Tam Logaritmik Fonksiyon

Tam Logaritmik FonksiyonTam Logaritmik Fonksiyon

X3

X2

Y1

Y2

0<2<1

2<0

Y

X2

2>1

(X3 sabit tutulduğunda)

uk321 e.XX.X.Y k32

lnY =ln1 + 2 lnX2+ 3 lnX3 + ... + k lnXk + u lne

Y* =1 *+ 2 X2

*+ 3 X3* + ... + k Xk

* + u

2*2

*1

***

*2

*1

*

Xb̂b̂XXY

Xb̂b̂nY

eXb̂b̂Y *2

*1

*

?b̂*1 ?b̂2


Y

X

X

1Y.2

2X.Y 1

121

2X..'Y X

1X.. 2

12

X

1Y.2

Y

X

X

YEyx

2

rsapmasızdı i tahminlerb̂ veb̂ 2*1

.sapmalıdır tahminib̂logantib *11

aynıdır. heryerindeeğrinin tahminib̂ 2


Uygulama 4.3 (207-210)Uygulama 4.3 (207-210)

X

4003002001000

Y 80

60

40

20

0



*Y n

Y*25

1449.101 = 4.0458

*Xn

X*25

0374.124 = 4.9615

x*2 =7.3986

y*x* =2.6911


2*

**

2 x

yxb̂

7.3986

2.6911

= 2.2413= 4.0458 - (0.3637) 4.9615

[ln(9.4046) = 2.2413]

= 0.3637


Üretim FonksiyonuÜretim Fonksiyonu

32 b3

b21 X.XbY Y= Üretim X2=Emek ; X3=Sermaye

22

2 X

Yb

X

Y

= Emeğin Marjinal Verimliliği

33

3 X

Yb

X

Y

= Sermayenin Marjinal Verimliliği

lnY = -3.4485 + 1.5255 lnX2 + 0.4858 lnX3

(t) (-1.43) (2.87) (4.82)

n=15 Düz-R2= 0.8738

Yarı-Logaritmik FonksiyonYarı-Logaritmik FonksiyonLog-Doğ Model(Üstel Model)Log-Doğ Model(Üstel Model)

Y

X(a)

Y = Aeb X2

Y

X(b)

Y = Aeb X2

A

A

b >0

b <02

2

Xbb 21eY Xbb 21ee Xb2e A

Yarı-Logaritmik FonksiyonYarı-Logaritmik FonksiyonLog-Doğ Model(Üstel Model)Log-Doğ Model(Üstel Model)

lnY = b1 +b2 X+ u

X d

Yln db2

X d

Y d.

Y

1

X d

Y/Y d

değişmemutlak dekiX'

değişme nisbi dekiY'

Y

X

X d

Y dEyx = ( b2Y )

Y

X= b2 X

Artış Hızı ModeliArtış Hızı ModeliLog-Doğ Model(Üstel Model)Log-Doğ Model(Üstel Model)

lnY = b1 +b2 t + u

r = (Antilog b2 - 1) . 100

Y= İş hacmi(1983-1988)

r = (Antilog 0.131 - 1) . 100

= (1.13997 - 1) . 100

= (0.13997 1) . 100

= % 14

Ücret ModeliÜcret ModeliLog-Doğ Model(Üstel Model)Log-Doğ Model(Üstel Model)

lnY = 1.19 + 0.033 X2 + 0.074 X3

Aşağıdaki ücret modeli Uygulama 9.3’den alınmıştır.(s.427)

Modelde:

Y:Haftalık Kazanç ($) ; X2: Tecrübe ; X3 : Eğitim Kategorisi

Yarı-Logaritmik Fonksiyon Yarı-Logaritmik Fonksiyon Doğ - Log ModelDoğ - Log Model

Y = b1 +b2 lnX+ u

Y

X(a)

Y = b + b lnX

Y

X(b)

b >0

b <02

2

21 Y = b + b lnX21

Yarı-Logaritmik Fonksiyon Yarı-Logaritmik Fonksiyon Doğ - Log ModelDoğ - Log Model

Y = b1 +b2 lnX+ u

lnX d

dYb2

)X/1(

1

X d

Y d

X/X d

Y d

değişme nisbi dekiX'

değişmemutlak dekiY'

