2012 CT6 Sept Q9

Discussion in 'CS2' started by Laura, Apr 10, 2022.

  1. Laura

    Laura Very Active Member

    Hi all,

    Could I check in part Ii, after obtaining the values of X and Y, why do we say that alpha and beta are roots of the quadratic eqn? Where do we get this quadratic eqn from?

    Also, in part III, where does the eqn y_101=0.0625y_100+0.6875y_99 come from?

    Thanks in advance for your help!
     
    Last edited: Apr 10, 2022
    Laura, Apr 10, 2022
    #1
  2. Andrew Martin

    Andrew Martin ActEd Tutor Staff Member

    Hello

    (ii)
    For a quadratic to have roots a and b, it must be able to be expressed as:

    \( f(x) = (x-a)(x-b) = x^2 - ax - bx + ab = x^2 - (a+b)x + ab \)

    we know that a + b = X = 0.0625 and ab = Y = -0.6875, so we have:

    \( f(x) = x^2 - 0.0625x - 0.6875. \)

    (iii)
    We have:

    \( (1-B^3)(1 - (a + b)B +abB^2)X_t = e_t \)

    Also, we know:

    \( Y_t = X_t - X_{t-3} = (1 - B^3)X_t \)

    Subbing into the equation above, this gives:

    \( (1 - (a + b)B +abB^2)Y_t = et \)

    Or:

    \( Y_t = (a+b)Y_{t-1} - abY_{t-2} + e_t \)

    Hope this helps!

    Andy
     
    Andrew Martin, Apr 11, 2022
    #2

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