Ekonometrika · 3-mavzu
Geteroskedastiklikni testlash va birinchi tartibli avtokorrelyatsiya
Breusch–Pagan White Goldfeld–Quandt Durbin–Watson Breusch–Godfrey
Toshkent davlat iqtisodiyot universiteti · Asosiy manba: Wooldridge, Introductory Econometrics, 7-nashr
01 Geteroskedastiklik Xatolar dispersiyasi omillarga bog‘liq bo‘lsa nima buziladi va buni qanday aniqlaymiz.
03
Grafikda bu yelpig‘ich shaklida ko‘rinadi
Daromad va oziq-ovqat xarajati, mln so‘m (simulyatsiya, n = 150) Qoldiq kvadrati û2 : o‘ng tomonda kattalashadi
05
Geteroskedastiklik qayerdan keladi
Turli o‘lchamlar Toshkent shahri va Sirdaryo viloyati bitta tanlanmada
Tanlov erkinligi Daromad oshsa, xarajat qarorlari xilma-xil bo‘ladi
O‘rganish samarasi Tajriba oshgan sari xatolar kamayadi
O‘rtacha qiymatlar Kichik guruhlar o‘rtachasi ko‘proq tebranadi
Noto‘g‘ri shakl Log o‘rniga daraja: «soxta» geteroskedastiklik
Chetga chiquvchilar Bir nechta katta qiymat dispersiyani buzadi
06
Wooldridge, 8.1
Koeffitsiyentlar to‘g‘ri, ishonch noto‘g‘ri
Saqlanadi β̂ siljimagan va asosliR2 talqini o‘zgarmaydi
Buziladi Standart xatoliklar noto‘g‘ri t, F testlar va p-qiymatlar ishonchsiz OLS endi eng samarali emas (BLUE emas)
«Bu omil 5% darajada ahamiyatli» degan xulosa noto‘g‘ri chiqishi mumkin .
07
Wooldridge, 8.3, 270-bet
Breusch–Pagan testi
Modelni OLS bilan baholang, û2 ni oling
û2 ni barcha x larga regress qiling, R2 ni oling
LM yoki F ni hisoblang: p < 0.05 bo‘lsa, geteroskedastiklik bor
LM = n · R 2 û²
taqsimoti χ2 k , k — omillar soni
Misol: n = 200, R² = 0.045 → LM = 9.0 > 7.81 (χ²₃, 5%)
09
Wooldridge, 8.4-misol, HPRICE1, n = 88
Logarifm geteroskedastiklikni yo‘qotdi
A · Darajalarda
price = β₀ + β₁lotsize + β₂sqrft + β₃bdrms
14.09 LM statistika
.003 p-qiymat
Geteroskedastiklik bor
B · Logarifmda
log(price) = β₀ + β₁log(lotsize) + …
4.22 LM statistika
.239 p-qiymat
Gomoskedastiklik rad etilmaydi
Narx, daromad, aylanma uchun logarifm — birinchi va eng arzon davo .
10
Wooldridge, 8.3a, 271–273-betlar
White testi: har qanday shaklni ushlaydi
Yordamchi regressiyaga kvadratlar va o‘zaro ko‘paytmalar qo‘shiladi. 3 ta omil bo‘lsa:
3 omil
+ 3 kvadrat
+ 3 ko‘paytma
= 9 regressor
Kamchiligi: 6 ta omilda 27 ta regressor, erkinlik darajasi tez tugaydi.
Qisqa shakl
û 2 ~ ŷ + ŷ 2
Omillar nechta bo‘lmasin, faqat 2 erkinlik darajasi. Diqqat: y emas, ŷ .
