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- 1. How do you write the SETAR model if the innovation is the same across regimes? $Cue$: Dummy variable
- 2. Do we observe the regime-switch in a Markov-Regime-Switching model?
- 3. In the density (1.14) normally distributed? Should you rewrite it? If yes, how? Cue : Use the probability laws that you know
- 4. What is $P(S_{t}=j|\Omega_{t-1}, \theta)$ ?
- 5. What do we mean by conditioning on a regime? What does it imply for the random variable $Y$ (how is it distributed? What is the mean? What is the variance?) ?
- 6. What is $P(S_{t-1}=i|\Omega_{t-1}, \theta)$ ?
- 7. How do you rewrite the transition probability $p_{ij}$ ?
- 8. Why do we breakdown the predicted probability for the likelhood function? Do you need the predicted probability for each iteration (conditional likelihood numerical optimization? Why? Cue: You still don't really know how the predicted probability behaves.
- 9. Express the predicted probability in terms of the filtered probability and the transition probability? Why is it obtained iteratively?
- 10. What's the filtered probability in plain english? Highlight the difference between the filtered and the transition probability?
- 11. How do you obtain $p_{i,t}^{filt}$ using Bayes' Theorem? Cue: It's a two step process, use conditioning formula. Break down the "events" (i.e. what is event $A$, $B$ in plain english? What is $P(A)$? What is $P(B)$? Also note that $A = y_{t}$ and $B = S_{t-1}$
- 12. What is the Hamilton filter? What does it allow us to do? Where is the loop (big picture)? Cue: getting to the likelihood function. You have 5 equations, a, b, c, d steps to estimate the likelihood function.
- Last step is to compute the likelihood function.
- 13. Definition of ergodicity?
- 14. What's the procedure with the unconditional probabilities (p.13)?
- *CHAPTER 1 COMPLETED*
- 15. What is stationarity?
- 16. What is a deterministic trend? Cue: deterministic function of time
- 17. Specify a stochastic process with a deterministic trend. Which part is stationary? Which part isn't?
- 18. What is $E[y_{t}]$
- 19. How would you run an OLS regression? Cue : $\hat{\delta}, \hat{\alpha}, \hat{\varepsilon}$ How do you obtain specifically $\hat{\varepsilon}$? What's the purpose of running this OLS regression?
- 20. What is a stochastic trend?
- 21. What is random walk in plain english?
- 22. Specify the randow walk?
- 23. What is the effect of the innovation $\varepsilon_t$ on the variable of interest?
- 24. Peform the recursive substitution? What is $y_0$? Explain in plain english? What is value of $\phi$?
- 25. Calculate the unconditional mean of the $y_t$? Cue: Use the moving average representation with T lags?
- 26. Calculate the unconditional variance of $y_t$? Why does $t$ pop? Cue: Cross products, try to highligh them?
- 27. What is a random walk with a drift?
- 28. Perform the recursive substitution? Show the deterministic trend and the stochastic trend. Cue: you should get a $\delta$
- 29. How do I remove the stochastic trend?
- 30. Can you perform a first difference to get rid of a deterministic trend? Justify. Cue: Invertibility
- 31. Can you run an OLS to get rid of a stochastic trend? Justify. Cue: You don't get rid of the stochastic trend.
- 32. What does the lap operator do? What are some particularities?
- 33. Where does the term unit root come form? Cue: Lag polynomial
- 34. What condition do we need for stationarity for the roots? Cue: the roots are not the $\phi$?
- 35. How do you compute the eigenvalues?
- 36. How do you compute the roots of the lag characteristic equation? How do you relate it to the eigenvalues method? Cue: $\frac{\lambda}{x}$
- 37. What should you do if you have more than one units?
- 38. Perform example p.20. Cue: $x_1 = 2$ and $x_2 = 1$
- 39. Show that the process for example p. 20 is stationary after performing $\Delta^1$
- 40. Suppose $\phi_1 = 2$ and $\phi_2 = -1$. Compute the roots. Cue: $x_1 = 1$ and $x_2 = 1$
- 41. Show whether the process is stationary after performing $\Delta^1$. What about $\Delta^2$?
- 42. What is an ARIMA(p,d,q) process?
- 43. Specify ARIMA(p,d,q).
- 44. What does the expression $y_t \sim I(d)$ mean?
- 45. Decompose the ARIMA model in plain english? In math with a stationary variable called $z_t$?
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