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第 8 章 非线性回归. 8.1 可化为线性回归的曲线回归 8.2 多项式回归 8.3 非线性模型 8.4 本章小结与评注. §8.1 可化为线性回归的曲线回归. 可线性化的曲线回归模型 , 也称为本质线性回归模型. y =β 0 +β 1 e x +ε ( 8.1 ). 只须令 x ′=e x 即可化为 y 对 x ′ 是线性的形式 y =β 0 +β 1 x ′+ε 需要指出的是,新引进的自变量只能依赖于原始变量,而不能与未知参数有关。. §8.1 可化为线性回归的曲线回归. - PowerPoint PPT Presentation
Citation preview
8 8.1 8.2 8.3 8.4
8.1 y=0+1ex + 8.1 , x=exyxy=0+1x+
8.1 y=0+1x+2x2++pxp + 8.2x1=x,x2=x2,xp=xpyx1,x2,xpy=0+1x1+2x2++pxp+ (8.2)xpp
8.1 y=aeb xe 8.3, lny=lna+bx+y=lny, 0=lna,1=b,yxy=0+1x+
8.1 , y= aeb x +8.4 b (8.3) (8.4)
8.1 8.3y=aeb xe y=aeb x
8.1 SPSS10
Linear
y=b0+b1t
Logarithm
y=b0+b1lnt
Inverse
y=b0+b1/t
Quadratic
y=b0+b1t+b2t2
Cubic
y=b0+b1t+b2t2+b3t3
Power
y=b0
Compound
y=b0
S
S
y=exp(b0+b1/t)
Logistic
u
Growth
y=exp(b0+b1t)
Exponent
y=b0exp(b1t)
_1030013797.unknown
_1030013951.unknown
_1030013744.unknown
8.1 SPSS,1.
8.1 (a>0, b>0)
8.1 2. S Sa>0b>0x x+y1/a ; x-y0 y=0y=1/a S S
8.1
8.1 8.1GDP()GDPGDPt1981t=11998 t=18198119988.1
8.1
tyeiy=lny198114862.44296.35566.058.489198225294.75123.04171.668.574198335934.56108.80-174.308.689198447171.07284.24-113.248.878198558964.48685.86278.549.1011986610202.210357.16-154.969.2301987711962.512350.06-387.569.3901988814928.314726.42201.889.6111989916909.217560.04-650.849.73619901018547.920938.89-2390.999.82819911121617.824967.89-3350.099.98119921226638.129772.14-3134.0410.19019931334634.435500.81-866.4110.45319941446759.442331.774427.6310.75319951558478.150477.138000.9710.97619961667884.660189.807694.8011.12619971774462.671771.352691.2511.21819981879395.785581.38-6185.6811.282
8.1 1. SPSSCurve EstimationGDP8.2
8.1
8.1
8.2
y=b0+b1t
Multiple R .92528
R Square .85615
Adjusted R Square .84716
Standard Error 9964.23063
Analysis of Variance:
DF Sum of Squares Mean Square F Signif F
Regression 1 9454779005.1 9454779005.1 95.22782 .0000
Residuals 16 1588574273.6 99285892.1
Variable B SE B Beta T Sig T
Time 4417.522807 452.685809 .925284 9.758 .0000
(Constant) -13374.922222 4900.032018 -2.730 .0148
8.1
8.3
y=b0
Multiple R .99593
R Square .99188
Adjusted R Square .99138
Standard Error .08760
Analysis of Variance:
DF Sum of Squares Mean Square F Signif F
Regression 1 15.004878 15.004878 1955.31315 .0000
Residuals 16 .122782 .007674
Variable B SE B Beta T Sig T
Time 1.192417 .004746 2.707250 251.269 .0000
(Constant) 3603.061130 155.215413 23.213 .0000
_1031938334.unknown
8.1 SSE =262467769=2.625108 R2=1-262467769/11043353279=0.97623
8.1 b0=3603.06,b1=1.192417 b1=1.192417=119.2417%GDP19.2417% GDPGDP
8.1 2. y=b0lny=lnb0+ln(b1) ty=lny, 0=lnb0,1=ln(b1),yty=0+1ty=lny8.4yt
8.1
8.1
=8.190
=0.176
EMBED Equation.3
SPSSCurve Estimation
_1031303507.unknown
_1031303538.unknown
_1031303610.unknown
_1031303491.unknown
8.2 yi=0+1xi+11+iyi=0+1xi+11111 yi=0+1xi+11+111+i
8.2
yi=0+1xi1+2xi2+11
+22
+12xi1xi2+i
12
112212
x1x2
_1031036694.unknown
_1031036715.unknown
8.2 8.2 8.5183544: x1 x2 y
8.2
yi=0+1xi1+2xi2+11
+22
+12xi1xi2+i
_1031036694.unknown
_1031036715.unknown
8.2
xi1xi2yi12345678910111213141516171866.29040.96472.99645.01057.20426.85238.12235.84075.79637.40854.37646.18646.13030.36639.06079.38052.76655.916751064546952743518619663252841261449492664910598771456245133133
8.2
x1x2
x1x2
yx1Block 1 of Next,
x1x2
Block 2 of Next,
x1x2
_1031039584.unknown
_1031041474.unknown
8.2
8.2
8.2 8.6
F
x1
x2|x1
|x1,x2
| x1,x2,
x1x2|x1,x2,
,
104474
2284
1238
3
6
3567
1283
45
42
36
1
2
11
22
12
-
-
1238/45/14=385
3/42/13=0.93
6/36/12=2.00
108005
5
_1031039610.unknown
_1031041474.unknown
_1031041519.unknown
_1031039584.unknown
8.2
=-62.349+0.840x1+5.685x2+0.0371
(0.164)(0.164) (0.785)
_1031045671.unknown
_1031045822.unknown
8.2 8.3 C3EDTAX1X2X3 7U7741236.9
8.2 6.9
EDTA
X1g
X2g
X3g
y
1/y
1
0.00
30
0.6
1.160
0.862
2
0.02
