3=0=+

41=0234

=+6

yi=a+b1xi1+b2xi2+b3xi3+bkxik+ui (i=1,2, ,n)=+ui

u^iuiui

uiuiuiuixiui

ui Vertical Error JumpsyixiyixixixiyixiErrorsin Variable Model

a+bxiyi=a+b(xi+i)+ui

Zero Mean Error DisplacementE(ui)=0 (i=1,2,.,n) E(u^i)=0E(ui)=0ui

Constant Error VarianceVar(ui)=2 (i=1,2,n) uiVar(ui)=2i (i=1,2,n) Var(ui)=[ui-E(ui)]2=2i (i=1,2,n) 2iI

x1 x2XuYxu a + b x

x1 x2XuxY

Error IndependentuiujijE(ui,uj)=02E(ui)=0E(uj)=0COV(ui,uj)=E[ui-E(ui)][uj-E(uj)]=E(ui,uj)4COV(ui,uj) =E(ui,uj)=0uiuji< >jE(ui,ui+1)= E(ui-1,ui)=0 E(ui,uj) < > 0Autocorrelation

xi uii,jCOV(xi,uj)=0The x are revealed and independent of uii,jCOV(xi,uj)=0uixjxixjGNP

GNPCi = a+bGNPi+ui 1GNPi = Ci+Ii+Gi 2CiIiGiGNPiuiuiCiGNPiuiGNPiGiIi12

Linearity of the Modelyi=a+b1xi1+b2xi2+b3xi3+bkxik+ui (i=1,2, ,n)yi=yiyiabyixij

yi

yi=yiyiabyixijyi

-

6uiXui=0Var(ui)=2 E(ui,uj)=0uiuj=0 (ij)xiui E(x,uj)=0xuj=0 Y=XB+u

yi6yi=a+b1xi1+b2xi2+b3xi3+bkxik+ui (i=1,2, ,n)=2=0ui i.i.d0, 2i.i.dIdentical Independent Distribution

yiEyi=Ea+b1xi1+b2xi2+b3xi3+bkxik+uia,b1b2,b3,bk, xi1,xi2,xi3,xik a,b1b2,b3,bkEyi=a+b1xi1+b2xi2+b3xi3+bkxikVaryi=Vara+b1xi1+b2xi2+b3xi3+bkxik+uiVaryi=2 yi i.i.da+b1xi1+b2xi2+b3xi3+bkxik2Cov(yi,yj)=0

Yi222E(yi) E(yi)=a+b1x1++bkxkE(yi)=a+b1x1++bkxk

yi=a+b1xi1+b2xi2+b3xi3+bkxik+ui (i=1,2, ,n)

123

yi=a+bxi+ui12wi3Eb^=b4Varb^= 5Ea^= a6Vara^=

1

2wi

3

4

5

-

yi=a+b1xi1+b2xi2+b3xi3+bkxik+ui

-

-

t

tf(t)t-t

-

yi=a+bxi+ui12

-yi=a+b1xi1+b2xi2+b3xi3+bkxik+uiuii.i.d(0,2),a^,b1^,b2^, ,bk^a,b1,b2, ,bkyiyi

yi=a+bxi+ui123

1

2

3LS

3

3

4123==>

4 1 yi23243LS5LS

2

41

42

43

==>==>==>==>==>12

50kg1000kg1030kg30kg

3040506070kg80090010001100900

12345

0Var(Xm)1

1121

Wi=a+b1Si+b2Ei+b3Ai+b4Ui+ui 1Wi= Si= Ei=Ai=Ui=R2=1Ai=7+Si+Ei 221Ai Wi=a+7 b3+b1+ b3Si+b2 + b3 Ei+ b4Ui+ uiWi=c+dSi+eEi+fUi+ ui

=1=11

R2==>1

R2

3

12

41

42

12

1R2F21233

12

-BLUEBLUEBest linear Unbias Estimator

BLUE

`

SS0 R2=1 R21 R2=1 R21

Sheet1

obsYX1X2

110.93900012976.03100013732.0409998894

29.50699996954.97300004961.8129999638

39.07800006877.86199998863.8819999695

47.69099998476.66900014883.5490000248

511.75300025944.40700006480.6869999766

68.65100002294.41599988941.5470000505

73.79500007639.45499992376.7049999237

812.08899974824.16099977490.4110000134

96.57299995425.02699995042.8269999027

108.06099987034.13700008391.7389999628

Sheet2

Sheet3

FLX3\HXQ89

LX3\HXQ105

LX3\HXQ133

LX3\hxq149

BlaisdellLX3\HXQ195

LX3\WB36

0-12312345

0-1231234

0-

1

2

1

2

3

KT1%0.5516%LT1%0.325%T/T=1/TT1KL0.11791/T

4