.44453 0.00000

17 0.95585 -0.00806 0.24853 -0.22940 -0.11504 -0.07879 0.03623 76.74746 0.00000

18 0.95103 0.00877 -0.05964 -0.02349 0.02328 -0.12075 -0.09161 79.82534 0.00000

19 0.94599 0.04039 0.06926 -0.18385 -0.03657 -0.16509 -0.13108 85.61249 0.00000

20 0.94079 0.09217 0.11325 0.46354 0.23239 0.03227 -0.03583 85.83488 0.00000

21 0.93520 0.14076 0.02009 -0.17497 -0.05755 0.07258 0.11405 86.96681 0.00000

22 0.92944 0.18120 -0.13266 0.06325 0.08141 0.08799 -0.09024 88.64023 0.00000

23 0.92343 0.21151 0.07075 -0.06137 0.11625 0.18726 0.04023 96.26624 0.00000

24 0.91723 0.21966 -0.05951 0.49256 0.00373 0.01709 -0.09781 96.33016 0.00000

BJDiff Table, Series M1NSA

Reg Diff Seas Diff Intercept Crit

0 0 No 0.202853

0 0 Yes -3.325071

0 1 No -8.128388

0 1 Yes -8.267776

1 0 No -8.151370

1 0 Yes -8.365352

1 1 No -8.808739*

1 1 Yes -8.781005

Estimation Range: 1961:04 to 1995:03

Forecast Range: 1995:04 to 2008:01

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 3 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1961:04 To 1995:03

Usable Observations 136

Degrees of Freedom 128

Centered R^2 0.8205491

R-Bar^2 0.8107354

Uncentered R^2 0.9189417

Mean of Dependent Variable 1.3878197059

Std Error of Dependent Variable 1.2643087248

Standard Error of Estimate 0.5500318407

Sum of Squared Residuals 38.724483302

Regression F(7,128) 83.6124

Significance Level of F 0.0000000

Log Likelihood -107.5552

Durbin-Watson Statistic 2.0038

Q(34-7) 22.3708

Significance Level of Q 0.7183058

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.351487121 0.301909009 4.47647 0.00001659

2. AR{1} 1.159915901 0.087275081 13.29034 0.00000000

3. AR{2} -0.494951526 0.128316005 -3.85729 0.00018076

4. AR{3} 0.446059748 0.131067297 3.40329 0.00088925

5. AR{4} -0.351758180 0.133567471 -2.63356 0.00949082

6. AR{5} 0.391453999 0.131261596 2.98224 0.00342675

7. AR{6} -0.466884011 0.128607847 -3.63029 0.00040798

8. AR{7} 0.159779881 0.088100655 1.81361 0.07208107

Estimation Range: 1961:03 to 2008:02

Forecast Range: 2008:03 to 2011:02

Model uses log transformation

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 9 Iterations. Final criterion was 0.0000054 <= 0.0000100

Dependent Variable M1NSA

Quarterly Data From 1961:03 To 2008:02

Usable Observations 188

Degrees of Freedom 185

Centered R^2 0.9998359

R-Bar^2 0.9998341

Uncentered R^2 0.9999976

Mean of Dependent Variable 6.2413897053

Std Error of Dependent Variable 0.7694693456

Standard Error of Estimate 0.0099094481

Sum of Squared Residuals 0.0181664747

Log Likelihood 602.2338

Durbin-Watson Statistic 2.0847

Q(36-2) 35.9096

Significance Level of Q 0.3790572

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT -0.000107905 0.000395783 -0.27264 0.78543643

2. AR{1} 0.527629512 0.062116602 8.49418 0.00000000

3. SMA{4} -0.755245102 0.049805820 -15.16379 0.00000000

BIC analysis of models for series SPREAD

AR 0 1 2 3 4 5 6 7

0 618.9641 436.7480 383.6948 343.2322 340.2751 334.8581 314.7369 319.9699

1 323.0264 313.4086 316.0286 316.7097 321.3585 325.1005 319.9753 324.6645

2 316.5067 313.3651* 322.3184 320.4113 324.1219 329.9807 363.2536 335.1889

3 320.8379 318.4922 321.2436 324.0077 329.2574 333.4293 335.7526 336.2029

4 319.6297 324.6967 321.6646 329.1025 326.6846 329.6273 330.9086 333.2572

5 324.8301 326.1888 327.6657 326.3371 325.8488 336.4870 334.7450 335.7013

6 321.8561 324.5037 322.7470 333.0741 339.7822 333.3788 347.7693 466.5056

7 322.8906 327.0667 329.8409 338.8530 338.1161 346.9846 356.0072 489.7948

Box-Jenkins - Estimation by LS Gauss-Newton

Convergence in 24 Iterations. Final criterion was 0.0000072 <= 0.0000100

Dependent Variable SPREAD

Quarterly Data From 1960:03 To 2008:01

Usable Observations 191

Degrees of Freedom 187

Centered R^2 0.8128791

R-Bar^2 0.8098772

Uncentered R^2 0.9187313

Mean of Dependent Variable 1.3779938901

Std Error of Dependent Variable 1.2105947978

Standard Error of Estimate 0.5278565561

Sum of Squared Residuals 52.104285693

Log Likelihood -146.9603

Durbin-Watson Statistic 1.9909

Q(36-3) 40.2507

Significance Level of Q 0.1800448

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. CONSTANT 1.4148170112 0.3127325433 4.52405 0.00001076

2. AR{1} 0.4598285309 0.1451145963 3.16873 0.00178881

3. AR{2} 0.3316941827 0.1374002146 2.41407 0.01673933

4. MA{1} 0.7134485121 0.1126034686 6.33594 0.00000000

Forecast Analysis for SPREAD

From 1995:04 to 2008:01

Mean Error 0.00013639

Mean Absolute Error 0.34935624

Root Mean Square Error 0.43896063

Mean Square Error 0.192686

Theil's U 0.947626