Y

X

X d

Y dEyx

Y

X

X

b2 Y

b2

Hedonik Model Hedonik Model Doğ - Log ModelDoğ - Log Model

Y = b1 +b2 lnX2+ b3 lnX3 + u

Fiyat = -1.749.97 + 299.97 ln(m2) - 145.09 ln(YatakOda)

(t) (-6.8) (7.5) (-1.7)

Prob. [0.1148]

Düz-R2= 0.826 sd=11

Polinomial Fonksiyonlar Polinomial Fonksiyonlar

Y =1 + 2 X + 3 X2 + 4 X3 + ... + k+1 Xk + u

Kuadratik Model:

Y =1 + 2 X + 3 X2 + u

dX

dY= 2 + 23 X =

X0= -2 / 23

Xd

Yd2

2

= 23

Eğer 3<0 ise X0 noktası maksimumdur

Eğer 3>0 ise X0 noktası minimumdur

Polinomial Fonksiyonlar Polinomial Fonksiyonlar Kuadratik ModelKuadratik Model

OM = 10.52 - 0.175 Çıktı + 0.0009 (Çıktı)2 + 0.02 GMİ

(t) (14.3) (-9.7) (7.8) (14.45)

Düz-R2=0.978 sd=16

OM= Ortalama Maliyet ; Çıktı =Üretimİndeksi

GMİ= Girdi Maliyetleri İndeksi

Polinomial Fonksiyonlar Polinomial Fonksiyonlar Kübik ModelKübik Model

TM= Toplam Maliyet ;Q =Üretim Miktarı

Polinomial Fonksiyonlar Polinomial Fonksiyonlar Kübik ModelKübik Model

Y =1 + 2 X + 3 X2 + 4 X3 + u

TM = 141.76 + 63.47 Q - 12.96 Q2 + 0.94 Q3

s(bi) (6.37) (4.78) (0.98) (0.059)

R2 =0.998 sd=6

DATA4-1: Data on single family homes in University City community of San Diego, in 1990.

price = sale price in thousands of dollars (Range 199.9 - 505)

sqft = square feet of living area (Range 1065 - 3000)

bedrms = number of bedrooms (Range 3 - 4)

baths = number of bath roooms (Range 1.75 - 3)

Dependent Variable: PRICE Included observations: 14

Variable Coefficient Std. Error t-Statistic Prob.

C 52.35091 37.28549 1.404056 0.1857

SQFT 0.138750 0.018733 7.406788 0.0000

R-squared 0.820522 Akaike info criterion 10.29774

Adjusted R-squared 0.805565 Schwarz criterion 10.38904

S.E. of regression 39.02304 F-statistic 54.86051

Sum squared resid 18273.57 Prob(F-statistic) 0.000008



C -1749.974 259.1410 -6.752980 0.0000

LOG(SQFT) 299.9724 39.97577 7.503856 0.0000

LOG(BEDRMS) -145.0942 84.71878 -1.712657 0.1148







C -28.60714 133.0212 -0.215057 0.8337

SQFT 0.224620 0.136493 1.645651 0.1281

SQFT*SQFT -2.10E-05 3.30E-05 -0.635444 0.5381

R-squared 0.826877 Mean dependent var 317.4929

Adjusted R-squared 0.795400 S.D. dependent var 88.49816

S.E. of regression 40.03014 Akaike info criterion 10.40455

Sum squared resid 17626.53 Schwarz criterion 10.54149

Log likelihood -69.83186 F-statistic 26.26930

Durbin-Watson stat 1.994036 Prob(F-statistic) 0.000065



C -1775.234 383.2026 -4.632625 0.0007

LOG(SQFT) 284.2528 60.41533 4.704979 0.0006

BATHS -18.18913 41.03884 -0.443217 0.6662





DATA6-1: Data on cost function Source: W.A. Spurr and C.P.Bonini, STATISTICAL ANALYSIS FOR BUSINESS

DECISIONS, Irwin, 1973, Page 535.

UNITCOST = Cost per unit dollars (Range 3.65 - 6.62)

OUTPUT = Index of output (Range 50 - 104)

INPCOST = Index of input costs (Range 80 - 150)

Dependent Variable: UNITCOST Included observations: 20

VariableCoefficien

t Std. Error t-Statistic Prob.

