Title | Var uc davis 4 - Grade: A++ |
---|---|
Author | 혜정 서 |
Course | Introduction to Econometrics |
Institution | Universiti Selangor |
Pages | 14 |
File Size | 288.7 KB |
File Type | |
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var uc davis problems set econometrics...
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
PROBLEM SET 4 – SOLUTIONS Part I – Analytical Questions Problem 1: Consider a stationary autoregressive process A(L)Xt = t and its corresponding moving average representation, Xt = C(L) t , where C( L) i 0 Ci Li . (a) Find the moving average coefficients for an VAR(1) process. Solution Because this is a VAR(1), calculation of the MA representation is quite easy. Thus, if Xt = A1Xt-1 + t, then Ci = A1i. (b) Show that the moving average coefficients for a VAR(2) can be found recursively by C 0 I ; C 1 A1; and Ci A1Ci 1 A2 Ci 2 for i 2,... Solution
A stationary VAR has a moving average representation given by X t i 0 Ci t i . Plugging this formula into that of a VAR(2) such as, X t A1 X t 1 A2 X t 2 t , we find,
i0
C i t i A1 i C i t i 1 A2 i Ci t i 2 t , 0 0
which can be rewritten as, 0 ( C0 I) t ( C1 A1C0 ) t 1 ( C2 A1C1 A2 C0 ) t 2 ... ,
which delivers the coefficients for each of the epsilon. Since the epsilon can take on any value in p , each of these coefficients must equal zero. Hence, C0 = I, C1 = A1, and C t A1Ct 1 A2 Ct 2
t 2,3,...
1
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
Problem 2: Consider the following bivariate VAR, y t yy y t 1 ymm t 1 u yt mt my yt 1 mm mt 1 umt
2y ( ' ) with E ut ut . 2 m (a) Find a matrix H, which is lower triangular and ensures that if Hut t , then E ( t t ' ) D where D is a diagonal matrix. Solution
1 For example, H 2y
0 1
(b) Given this matrix H calculate the structural representation of this VAR. Solution yt mt
y t ( my y2
yy y t 1 ym m t 1 yt 2 yy ) yt 1 ( mm 2 ym )m t 1 mt y y
(c) Calculate the VMA representation for the reduced form of this VAR (notice that it is very simple in this case – don’t apply the usual formulas mechanically!) Solution yt u yt yy mt umt my
ym u yt 1 ym ... yy mm u mt 1 mm my
k
u yt k ... u mt k
2
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
(d) Calculate the VMA representation of the structural form of the VAR. Solution
1 0 1 0 u yt y t yy ym u yt 1 ... 1 u 2 1 my mm umt 1 mt mt y2 y k 0 1 yy ym u yt k ... 1 y2 my mm u mt k (e) Under what conditions will the reduced form and the structural form produce identical impulse response functions? Solution: The obvious one is = 0. Less obvious, yy ym 0. (f) Suppose you obtained the structural form as in part (a) but for a system that had the variable m ordered first. Under what conditions would these two structural identification schemes deliver the same impulse responses? Solution: Notice that the matrix H is in this case,
0 1 H 1 2 m Naturally, = 0, would work, but also, either yy ym 0 or my mm 0. Problem 3: Consider the following bivariate VAR y 1t 0.3y 1t 1 0.8 y 2t 1 1t y 2t 0.9 y 1t 1 0.4 y 2t 1 2t
with E ( 1 t 1 ) 1 for t = and 0 other wise, E ( 2 t 2 ) 2 for t = and 0 other wise, and E ( 1t 2 ) 0 for all t, and . Answer the following questions:
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ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
(a) Is this system covariance-stationary? Solution To answer this question, verify the roots of the polynomial 1 0
0 1
0 .3 0. 9
0.8 z (1 0.3 z)(1 0.4 z ) ( 0.8 z )( 0.9 z ) 0 .4
1 0. 7z 0 .6z 2
The roots are 0.833 and 2, hence the system is not-stationary.
(b) Calculate s
yt s for s = 0, 1, and 2. What is the limit as s ? t '
Solution 1 0
0
Clearly,
0.3
since s 0.9
0 0.3 ; 1 1 0.9 0.8 0.4
0.8 0. 81 ; 2 0.4 0.63
0. 56 0. 88
s
and the process is not stationary, then s .