8.5-misol: LM = 3.45, p = .178
11
Ma’ruza slaydlari, 19–23; Демидова, Малахов
Goldfeld–Quandt: ikki chekkani solishtiramiz
Ko‘k: kichik x guruhi · qizil: katta x guruhi · sariq: tashlangan o‘rta qism
Gumon qilingan omil bo‘yicha tartiblang
O‘rtadagi ~1/4 qismni tashlang
Ikki guruhga alohida OLS
F = RSS2 / RSS1
28 101 / 321 = 87.5 > Fkr 3.23 → bor
12
Qaysi testni qachon tanlash kerak
Statistika n·R² ~ χ²(k) n·R² ~ χ²(q) RSS₂ / RSS₁ ~ F Ushlaydi Chiziqli bog‘liqlik Har qanday shakl Bitta omil bo‘yicha o‘sish Kuchli tomoni Oddiy, tez Eng umumiy Kichik tanlanmada aniq F Zaif tomoni Chiziqsiz shaklni o‘tkazadi Ko‘p regressor Omilni bilish kerak
13
statsmodels · HPRICE1 · HC1
Robust xatolik xulosani o‘zgartirdi
×1.95 yer maydoni (lotsize) xatoligi o‘sdi
.002 → .102 p-qiymat: ahamiyatli → ahamiyatsiz
|t| statistikalar: kulrang — odatiy, ko‘k — robust. Qizil chiziq: 1.96 (5%)
Koeffitsiyent o‘sha-o‘sha, lekin unga ishonch ikki baravar kamaydi .
15
Wooldridge, 273-bet
Test «bor» desa ham, sabab boshqa bo‘lishi mumkin
Funksional shakl noto‘g‘ri bo‘lsa, test dispersiya o‘zgarmas bo‘lsa ham rad etadi.
Qoida Avval funksional shakl (RESET), keyin geteroskedastiklik
Ish haqi va tajriba teskari U shaklida bog‘liq. Modelga faqat exper kiritilsa, chekkalarda qoldiqlar katta bo‘ladi.
Davo robust xatolik emas, exper2 ni qo‘shish.
16
Simulyatsiya, n = 120
Tarqoqlik kuchaygan sari testlar uni sezadi
17
xarajat = 2 + 0.6·daromad + u, sd(u) = 0.6·daromadγ
Testlar real vaqtda
02 Birinchi tartibli avtokorrelyatsiya Bugungi xato kechagisiga bog‘liq: baland davrlar ketidan yana baland, past ketidan yana past.
18
Wooldridge, 12.1
Musbat ρ t-statistikani sun’iy oshiradi
Bir xil shoklar: kulrang ρ = 0, ko‘k ρ = 0.85
Saqlanadi Regressorlar qat’iy ekzogen bo‘lsa, β̂ siljimagan
Buziladi Standart xatoliklar kichrayadi Ahamiyatsiz omil «ahamiyatli» ko‘rinadi yt−1 bo‘lsa, OLS asossiz
20
Wooldridge, 12.3a, 402–403-betlar
Qoldiqni o‘z lagiga regress qilamiz
ût = ρ û t −1
ρ̂ ning t-statistikasi bilan H0 : ρ = 0 tekshiriladi. Shart: regressorlar qat’iy ekzogen.
12.2-misol · Fillips egri chizig‘i
.571 ρ̂, statik model
5.48 t-statistika
Kuchli musbat AR(1)
Kengaytirilgan modelda ρ̂ = −.033, p = .773: dalil yo‘q
21
Wooldridge, 12.3b; Savin, White (1977)
Durbin–Watson: 2 atrofida bo‘lsa yaxshi
DW = Σ(ût − û t −12 Σût 2 ρ̂ )
ρ̂ = 0 → DW ≈ 2 · ρ̂ → 1 → DW → 0 · ρ̂ → −1 → DW → 4
Kamchiligi: noaniqlik zonasi; lag qilingan y bilan yaroqsiz
n = 45, k = 4, 5%: dL = 1.336, dU = 1.720
Rad etiladiρ > 0
?
