ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 10: VAR Models – II
We continue with the data file money dem.csv we were using last week. • RGDP: real US GDP;
• GDP: nominal GDP;
• M2: Money supply;
• Tb3mo: Three-month rate on US Treasury Bills.
Load the data in Stata and generate the following variables:
• Real GDP Growth: dlrgdpt = ln(RGDPt/RGDPt−1);
• GDP Deflator: pricet = GDPt/RGDPt;
• Real Money supply growth: dlrm2t = ln(M2t/pricet) − ln(M2t−1/pricet−1); • Changes in the short-term interest rate: drst = tb3mot − tb3mot−1.
We had determined the lag length of the (reduced) VAR model for the system of vari- ables (dlrgdpt, dlrm2t, drst) to be p = 8. We will continue using this lag length.
1. Using the VAR(8) model estimated in Tutorial 9, we now study the impulse re- sponse function (IRF) and forecast error variance decomposition (FEVD) to ex- plore the dynamic effects in the system. We first use a Cholesky decomposition and thus we need to decide on an ordering.
(a) Write the VAR(p) in the notation A(L)xt = a0 + et.
(b) Compute the IRFs for all the possible orderings of the system and study the responses. Is the system sensitive to ordering? Choose the most reasonable ordering and explain your answer.
(c) Using the ordering chosen in Part (b), compute the FEVDs and comment on your findings.
2. Consider the following set of restrictions to identify the structural system.
e1t 1 0 0ε1t e2t =c21 1 c23 ε2t ,
e3t 001 ε3t
where c21 and c23 are the corresponding elements of the B−1 matrix. Provide a
plausible economic interpretation for these restrictions. Is this system identified? 1
3. Test the existence of Granger causality among (dlrgdpt, dlrm2t, drst). What is the null hypothesis for the test? State your conclusions.
Solution:
1. See the do-file tutorial10.do.
(a) Let xt = (dlrgdpt, dlrm2t, drst)′. The VAR(8) model can be expressed as
A11(L) A21(L) A31 (L)
A12(L) A13(L)x1t a10 e1t A22(L) A23(L) x2t = a20 + e2t ,
A32 (L)
A(L)
A33 (L)
x3t a30 e3t
xt a0 et
where Aii(L) = 1 − aii,1L − aii,2L2 − · · · − aii,8L8 and Aij (L) = −aij,1L − aij,2L2 − · · · − aij,8L8 for all i ̸= j. Accordingly, writing A(L) = I3 − A1L−A2L2 −···−A8L8,the(i,j)th-elementofmatrixAl,l=1,…,8is denoted by aij,l.
(b) We compare IRFs of 6 orderings. The following are some interesting examples.
i. (dlrgdpt, dlrm2t, drst) vs. (dlrgdpt, drst, dlrm2t): We observe simi- lar patterns in impulse responses, but note that the response of in- terest rates to a change in money supply is fairly different within the two orderings. When money supply is ordered prior to inter- est rates, there is a significant contemporaneous response of interest rates to a change in money supply; no significant response is ob- served in the alternative case (money supply prior to interest rates).
ii. (dlrgdpt, dlrm2t, drst) vs. (dlrm2t, drst, dlrgdpt): These two order- ings exhibit patterns that are very similar to IRFs in part (i). How- ever, the response of interest rates to a change in GDP is larger when GDP is ordered prior to interest rates, than the other way around.
iii. (drst, dlrgdpt, dlrm2t) vs. (drst, dlrm2t, dlrgdpt): The IRFs are very similar for these two orderings and do not show any significant sen- sitivity to switching the order of GDP and money supply when in- terest rates or ordered first.
Data is not informative on which ordering is most suitable, so we need to draw on economic theory if we are to focus on one particular order- ing. There are a number of ways to reason in the present setting. A conventional approach used in analysing dynamic responses to mone- tary policy shocks (e.g., interest rates) separates all non-policy variables into fast-moving and slow-moving variables. Then, all slow-moving vari- ables are ordered prior to the interest rate variable and all fast-moving variables are placed after.
Typically, fast-moving variables are taken to be financial indicators and asset prices, whereas slow-moving variables are those related real eco- nomic activity. In the literature, it has been shown that under reason- able conditions the particular ordering within groups of slow-moving
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and fast-moving variables is not important for the purpose of drawing inference on the response of economic variables to a change in interest rates.
In our setting, one might argue that GDP is certainly a slow-moving vari- able, so it should definitely be ordered before interest rates. It is also rea- sonable to assume that GDP does not respond to money supply within one quarter. It may be less clear on how to classify money supply. Fortu- nately, the comparison of IRFs carried out in part (i) suggests that it may not matter much on whether money supply is ordered prior to interest rates or vice-versa.
If we focus on two orderings that put drs last, i.e., (dlrgdpt, dlrm2t, drst) and (dlrm2t, dlrgdpt, drst), we observe that (as theory suggests) all re- sponses to interest rates are indistinguishable, and the IRFs for the other impulses / responses are also “qualitatively” the same. Therefore, in- ference drawn based on these IRFs may be considered to be robust to changes in the ordering of GDP and money supply (which is reassuring in case our intuition that GDP does not respond to money supply within one quarter fails).
This is the type of approach macroeconomists frequently undertake in policy analysis. One word of caution: it is tempting to look at the IRFs and choose an ordering (or more generally set of identifying restrictions) based on what yields the “most reasonable” results. This, however, is circular reasoning—by undertaking such an approach we are simply finding “the right method” that confirms what we hypothesised before seeing the data. Such an approach has been shown to lead to very dan- gerous conclusions!
