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S
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UN
E R SIT
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Universität des Saarlandes
Statistik und Ökonometrie
PD Dr. Stefan Klößner
IV
A VIE N
3rd Tutorial to
Econometrics II: Time Series Analysis SS 2017
Exercise 14 [2.5%]
For random vectors X and Y of dimension n and m, resp., and a univariate random
variable Z, show that
(a) Cov(a0 X, Z) = a0 Cov(X, Z) (a ∈ Rn ),
(b) Cov(Z, b0 Y ) = Cov(Z, Y )b (b ∈ Rm ),
(c) Cov(X, Y ) = Cov(Y, X)0 .
Exercise 15 [2.5%]
For a random variable W with positive variance and some other random variable Y ,
show that:
(a) P (Y |W ) = βb0 + βb1 W , with βb1 =
(b) E(Y − Yb )2 = Var(Y ) −
Cov(W,Y )2
Var(W )
Cov(W,Y )
Var(W )
and βb0 = E(Y ) − βb1 E(W ).
= Var(Y ) − βb1 Cov(W, Y ).
Exercise 16 [5%]
For a white noise ε and |θ| < 1, we consider the MA(1)-Prozess Xt = εt + θεt−1 .
Derive the following formulas:
(a) P (X3 |X2 , X1 ) =
θ(1+θ2 )
X
1+θ2 +θ4 2
−
θ2
X,
1+θ2 +θ4 1
(b) E(X3 − P (X3 |X2 , X1 ))2 = σε2 (1 +
θ6
),
1+θ2 +θ4
(c) P (X2+h |X2 , X1 ) = 0 for all h > 2.
Show the inequality
E(X3 − E(X3 |X2 , X1 , . . .))2 < E(X3 − P (X3 |X2 , X1 ))2 < E(X3 − P (X3 |X2 ))2
and give an interpretation.
Exercise 17 [2.5%]
For an independent white noise ε and a related random walk (Xt )t∈N0 with drift α0
and initial value x ∈ R, show for t ≥ 1:
(a) P (Xt |Xt−1 ) = Xt−1 + α0 ,
(b) E(Xt − P (Xt |Xt−1 ))2 = σε2 .
Exercise 18 [2.5%]
For a weakly stationary process X with mean µ, autocovariance function γX and
ACF ρX , derive the following formulas:
(a) P (X2 |X1 ) = µ + ρX (1)(X1 − µ),
(b) E(X2 − P (X2 |X1 ))2 = γX (0)(1 − ρX (1)2 ).
Exercise 19 [2.5%]
For a Gaussian white noise ε and |φ| < 1, we consider the AR(1) process Xt =
φXt−1 + εt .
(a) Use the preceding exercise to show that P (X2 |X1 ) = φX1 .
(b) Compare P (X2 |X1 ) to E(X2 |X1 , X0 , . . .). What can you conclude for
P (X2 |X1 , X0 ), P (X2 |X1 , X0 , X−1 ) etc.?
Exercise 20 [5%]
Show that the partial correlation of X1 and X2 given X3 can be written as
Corr(X1 , X2 ) − Corr(X1 , X3 ) Corr(X2 , X3 )
p
.
(1 − Corr(X1 , X3 )2 )(1 − Corr(X2 , X3 )2 )
Hint: use the properties of the forecast operator P (·|X3 ).
Exercise 21 [2.5%]
For a Gaussian white noise ε and |φ| < 1, we consider the AR(1) process Xt =
φXt−1 + εt . Determine the partial correlation between X0 and X50 given X30 .
Exercise 22 [2.5%]
For a Gaussian white noise ε and |φ| < 1, we consider the AR(1) process Xt =
φXt−1 + εt . Use the preceding exercises to show that
(a) P (X0 |X1 ) = φX1 ,
(b) P (X2 |X1 ) = φX1 ,
(c) Corr(X0 , X2 |X1 ) = 0.