38
1.2
0.312
3.205
3
0.04
46
0.4
0.306
3.263
4
0.06
26
1.0
1.318
0.759
5
0.08
34
0.2
0.877
1.140
6
0.10
42
0.8
0.147
6.803
7
0.12
50
1.4
0.204
4.902
8.2 6.1y = 2.63 + 0.77 X1 - 0.0524 X2 - 0.087 X3P=0.1040R-square= 83.9% AdjR-sq= 67.8%
8.2 9
8.2 6.10
X1
X2
X3
X11
X22
X33
X12
X13
X23
y
0.00
0.02
0.04
0.06
0.08
0.10
0.12
30
38
46
26
34
42
50
0.6
1.2
0.4
1.0
0.2
0.8
1.4
0.0000
0.0004
0.0016
0.0036
0.0064
0.0100
0.0144
900
1444
2116
676
1156
1764
2500
0.360
1.440
0.160
1.000
0.040
0.640
1.960
0.00
0.76
1.84
1.56
2.72
4.20
6.00
0.000
0.024
0.016
0.060
0.016
0.080
0.168
18.0
45.6
18.4
26.0
6.8
33.6
70.0
1.160
0.312
0.306
1.318
0.877
0.147
0.204
8.2 710 SPSSP=0.05P=0.10x2 OptionP0.30P0.50
8.2 6.12 2
Step
1
2
3
4
5
Constant
2.579
5.957
7.311
7.873
9.165
X2
Prob>F
-0.0516
0.004
-0.2376
0.053
-0.3034
0.021
-0.3126
0.030
-0.378
0.016
X22
Prob>F
0.00245
0.100
0.00336
0.033
0.00323
0.048
0.0046
0.019
X3
Prob>F
-0.292
0.107
-1.115
0.168
-1.430
0.033
X23
Prob>F
0.0206
0.251
0.0317
0.039
X13
Prob>F
-2.33
0.058
R-square
83.14
92.12
97.11
98.73
99.99
8.2 5X2 X22= X3X23=X2 X3X13= X1 X355: X2 X22= yX2 X2=48.548y=0.1976y=0.147
8.2 yX3X3=1.4X2X3X2=45.145X3=1.4y=0.0746y=0.147
8.2 X31.115 X3+0.0206 X2X3X254X3X24845X254yX3X3yX3=1.4 X3=1.4X2X2=43.944X2=44X3=1.4y=0.080
8.2 3 97.1198.7399.99% 6y7 0.0806y=0.1476.13
8.2 6.13
X1g
X2g
X3g
0.00
0.00
0.00
0.12
48
45
44
41
0.0
1.4
1.4
1.4
0.197
0.074
0.080
0.000
8.2 [17]yX1=0.12X2=38X3=1.4
8.3 yi = f (xi,)+i , i=1,2,,n 8.9yi xi=(xi1,xi2,xik) =(0,1,p ) i
8.3
,
_1030953497.unknown
8.3
f
Q()
fj0p+1
_1030953513.unknown
Q()
_1030953513.unknown
8.3 SST=SSR+SSE
8.3 8.4 x y 3c0c1c2c0100% 3c0=100c1=5c2=4.89
8.3 8.3
x 1 2 3 4 5 6 7 8 9y(%)0.5 2.3 3.4 24.0 54.7 82.1 94.8 96.2 96.4
8.3 SPSSRegressionNonlinearymodel Expressionc0-c0/(1+(x/c2)**c1),Parameters
8.3
Iteration Residual SS C0 C1 C2 1 172.7877170 100.000000 5.00000000 4.80000000 1.1 32.60704344 97.7943996 6.57938197 4.74460195 2 32.60704344 97.7943996 6.57938197 4.74460195 2.1 20.20240372 99.5785656 6.73691756 4.80074972 3 20.20240372 99.5785656 6.73691756 4.80074972 3.1 20.18814307 99.5334852 6.76307026 4.79941696 4 20.18814307 99.5334852 6.76307026 4.79941696 4.1 20.18803580 99.5411768 6.76104089 4.79966204 5 20.18803580 99.5411768 6.76104089 4.79966204 5.1 20.18803473 99.5404448 6.76127044 4.79964160 6 20.18803473 99.5404448 6.76127044 4.79964160 6.1 20.18803472 99.5405197 6.76124802 4.79964382
8.3
Nonlinear Regression Summary Statistics
Source DF Sum of Squares Mean Square
Regression 3 37839.85197 12613.28399
Residual 6 20.18803 3.36467
Uncorrected Total 9 37860.04000
(Corrected Total) 8 14917.88889
R squared = 1 - Residual SS / Corrected SS = .99865
8.3
Asymptotic 95 %
Asymptotic Confidence Interval
Parameter Estimate Std. Error Lower Upper
C0 99.540519687 1.567325937 95.705411276 103.37562810
C1 6.761248019 .421980049 5.728700036 7.793796003
C2 4.799643816 .050165521 4.676893208 4.922394423
8.3
xye110.500.5-50.48889222.30.272.03-50.21889333.43.98-0.58-46.50889442422.481.52-28.008895554.756.61-1.916.121116682.181.520.5831.031117794.892.342.4641.851118896.296.49-0.2946.001119996.498.14-1.7447.65111550.4888950.203330.285556-0.285566014917.8915156.5519.4316215156.5528537860.0437839.8520.1880315157.28
8.3 yx
8.3 8.4 Gompertz k k8.12
8.3 8.8
tyty1980111.4119931468.211981213.5519941569.371982313.2819951678.081983412.9219961778.021984515.2819971892.061985617.12199819100.141986721.6719992099.891987824.0220002199.451988924.55200122103.6719891030.55200223106.3219901134.04200324103.4219911238.17200425115.5219921353.36
8.3 SPSS AnalyzeRegressionNonlinear k*a**(b**t),k115.52k120ab010.531
8.3
8.3
Dependent variable: y
a R squared = 1 - (Residual Sum of Squares) / (Corrected Sum of Squares) = .976.
