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  1. 1. How do you write the SETAR model if the innovation is the same across regimes? $Cue$: Dummy variable
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  3. 2. Do we observe the regime-switch in a Markov-Regime-Switching model?
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  5. 3. In the density (1.14) normally distributed? Should you rewrite it? If yes, how? Cue : Use the probability laws that you know
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  7. 4. What is $P(S_{t}=j|\Omega_{t-1}, \theta)$ ?
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  9. 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?) ?
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  11. 6. What is $P(S_{t-1}=i|\Omega_{t-1}, \theta)$ ?
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  13. 7. How do you rewrite the transition probability $p_{ij}$ ?
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  15. 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.
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  17. 9. Express the predicted probability in terms of the filtered probability and the transition probability? Why is it obtained iteratively?
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  19. 10. What's the filtered probability in plain english? Highlight the difference between the filtered and the transition probability?
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  21. 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 $B = S_{t-1}$
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  23. 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.
  24. Last step is to compute the likelihood function.
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  26. 13. Definition of ergodicity?
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  28. 14. What's the procedure with the unconditional probabilities (p.13)?
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  30. *CHAPTER 1 COMPLETED*
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  32. 15. What is stationarity?
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  34. 16. What is a deterministic trend? Cue: deterministic function of time
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  36. 17. Specify a stochastic process with a deterministic trend. Which part is stationary? Which part isn't?
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  38. 18. What is $E[y_{t}]$
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  40. 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?
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  42. 20. What is a stochastic trend?
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  44. 21. What is random walk in plain english?
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  46. 22. Specify the randow walk?
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  48. 23. What is the effect of the innovation $\varepsilon_t$ on the variable of interest?
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  50. 24. Peform the recursive substitution? What is $y_0$? Explain in plain english? What is value of $\phi$?
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  52. 25. Calculate the unconditional mean of the $y_t$? Cue: Use the moving average representation with T lags?
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  54. 26. Calculate the unconditional variance of $y_t$? Why does $t$ pop? Cue: Cross products, try to highligh them?
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  56. 27. What is a random walk with a drift?
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  58. 28. Perform the recursive substitution? Show the deterministic trend and the stochastic trend. Cue: you should get a $\delta$
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  60. 29. How do I remove the stochastic trend?
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  62. 30. Can you perform a first difference to get rid of a deterministic trend? Justify. Cue: Invertibility
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  64. 31. Can you run an OLS to get rid of a stochastic trend? Justify. Cue: You don't get rid of the stochastic trend.
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  66. 32. What does the lap operator do? What are some particularities?
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  68. 33. Where does the term unit root come form? Cue: Lag polynomial
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  70. 34. What condition do we need for stationarity for the roots? Cue: the roots are not the $\phi$?
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  72. 35. How do you compute the eigenvalues?
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  74. 36. How do you compute the roots of the lag characteristic equation? How do you relate it to the eigenvalues method? Cue: $\frac{\lambda}{x}$
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  76. 37. What should you do if you have more than one units?
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  78. 38. Perform example p.20. Cue: $x_1 = 2$ and $x_2 = 1$
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  80. 39. Show that the process for example p. 20 is stationary after performing $\Delta^1$
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  82. 40. Suppose $\phi_1 = 2$ and $\phi_2 = -1$. Compute the roots. Cue: $x_1 = 1$ and $x_2 = 1$
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  84. 41. Show whether the process is stationary after performing $\Delta^1$. What about $\Delta^2$?
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  86. 42. What is an ARIMA(p,d,q) process?
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  88. 43. Specify ARIMA(p,d,q).
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  90. 44. What does the expression $y_t \sim I(d)$ mean?
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  92. 45. Decompose the ARIMA model in plain english? In math with a stationary variable called $z_t$?
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