Mimi, an ECMT3150 student, studies the following MA(1) process y t
​
=ε t
​
+0.9ε t−1
​
, where ε t
​
∼ iid N(0,0.09) (normal distribution with mean 0 and variance 0.09 ). (a) [3 marks] Is {y t
​
} a martingale difference sequence? Justify your answer with a proof. (b) [3 marks] Is {y t
​
} stationary? Why or why not? (c) [3 marks] Is {y t
​
} invertible? Why or why not? (d) [3 marks] Compute the unconditional mean and variance of {y t
​
}. (e) [4 marks] Derive the autocorrelation function (ACF) of {y t
​
}. (f) [4 marks] Plot the ACF and partial autocorrelation function (PACF) of {y t
​
}. (g) [4 marks] Derive the AR representation of {y t
​
}. Show your steps. (h) Little Bob studies the following AR(1) model instead: z t
​
=0.9z t−1
​
+ε t
​
, where ε t
​
∼ iid N(0,0.09). (i) [2 marks] Plot the ACF and PACF of {z t
​
}. (ii) [4 marks] Compare and discuss how a negative shock today will have an impact on the future values of y t
​
and z t
​
.