Multicollinearity (cont)

• Special case: polynomial regression (cont)
• Ridge Regression (8.4)
  • For a given shrinkage parameter, k, how it is done.
  • Why it reduces the impact of multicollinearity on variances of coefficients.
  • Magnitude of the bias.
  • Scaling vs scaling and centering: if center, don't shrink intercept.
  • Selecting shrinkage parameter
    • Graphical device: ridge trace, coefficients vs k
    • C_k measure (related to Mallow's C_p)
    • Leave-one-out cross validation of prediction errors
      • An approximation, PR(Ridge)
    • Generalized cross validation, GCV
    • Examining VIF's as function of k
    • df-trace criterion

  • SAS code for fitting ridge and drawing trace

  • S-Plus code for fitting ridge model, drawing trace, PRESS, C_k

原始数据

. . .gdp . .nsow mrl orl

1978.00 100.00 22.37 . 9.80

1979.00 107.60 21.53 . 11.26

1980.00 116.00 24.03 . 12.26

1981.00 122.00 25.24 59.20 15.12

1982.00 133.30 26.64 63.00 14.57

1983.00 148.20 26.64 63.90 14.49

1984.00 170.90 30.91 62.90 16.75

1985.00 193.50 35.14 63.90 23.05

1986.00 209.50 37.73 64.30 25.29

1987.00 234.10 40.28 63.90 25.78

1988.00 260.50 43.21 69.70 25.60

1989.00 271.50 43.93 74.40 24.58

1990.00 283.00 45.39 71.70 29.98

1991.00 308.80 43.84 69.70 33.43

1992.00 352.20 48.48 68.30 34.24

1993.00 398.40 53.05 72.80 32.54

1994.00 448.70 61.96 74.60 43.59

1995.00 489.10 66.03 76.50 40.19

1996.00 536.80 71.52 77.80 35.55

1997.00 582.90 74.48 78.30 36.22

1998.00 628.40 78.43 76.50 34.28

1999.00 673.50 79.69 74.50 36.43

2000.00 729.60 77.97 73.30 43.93

2001.00 780.60 81.95 75.78 44.07

2002.00 848.90 84.41 75.92 49.62

这是最小二乘的结果，它把参数都给出来了。

可是岭回归给出的结果只有岭迹图，和下表

R-SQUARE AND BETA COEFFICIENTS FOR ESTIMATED VALUES OF K

K RSQ NSOW MRL ORL

______ ______ ________ ________ ________

.00000 .97570 1.108390 -.238223 .092451

.05000 .96431 .809517 -.069899 .213568

.10000 .95084 .672341 .012790 .254793

.15000 .93994 .591961 .061931 .271769

.20000 .93098 .538219 .094082 .278904

.25000 .92328 .499162 .116356 .281345

.30000 .91639 .469095 .132365 .281308

.35000 .91003 .444956 .144154 .279874

.40000 .90403 .424948 .152973 .277619

.45000 .89826 .407947 .159628 .274871

.50000 .89264 .393212 .164665 .271829

.55000 .88713 .380234 .168466 .268613

.60000 .88169 .368651 .171306 .265304

.65000 .87628 .358199 .173388 .261953

.70000 .87091 .348680 .174866 .258596

.75000 .86556 .339943 .175855 .255256

.80000 .86022 .331869 .176449 .251949

.85000 .85489 .324365 .176718 .248687

.90000 .84957 .317355 .176720 .245476

.95000 .84426 .310779 .176502 .242323

1.0000 .83895 .304585 .176101 .239229

_

请问如何求自变量参数？