Rad etilmaydidalil yo‘q
0 1.336 1.720 … 2
Fillips: DW = 0.80 < dL → rad etiladi Tekshiruv: 2(1 − .571) = 0.86
22
statsmodels · PHILLIPS (Wooldridge) · n = 56
Inflyatsiya xatolari 9 yil bir tomonda turdi
AQSh, 1948–2003. Sariq: 1958–66 ketma-ket manfiy, 1973–81 (neft shoklari) ketma-ket musbat
0.80 Durbin–Watson
.572 ρ̂ · t = 5.32
20.9 Breusch–Godfrey LM · p < .001
23
Wooldridge, 12.3c–d, 404–407-betlar
Lag qilingan y bo‘lsa: Breusch–Godfrey
Modelda yt−1 bo‘lsa, DW va oddiy t-test noto‘g‘ri. Yordamchi regressiyaga barcha omillar qo‘shiladi:
ût ~ û t −1û t −q x t 1xtk
LM = (n − q ) R 2 ~ χ 2 q
12.3-misol · minimal ish haqi
ρ̂ = .481, t = 2.89 → musbat AR(1)
12.4-misol · oylik ma’lumot
AR(3): F = 5.12, p = .0023
24
Simulyatsiya, 60 oy · dL = 1.549, dU = 1.616
ρ o‘zgarsa, DW 0 dan 4 gacha suriladi
25
savdo = 5 + 0.8·reklama + u, u = ρu(t−1) + e · 60 oy
ρ ni suring, DW javob beradi
Dasturda bir qatorda
library(lmtest); library(sandwich)
m <- lm(price ~ lotsize + sqrft + bdrms)
bptest(m) # Breusch–Pagan
gqtest(m, order.by = ~ bdrms) # G–Q
coeftest(m, vcovHC(m, "HC1")) # robust
dwtest(mt) # Durbin–Watson
bgtest(mt, order = 1) # BG
coeftest(mt, NeweyWest(mt)) # HAC
Excel · BP testi
Regression → Residuals
=E2^2 → û²
û² ni x larga regress → R²
=CHISQ.DIST.RT(n*R²; k)
27
Test tamg‘a emas: u xulosamizga qanchalik ishonish mumkinligini ko‘rsatadi
Tartib muhim Avval grafik va funksional shakl, keyin test
Bizning ma’lumot Hududlar kesimida geteroskedastiklik, oylik makroda avtokorrelyatsiya deyarli muqarrar
DW o‘rniga BG Noaniq zona va lag qilingan y: amalda BG universalroq
Robust — odat HC va HAC xatoliklarni doimo hisobot qilish kerak
28
Bir sahifada
Test H₀ Statistika Muammo bor, agar
Breusch–Pagan Var(u|x) = σ² n·R² ~ χ²(k) p < α White Var(u|x) = σ² n·R² ~ χ²(q) p < α Goldfeld–Quandt σ²₁ = σ²₂ RSS₂ / RSS₁ F > Fkr t-test, AR(1) ρ = 0 t(ρ̂) |t| > tkr Durbin–Watson ρ = 0 ≈ 2(1 − ρ̂) DW < dL Breusch–Godfrey ρ₁ = … = ρq = 0 (n − q)R² ~ χ²(q) p < α
29
Foydalanilgan adabiyotlar
1 Wooldridge J.M. Introductory Econometrics: A Modern Approach , 7th ed. Cengage Learning, 2019. 8 va 12-boblar.2 Демидова О.А., Малахов Д.И. Эконометрика: учебник и практикум. М.: Юрайт.3 «Гетероскедастичность», Гл. 8. Ekonometrika fanidan ma’ruza slaydlari.4 Breusch T.S., Pagan A.R. A Simple Test for Heteroscedasticity and Random Coefficient Variation. Econometrica , 47(5), 1979, 1287–1294.5 Koenker R. A Note on Studentizing a Test for Heteroscedasticity. Journal of Econometrics , 17(1), 1981, 107–112.6 White H. A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity. Econometrica , 48(4), 1980, 817–838.7 Goldfeld S.M., Quandt R.E. Some Tests for Homoscedasticity. JASA , 60(310), 1965, 539–547.8 Durbin J., Watson G.S. Testing for Serial Correlation in Least Squares Regression I. Biometrika , 37(3/4), 1950, 409–428.9 Breusch T.S.; Godfrey L.G. Australian Economic Papers , 17(31), 1978, 334–355; Econometrica , 46(6), 1978, 1293–1301.10 Savin N.E., White K.J. The Durbin–Watson Test for Serial Correlation with Extreme Sample Sizes or Many Regressors. Econometrica , 45(8), 1977, 1989–1996.Ma’lumotlar: HPRICE1 va PHILLIPS to‘plamlari (Wooldridge, Python wooldridge paketi). Hisob-kitoblar: Python statsmodels, R lmtest.
E’tiboringiz uchun rahmat!
Savollaringiz bo‘lsa, marhamat. Endi bilimni jonli testda sinaymiz.
Toshkent davlat iqtisodiyot universiteti · Ekonometrika 30