8.3 k=150.0a=0.012 43b=0.892 7k=150.0 8.4Excel8.4
Chart3
11.412.99
13.554.55
13.286.62
12.929.25
15.2812.47
17.1216.29
21.6720.67
24.0225.57
24.5530.92
30.5536.63
34.0442.62
38.1748.78
53.3655.03
68.2161.28
69.3767.46
78.0873.5
78.0279.35
92.0684.96
100.1490.3
99.8995.35
99.448100.1
103.6737104.54
106.32108.67
103.42112.49
115.52116.02
2005119.26
2006122.23
2007124.94
2008127.41
2009129.66
2010131.7
2.1
yx
26.23.4
17.81.8
31.34.6
23.12.3
27.53.1
365.5
14.10.7
22.33
19.62.6
31.34.3
242.1
17.31.1
43.26.1
36.44.8
26.13.8
2.2
xy
1980460234.75
1981489259.26
1982525280.58
1983580305.97
1984692347.15
1985853433.53
1986956481.36
19871104545.4
19881355687.51
19891512756.27
19901634797.08
19911879890.66
199222871063.39
199329391323.22
199439231736.32
199548542224.59
199655762627.06
199760532819.36
199863922958.18
3.1
x1x2x3x4x5x6x7x8x9x10x11x12y
1.944.5154.45207.33246.87277.64135.7930.58110.6780.8351.8314.092384
0.336.49133.16127.29120.17114.8881.2114.0535.71627.12.93202
6.1617.18313.4386.96202.98204.2279.4332.4279.3814.54128.1342.15100
5.359.3123.8122.94101.5996.8434.6713.9937.285.9363.913.1238
3.784.26106.0595.4927.5822.7534.2414.0628.24.6935.729.51126
11.178.17271.96533.15164.4123.78187.758.6390.5231.7184.0511.61262
2.843.61109.37130.852.4962.2638.1521.8244.5325.7848.4914.2238
8.6411.41160.06246.57109.18115.3268.7134.5558.0813.5272.0521.17121
3.646.67244.42412.04459.63512.21160.4543.5189.9348.5548.637.051218
30.8919.08435.77724.85376.04381.81210.3971.82150.6423.74188.2819.65529
6.266.3321.75665.8157.94172.19147.1652.4478.1610.993.059.45361
4.138.87152.29258.683.4285.175.7426.7563.475.8947.022.6651
5.855.61347.25332.59157.32172.48115.1633.877.278.6979.018.24651
6.76.8145.4143.5497.4100.543.2817.7151.035.4162.0318.2543
10.811.73442.2665.33411.89429.88115.0787.45145.2521.39187.77110.2220
4.1622.51299.63316.81132.57139.7684.7953.9384.2312.36116.8910.38101
4.647.65195.56373.04161.84180.14101.585880.5321.61100.695.1688
7.0810.99216.49291.73119.22125.6247.0548.1997.9712.07139.3916.67156
16.324.1688.83827.16271.07268.2331.5571.44146.1523.38145.7716.522942
4.014125.04243.552.0631.2247.2525.5955.274.4960.1313.64156
0.82.0735.0360.929.230.1420.224.2212.191.39.290.2796
4.422.1178.93138.4368.3173.8479.9818.4243.320.0148.480.7288
11.189.42196.27328.46204.49144.45101.2143.0174.2215.8590.611.0584
2.012.0325.0469.9740.8636.4527.0213.826.832.8625.636.7648
6.436.0888.9170.1588.8689.8433.6629.251.258.640.474.81261
1.910.985.0811.130.671.691.942.955.020.897.590.1733
5.499.9115.4294.6376.5753.1447.8822.0856.9714.0248.6438.17247
3.977.839.3299.2341.6450.5511.418.8115.986.3316.467.0230
1.313.0813.6718.7918.3718.573.153.148.661.2614.31.23
1.12.116.1119.6417.8516.524.163.036.761.067.523.181
4.5810.3592.03103.3449.1950.228.1411.8237.954.5239.493.5382
3.2
x1x2y
253547.79553.96
20896.34208.55
6750.323.1
10012087.052815.4
5251639.311052.12
8253357.73427
120808.47442.82
28520.2770.12
7671.13122.24
5322863.321400
751160464
40862.757.5
187672.99224.18
122901.76538.94
743546.182442.79
2949158.153600
195015.51965.6511
195125.635.7025
195235.74825.7704
195345.87965.8517
195456.02665.9443
195566.14656.0466
195676.28286.1577
195786.46536.2764
195896.59946.4020
1959106.72076.5337
1960116.62076.6709
1961126.58596.8129
1962136.72956.9592
1963146.91727.1093
1964157.04997.2628
1965167.25387.4191
1966177.45427.5778
1967187.63687.7386
1968197.85347.9012
1969208.06718.0650
1970218.29928.2299
1971228.52298.3955
1972238.71778.5615
1973248.92118.7277
1974259.08598.8937
1975269.2429.0594
1976279.37179.2246
1977289.49749.3889
1978299.62599.5523
1979309.75429.7144
1980319.87059.8753
19813210.007210.0346
19823310.154110.1923
19833410.249510.3481
19843510.347510.5021
19853610.453210.6540
19863710.572110.8038
19873810.72410.9513
19883910.897811.0965
19894011.270411.2392
19904111.433311.3795
19914211.582311.5173
19924311.717111.6524
19934411.851711.7849
19944511.98511.9148
19954612.112112.0419
19964712.238912.1662
19974812.362612.2878
