Formulas
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OLS estimator in Matrix Format: 𝛽 = (𝑋′𝑋)−1 𝑋′𝑦.
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t test statistics=𝑆𝐸(𝛽̂)
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(1-significance level) % confidence interval:
̂ −𝛽 ∗
𝛽
[𝛽̂ − 𝑡𝑐 ∗ 𝑆𝐸(𝛽̂ ),
𝛽̂ + 𝑡𝑐 ∗ 𝑆𝐸(𝛽̂ ) ]
where tc is the critical value of a given significance level (10%, 5%, 1%) for a t
distribution or a standardized normal distribution.
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The non-rejection region for the partial autocorrelation coefficient 𝜃̂𝑝 obtained from an
AR(p) model is:
[−𝑡𝑐 /√𝑇,
𝑡𝑐 /√𝑇 ].
̂
𝜓
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Dicky Fuller test statistics= 𝑆𝐸(𝜓̂)
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The Likelihood ratio (LR) test statistics is: 𝐿𝑅 = −2(𝑙𝑛𝐿∗ − 𝑙𝑛𝐿)
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The Wald test statistics is: 𝐽0 = 𝛼̂ ′ [𝑉𝑎𝑟(𝛼̂)]−1 𝛼̂
•
1
For event studies, the test statistics=𝑆𝐷(𝐶𝐴𝑅
̅̅̅̅̅̅ (𝑠
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The critical values for a standardized normal distribution are ±1,65, ±1,96, ±2,58 at
̅̅̅̅̅̅
𝐶𝐴𝑅 (𝑠 ,𝑠2 )
1 ,𝑠2 ))
10%, 5%, 1% significance level, respectively.
Tables
ADF regression models with one lag for ∆𝑦𝑡 :
Model i: ∆𝑦𝑡 = 𝜓𝑦𝑡−1 + 𝜃1 ∆𝑦𝑡−1 + 𝑢𝑡
Model ii: ∆𝑦𝑡 = 𝜇 + 𝜓𝑦𝑡−1 + 𝜃1 ∆𝑦𝑡−1 + 𝑢𝑡
Model iii: ∆𝑦𝑡 = 𝜇 + 𝜓𝑦𝑡−1 + 𝜃1 ∆𝑦𝑡−1 + 𝜆𝑡 + 𝑢𝑡
The table for Critical Values for the DF or ADF test is:
1