THE MARSH METHOD OF IMPOSING SPLINE REGRESSION RESTRICTIONS 1
Variable N Mean Std Dev Minimum Maximum
----------------------------------------------------------------------------------
INTRATE 87 4.1648276 2.1930291 0.5400000 9.8700000
N 87 44.0000000 25.2586619 1.0000000 87.0000000
YEAR 87 1933.00 25.2586619 1890.00 1976.00
--------------------------------------------------------------------
Plot of INTRATE*YEAR. Symbol used is '*'.
(NOTE: 20 obs hidden.)
10 + *
|
INTRATE | * * * *
| * * * *
| * * **** * * * * ** *
5 + ** * ******* **** * * * *
| * **** ** **
| * ** *
| * ****
| *********
0 +
---+---------+---------+---------+---------+---------+--
1880 1900 1920 1940 1960 1980
YEAR
General Linear Models Procedure
Number of observations in data set = 87
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 29.37934187 29.37934187 6.50 0.0126
Error 85 384.22703055 4.52031801
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.071032 51.04903 2.1261040 4.1648276
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
YEAR 1 29.37934187 29.37934187 6.50 0.0126
Source DF Type III SS Mean Square F Value Pr > F
YEAR 1 29.37934187 29.37934187 6.50 0.0126
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 48.89425312 2.79 0.0066 17.54661842
OLS 12:04 Wednesday, March 19, 1997 6
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
YEAR -0.02313990 -2.55 0.0126 0.00907664
SPLINE 7
General Linear Models Procedure
Number of observations in data set = 87
SPLINE 8
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 272.77229241 136.38614620 81.35 0.0001
Error 84 140.83408001 1.67659619
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.659497 31.08975 1.2948344 4.1648276
SPLINE 9
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
DUM1 1 114.40372919 114.40372919 68.24 0.0001
DUM2 1 158.36856322 158.36856322 94.46 0.0001
Source DF Type III SS Mean Square F Value Pr > F
DUM1 1 258.68119921 258.68119921 154.29 0.0001
DUM2 1 158.36856322 158.36856322 94.46 0.0001
SPLINE 10
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 250.4568001 12.62 0.0001 19.85080277
DUM1 -0.1285116 -12.42 0.0001 0.01034604
DUM2 0.1485410 9.72 0.0001 0.01528361
DUMMY 11
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
DUMMY 12
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 305.32823231 101.77607744 78.02 0.0001
Error 83 108.27814010 1.30455590
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.738210 27.42422 1.1421716 4.1648276
DUMMY 13
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
D2 1 31.90927874 31.90927874 24.46 0.0001
YEAR 1 1.27398760 1.27398760 0.98 0.3259
D2*YEAR 1 272.14496597 272.14496597 208.61 0.0001
Source DF Type III SS Mean Square F Value Pr > F
D2 1 272.86840604 272.86840604 209.17 0.0001
YEAR 1 96.23539502 96.23539502 73.77 0.0001
D2*YEAR 1 272.14496597 272.14496597 208.61 0.0001
DUMMY 14
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 188.7398084 8.81 0.0001 21.42996926
D2 -586.5807994 -14.46 0.0001 40.55856338
YEAR -0.0961369 -8.59 0.0001 0.01119319
D2*YEAR 0.3010918 14.44 0.0001 0.02084636
DUMMY 15
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
DUMMY 16
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 4 1814.4118599 453.6029650 347.71 0.0001
Error 83 108.2781401 1.3045559
Uncorrected Total 87 1922.6900000
R-Square C.V. Root MSE INTRATE Mean
0.943684 27.42422 1.1421716 4.1648276
NOTE: No intercept term is used: R-square is not corrected for the mean.