19984912.476112.4066
19995012.578612.5227
20005112.674312.6359
20015212.762712.7463
20025312.845312.8540
20035412.922712.9588
20045512.998813.0609
20055613.075613.1603
20065713.2569
20075813.3508
20085913.4421
20096013.5306
20106113.6166
20116213.7000
20126313.7809
20136413.8592
20146513.9351
20156614.0086
20166714.0796
20176814.1484
20186914.2148
20197014.2791
20207114.3411
20217214.4010
20227314.4588
20237414.5145
20247514.5682
20257614.6200
20267714.6699
20277814.7180
20287914.7642
20298014.8087
20308114.8515
20318214.8926
20328314.9321
20338414.9701
20348515.0066
20358615.0416
20368715.0752
20378815.1074
20388915.1383
20399015.1679
20409115.1962
20419215.2234
20429315.2494
20439415.2743
20449515.2981
20459615.3209
20469715.3426
20479815.3634
20489915.3833
204910015.4023
205010115.4204
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li8.
ty
1980111.412.99
1981213.554.55
1982313.286.62
1983412.929.25
1984515.2812.47
1985617.1216.29
1986721.6720.67
1987824.0225.57
1988924.5530.92
19891030.5536.63
19901134.0442.62
19911238.1748.78
19921353.3655.03
19931468.2161.28
19941569.3767.46
19951678.0873.5
19961778.0279.35
19971892.0684.96
199819100.1490.3
19992099.8995.35
20002199.448100.1
200122103.6737104.54
200223106.32108.67
200324103.42112.49
200425115.52116.02
200526119.26
200627122.23
200728124.94
200829127.41
200930129.66
201031131.7
li8.
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
Sheet1
199319.546.633.919.947.432.7-0.4-0.81.2100.019.546.633.9
199419.746.533.820.347.831.9-0.6-1.31.9100.019.746.533.8
199519.747.233.120.548.830.7-0.8-1.62.4100.019.747.233.1
199619.547.533.020.449.530.1-0.9-2.02.9100.019.547.533.0
199718.147.534.419.150.030.9-1.0-2.53.5100.018.147.534.4
199817.346.236.518.649.332.1-1.3-3.14.4100.017.346.236.5
199916.245.838.017.649.432.9-1.4-3.75.199.916.245.838.0
200014.845.939.316.450.233.4-1.5-4.35.8100.014.845.939.3
200114.145.240.715.850.134.1-1.7-4.96.6100.014.145.240.7
200213.544.841.715.350.434.3-1.8-5.67.4100.013.544.841.7
200312.546.041.514.452.233.4-1.9-6.28.1100.012.546.041.5
200413.146.240.715.252.931.9-2.1-6.78.8100.013.146.240.7
GDP
199018547.95017.07717.45813.5
199121617.85288.69102.27227.0
199226638.15800.011699.59138.6
199334634.46882.116428.511323.8
199446759.49457.222372.214930.0
199558478.111993.028537.917947.2
199667884.613844.233612.920427.5
199774462.614211.237222.723028.7
199878345.214552.438619.325173.5
199982067.514472.040557.827037.7
200089468.114628.244935.329904.6
200197314.815411.848750.033153.0
2002104790.616117.353540.735132.6
SUMMARY OUTPUT
Multiple R0.9998190104
R Square0.9996380536
Adjusted R Square0.9996051494
586.4816333583
13
dfSSMSFSignificance F
110449609223.34810449609223.34830380.24121059550
113783567.76893328343960.706266662
1210453392791.1169
Coefficientst StatP-valueLower 95%Upper 95% 95.0% 95.0%
Intercept4134.2489719396367.720989829611.24289634340.00000022663324.90053077424943.5974131053324.90053077424943.597413105
X Variable 11.90096847330.0109063469174.299286316901.87696376571.92497318091.87696376571.9249731809
199018547.950177717.45813.518547.9
199121617.85288.69102.2722721617.8
199226638.1580011699.59138.626638.1
199334634.46882.116428.511323.834634.4
199446759.49457.222372.21493046759.4
199558478.11199328537.917947.258478.1
199667884.613844.233612.920427.567884.6
199774462.614211.237222.723028.774462.6
199878345.214552.438619.325173.578345.2
199982067.51447240557.827037.782067.5
200089468.114628.244935.329904.689468.1
200197314.815411.8487503315397314.8
2002104790.616117.353540.735132.6104790.6
SUMMARY OUTPUTSUMMARY OUTPUT
Multiple R1Multiple R0.9986922681
R Square1R Square0.9973862464
Adjusted R Square1Adjusted R Square0.9968634956
0730.238812478
77
dfSSMSFSignificance FdfSSMSFSignificance F
31020082220.85429340027406.95142938496757177629394000000000000000011017415977.238041017415977.238041907.95764317750.0000001187
30052666243.61624623533248.723249246
61020082220.8542961020082220.85429