(c) We use the ordering (dlrgdpt , dlrm2t , drst ) to compute the FEVDs. Some implications of the estimated FEVDs are the following.
i. Asmallbutsignificantpercentageofthemediumandlong-runfore- cast error variance in the growth of Real GDP is attributed to changes in money supply and interest rate.
ii. Asmallbutsignificantpercentageofthemediumandlong-runfore- cast error variance in the change in money supply is attributed to changes in the interest rate, but GDP does not explain any variation in money supply at any horizon.
iii. Compared to changes in money supply, changes in GDP explain a larger proportion of the forecast error variance in the changes in interest rates.
2. Real GDP growth and changes in the interest rate do not respond contempo- raneously to changes in any of the other variables in the system. In contrast, the change real money supply responds contemporaneously to all three vari- ables.
These restrictions are largely consistent with the reasoning outlined in Quet- sion 1, part b. In fact we can view the resulting SVAR as restricted version
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of the SVAR identified with the Cholesky decomposition and the ordering (dlrgdpt, dlrm2t, drst). If we denote the corresponding B ̃−1 by
100 B ̃−1=c ̃21 1 0,
c ̃32 c ̃32 1
then we se that B−1 can be obtained by interchanging rows 2 and 3 along with
columns 2 and 3 of B ̃−1, then setting
c21 =c ̃32, c23 =c ̃32, c31 =c ̃21 =0.
Conversely, if we re-order the variables as (dlrgdpt, drst, dlrm2t) and im- pose the restrictions stated in this question, it is equivalent to applying the Cholesky decomposition to the ordering (dlrgdpt, drst, dlrm2t), then impos- ing the additional restriction c31 = c ̃21 = 0. Therefore, a general intuition behind this set of identifying restrictions follows the arguments outlined in Question 1, part b. The additional justification needed in the present case is that related to restricting c31 = c ̃21 = 0.
From an economic point of view, the additional restriction simply states that the interest rate does not respond contemporaneously to changes in GDP. Given that “contemporaneously” in this case is interpreted as “within one quarter”, one might question the validity of such a restriction. In fact the Taylor rule suggests that interest rates are set in response to output and infla- tion, which would contradict it.
Fortunately, this set of restrictions involves more restrictions on the model parameters than what is strictly necessary, meaning that the model is over- identified. One useful feature of an over-identified model is that the “extra” restriction(s) are now testable. That is, if we assume the overall Cholesky structure of the VAR is valid, then we can set up a hypothesis test to verify the validity of c31 = c ̃21 = 0.
A brief outline of how this is done is the following. Starting with the more general specification, i.e. the Cholesky approach on the variables ordered as xt = (dlrgdpt, drst, dlrm2t), we see that
1 0 0 1 0 0 −1 B ̃ = ̃b21 1 0 = c ̃21 1 0
̃b32 ̃b32 1 c ̃32 c ̃32 1
is also lower triangular (inverting a lower-triangular matrix yields a lower- triangular matrix, which is a general property). Moreover, the restriction c ̃21 = 0 corresponds to ̃b21 = 0.
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Now the implied SVAR is:
1 0 0x1t b10 γ11,1 γ12,1 γ13,1 x1,t−1 ̃b21 1 0 x2t = b20 + γ21,1 γ22,1 γ23,1 x2,t−1 ̃b32 ̃b32 1 x3t b30 γ31,1 γ32,1 γ33,1 x3,t−1
γ11,8 γ12,8 γ13,8 x1,t−8 ε1t +···+ γ21,8 γ22,8 γ23,8 x2,t−8 + ε2t ,
γ31,8 γ32,8 γ33,8 x3,t−8 ε3t from which we obtain the ARDL:
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x2t = b20 − ̃b21x1t + γ11,lx1,t−l + γ21,lx2,t−l + γ31,lx3,t−l + ε2t,
l=1 l=1 l=1
where x1t is exogenous (so the ARDL is valid). All that remains is to estimate this ARDL using standard methods and test the significance of b20. Indeed, we find that the restriction is rejected at 1% significance level.
Nevertheless, if we have strong reasons to believe that the additional restric- tion is important, we may proceed with estimation and computation of IRFs / FEVDs and impose the restriction outright. Because the overall struture of the SVAR in this case is still recursive, the SVAR can be estimated by running OLS on each of the three implied ARDL models, with b20 = 0 being enforced in the ARDL for x2t specified above.
More generally, however, the over-identified nature of the SVAR implies it must be estimated directly using non-linear methods. Stata (and many other statistical software) provide functionality for specifying restrictions and esti- mating SVARs directly.
3. Here we discuss the general case. Suppose we have an n-variate VAR(p)
model:
p
xt =a0 +Alxt−l +et,
l=1
where xt = (x1t, …xnt)′. To test if xjt Granger causes xit based on this model, we have H0 : aij,1 = ··· = aij,p = 0, where aij,l is the (i,j)th-element of the matrix Al. Similarly, we can test if (xjt,xkt) Granger cause xlt with H0 : aij,1 = ··· = aij,p = 0 and aik,1 = ··· = aik,p = 0. This can be generalized to any subvector of xt not including xit as a component. Note that the null hypothesis is “no Granger causality”. For this question, we reject most of the tests at the 5% significance level except “GDP not Granger causing money supply” or “money supply not Granger causing interest rates”.
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