Final Exam, due on April 20th at 23:59pm
Question 1
Assume we have the following bi-variate VAR(1) process
We are interesting in understanding the relationship between the growth rate of GDP of the U.K.,
Canada and the United U.S.. To do so, we model these three variables as a Vector Autoregression
process VAR(p), which with generic rerpresentation
# " #
#"
# "
"
#" # "
εyt
γ11,1 γ12,1 yt−1
γ1,0
1 b12 yt
+
+
=
εzt
γ21,1 γ22,1 zt−1
γ2,0
b21 1
zt
(1)
or, in matrix form,
Bxt = Γ0 + Γ1 xt−1 + εt
(2)
with εt being a white noise process.
Question 1.1: Explain why equation (2) is a structural VAR(1) model.
Question 1.2: Explain what it means for εt to be a white noise process?
Call
"
Σε =
σy
σy,z
σz,y
σz
#
(3)
the covariance matrix of the structural shocks ε.
Question 1.3: Given what you know of the model so far, and in particular about εyt
and εzt being structural shocks, what is the value of σy,z ? Is it any different than the
value of σz,y ? Explain why.
Question 1.4: Opening up any introductory textbook on matrix algebra, find the
algebraic representation of B −1 .
The reduced-form VAR associated with the structural representation given in (2) is given by
xt = A0 + A1 xt−1 + et
where A0 = B −1 Γ0 , A1 = B −1 Γ1 and et = B −1 εt and
" #
e1t
et =
e2t
1
(4)
Question 1.5: Using your results from 1d, find the algebraic representation for A0 , A1
and et .
Question 1.6: Using your results in 1e, write in matrix form the reduced-form representation of the VAR model.
Question 1.7: Using your result from 1e, we know that e1t = f (εyt , εzt ) and that e2t =
f (εyt , εzt ). Show explictly how e1t is related to εyt , εzt and how e2t is explicity related to
εyt , εzt .
Now, we call
"
Σe =
σ12
σ12
σ21
σ22
#
Question 1.8: Remembering that εt is a white noise process with Σε given in (3),
calculate explicitly E(e1t ), E(e2t ) and σ12 . What condition do you need for σ12 to be
equal to zero?
Assume now that we have estimated the system (4) by OLS and that we obtain the following
estimates
" #
yt
zt
=
" #
0
0
"
+
0.7 0.2
#"
#
yt−1
0.2 0.7
zt−1
"
+
e1t
#
e2t
(5)
with
Σe =
"
#
3.06 1.08
1.08 1.44
(6)
Question 1.9: Using a software of your choice (it is very easy with Excel!), trace the
reponse of yt and zt to a unit shock e11 = 1. Do the same for e21 = 1. For both cases,
trace the response of yt and zt for t = 1 to t = 10.
Assume now that the B matrix in (1) is given by
"
B=
#
1 −0.75
0
1
(7)
Question 1.10: Given the coefficient estimates obtained in (5) and (6) and your knowledge of the B matrix, recover the structural parameters Γ0 , Γ1 and Σε .
Question 1.11: Write the matrix representation of the structural model using the
numerical values obtained for Γ0 , Γ1 and Σε .
2
Question 1.12: Does yt responds contemporaneously to a structural shock εzt ? Does
zt responds contemporaneously to a structural shock εyt ?
QUESTION 2: ERROR-CORRECTION MODEL
Assume that the relationship between variables yt and xt is well represented by an autoregressivedistributed lag model ADL(1,1).
yt = α0 + α1 yt−1 + β0 xt + β1 xt−1 + εt
(8)
with εt is a white noise process with E(εt ) = 0 and V (εt ) = σε2 . This is an ADL(1,1) model because
there is 1 lag of yt and 1 lag of xt . Further, let’s assume that yt and xt are both I(1) variables.
Question 2.1: Given what we assume about εt , are the residuals from this equation
I(1) or I(0)?
Question 2.2: Provide a definition of cointegration between two I(1) variables.
Question 2.3: Given your answers in questions 2.1 and 2.2, does model (8) suggests
that variables yt and xt are cointegrated?
Cointegration between two I(1) variables imply that there is a stable long-run equilibrium
relationship between these variables. Specifically, that relationship is represented by
yt = ϕ0 + ϕ1 x t
(9)
The long-run relationship between yt and xt is only implicit in the ADL(1,1) model. However, it
is possible to recover it from the ADL(1,1) model.
Indeed, an equilibrium is defined as a situation such that once reached, there is no tendancy
for the system to change. That is, once the equilibrium is reached, yt = yt−1 = yt−2 = . . .,
xt = xt−1 = xt−2 = . . . and εt = εt−1 = . . . = 0.
Question 2.4: Given the definition of an equilibrium, use equation (8) to recover the
long-run relation (9), that is, express ϕ0 as a function of α0 , α1 , β0 and β1 . Similarly,
find ϕ1 = f (α0 , α1 , β0 , β1 ).
Having recovered the long-run relationship, it is now possible to re-express the ADL(1,1) and an
error-correction model (ECM)
∆yt = θ0 + θ1 ∆xt + θ2 (yt−1 − ϕ1 xt−1 ) + εt
3
(10)
The ECM is an equivalent way (i.e. it’s the very same model) of representing the ADL(1,1) model
but has the advantage of showing explicitly the long-run relationship that exists between the two
variables.
Question 2.5: Show how you can rewrite the ADL(1,1) model as the ECM model (10).
Specifically, start from (8) and substract yt−1 from both side of the equation. Then, add
and substract β0 xt−1 from the right-hand side of (8). Continue with similar algebrais
operations you recover (10). Show the relationship between that exists between θ0 , θ1 ,
θ2 and α0 , α1 , β0 , and β1 .
A researcher is interested in the relationship between yt and xt . She has a sample of N = 250 observations for both series. As a first step, she tests for non-stationarity for both series by estimating
the following relationships
∆yt = 0.210 − 0.0103yt−1
(11)
∆xt = 0.307 − 0.0160xt−1
(12)
(0.108)
(0.006)
and
(0.146)
(0.0088)
where the number in parenthesis under each coefficient parameter is the coefficient estimated standard error. Using a 5% confidence level, she concludes from these results that both series are
I(1).
Question 2.6: Write explicitly the null and the alternative hypothesis that underlies
the test of non-stationarity that our research conducted.
What is the t-statistics
associated with the coefficient of yt−1 and xt−1 ?
Question 2.7:The student-t 5% critical value for a one-sided test is -1.651. Why, then,
did our research concluded that both series were I(1)?
Having concluded that both series are I(1), she estimates 2 models for capturing the relationship
between yt and xt . The first estimated model is
yt = 4.638 + 0.536 yt−1 + 0.145 xt + 0.087 xt−1
(0.263)
(0.026)
(0.013)
(13)
(0.019)
For the second model, she first estimates
yt = 10.002 + 0.496 xt
(0.073)
(14)
(0.004)
From which she recovers a vector or disequilibrium
êt = yt − 10.002 − 0.496xt
4
(15)
that she uses to estimate the following ecm
∆yt = 0.144 ∆xt − 0.464 êt−1
(0.013)
(16)
(0.026)
Question 2.8: Given your answer in question 2.5, are the models estimated in (13) on
the one hand and (14) and (16) on their other are consistent with one another? tip:
if the relationships identified in 2.5 hold, you should not expect them to hold exactly
given these estimations are based on sample data.
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