Coefficientst StatP-valueLower 95%Upper 95% 95.0% 95.0%Coefficientst StatP-valueLower 95%Upper 95% 95.0% 95.0%
Intercept0.00000000010.00000000010.88933388360.4393368096-0.00000000020.0000000003-0.00000000020.0000000003Intercept5716.32804942921833.8045625953.11719589210.02633557071002.383352437210430.27274642121002.383352437210430.2727464212
X Variable 11015104323513538401111X Variable 11.8648954650.042694322743.68017448660.00000011871.75514621461.97464471541.75514621461.9746447154
X Variable 21055881074840137701111
X Variable 31054596202838836201111
4.3
yx
12648777
21059210
3909954
413110508
512210979
610711912
740612747
850313499
943114269
1058815522
1189816730
1295017663
1377918575
1481919635
15122221163
16170222880
17157824127
18165425604
19140026500
20182927670
21220028300
22201727430
23210529560
24160028150
25225032100
26242032500
27257035250
28172033500
29190036000
30210036200
31230038200
3.3
yx1x2x3x4x5
1978231301018888149114.89180.92
1979298335021958638916.00420.39
1980343368825319220419.53570.25
1981401394127999530021.82776.71
1982445425830549992223.27792.43
19833914736335810604422.91947.70
19845545652390511035326.021285.22
19857447020487911211027.721783.30
19869977859555210857932.432281.95
198713109313638611242938.912690.23
1988144211738803812264537.383169.48
1989128313176900511380747.192450.14
199016601438496639571250.682746.20
1991217816557109699508155.913335.65
1992288620223129859969383.663311.50
19933383248821594910545896.084152.70
8.3 Constraints k130k>116
8.3 [15]13
8.3 8.6 8.919502005Weibullka >0, 0< b 0
8.3 8.9
tyty195015.51961978299.6259195125.631979309.7542195235.74821980319.8705195345.879619813210.0072
1975269.24220035412.92271976279.371720045512.99881977289.497420055613.0756
8.3 16k=16b=0.5c=1 t=11950=5.5196
8.7k=15.76168.5Excel
8.3
8.7Parameter Estimates
Parameter
Estimate
Std. Error
95% Confidence Interval
Lower Bound
Upper Bound
k
15.760
.650
14.455
17.064
a
10.135
.693
8.746
11.525
b
.997
.000
.996
.998
c
1.551
.071
1.408
1.694
8.3
ANOVA(a)
Source
Sum of Squares
df
Mean Squares
Regression
5266.738
4
1316.685
Residual
.884
52
.017
Uncorrected Total
5267.622
56
Corrected Total
319.677
55
Dependent variable: y
a R squared = 1 - (Residual Sum of Squares) / (Corrected Sum of Squares) = .997.
8.3 8.5
Chart2
5.51965.6511
5.635.7025
5.74825.7704
5.87965.8517
6.02665.9443
6.14656.0466
6.28286.1577
6.46536.2764
6.59946.402
6.72076.5337
6.62076.6709
6.58596.8129
6.72956.9592
6.91727.1093
7.04997.2628
7.25387.4191
7.45427.5778
7.63687.7386
7.85347.9012
8.06718.065
8.29928.2299
8.52298.3955
8.71778.5615
8.92118.7277
9.08598.8937
9.2429.0594
9.37179.2246
9.49749.3889
9.62599.5523
9.75429.7144
9.87059.8753
10.007210.0346
10.154110.1923
10.249510.3481
10.347510.5021
10.453210.654
10.572110.8038
10.72410.9513
10.897811.0965
11.270411.2392
11.433311.3795
11.582311.5173
11.717111.6524
11.851711.7849
11.98511.9148
12.112112.0419
12.238912.1662
12.362612.2878
12.476112.4066
12.578612.5227
12.674312.6359
12.762712.7463
12.845312.854
12.922712.9588
12.998813.0609
13.075613.1603
200613.2569
200713.3508
200813.4421
200913.5306
201013.6166
201113.7
201213.7809
201313.8592
201413.9351
201514.0086
201614.0796
201714.1484
201814.2148
201914.2791
202014.3411
202114.401
202214.4588
202314.5145
202414.5682
202514.62
202614.6699
202714.718
202814.7642
202914.8087
203014.8515
14.8926
14.9321
14.9701
15.0066
15.0416
15.0752
15.1074
15.1383
15.1679
15.1962
15.2234
15.2494
15.2743
15.2981
15.3209
15.3426
15.3634
15.3833
15.4023
15.4204
2.1
yx
26.23.4
17.81.8
31.34.6
23.12.3
27.53.1
365.5
14.10.7
22.33
19.62.6
31.34.3
242.1
17.31.1
43.26.1
36.44.8
26.13.8
2.2
xy
1980460234.75
1981489259.26
1982525280.58
1983580305.97
1984692347.15
1985853433.53
1986956481.36
19871104545.4
19881355687.51
19891512756.27
19901634797.08
19911879890.66
199222871063.39
199329391323.22
199439231736.32
199548542224.59
199655762627.06
199760532819.36
199863922958.18
3.1
x1x2x3x4x5x6x7x8x9x10x11x12y