DUMMY 17
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
D1 1 1097.8360820 1097.8360820 841.54 0.0001
D2 1 443.1568243 443.1568243 339.70 0.0001
D1*YEAR 1 96.2353950 96.2353950 73.77 0.0001
D2*YEAR 1 177.1835586 177.1835586 135.82 0.0001
Source DF Type III SS Mean Square F Value Pr > F
D1 1 101.19210224 101.19210224 77.57 0.0001
D2 1 174.13551017 174.13551017 133.48 0.0001
D1*YEAR 1 96.23539502 96.23539502 73.77 0.0001
DUMMY 18
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type III SS Mean Square F Value Pr > F
D2*YEAR 1 177.18355856 177.18355856 135.82 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
D1 188.7398084 8.81 0.0001 21.42996926
D2 -397.8409910 -11.55 0.0001 34.43477140
D1*YEAR -0.0961369 -8.59 0.0001 0.01119319
D2*YEAR 0.2049550 11.65 0.0001 0.01758645
OLS 12:04 Wednesday, March 19, 1997 19
General Linear Models Procedure
Number of observations in data set = 87
OLS 12:04 Wednesday, March 19, 1997 20
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 204.73152867 102.36576434 41.17 0.0001
Error 84 208.87484374 2.48660528
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.494991 37.86225 1.5768974 4.1648276
OLS 12:04 Wednesday, March 19, 1997 21
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
YEAR 1 29.37934187 29.37934187 11.82 0.0009
YEAR*YEAR 1 175.35218680 175.35218680 70.52 0.0001
Source DF Type III SS Mean Square F Value Pr > F
YEAR 1 176.18111325 176.18111325 70.85 0.0001
YEAR*YEAR 1 175.35218680 175.35218680 70.52 0.0001
OLS 12:04 Wednesday, March 19, 1997 22
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 9453.218441 8.44 0.0001 1119.965764
YEAR -9.755071 -8.42 0.0001 1.158922
YEAR*YEAR 0.002517 8.40 0.0001 0.000300
SQUARED SPLINE 23
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
SQUARED SPLINE 24
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 300.10998300 100.03666100 73.16 0.0001
Error 83 113.49638941 1.36742638
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.725593 28.07727 1.1693701 4.1648276
SQUARED SPLINE 25
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
DUM1SQ 1 29.37934187 29.37934187 21.49 0.0001
DUM2SQ 1 104.63640402 104.63640402 76.52 0.0001
DUM3SQ 1 166.09423712 166.09423712 121.46 0.0001
Source DF Type III SS Mean Square F Value Pr > F
DUM1SQ 1 7.62424738 7.62424738 5.58 0.0206
DUM2SQ 1 7.97512149 7.97512149 5.83 0.0179
DUM3SQ 1 166.09423712 166.09423712 121.46 0.0001
SQUARED SPLINE 26
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT -4236.214834 -2.31 0.0236 1837.527481
DUM1SQ 4.516602 2.36 0.0206 1.912781
DUM2SQ -0.001202 -2.41 0.0179 0.000498
DUM3SQ 0.009374 11.02 0.0001 0.000851
SQUARED DUMMY 27
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
SQUARED DUMMY 28
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 4 320.64841378 80.16210344 70.71 0.0001
Error 82 92.95795864 1.13363364
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.775250 25.56462 1.0647223 4.1648276
SQUARED DUMMY 29
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
D2 1 31.90927874 31.90927874 28.15 0.0001
YEAR 1 1.27398760 1.27398760 1.12 0.2922
D2*YEAR 1 272.14496597 272.14496597 240.06 0.0001
YEAR*YEAR 1 15.32018146 15.32018146 13.51 0.0004
D2*YEAR*YEAR 0 0.00000000 . . .
Source DF Type III SS Mean Square F Value Pr > F
D2 0 0.00000000 . . .
YEAR 1 15.03536824 15.03536824 13.26 0.0005
SQUARED DUMMY 30
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type III SS Mean Square F Value Pr > F
D2*YEAR 0 0.00000000 . . .
YEAR*YEAR 0 0.00000000 . . .
D2*YEAR*YEAR 0 0.00000000 . . .
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT -9672.037625 -3.61 0.0005 2682.425126
D2 -1040.050939 B -8.06 0.0001 129.018101
SQUARED DUMMY 31
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
YEAR 10.205600 3.64 0.0005 2.802321
D2*YEAR 0.535161 B 8.04 0.0001 0.066571
YEAR*YEAR -0.002690 B -3.68 0.0004 0.000732
D2*YEAR*YEAR 0.000000 B . . .
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
SQUARED DUMMY 32
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
SQUARED DUMMY 33
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 5 1839.6257230 367.9251446 363.21 0.0001
Error 82 83.0642770 1.0129790
Uncorrected Total 87 1922.6900000
R-Square C.V. Root MSE INTRATE Mean
0.956798 24.16591 1.0064686 4.1648276
NOTE: No intercept term is used: R-square is not corrected for the mean.
SQUARED DUMMY 34
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
D1 1 1097.8360820 1097.8360820 1083.77 0.0001
D2 1 443.1568243 443.1568243 437.48 0.0001
D1*YEAR 1 96.2353950 96.2353950 95.00 0.0001
D2*YEAR 1 177.1835586 177.1835586 174.91 0.0001
D1*YEAR*YEAR 1 25.2138631 25.2138631 24.89 0.0001
D2*YEAR*YEAR 0 0.0000000 . . .
Source DF Type III SS Mean Square F Value Pr > F
D1 1 24.53660371 24.53660371 24.22 0.0001
SQUARED DUMMY 35
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type III SS Mean Square F Value Pr > F
D2 0 0.00000000 . . .
D1*YEAR 1 24.88277206 24.88277206 24.56 0.0001
D2*YEAR 0 0.00000000 . . .
D1*YEAR*YEAR 1 25.21386311 25.21386311 24.89 0.0001
D2*YEAR*YEAR 0 0.00000000 . . .
SQUARED DUMMY 36
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
D1 -13792.69019 -4.92 0.0001 2802.476474
D2 -397.84099 B -13.11 0.0001 30.343528
D1*YEAR 14.51052 4.96 0.0001 2.927749
D2*YEAR 0.20495 B 13.23 0.0001 0.015497
D1*YEAR*YEAR -0.00381 -4.99 0.0001 0.000765
D2*YEAR*YEAR 0.00000 B . . .