(c) Calculate the fraction of the MSE of the two period-ahead forecast error for 2 ˆ ( y t | yt , y t ,...) , that is due to 1,t 1 and 1, t 2 variable 1, E y1,t 2 E 1, 2 1
Solution
ˆ ( y1, t 2 | yt, yt 1,...) E y1, t 2 E
2
E 1, t 2 0.3 1, t 1 0.8 2, t 1
2
1 0.3 2 0.8 2 2 2.37
The fraction due to 1 is (1 + 0.32)/2.37 = 0.46 or 46%. Problem 4: Consider the process yt zt yt 1 1t | | 1 zt zt 1 2 t | | 1
12 0 t ~ N ; 11 0 12 22
(a) Derive E ( x t | xt 1 ); V ( xt | x t 1 ) and D( x t | xt 1 ) where D denotes the density function and xt ' ( y t z t ). Hint: the system can be rewritten in matrix form as
4
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
1 yt 0 yt 1 1t 0 1 z t 0 zt 1 2t
Solution 1 , the inverse of the contemporaneous 0 1
Pre-multiply the system by A = correlation matrix, to obtain,
yt y t 1 u1 t ; z t 0 z t 1 u2 t
u1 t 1 1 t u2 t 0 1 2 t
Thus, y t 1 xt 1 ; E ( xt | xt 1 ) 0 zt 1 E( ut ut ' ) E( At t ' A' ) 1 11 12 1 0 1 12 22
0 11 2 12 2 22 12 22 W . 1 12 22 22
Given this expressions for the conditional mean and the variance, and noting that the u’s are linear combinations of the and therefore are normally distributed, the conditional distribution D(xt|xt-1) is multivariate normal with conditional mean and variance given by the expressions derived above. (b) Assume that xt is stationary. Derive E ( x t ); V ( xt ) and show that V (x t ) V (x t | x t 1 ) is positive definite. What are the implications of this result? Solution By stationarity, E ( x t ) E ( x t 1 ) 0 and therefore, E ( x t ) 0 . Similarly, V ( xt ) V ( x t 1 ) ' W . Since in (a) we calculated that V (xt | x t 1 ) W , it follows that V ( x t ) V ( x t | x t 1 ) V ( xt ) ' which is a quadratic form and therefore positive definite for V ( xt ) 0 . The rational is that the conditional variance uses “more information” (hence the conditioning) than the unconditional variance.
5
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
Problem 5: Consider the Gaussian linear regression model, yt x t ' ut
with ut ~ i.i.d. N(0, 2) and ut independent of x for all t and . Define ( ' , 2 )'. The log of the likelihood of (y1, …, yT) conditional on (x1,…,xT) is given by L ( ) (T / 2) log(2 ) (T / 2 ) log( 2 )
T
(y
t
x t ' ) 2 /(2 2 )
t 1
(a) Show that the estimate ˆ ' g( ; YT ) D T '
T
ˆ T
(1 / T ) t 1
h ( , YT ) ' ˆ
T
(1 / T ) t 1
2 log f ( y t | Y t 1 ; ) '
T
1T xt xt ' / ˆT2 0 T t1 ˆ is given by Dˆ T ' 2 ’ where uˆ ( y x ' T 1 1 uˆt 0 4 6 T t 1 2ˆ T ˆT t
t
t
T
ˆ T
) and ˆT and
ˆ T2 denote the maximum likelihood estimates. Solution The proof is straightforward by direct differentiation of the likelihood and noticing T
that
uˆ
t
0 from the first order conditions.
t1
T
(b) Show that the estimate SˆT (1 / T) h( ˆ , Yt ) h( ˆ, Yt ) ' is given by t 1
1 T 2 1 T uˆt3 x t 4 6 uˆt x t x t ' / ˆT T t 1 2 ˆT ˆS T t1 2 . T 3 2 T T 1 uˆ x 1 uˆ 1 t 6t t 4 2 T t1 2ˆ T T t1 2ˆT 2 ˆT
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ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
Solution: this proof is also straightforward once you realize h( ˆ, Yt ) is the score of the log-likelihood evaluated at the MLE estimates.