1.944.5154.45207.33246.87277.64135.7930.58110.6780.8351.8314.092384
0.336.49133.16127.29120.17114.8881.2114.0535.71627.12.93202
6.1617.18313.4386.96202.98204.2279.4332.4279.3814.54128.1342.15100
5.359.3123.8122.94101.5996.8434.6713.9937.285.9363.913.1238
3.784.26106.0595.4927.5822.7534.2414.0628.24.6935.729.51126
11.178.17271.96533.15164.4123.78187.758.6390.5231.7184.0511.61262
2.843.61109.37130.852.4962.2638.1521.8244.5325.7848.4914.2238
8.6411.41160.06246.57109.18115.3268.7134.5558.0813.5272.0521.17121
3.646.67244.42412.04459.63512.21160.4543.5189.9348.5548.637.051218
30.8919.08435.77724.85376.04381.81210.3971.82150.6423.74188.2819.65529
6.266.3321.75665.8157.94172.19147.1652.4478.1610.993.059.45361
4.138.87152.29258.683.4285.175.7426.7563.475.8947.022.6651
5.855.61347.25332.59157.32172.48115.1633.877.278.6979.018.24651
6.76.8145.4143.5497.4100.543.2817.7151.035.4162.0318.2543
10.811.73442.2665.33411.89429.88115.0787.45145.2521.39187.77110.2220
4.1622.51299.63316.81132.57139.7684.7953.9384.2312.36116.8910.38101
4.647.65195.56373.04161.84180.14101.585880.5321.61100.695.1688
7.0810.99216.49291.73119.22125.6247.0548.1997.9712.07139.3916.67156
16.324.1688.83827.16271.07268.2331.5571.44146.1523.38145.7716.522942
4.014125.04243.552.0631.2247.2525.5955.274.4960.1313.64156
0.82.0735.0360.929.230.1420.224.2212.191.39.290.2796
4.422.1178.93138.4368.3173.8479.9818.4243.320.0148.480.7288
11.189.42196.27328.46204.49144.45101.2143.0174.2215.8590.611.0584
2.012.0325.0469.9740.8636.4527.0213.826.832.8625.636.7648
6.436.0888.9170.1588.8689.8433.6629.251.258.640.474.81261
1.910.985.0811.130.671.691.942.955.020.897.590.1733
5.499.9115.4294.6376.5753.1447.8822.0856.9714.0248.6438.17247
3.977.839.3299.2341.6450.5511.418.8115.986.3316.467.0230
1.313.0813.6718.7918.3718.573.153.148.661.2614.31.23
1.12.116.1119.6417.8516.524.163.036.761.067.523.181
4.5810.3592.03103.3449.1950.228.1411.8237.954.5239.493.5382
3.2
x1x2y
253547.79553.96
20896.34208.55
6750.323.1
10012087.052815.4
5251639.311052.12
8253357.73427
120808.47442.82
28520.2770.12
7671.13122.24
5322863.321400
751160464
40862.757.5
187672.99224.18
122901.76538.94
743546.182442.79
2949158.153600
195015.51965.6511
195125.635.7025
195235.74825.7704
195345.87965.8517
195456.02665.9443
195566.14656.0466
195676.28286.1577
195786.46536.2764
195896.59946.4020
1959106.72076.5337
1960116.62076.6709
1961126.58596.8129
1962136.72956.9592
1963146.91727.1093
1964157.04997.2628
1965167.25387.4191
1966177.45427.5778
1967187.63687.7386
1968197.85347.9012
1969208.06718.0650
1970218.29928.2299
1971228.52298.3955
1972238.71778.5615
1973248.92118.7277
1974259.08598.8937
1975269.2429.0594
1976279.37179.2246
1977289.49749.3889
1978299.62599.5523
1979309.75429.7144
1980319.87059.8753
19813210.007210.0346
19823310.154110.1923
19833410.249510.3481
19843510.347510.5021
19853610.453210.6540
19863710.572110.8038
19873810.72410.9513
19883910.897811.0965
19894011.270411.2392
19904111.433311.3795
19914211.582311.5173
19924311.717111.6524
19934411.851711.7849
19944511.98511.9148
19954612.112112.0419
19964712.238912.1662
19974812.362612.2878
19984912.476112.4066
19995012.578612.5227
20005112.674312.6359
20015212.762712.7463
20025312.845312.8540
20035412.922712.9588
20045512.998813.0609
20055613.075613.1603
20065713.2569
20075813.3508
20085913.4421
20096013.5306
20106113.6166
20116213.7000
20126313.7809
20136413.8592
20146513.9351
20156614.0086
20166714.0796
20176814.1484
20186914.2148
20197014.2791
20207114.3411
20217214.4010
20227314.4588
20237414.5145
20247514.5682
20257614.6200
20267714.6699
20277814.7180
20287914.7642
20298014.8087
20308114.8515
20318214.8926
20328314.9321
20338414.9701
20348515.0066
20358615.0416
20368715.0752
20378815.1074
20388915.1383
20399015.1679
20409115.1962
20419215.2234
20429315.2494
20439415.2743
20449515.2981
20459615.3209
20469715.3426
20479815.3634
20489915.3833
204910015.4023
205010115.4204
00
00
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li8.
