ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 9: VAR Models – I
The data file money dem.csv contains quarterly observations for the following vari- ables from 1959Q1 to 2001Q1
• 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.
Answer the following questions:
1. FitaVAR(p)modeltothesystem(dlrgdpt,dlrm2t,drst)withaninterceptvector. In this question, we conduct tests to decide on an adequate lag order.
(a) Use the AIC, BIC and LR test to select the lag order of the VAR up to a maximum of 16 lags.
(b) Fit the VAR(4) and VAR(8) models to the data and compute the residual autocorrelations.
(c) As a final check, estimate the VAR(4) and VAR(8) and provide the AIC, BIC, and log-likelihood ratio.
(d) Check the stationarity of the chosen VAR(p).
(e) How many intercept and slope coefficients do you estimate for the the cho-
sen VAR(p)?
(f) Write out the companion form of the chosen VAR(p)? What is the advantage
of working with the VAR companion form?
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Solution: See the do-file tutorial9.do.
1. Notice that we are analyzing quarterly data. Standard practice is to work with 4 or 8 lags (in less frequent situations, 12 lags are sometimes used, be we will not worry about that here).
(a) TheLRtestandAICchoosep = 11,butBICchoosesp = 1. Thisisa wide range of values, indicating more work is needed to narrow down the set of VAR models. Analysing the autocorrelation in VAR residuals as well as the parsimony principle are two important considerations that we typically rely on.
(b) The VAR(4) residual series exhibits substantial autocorrelation and we can easily reject it being white noise. Although the VAR(8) still shows some autocorrelations in the residual, there is a noticeable (albeit ad-hoc) improvement over the VAR(4).
(c) BIC is smaller for VAR(4) compared to the VAR(8), while the AIC and log-likelihood favor the VAR(8), with the LR test clearly choosing the VAR(8) over the VAR(4). Putting all the evidence together (including autocorrelations from Part (b)), we would choose the VAR(8).
(d) For the estimated VAR(8), all the eigenvalues lie inside the unit circle. The model satisfies the stability/stationarity condition.
(e) The number of intercept and slope coefficients estimated in a n-variate VAR(p) is n+p×n2 = n(1+np). Hence, we estimate 3×(1+3×8) = 75 coefficients.
(f) Let xt denote (dlrgdpt, dlrm2t, drst). The VAR(8) model can be expressed
as
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xt = a0 + Aj xt−j + et
j=1 The companion form of this model is
xt a0 A1 A2 ··· A8xt−1 et xt−1 0 I3 0xt−2 0
. = . + .. . . + . .. . ...
xt−7 0 I3 0 xt−8 0
The companion form, which is equivalent to a VAR(1), is useful in an- alyzing VAR(p) models because the stationarity of the dynamic system can be assessed by checking the eigenvalues of the matrix
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A1 A2 ···A8 I3 0
.. . ..
I3 0
Moreover, we will see in the next lecture that it is much easier to work with the VAR(1) companion form when deriving the VMA(∞) repre- sentation of a VAR(p). The VMA(∞) what we use to compute the IRFs and FEVDs (dynamic effects) of interest.
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