This will provide you with a practical overview of all the tests discussed during the first part of Econometrics 2.
In table format, for each test, a short description is provided as well as the null and alternative hypothesis, statistic and distribution of the test statistic. Hence, a very usefu...
Assumptions tests Econometrics 2 (Part I of course)
Misspecification Test Setup H0 H1 Statistic Distribution (under
H0 )
Heteroskedasticity Goldfeld- Split data in 2 (or 3) groups, then Homoskedasticity Heteroskedasti 𝑆𝑆𝑅2 /(𝑛2 − 𝑘)𝜎22 F(n2 – k, n1 – k)
Quandt compare error terms with F-test. (σ2)1 = (σ2)2 city in the 𝑆𝑆𝑅1/(𝑛1 − 𝑘)𝜎12
Reject if statistic large. sense of
(σ2)1 < (σ2)2 𝑠22 /𝜎22 𝑠 2
= = ( 2)
𝑠12 /𝜎12 𝑠1
under H0
Breusch- We have the model 𝜎𝑖2 = ℎ(𝑧𝑖′ 𝛾) Homoskedasticity H0 not true LM = nR2 χ2 (p-1)
Pagan 𝛾2 = ⋯ = 𝛾𝑝 = 0
LM-test, estimate restricted model in 𝜎𝑖2 = ℎ(𝑧𝑖′ 𝛾) where
and obtain residuals. Then perform 𝑧𝑖′ = (1, 𝑧2𝑖 , … , 𝑧𝑝𝑖 )
a auxiliary regression of these
residuals:
𝑒𝑖2 = 𝛾1 + 𝛾2 𝑧2𝑖 + ⋯ + 𝛾2 𝑧2𝑖 + 𝜂𝑖
Reject if statistic large.
White Similar to Breusch-Pagan, but now Homoskedasticity H0 not true LM = nR2 χ2 (2k-2)
z consists of the original regressors 𝛾2 = ⋯ = 𝛾𝑝 = 0
+ original regressors squared. in 𝜎𝑖2 = ℎ(𝑧𝑖′ 𝛾) where since 2(k-1) error-
𝑧𝑖′ = (1, 𝑥2𝑖 , … , explaining variables
Cross terms could be added 2
𝑥𝑘𝑖 , 𝑥2𝑖 2
, … , 𝑥𝑘𝑖 )
(White with/without cross-terms) (without cross-terms)
LR Test significance of 𝛾 in 𝜎𝑖2 = Homoskedasticity H0 not true 2(log(L1) – log(L0)) χ2 (k)
ℎ(𝑧𝑖′ 𝛾) using a LR statistic, so have
to estimate model under both H0
and H1
Function h is important.
Serial correlation Durbin- Under no serial correlation, DW is No serial correlation, H0 not true ∑𝑛𝑖=2(𝑒𝑖 −𝑒𝑖−1 )2 Depends on
Watson approximately 2. Test whether DW thus rk = 0 for k = 1, 2, 𝐷𝑊 = properties of
∑𝑛𝑖=1 𝑒𝑖2
is significant different from 2 to .. regressors
show serial correlation. ≈ 2(1 − 𝑟1 )
, Assumptions tests Econometrics 2 (Part I of course)
∑𝑛𝑖=𝑘+1 𝑒𝑖 𝑒𝑖−𝑘 DW≈2 0≤DW≤4
𝑟𝑘 = Box-Pierce We use the autocorrelation of the No serial correlation, H0 not true 𝑝
χ2(p)
∑𝑛𝑖=1 𝑒𝑖2
residuals 𝑟𝑘 thus rk = 0 for k = 1, 2, 𝐵𝑃 = 𝑛 ∑ 𝑟𝑘2
.. 𝑘=1
𝑝
Ljung-Box Same as Box-Pierce but test No serial correlation, H0 not true 𝑛+2 2 χ2(p)
statistic includes a weight for thus rk = 0 for k = 1, 2, 𝐿𝐵 = 𝑛 ∑ 𝑟
higher order autocorrelations .. 𝑛−𝑘 𝑘
𝑘=1
Breusch- LM-test, use model No serial correlation, H0 not true LM = nR2 χ2(p)
Godfrey 𝑦𝑖 = 𝑥𝑖′ 𝛽 + 𝜀𝑖 𝛾1 = ⋯ = 𝛾𝑝 = 0
𝜀𝑖 = 𝛾1 𝜀𝑖−1 + ⋯ + 𝛾𝑝 𝜀𝑖−𝑝 + 𝜂𝑖
Estimate restricted model, which is
𝑦𝑖 = 𝑥𝑖′ 𝛽 + 𝜂𝑖
Then perform an auxiliary
regression of these residuals:
𝑒𝑖 = 𝛾1 𝑒𝑖−1 + ⋯ + 𝛾𝑝 𝑒𝑖−𝑝 + 𝑥𝑖′ 𝛿 + 𝜔𝑖
➔ Most generally applicable,
best suited to test serial
correlation
Error terms not Jarque-Bera Testing whether its kurtosis and Normality : H0 not true 𝐽𝐵 χ2(2)
normally skewness is different from the 𝑛 2
√ (𝐾 − 3) ~ 𝑁(0,1) 𝑛
distributed normal distribution. 24 = [√ (𝐾 − 3)] (sum of 2 normally
24
Normal: K ≈ 3 and S ≈ 0 𝑛 2 squared)
√ 𝑆 ~ 𝑁(0,1) 𝑛
6 + [√ 𝑆]
6
Endogeneity
• X&ε Comparison Test whether bOLS significantly Exogeneity, so bOLS = H0 not true (bIV - bOLS )’[ var(bIV) – χ2(k)
correlated test OLS differs from bIV, considering bOLS var(bOLS)]-1 (bIV - bOLS)
and IV d = bIV - bOLS E[d] = 0
Which should be approximately Vard(d) = Var(bIV) –
zero by H0 Var(bOLS)
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