ty
1980111.41
1981213.55
1982313.28
1983412.92
1984515.28
1985617.12
1986721.67
1987824.02
1988924.55
19891030.55
19901134.04
19911238.17
19921353.36
19931468.21
19941569.37
19951678.08
19961778.02
19971892.06
199819100.14
19992099.89
20002199.448
200122103.6737
200223106.32
200324103.42
200425115.52
Sheet1
199319.546.633.919.947.432.7-0.4-0.81.2100.019.546.633.9
199419.746.533.820.347.831.9-0.6-1.31.9100.019.746.533.8
199519.747.233.120.548.830.7-0.8-1.62.4100.019.747.233.1
199619.547.533.020.449.530.1-0.9-2.02.9100.019.547.533.0
199718.147.534.419.150.030.9-1.0-2.53.5100.018.147.534.4
199817.346.236.518.649.332.1-1.3-3.14.4100.017.346.236.5
199916.245.838.017.649.432.9-1.4-3.75.199.916.245.838.0
200014.845.939.316.450.233.4-1.5-4.35.8100.014.845.939.3
200114.145.240.715.850.134.1-1.7-4.96.6100.014.145.240.7
200213.544.841.715.350.434.3-1.8-5.67.4100.013.544.841.7
200312.546.041.514.452.233.4-1.9-6.28.1100.012.546.041.5
200413.146.240.715.252.931.9-2.1-6.78.8100.013.146.240.7
GDP
199018547.95017.07717.45813.5
199121617.85288.69102.27227.0
199226638.15800.011699.59138.6
199334634.46882.116428.511323.8
199446759.49457.222372.214930.0
199558478.111993.028537.917947.2
199667884.613844.233612.920427.5
199774462.614211.237222.723028.7
199878345.214552.438619.325173.5
199982067.514472.040557.827037.7
200089468.114628.244935.329904.6
200197314.815411.848750.033153.0
2002104790.616117.353540.735132.6
SUMMARY OUTPUT
Multiple R0.9998190104
R Square0.9996380536
Adjusted R Square0.9996051494
586.4816333583
13
dfSSMSFSignificance F
110449609223.34810449609223.34830380.24121059550
113783567.76893328343960.706266662
1210453392791.1169
Coefficientst StatP-valueLower 95%Upper 95% 95.0% 95.0%
Intercept4134.2489719396367.720989829611.24289634340.00000022663324.90053077424943.5974131053324.90053077424943.597413105
X Variable 11.90096847330.0109063469174.299286316901.87696376571.92497318091.87696376571.9249731809
199018547.950177717.45813.518547.9
199121617.85288.69102.2722721617.8
199226638.1580011699.59138.626638.1
199334634.46882.116428.511323.834634.4
199446759.49457.222372.21493046759.4
199558478.11199328537.917947.258478.1
199667884.613844.233612.920427.567884.6
199774462.614211.237222.723028.774462.6
199878345.214552.438619.325173.578345.2
199982067.51447240557.827037.782067.5
200089468.114628.244935.329904.689468.1
200197314.815411.8487503315397314.8
2002104790.616117.353540.735132.6104790.6
SUMMARY OUTPUTSUMMARY OUTPUT
Multiple R1Multiple R0.9986922681
R Square1R Square0.9973862464
Adjusted R Square1Adjusted R Square0.9968634956
0730.238812478
77
dfSSMSFSignificance FdfSSMSFSignificance F
31020082220.85429340027406.95142938496757177629394000000000000000011017415977.238041017415977.238041907.95764317750.0000001187
30052666243.61624623533248.723249246
61020082220.8542961020082220.85429
Coefficientst StatP-valueLower 95%Upper 95% 95.0% 95.0%Coefficientst StatP-valueLower 95%Upper 95% 95.0% 95.0%
Intercept0.00000000010.00000000010.88933388360.4393368096-0.00000000020.0000000003-0.00000000020.0000000003Intercept5716.32804942921833.8045625953.11719589210.02633557071002.383352437210430.27274642121002.383352437210430.2727464212
X Variable 11015104323513538401111X Variable 11.8648954650.042694322743.68017448660.00000011871.75514621461.97464471541.75514621461.9746447154
X Variable 21055881074840137701111
X Variable 31054596202838836201111
4.3
yx
12648777
21059210
3909954
413110508
512210979
610711912
740612747
850313499
943114269
1058815522
1189816730
1295017663
1377918575
1481919635
15122221163
16170222880
17157824127
18165425604
19140026500
20182927670
21220028300
22201727430
23210529560
24160028150
25225032100
26242032500
27257035250
28172033500
29190036000
30210036200
31230038200
3.3
yx1x2x3x4x5
1978231301018888149114.89180.92
1979298335021958638916.00420.39
1980343368825319220419.53570.25
1981401394127999530021.82776.71
1982445425830549992223.27792.43
19833914736335810604422.91947.70
19845545652390511035326.021285.22
19857447020487911211027.721783.30
19869977859555210857932.432281.95