C 10.39073 1.330698 7.808484 0.0000

OUTPUT -0.173953 0.019145 -9.085969 0.0000

OUTPUT*OUTPUT 0.000891 0.000122 7.278983 0.0000

INPCOST 0.022162 0.016613 1.333987 0.2021

INPCOST*INPCOST -8.57E-06 7.11E-05 -0.120474 0.9057

R-squared 0.981884 Akaike info criterion -1.236679

Adjusted R-squared 0.977053 Schwarz criterion -0.987746



Dependent Variable: UNITCOST Included observations: 20


C 10.52230 0.736531 14.28630 0.0000

OUTPUT -0.174473 0.018069 -9.655786 0.0000

OUTPUT*OUTPUT 0.000895 0.000115 7.756616 0.0000

INPCOST 0.020168 0.001395 14.45369 0.0000

R-squared 0.981866 Akaike info criterion -1.335712

Adjusted R-squared 0.978466 Schwarz criterion -1.136566



d(unit cos t)

d(output)- 0.174473+2*0.000895*Output

= - 0.174473+0.00179*Output

= - 0.174473+0.00179*(1) = - 0.173

0

1

2

3

4

5

6

7

8

40 50 60 70 80 90 100 110

unitcost-tah1 unitcost-tah2 unitcost-tah3

Polinom (unitcost-tah1) Polinom (unitcost-tah2) Polinom (unitcost-tah3)

DATA6-2: Data on the white tuna (Thunnus Alalunga) fishery production in the Basque region of Spain. Data compiled by Felix Telleria.

catch = total catch in thousands of tonnes, Range 16.608 - 51.8

effort = total days of fishing in thousands, Range 10.31185 - 61.24754

Üretim Fonksiyonu Tahmini

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

CATCH

Üretim Fonksiyonu Tahmini

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70

CATCH Polinom (CATCH)

Dependent Variable: CATCH Sample: 1961 1994


C 2.339564 5.785686 0.404371 0.6887

EFFORT 1.491794 0.383074 3.894275 0.0005

EFFORT*EFFORT -0.014521 0.005386 -2.696384 0.0112







Dependent Variable: CATCHSample: 1961 1994


EFFORT 1.641621 0.095991 17.10185 0.0000

EFFORT*EFFORT -0.016530 0.002056 -8.041338 0.0000





Log likelihood -105.3027 Durbin-Watson stat 1.512102

DATA6-3: United Kingdom Annual data.

Cons = Per capita consumption expenditure in British pounds (Range 1858 - 4744)

DI = Per capita personal disposable income in British pounds (Range 1875 - 5084)

Source: Economic Trends, Annual Supplement 1991 Edition. A publication of the Government Statistical Service, London, UK.

Model A:

Dependent Variable: CONS Sample: 1948 1989


C 168.3149 43.27960 3.889013 0.0004

DI 0.864323 0.013300 64.98492 0.0000







Model B:

Dependent Variable: CONS Sample (adjusted): 1949 1989


C -56.09466 29.93634 -1.873798 0.0689

CONS(-1) 1.068312 0.097298 10.97983 0.0000

DI 0.684000 0.084986 8.048355 0.0000

DI(-1) -0.723031 0.080315 -9.002388 0.0000







Wald Test:

Null Hypothesis :

C(3)=0

C(4)=0

Test Statistic Value df Probability

F-statistic 48.43943 (2, 37) 0.0000

Chi-square 96.87885 2 0.0000

Model C:

Dependent Variable: CONS Sample (adjusted): 1949 1989


C -46.80204 22.56606 -2.074001 0.0449

CONS(-1) 1.021883 0.008307 123.0201 0.0000

DI-DI(-1) 0.705785 0.071060 9.932268 0.0000







-80

-40

0

40

80

1000

2000

3000

4000

5000

1950 1955 1960 1965 1970 1975 1980 1985

Residual Actual Fitted

DATA3-3: Annual data on U.S. Patents and R&D expenditures

YEAR = 1960-93, 34 observations.

PATENTS = Number of patent applications filed, in thousands (Range 84.5 - 189.4)

Source: Statistical Abstract of the U.S., various years.

