Y
|
14
|
15
|
18
|
19
|
22
|
X1
|
4
|
8
|
12
|
13
|
15
|
X2
|
8
|
9
|
11
|
15
|
17
|
Bir omilli ekonometrik modelini tuzish uchun berilgan ma'lumotlar asosida quyidagi formuladan foydalanamiz:
Y = β0 + β1X1 + β2X2 + ε
Y: 14, 15, 18, 19, 22
X1: 4, 8, 12, 13, 15
X2: 8, 9, 11, 15, 17
Modelni tahlil qilish uchun koeffitsiyentlarni hisoblashimiz:
β0, β1, β2 larni hisoblash:
β1 = ((Σ(X1Y) - nX1_meanY_mean) / (Σ(X1^2) - nX1_mean^2))
β2 = ((Σ(X2Y) - nX2_meanY_mean) / (Σ(X2^2) - nX2_mean^2))
β0 = Y_mean - β1X1_mean - β2X2_mean
Y, X1 va X2 larni qiymatlarini hisoblash:
Y_mean = (14 + 15 + 18 + 19 + 22) / 5 = 17.6
X1_mean = (4 + 8 + 12 + 13 + 15) / 5 = 10.4
X2_mean = (8 + 9 + 11 + 15 + 17) / 5 = 12
Σ(X1Y), Σ(X2Y), Σ(X1^2), Σ(X2^2) hisoblash:
Σ(X1Y) = (414 + 815 + 1218 + 1319 + 1522) = 892
Σ(X2Y) = (814 + 915 + 1118 + 1519 + 1722) = 1149
Σ(X1^2) = (4^2 + 8^2 + 12^2 + 13^2 + 15^2) = 706
Σ(X2^2) = (8^2 + 9^2 + 11^2 + 15^2 + 17^2) = 674
β0, β1, β2 ni hisoblash:
β1 = ((892 - 510.417.6) / (706 - 510.4^2)) = 1.018
β2 = ((1149 - 51217.6) / (674 - 512^2)) = 0.704
β0 = 17.6 - 1.01810.4 - 0.70412 = 10.931
Modelimizning shakli:
Y = 10.931 + 1.018X1 + 0.704X2 + ε
Fisher mezoni qiymatini hisoblash uchun quyidagi formuladan foydalanamiz:
F = (R^2 / k) / ((1 - R^2) / (n - k - 1))
R^2 = 1 - (Σ(Y - Yhat)^2 / Σ(Y - Y_mean)^2)
Yhat = β0 + β1X1 + β2X2
k = 2 (X1 va X2 uchun koeffitsiyentlar soni)
n = 5 (ma'lumotlar soni)
Yhat, Y - Y_mean, Y - Yhat ni hisoblash:
Yhat = 10.931 + 1.018X1 + 0.704X2
Y - Y_mean = (14 - 17.6, 15 - 17.6, 18 - 17.6, 19 - 17.6, 22 - 17.6) = (-3.6, -2.6, 0.4, 1.4, 4.4)
Y - Yhat = (14 - Yhat, 15 - Yhat, 18 - Yhat, 19 - Yhat, 22 - Yhat)
Σ(Y - Yhat)^2, Σ(Y - Y_mean)^2, R^2 hisoblash:
Σ(Y - Yhat)^2 = (-3.6^2 + -2.6^2 + 0.4^2 + 1.4^2 + 4.4^2) = 473. Σ(Y - Y_mean)^2 = ((14 - 17.6)^2 + (15 - 17.6)^2 + (18 - 17.6)^2 + (19 - 17.6)^2 + (22 - 17.6)^2) = 47.2
R^2 = 1 - (47 / 47.2) = 0.0034
Fisher mezoni qiymatini hisoblash:
F = (0.0034 / 2) / ((1 - 0.0034) / (5 - 2 - 1)) = 0.0051 / 0.9966 = 0.0051
Iqtisodiy tahlilning natijasi:
Bu omilli ekonometrik model yaxshi to'g'ridan-to'g'ri tasvirlashga qobiliyatga ega emas. R^2 qiymati juda yaxshi emas, ya'ni modelning o'rniga ma'lumotlar to'plamining faolligi juda kam hisoblanadi. Fisher mezoni qiymati ham juda kichik bo'lib, modelning umumiy aniq qilish qobiliyatini ko'rsatmaydi.
Shu sababli, berilgan ma'lumotlar asosida tuzilgan model iqtisodiy tahlilga qaraganda yaxshi natijalar beradi. Boshqa o'zgaruvchilar yoki faktorlar qo'shilishi va to'g'rimoqchi bo'lgan ma'lumotlar olganda, modelning aniqlash sifati oshirilishi mumkin.
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