198713109313638611242938.912690.23
1988144211738803812264537.383169.48
1989128313176900511380747.192450.14
199016601438496639571250.682746.20
1991217816557109699508155.913335.65
1992288620223129859969383.663311.50
19933383248821594910545896.084152.70
8.3 8.6 C-DCobbDouglas y,K()L()A>0abKLAab
8.3 a1% b1% a+b returns to scalea+b =111a+b 111a+b 111
8.3
(1)
(2)
lny=lnA+ lnK+lnL
y=lny,0=lnA,x1=lnK,x2=lnL,y=0+ x1+x2+
_1239827757.unknown
_1239827781.unknown
8.3
tGDPKL lnGDPlnKlnL197813624.1 1377.9 401528.1953617.22831610.60043197924038.2 1474.2 410248.3035547.29587110.62191198034517.8 1590.0 423618.4157807.37148910.65398198144862.4 1581.0 437258.4892877.36581310.68568198255294.7 1760.2 452958.5744627.47318310.72095198365934.5 2005.0 464368.6885387.60339910.74583198477171.0 2468.6 481978.8778007.81140610.78305198588964.4 3386.0 498739.1010168.12740510.817241986910202.2 3846.0 512829.2303598.25478910.8451019871011962.5 4322.0 527839.3895328.37147410.8739419881114928.3 5495.0 543349.6110148.61159410.9029119891216909.2 6095.0 553299.7356138.71522410.9210519901318547.9 6444.0 647499.8281128.77090511.0782719911421617.8 7517.0 654919.9812728.92492211.0896719921526638.1 9636.0 6615210.190109.17326111.0997119931634634.4 14998.0 6680810.452609.61567211.1095819941746759.4 19260.6 6745510.752779.86581711.1192219951858478.1 23877.0 6806510.9764110.0806711.1282219961967884.6 26867.2 6895011.1255610.1986611.1411419972074462.6 28457.6 6982011.2180510.2561711.1536819982178345.2 29545.9 7063711.2688810.293711.1653119992282067.5 30701.6 7139411.3153010.3320711.1759720002389468.1 32611.4 7208511.4016410.3924211.1856020012497314.8 37460.8 7302511.4857110.5310511.19856200225105172.3 42355.4 7374011.5633610.6538511.20830
8.3 yGDP () K () L() (1)8.158.14SPSS
8.3
ANOVA
Model
Sum of Squares
df
Mean Square
F
Sig.
1
Regression
32.236
2
16.118
5917.774
.000
Residual
.060
22
.003
Total
32.296
24
Coefficients(a)
Unstandardized Coefficients
Standardized Coefficients
t
Sig
B
Std. Error
Beta
(Constant)
-2.086
1.903
-1.096
.285
lnK
.902
.035
.936
25.863
.000
lnL
.361
.201
.065
1.794
.087
a Dependent Variable: lnGDP
8.3 a=0.902b =0.361a+b =0.902+0.361=1.2631A==0.1242b P=0.087C-D
8.3 ,818.95%-0.555 1.56500C-D
8.3
Parameter Estimates
Parameter
Estimate
Std. Error
95% Confidence Interval
Lower Bound
Upper Bound
A
.020
.104
-.196
.236
alpha
.922
.064
.789
1.056
beta
.505
.511
-.555
1.565
8.3
yt l nyt
yt
ytyt
8.3 VIF=15.5k=0.20R20.998 140.980 58****** Ridge Regression with k = 0.20 ****** B SE(B) Beta B/SE(B)lnK .49700385 .02048319 .51558506 24.26398868lnL 2.18274631 .11798929 .39309616 18.49952910Constant -18.43784255 1.27336521 .00000000 -14.47961853
8.3 a=0.4970b =2.183A= b =2.183
8.3 ,
8.3 SPSS3.2SPSS
8.3 1yModel Expressionsb0+b1*x1+b2*x2 2.b0=-213.7,b1=2.185,b2=0.368Continue; 3.Options,OptionsLevenberg-Marquardt,Sequential quadratic programContinue 4LossLoss FunctionSum of squared residuals,User-defiend loss functionABS(y-b0-b1*x1-b2*x2)Continue 5Save
8.3
Iteration Loss funct B0 B1 B2 0.1 4511.684440 -213.70000 2.18500000 .368000000 1.1 4393.393596 -213.69997 2.19431077 .403145685 2.1 4362.671030 -213.69984 2.13568898 .431261430 3.1 4354.739136 -213.78915 2.12883998 .429074345 4.1 4352.083704 -213.78515 2.13523424 .427309206
=-213.79+2.1352x1+0.4273x2
_1031994335.unknown
8.3
x1
x2
yi
eia
ei
1
25
3547.79
553.96
1355.6
-801.64
1385.63
-831.67
2
20
896.34
208.55
211.93
-3.38
133.52
75.03
3
6
750.32
3.10
119.64
-116.54
36.62
-33.52
4
1001
2087.05
2815.40
2815.4
0
2688.57
126.83
5
525
1639.31
1052.12
1607.71
-555.59
1509.71
-457.59
6
825
3357.70
3427.00
2982.56
444.44
2925.4
501.6
7
120
808.47
442.82
387.91
54.91
295.96
146.86
8
28
520.27
70.12
68.32
1.8
-26.34
96.46
9
7
671.13
122.24
87.94
34.3
1.57
120.67
10
532
2863.32
1400.00
2145.68
-745.68
2097.28
-697.28
11
75
1160.00
464.00
442.04
21.96
369
95
12
40
862.75
7.50
240.29
-232.79
158.51
-151.01
13
187
672.99
224.18
473.08
-248.9
368.92
-144.74
14
122
901.76
538.94
432.04
106.9
343.73
195.21
15
74
3546.18
2442.79
1459.54
983.25
1484.64
958.15
_1031995501.unknown
_1257582779.unknown