AR(p)

Question 1

What is the Wald Test?

Answer 1

W=((Rβˆ_1)′[R(X′_1X_1)^(-1)R′]^(-1)(Rβˆ_1))/σ2_1a∼χ2(k)

Question 2

What is the big difference between the ML ratio test and the Wald test?

Answer 2

The ML test has to estimate the restricted and unrestricted regressions, so H_0 and H_1, and test with the restrictions. The Wald Test only estimates the unrestricted one and Test in the restrictions.

Question 3

What is the Maximum Likelihood ratio test?

Answer 3

LR= 2(logLˆ1−logLˆ0) = (T−p)(log ˆσ2_0−log ˆσ2_1)∼χ2(k)

With

Question 4

What is logLˆ_0?

Answer 4

−((T−p)/2)log(2π)−((T−p)/2)log(ˆσ^2)−(1/2)(Y-X_0βˆ_0)′(Y−X_0βˆ_0)/σˆ^2

Question 5

What is the main difference between Akaike and Bayesian information criteria?

Answer 5

The Bayesian Criteria penalizes more the the models with more coefficients, as they can be overspecified.

Question 6

What is the penalty of having more parameters in the AIC and in the BIC?

Answer 6

AIC: 2(p+1)
BIC: log(T)(p=1)

Question 7

What are the equations for AIC and BIC? How can we tell from AIC/BIC which model is the best?

Answer 7

AIC= -2log(L) + 2(p+1)
BIC= -2log(L) + log(T)(p+1)

We are looking for the model that minimizes AIC / BIC.

[-2*log(L)] captures the fit of the model on the data.

Question 8

What is the joint density function applying the Bayesian theorem?

Answer 8

f(yT, . . . , y2, y1|δ, φ1, φ2, σ2) =(PI)^YT_t=3[f(yt|yt−1,yt−2,δ,φ,σ2)]*f(y2, y1|δ, φ, σ2)

Question 9

What is McLeod’s test statistic? What’s the decision rule?

Answer 9

T(T+ 2)SUM_i=1^k(Corrhat[u^2_t - u^2_t−i]2)/(T−i)

If this statistic is smaller (larger) than the critical value associated with the χ2(k) distribution, then there is conditional homoscedasticity (heteroscedasticity)

Question 10

What are the Mcleod’s test hypothesis?

Answer 10

H_0:corr(u^2_t, u^2_t−1) =· · ·=corr(u^2_t, u^2_t−k) = 0
H_1:corr(u^2_t, u^2_t−1)=/=· · · =/=corr(u^2_t, u^2_t−k)=/= 0

Question 11

Quelle est la procédure du test McLeod?

Answer 11

La procédure est la suivante:

1) Estimer par MCO le AR(1) et calculer le carré des résidus,

2)Calculer l’autocorrélation du carré des résidus à partir de l’estimateur d’échantillon,

3) Calculer l’analogue de la statistique Q de Ljung-Box à partir de l’estimation des autocorrélations du carré des résidus

4) Confronter cette statistique à la valeur critique de la distribution χ2(k)