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REPLICATION OF TIME-SERIES ANALYSIS OF BLAISDELL DATA (NKNW pp. 504-516)
USING COCHRANE-ORCUTT, HILDRETH-LU, & FIRST DIFFERENCES PROCEDURES
MON 3/22/99 9:10:04 AM
SYSTAT VERSION 7.0.1
COPYRIGHT (C) 1997, SPSS INC.
Welcome to SYSTAT!
>USE 'C:\SYSTAT7\S209\BLAIS.SYD'
SYSTAT Rectangular file C:\SYSTAT7\S209\BLAIS.SYD,
created Fri Mar 19, 1999 at 16:06:42, contains variables:
Y X
>mglh
>model y=constant+x
>save blaisres/resid data
>estimate
Dep Var: Y N: 20 Multiple R: 0.999396 Squared multiple R: 0.998792
Adjusted squared multiple R: 0.998725 Standard error of estimate: 0.086056
Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)
CONSTANT -1.454750 0.214146 0.0 . -6.79326 0.00000
X 0.176283 0.001445 0.999396 1.000000 1.22E02 0.00000
Analysis of Variance
Source Sum-of-Squares df Mean-Square F-ratio P
Regression 110.256878 1 110.256878 1.48881E+04 0.000000
Residual 0.133302 18 0.007406
Durbin-Watson D Statistic 0.735
First Order Autocorrelation 0.626
Residuals have been saved.
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Estimated rho (0.626) differs slightly from the one in NKNW (0.631166; Table 12.3 p. 510);
Estimated D-W statistic (0.735) is the same.
Next switch to file of residuals to do an index plot, then back to original
file
>USE 'C:\SYSTAT7\S209\BLAISRES.SYD'
SYSTAT Rectangular file C:\SYSTAT7\S209\BLAISRES.SYD,
created Mon Mar 22, 1999 at 09:11:12, contains variables:
ESTIMATE RESIDUAL LEVERAGE COOK STUDENT SEPRED
Y X
>plot residual/stick ylimit=0 line dash=11
>USE 'C:\SYSTAT7\S209\BLAIS.SYD'
SYSTAT Rectangular file C:\SYSTAT7\S209\BLAIS.SYD,
created Fri Mar 19, 1999 at 16:06:42, contains variables:
Y X
>rem Cochrane-Orcutt procedure
>basic
File in use is C:\SYSTAT7\S209\BLAIS.SYD.
Variables in the SYSTAT Rectangular file are:
Y X
BASIC statements cleared.
>let y1 = lag(y)
>let x1 = lag(x)
>let yp = y - 0.626*y1
>let xp = x - 0.626*x1
>run
SYSTAT file created.
20 cases and 6 variables processed.
BASIC statements cleared.
>page wide
>list x x1 xp y y1 yp
Case number X X1 XP Y Y1 YP
1 127.300000 . . 20.960000 . .
2 130.000000 127.300000 50.310200 21.400000 20.960000 8.279040
3 132.700000 130.000000 51.320000 21.960000 21.400000 8.563600
4 129.400000 132.700000 46.329800 21.520000 21.960000 7.773040
5 135.000000 129.400000 53.995600 22.390000 21.520000 8.918480
6 137.100000 135.000000 52.590000 22.760000 22.390000 8.743860
7 141.200000 137.100000 55.375400 23.480000 22.760000 9.232240
8 142.800000 141.200000 54.408800 23.660000 23.480000 8.961520
9 145.500000 142.800000 56.107200 24.100000 23.660000 9.288840
10 145.300000 145.500000 54.217000 24.010000 24.100000 8.923400
11 148.300000 145.300000 57.342200 24.540000 24.010000 9.509740
12 146.400000 148.300000 53.564200 24.300000 24.540000 8.937960
13 150.200000 146.400000 58.553600 25.000000 24.300000 9.788200
14 153.100000 150.200000 59.074800 25.640000 25.000000 9.990000
15 157.300000 153.100000 61.459400 26.360000 25.640000 10.309360
16 160.700000 157.300000 62.230200 26.980000 26.360000 10.478640
17 164.200000 160.700000 63.601800 27.520000 26.980000 10.630520
18 165.600000 164.200000 62.810800 27.780000 27.520000 10.552480
19 168.700000 165.600000 65.034400 28.240000 27.780000 10.849720
20 171.700000 168.700000 66.093800 28.780000 28.240000 11.101760
>mglh
>model yp = constant + xp
>estimate
1 case(s) deleted due to missing data.
