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1.4) Answer the following questions about the Granger-causality test between two variables, x, and y,. If the two variables are stationary, we can use the following VAR model where the lag order is 3. x, = 0, +2,1*-1 +0,2%,-2 +0,3*,-3 +0,19:- +0,2.Y-2 +a, 39-3 + u,, y; = 3, + B12- + 8x2*-2 + 3x3*7-3 +3 17:24 + 3,29-2 +373-3 + 42, (a) Write the null hypothesis that x does not cause y. (b) Write the null hypothesis that y does not cause x. Suppose that the two variables are integrated of order one, 7(1), and cointegrated. Using the cointegrating relation of z, = y, - 1x, (4>0), we can modify the above VAR model into the following vector error correction model (VECM). Ax, = 4x4x,-1+%24x,-2 + 7y4y1 + 7,2Ay-2 +0,2- + Eu Ay, = & Ax__+$24.x,-2 +8,Ay-1 +8,24y-2 +0,2-1 + E21 (c) Write the null hypothesis that y does not cause x, both in the short and long run. (d) It is expected that the two series will move toward the long-run equilibrium. The VECM specifies that any deviation from the long-run equilibrium is partly corrected in the following period. If so, what do you expect the sign of e,? Explain. (e) What do you expect the sign of 0,? Suppose that the two variables are integrated of order one, I(1), but not cointegrated. (f) In order to test the Granger-causality, what form of VAR model can we use?