R&D = R&D expenditures, billions of 1992 dollars obtained as the ratio of expenditure on current dollars divided by the GDP price deflator. (Range 57.94 - 166.7)

Source for R&D expenditures, 1995 Statistical Abstract of the U.S., Table 979, Page 611, and earlier issues Source for GDP deflator, 1996 Economic Report of the President, Table B-3, Page 284.

80

100

120

140

160

180

200

40 60 80 100 120 140 160 180

RD

PATENTS

Dependent Variable: PATENTS Sample: 1960 1993


C 34.57106 6.357873 5.437521 0.0000

RD 0.791935 0.056704 13.96621 0.0000







Dependent Variable: PATENTS

Sample (adjusted): 1964 1993


C 85.35255 22.10268 3.861639 0.0011

RD -0.047681 1.125105 -0.042379 0.9666

RDG1 0.603316 2.056192 0.293414 0.7724

RDG2 0.000179 2.185027 8.21E-05 0.9999

RDG3 -0.586882 2.052188 -0.285979 0.7780

RDG4 -0.183709 1.099382 -0.167102 0.8691

RD*RD -0.000733 0.004898 -0.149577 0.8827

RDG1*RDG1 -0.001754 0.008900 -0.197047 0.8459

RDG2*RDG2 0.001736 0.009822 0.176776 0.8616

RDG3*RDG3 -0.000756 0.009238 -0.081882 0.9356

RDG4*RDG4 0.007144 0.005085 1.404834 0.1762










C 84.84085 19.05788 4.451746 0.0002

RDG1 0.604292 0.635069 0.951537 0.3517

RDG3 -0.735232 0.523331 -1.404909 0.1740

RDG4 -0.074464 0.513419 -0.145035 0.8860

RD*RD -0.000949 0.001151 -0.824413 0.4186

RDG1*RDG1 -0.001693 0.003414 -0.495939 0.6249

RDG2*RDG2 0.001628 0.002538 0.641485 0.5278

RDG4*RDG4 0.006610 0.001965 3.364449 0.0028









Included observations: 30 after adjustments


C 82.85449 12.03555 6.884148 0.0000

RDG1 0.477055 0.327782 1.455404 0.1580

RDG3 -0.637010 0.238843 -2.667066 0.0132

RD*RD -0.001146 0.001000 -1.146413 0.2625

RDG4*RDG4 0.006519 0.000678 9.608783 0.0000










C 91.34639 6.404627 14.26256 0.0000

RDG3 -0.295068 0.117463 -2.512013 0.0183

RDG4*RDG4 0.005856 0.000549 10.67451 0.0000







80

100

120

140

160

180

200

1960 1965 1970 1975 1980 1985 1990 1995

Actual Fitted

Beşeri Sermaye ModeliBeşeri Sermaye Modeli

r= Her ilave eğitim yılının getiri oranı

Bir yıl ilave eğitimin getirisi: w1= (1+r).w0

İki yıl ilave eğitimin getirisi: w2= (1+r)2.w0

s yıl ilave eğitimin getirisi: ws= (1+r)s.w0

ln( ws) = s.ln (1+r) + ln(w0)

ln( ws) = b1 + b2.s

DATA6-4: Data on salaries and employment characteristics of 49 employees in a certain company. Data compiled by Susan Wong

WAGE = Wage rate per month (Range 981 - 3833)

EDUC = Years of education beyond 8th grade when hired (Range 1 - 11)

EXPER = Number of years at the company (Range 1 - 23)

AGE = Age of employee (25 - 64)

Dependent Variable: LOG(WAGE) Included observations: 49


EDUC 0.064553 0.016750 3.853948 0.0004

EXPER 0.022700 0.006873 3.302822 0.0019

AGE 0.000392 0.004033 0.097144 0.9230

C 6.835956 0.203431 33.60335 0.0000







Redundant Variables: AGE

F-statistic 0.009437 Probability 0.923043

Log likelihood ratio 0.010275 Probability 0.919261

Test Equation:

Dependent Variable: LOG(WAGE)