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
OLS 12:04 Wednesday, March 19, 1997 37
General Linear Models Procedure
Number of observations in data set = 87
OLS 12:04 Wednesday, March 19, 1997 38
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 2 204.73152867 102.36576434 41.17 0.0001
Error 84 208.87484374 2.48660528
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.494991 37.86225 1.5768974 4.1648276
OLS 12:04 Wednesday, March 19, 1997 39
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
YEAR 1 29.37934187 29.37934187 11.82 0.0009
YEAR*YEAR 1 175.35218680 175.35218680 70.52 0.0001
YEAR*YEAR*YEAR 0 0.00000000 . . .
Source DF Type III SS Mean Square F Value Pr > F
YEAR 0 0 . . .
YEAR*YEAR 0 0 . . .
YEAR*YEAR*YEAR 0 0 . . .
OLS 12:04 Wednesday, March 19, 1997 40
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 9453.218441 B 8.44 0.0001 1119.965764
YEAR -9.755071 B -8.42 0.0001 1.158922
YEAR*YEAR 0.002517 B 8.40 0.0001 0.000300
YEAR*YEAR*YEAR 0.000000 B . . .
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
CUBIC SPLINE 41
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
CUBIC SPLINE 42
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 264.96864425 88.32288142 49.32 0.0001
Error 83 148.63772817 1.79081600
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.640630 32.13131 1.3382137 4.1648276
CUBIC SPLINE 43
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
DCUBE1 1 29.37934187 29.37934187 16.41 0.0001
DCUBE2 1 175.35218680 175.35218680 97.92 0.0001
DCUBE3 0 0.00000000 . . .
DCUBE4 1 60.23711557 60.23711557 33.64 0.0001
Source DF Type III SS Mean Square F Value Pr > F
DCUBE1 0 0.00000000 . . .
DCUBE2 0 0.00000000 . . .
DCUBE3 0 0.00000000 . . .
CUBIC SPLINE 44
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type III SS Mean Square F Value Pr > F
DCUBE4 1 60.23711557 60.23711557 33.64 0.0001
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT 1556.758691 B 0.94 0.3512 1660.450843
DCUBE1 -1.529800 B -0.89 0.3780 1.725870
DCUBE2 0.000376 B 0.84 0.4047 0.000448
DCUBE3 0.000000 B . . .
CUBIC SPLINE 45
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
DCUBE4 0.000187 5.80 0.0001 0.000032
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
CUBIC DUMMY 46
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Number of observations in data set = 87
CUBIC DUMMY 47
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 5 332.71216040 66.54243208 66.63 0.0001
Error 81 80.89421202 0.99869398
Corrected Total 86 413.60637241
R-Square C.V. Root MSE INTRATE Mean
0.804417 23.99491 0.9993468 4.1648276
CUBIC DUMMY 48
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type I SS Mean Square F Value Pr > F
D2 1 31.90927874 31.90927874 31.95 0.0001
YEAR 1 1.27398760 1.27398760 1.28 0.2620
D2*YEAR 1 272.14496597 272.14496597 272.50 0.0001
YEAR*YEAR 1 15.32018146 15.32018146 15.34 0.0002
D2*YEAR*YEAR 0 0.00000000 . . .
YEAR*YEAR*YEAR 0 0.00000000 . . .
D2*YEAR*YEAR*YEAR 1 12.06374662 12.06374662 12.08 0.0008
CUBIC DUMMY 49
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
Source DF Type III SS Mean Square F Value Pr > F
D2 0 0.00000000 . . .
YEAR 0 0.00000000 . . .
D2*YEAR 0 0.00000000 . . .
YEAR*YEAR 0 0.00000000 . . .
D2*YEAR*YEAR 0 0.00000000 . . .
YEAR*YEAR*YEAR 0 0.00000000 . . .
D2*YEAR*YEAR*YEAR 1 12.06374662 12.06374662 12.08 0.0008
CUBIC DUMMY 50
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
Dependent Variable: INTRATE
T for H0: Pr > |T| Std Error of
Parameter Estimate Parameter=0 Estimate
INTERCEPT -13790.54086 B -4.96 0.0001 2782.645350
D2 14606.55150 B 3.24 0.0017 4503.521290
YEAR 14.50828 B 4.99 0.0001 2.907032
D2*YEAR -11.49928 B -3.32 0.0013 3.463153
YEAR*YEAR -0.00381 B -5.02 0.0001 0.000759
D2*YEAR*YEAR 0.00000 B . . .
YEAR*YEAR*YEAR 0.00000 B . . .
D2*YEAR*YEAR*YEAR 0.00000 3.48 0.0008 0.000000
CUBIC DUMMY 51
12:04 Wednesday, March 19, 1997
General Linear Models Procedure
NOTE: The X'X matrix has been found to be singular and a generalized inverse
was used to solve the normal equations. Estimates followed by the
letter 'B' are biased, and are not unique estimators of the parameters.
<\PRE>