7
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
Q / 2 0 for 1 /( 2 4 ) 0
(c) Show that the p lim( SˆT ) p lim(Dˆ T ) where Q p lim(1 / T ) t1 x tx t'. T
Solution: The proof requires the following intermediate results: p lim p lim
1 T
T
1 T
T t 1
uˆ 2t 2 ;
uˆt 0 . Direct application of conventional asymptotic results delivers
t 1
the desired result. (d) Consider a set of m linear restrictions on of the form R = r for R a known ˆ Dˆ , the ( m k ) matrix and r a known ( m 1) vector. Show that for T T Wald test statistic given by
g ( ) T g (ˆT ) ' '
ˆT
1 g ( ) ˆT '
1
ˆ T
'
g (ˆ ) T
is identical to the Wald test form of the OLS 2 test given by 1 (R ˆ r )' s 2 R (x ' x ) 1 R ' (R ˆ r ) with the OLS estimate of the variance T
T
T
T
T
2 T
sT replaced with the MLE ˆ . (e) Show that when the lower left and upper right blocks of Sˆ T are set to their plim of zero, then the quasi-maximum likelihood Wald test of R = r is identical to the heteroskedasticity-consistent form of the OLS 2 test given by ˆ 1 ˆ Q ˆ 1 / T ) R' 1 ( R ˆ r) (R ˆ r )' R (Q 2
T
T
T
T
T
Problem 6: Consider the following DGP for the cointegrated random variables z and y y 1 0.8 L (1 L ) t (1 0.4 L) 1 zt 0.1L
0.8 L 1 t 1 0.6L 2t
where ~ N(0, I) with z0 = y0 = 0. (a) Obtain the autoregressive representation of this DGP. (b) Obtain the error-correction representation of this DGP. (c) Deduce the long-run relation between z and y. (a) Directly inverting the lagged matrices on the right-hand side, we get
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ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
1 0.6L 0.8L y t 1t 0.1L 1 0.8L z t 2t
(b) From the autoregressive representation 0.8 L 0.8 L 1 L 0.4 L 1 0.6 L 1 L 0.2 L 0.1L 0.1L 1 0.8L 0 0.4 L 0.8 L 0.4 1 L 1 2 L I 1 L 0.1L 0.2 L 0.1 0 Hence (0.4
0.1)' ; (1
2 )'
(c) From the ECM, the long run solution is y = 2z. Problem 7: Consider the following DGP xt yt u1t ; where u1t u1t 1 1t xt yt u 2t ; where u 2t u2t 1 2t
with | | < 1, and
0 2 1t ~ D , 1 2 0 t 2 2 where D denotes a generic distribution. (a) Derive the degree of integratedness of the two series, xt and yt. Do your results depend on any restrictions on the values of , , and ? Discuss how. If θ = 1, then xt and yt are I(1) since x t y t u1t ~ I (1) . In addition, given | | < 1 one needs to impose the condition . The same linear combination of xt and yt cannot be simulatenously I(0) and I(1). (b) Under what coefficient restrictions are xt and yt cointegrated? What are the cointegrating vectors in such cases? 1; 1 . Cointegrating vector (1 ).
(c) Choose a particular set of coefficients that ensures xt and yt are cointegrated and derive the following representations: I. The moving-average. II. The autoregressive. III. The error-correction.
9
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
I. MA 1 1 x t 1 L 1t xt 1 ; 1 y t 1 2t y t 1 1 L Note : 1
1 1 Hence :
1
1
1 1 1 L 1t 1 1 L 2 t
1
i xt i 0 1t i i 0 2t i i y t i 0 1t i i 0 2t i Aside : (1 ) is the cointegrating vector, while (1 ) is a serial correlatio n feature
II. AR
x t y t 1t 1 L x t y t 2 t 1 L xt 1 yt 1
xt 1 1t 1 x t 1 1 y t 1 ( 1) y t 1 2t 1 x t 1 1 1 1 ( 1) y t 1 1
1t 1 2t
III. ECM Define the error correction term zt 1 xt 1 y t 1 , then ( 1) zt 1 u1t (1 ) yt zt 1 u2t with xt
1 ( 1t 2t ) 1 ( 1t 2t )
u1t u2t
(d) Can all the cointegrated systems be represented as an error-correction model? What are the problem/s of analyzing a VAR in the differences when the system is cointegrated?