Included observations: 49


EDUC 0.064505 0.016561 3.894924 0.0003

EXPER 0.022954 0.006285 3.652488 0.0007

C 6.850604 0.135079 50.71544 0.0000







C 7.329325 0.809178 9.057746 0.0000

EDUC -0.093041 0.086385 -1.077050 0.2876

EXPER 0.013863 0.024484 0.566210 0.5743

AGE -0.000426 0.033821 -0.012598 0.9900

EDUC^2 0.011525 0.006274 1.836991 0.0733

EXPER^2 0.000429 0.001119 0.383709 0.7031

AGE^2 2.11E-05 0.000381 0.055452 0.9560









C 7.319851 0.295150 24.80047 0.0000

EDUC -0.092871 0.084324 -1.101355 0.2769

EXPER 0.013886 0.024130 0.575473 0.5680

EDUC^2 0.011511 0.006112 1.883367 0.0664

EXPER^2 0.000428 0.001098 0.389398 0.6989

AGE^2 1.64E-05 4.51E-05 0.363167 0.7183









C 7.330007 0.290909 25.19693 0.0000

EDUC -0.088259 0.082536 -1.069337 0.2907

EXPER 0.014847 0.023747 0.625192 0.5351

EDUC^2 0.011154 0.005972 1.867523 0.0685

EXPER^2 0.000424 0.001087 0.390167 0.6983









C 7.286796 0.266457 27.34701 0.0000

EDUC -0.085943 0.081543 -1.053958 0.2975

EXPER 0.023791 0.006134 3.878702 0.0003

EDUC^2 0.011134 0.005916 1.882161 0.0663









C 7.023367 0.092457 75.96326 0.0000

EXPER 0.023681 0.006140 3.856595 0.0004

EDUC^2 0.005023 0.001171 4.289093 0.0001







DATA4-4: Demand for bus travel and its determinantsAll data are for the year 1988 for 40 cities across the U.S. Data compiled by Sean Naughton

BUSTRAVL = Demand for urban tranportation by bus in thousands of passenger hours (Range 18.1 - 1310.3)

FARE = Bus fare in dollars (Range 0.5 - 1.5)

GASPRICE = Price of a gallon of gasoline, in $ (Range 0.79 - 1.03)

INCOME = Average income PER CAPITA (Range 12349 - 21886)

POP = Population of city in thousands (Range 167 - 7323.3)

DENSITY = Density of city in persons/sq. mile (Range 1551 - 24288)

LANDAREA = Land area of the city in sq. miles (Range 18.9 - 556.4)

Dependent Variable: LOG(BUSTRAVL) Included observations: 40


C 44.71498 20.74953 2.154988 0.0386

LOG(FARE) 0.476381 0.425336 1.120010 0.2708

LOG(DENSITY) 0.275586 2.662865 0.103492 0.9182

LOG(GASPRICE) -1.732969 2.495004 -0.694576 0.4922

LOG(INCOME) -4.852523 1.047345 -4.633168 0.0001

LOG(LANDAREA) -0.816762 2.713267 -0.301025 0.7653

LOG(POP) 1.686758 2.695821 0.625694 0.5358









C 46.60735 9.664370 4.822596 0.0000

LOG(FARE) 0.492059 0.391617 1.256479 0.2175

LOG(GASPRICE) -1.710396 2.449026 -0.698398 0.4897

LOG(INCOME) -4.850320 1.031782 -4.700917 0.0000

LOG(LANDAREA) -1.096442 0.238906 -4.589419 0.0001

LOG(POP) 1.964220 0.278327 7.057246 0.0000









C 46.20138 9.576020 4.824696 0.0000

LOG(FARE) 0.438911 0.381331 1.150999 0.2575

LOG(INCOME) -4.765488 1.017081 -4.685453 0.0000

LOG(LANDAREA) -1.019002 0.210062 -4.850956 0.0000

LOG(POP) 1.865398 0.237915 7.840611 0.0000







Omitted Variables: LOG(FARE) LOG(DENSITY) LOG(GASPRICE)

F-statistic 0.583902 Probability 0.629794

Log likelihood ratio 2.068843 Probability 0.558242



C 45.84568 9.614110 4.768582 0.0000

LOG(INCOME) -4.730082 1.021192 -4.631923 0.0000

LOG(LANDAREA) -0.970997 0.206807 -4.695192 0.0000

LOG(POP) 1.820371 0.235733 7.722176 0.0000







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Tam Logaritmik Fonksiyon