Dep Var: YP N: 19 Multiple R: 0.997601 Squared multiple R: 0.995208
Adjusted squared multiple R: 0.994926 Standard error of estimate: 0.067195
Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)
CONSTANT -0.403282 0.167675 0.0 . -2.40514 0.02784
XP 0.173821 0.002925 0.997601 1.000000 59.41838 0.00000
Analysis of Variance
Source Sum-of-Squares df Mean-Square F-ratio P
Regression 15.941142 1 15.941142 3530.543519 0.000000
Residual 0.076759 17 0.004515
----------------------------------------------------------------------------------------------------------------------------------
Durbin-Watson D Statistic 1.644
First Order Autocorrelation 0.150
Now redo the D-W test; if D-W still significant one can iterate
the Cochrane-Orcutt procedure (see NKNW p. 510)
>rem Hildreth-Lu procedure using nonlin
>nonlin
>model y = rho*y1 + b0 + b1*(x-rho*x1)
>estimate
Iteration
No. Loss RHO B0 B1
0 .631740D+02 .101000D+01-.102000D+01 .103000D+01
1 .133054D+00 .101044D+01 .848852D-01 .158714D+00
2 .722882D-01 .967085D+00 .768116D-01 .159566D+00
3 .716819D-01 .961296D+00 .718397D-01 .160309D+00
4 .716707D-01 .959414D+00 .716441D-01 .160463D+00
5 .716704D-01 .958978D+00 .716043D-01 .160508D+00
6 .716704D-01 .958861D+00 .716071D-01 .160519D+00
7 .716704D-01 .958831D+00 .716075D-01 .160522D+00
Dependent variable is Y
Zero weights, missing data or estimates reduced degrees of freedom
Source Sum-of-Squares df Mean-Square
Regression 1.17437E+04 3 3914.570710
Residual 0.071670 16 0.004479
Total 1.17438E+04 19
Mean corrected 96.679779 18
Raw R-square (1-Residual/Total) = 0.999994
Mean corrected R-square (1-Residual/Corrected) = 0.999259
R(observed vs predicted) square = 0.999263
Wald Confidence Interval
Parameter Estimate A.S.E. Param/ASE Lower < 95%> Upper
RHO 0.958831 0.080054 11.977261 0.789123 1.128538
B0 0.071607 0.071555 1.000733 -0.080082 0.223297
B1 0.160522 0.007931 20.240415 0.143710 0.177335
>rem compare with NKNW Table 12.5 p. 513
>rem first differences procedure
>mglh
>let dy = y - y1
>let dx = x - x1
>list x x1 dx y y1 dy
Case number X X1 DX Y Y1 DY
1 127.300000 . . 20.960000 . .
2 130.000000 127.300000 2.700000 21.400000 20.960000 0.440000
3 132.700000 130.000000 2.700000 21.960000 21.400000 0.560000
4 129.400000 132.700000 -3.300000 21.520000 21.960000 -0.440000
5 135.000000 129.400000 5.600000 22.390000 21.520000 0.870000
6 137.100000 135.000000 2.100000 22.760000 22.390000 0.370000
7 141.200000 137.100000 4.100000 23.480000 22.760000 0.720000
8 142.800000 141.200000 1.600000 23.660000 23.480000 0.180000
9 145.500000 142.800000 2.700000 24.100000 23.660000 0.440000
10 145.300000 145.500000 -0.200000 24.010000 24.100000 -0.090000
11 148.300000 145.300000 3.000000 24.540000 24.010000 0.530000
12 146.400000 148.300000 -1.900000 24.300000 24.540000 -0.240000
13 150.200000 146.400000 3.800000 25.000000 24.300000 0.700000
14 153.100000 150.200000 2.900000 25.640000 25.000000 0.640000
15 157.300000 153.100000 4.200000 26.360000 25.640000 0.720000
16 160.700000 157.300000 3.400000 26.980000 26.360000 0.620000
17 164.200000 160.700000 3.500000 27.520000 26.980000 0.540000
18 165.600000 164.200000 1.400000 27.780000 27.520000 0.260000
19 168.700000 165.600000 3.100000 28.240000 27.780000 0.460000
20 171.700000 168.700000 3.000000 28.780000 28.240000 0.540000
The first differences model is run without a constant term
>model dy = dx
>estimate
1 case(s) deleted due to missing data.
Model contains no constant
Dep Var: DY N: 19 Multiple R: 0.991867 Squared multiple R: 0.983800
Adjusted squared multiple R: 0.983800 Standard error of estimate: 0.069392
Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)
DX 0.168488 0.005096 0.991867 1.000000 33.06272 0.00000
Analysis of Variance
Source Sum-of-Squares df Mean-Square F-ratio P
Regression 5.263726 1 5.263726 1093.143324 0.000000
Residual 0.086674 18 0.004815
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Durbin-Watson D Statistic 1.739
First Order Autocorrelation 0.122
But to calculate the D-W statistic one must run the same model WITH a
constant
>model dy = constant + dx
>estimate
1 case(s) deleted due to missing data.
Dep Var: DY N: 19 Multiple R: 0.982747 Squared multiple R: 0.965791
Adjusted squared multiple R: 0.963778 Standard error of estimate: 0.065498
Effect Coefficient Std Error Std Coef Tolerance t P(2 Tail)
CONSTANT 0.040528 0.022642 0.0 . 1.78996 0.09129
DX 0.158783 0.007248 0.982747 1.000000 21.90756 0.00000
Analysis of Variance
Source Sum-of-Squares df Mean-Square F-ratio P
Regression 2.058923 1 2.058923 479.941009 0.000000
Residual 0.072929 17 0.004290
----------------------------------------------------------------------------------------------------------------------------------
*** WARNING ***
Case 4 has large leverage (Leverage = 0.441713)
Durbin-Watson D Statistic 1.749
First Order Autocorrelation 0.116
Last modified 22 April 1999