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ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
From the Granger representation theorem we know the answer is yes. Analyzing the VAR in the differences omits the error correction term in the specification. Therefore we have the classical problems of omitted variable bias. (e) Suppose that economic theory suggests that xt and yt should be cointegrated with cointegrating vector [1 + 0.5t]. Describe: I. How would you test whether this is indeed a cointegrating vector? Run the OLS regression x t ( 0. 5 ) y t v t t
and test the residuals with an ADF test. II. What is the likely outcome of the test in short samples? Why? In short-samples, the cointegrating vector (1 +0.5t) will differ from the cointegrating vector (1 ). However, as the sample size gets larger, note that the bias 0.5t disappears very quickly. III. What is the likely outcome of the test asymptotically? Why? Asymptotically the bias disappears sufficiently quickly. Problem 8: Consider the bivariate VECM yt c ' yt 1 t
iid
it ~ ( 0, 2 )
where (1 ,0)' and (1, 2 )'. Equation by equation, the system is given by y 1t c 1 1 (y 1t 1 2 y 2t 1 ) 1t y2t c2 2 t
Answer the following questions: (a) From the VECM representation above, derive the VECM representation yt c y t 1 t
and the VAR(1) representation y t c Ay t 1 t
1 2 1 1 2 1 ; A 1 1 0 0 0
11
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
(b) Based on the given values of the elements in and , determine , , such that ' 0 and '
0.
0 k ; 2 ; k 0 k k
(c) Using the Granger representation theorem determine that (1) ( ' I 2 ' ) 1 , where ( L) is the moving average polynomial corresponding to the VECM system above and I2 is the identity matrix of order 2. Hint: you may show this result by showing that (1) is orthogonal to the cointegrating space. Using the hint: (1)’ = 0. It is easy to show that 0 2 ( ' I2 ' ) 1 0 1
and therefore 1 0
1 2 0 0 0
2 0 1
(d) Using the Beveridge-Nelson decomposition and the result in (c), determine the common trend in the VECM system. All you need to remember is that from the B-N decomposition, the trends are the linear combinations captured in (1)yt, which in this case turns out to be 2y1t + y2t. Notice that this combination is orthogonal to the cointegrating vector. (e) Show that ' y t follows an AR(1) process and show that this AR(1) is stable provided that 2 1 0 . What can you say about the system when 1 = 0? Let z t y1t 2 y2t be the cointegrating vector. From the equations for y1 and y2 we have z t c 1 ( 1 1)y 1t 1 1 2 y 2t 1 1t 2c 2 2 y 2t 1 2 2t
Combining terms z t (c 1 2c 2 ) ( 1 1)zt 1 v t ; vt 1t 2 2t
which is an AR(1) whose stationarity requires that | 1 + 1| < 1 or the equivalent condition -2 < 1 < 0. When 1 = 0, zt is no longer stationary, so there is no cointegration for any value of 2. y1 and y2 are in this case two independent random walks. 12
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
Problem 9: Consider the following VAR yt (1 ) yt 1 xt 1 1t xt yt 1 (1 ) xt 1 2 t
(a) Show that this VAR is not-stationary. Stationarity requires that the values of z satisfying 1 0
0 1 1
z 0 (1 )
lie outside the unit circle. For z = 1, notice
0
(b) Find the cointegrating vector and derive the VECM representation. Notice that (1)
(
)' (1
)
so that yt ( yt 1 xt 1 ) 1t xt ( yt 1 xt 1 ) 2 t
(c) Transform the model so that it involves the error correction term (call it z) and a difference stationary variable (call it wt). w will be a linear combination of x and y but should not contain z. Hint: the weights in this linear combination will be related the coefficients of the error correction terms. Given the ECM in part (b), notice yt xt zt 1 1t zt 1 2 t wt y t x t 1t 2 t
Next
13
ARE/ECN 240C Time Series Analysis Winter 2004
Professor Òscar Jordà Economics, U.C. Davis
yt y t 1 ( yt 1 xt 1 ) 1t xt xt 1 ( yt 1 xt 1 ) 2 t ( y t x t ) ( y t 1 x t 1 ) z t 1 ...