Econ 240C
Lecture 16
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Part I. VAR
• Does the Federal Funds Rate Affect Capacity Utilization?
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• The Federal Funds Rate is one of the principal monetary instruments of the Federal Reserve
• Does it affect the economy in “real terms”, as measured by capacity utilization
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Preliminary Analysis
5The Time Series, Monthly, January 1967through May 2003
6Federal Funds Rate: July 1954-April 2006
7Capacity Utilization Manufacturing:
Jan. 1972- April 2006
8Changes in FFR & Capacity Utilization
9Contemporaneous Correlation
10Dynamics: Cross-correlation
11Granger Causality
12Granger Causality
13Granger Causality
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Estimation of VAR
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Estimation Results
• OLS Estimation
• each series is positively autocorrelated– lags 1 and 24 for dcapu– lags 1, 2, 7, 9, 13, 16
• each series depends on the other– dcapu on dffr: negatively at lags 10, 12, 17, 21– dffr on dcapu: positively at lags 1, 2, 9, 10 and
negatively at lag 12
24Correlogram of DFFR
25Correlogram of DCAPU
26We Have Mutual Causality, But
We Already Knew That
DCAPU
DFFR
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Interpretation
• We need help
• Rely on assumptions
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What If
• What if there were a pure shock to dcapu– as in the primitive VAR, a shock that only
affects dcapu immediately
Primitive VAR
(1)dcapu(t) = dffr(t) +
dcapu(t-1) + dffr(t-1) + x(t)
+ edcapu(t)
(2) dffr(t) = dcapu(t) +
dcapu(t-1) + dffr(t-1) + x(t)
+ edffr(t)
30The Logic of What If• A shock, edffr , to dffr affects dffr immediately,
but if dcapu depends contemporaneously on dffr, then this shock will affect it immediately too
• so assume is zero, then dcapu depends only on its own shock, edcapu , first period
• But we are not dealing with the primitive, but have substituted out for the contemporaneous terms
• Consequently, the errors are no longer pure but have to be assumed pure
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DCAPU
DFFR
shock
32Standard VAR
• dcapu(t) = (/(1- ) +[ (+ )/(1- )] dcapu(t-1) + [ (+ )/(1- )] dffr(t-1) + [(+ (1- )] x(t) + (edcapu(t) + edffr(t))/(1- )
• But if we assume
• thendcapu(t) = + dcapu(t-1) + dffr(t-1) + x(t) + edcapu(t) +
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• Note that dffr still depends on both shocks
• dffr(t) = (/(1- ) +[(+ )/(1- )] dcapu(t-1) + [ (+ )/(1- )] dffr(t-1) + [(+ (1- )] x(t) + (edcapu(t) + edffr(t))/(1- )
• dffr(t) = (+[(+ ) dcapu(t-1) + (+ ) dffr(t-1) + (+ x(t) + (edcapu(t) + edffr(t))
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DCAPU
DFFR
shock
edcapu(t)
edffr(t)
Reality
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DCAPU
DFFR
shock
edcapu(t)
edffr(t)
What If
36EVIEWS
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38Interpretations• Response of dcapu to a shock in dcapu
– immediate and positive: autoregressive nature
• Response of dffr to a shock in dffr– immediate and positive: autoregressive nature
• Response of dcapu to a shock in dffr– starts at zero by assumption that – interpret as Fed having no impact on CAPU
• Response of dffr to a shock in dcapu– positive and then damps out– interpret as Fed raising FFR if CAPU rises
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Change the Assumption Around
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DCAPU
DFFR
shock
edcapu(t)
edffr(t)
What If
41Standard VAR• dffr(t) = (/(1- ) +[(+ )/(1-
)] dcapu(t-1) + [ (+ )/(1- )] dffr(t-1) + [(+ (1- )] x(t) + (edcapu(t) + edffr(t))/(1- )
• if
• then, dffr(t) = dcapu(t-1) + dffr(t-1) + x(t) + edffr(t))
• but, dcapu(t) = ( + (+ ) dcapu(t-1) + [ (+ ) dffr(t-1) + [(+ x(t) + (edcapu(t) + edffr(t))
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43Interpretations• Response of dcapu to a shock in dcapu
– immediate and positive: autoregressive nature
• Response of dffr to a shock in dffr– immediate and positive: autoregressive nature
• Response of dcapu to a shock in dffr– is positive (not - ) initially but then damps to zero– interpret as Fed having no or little control of CAPU
• Response of dffr to a shock in dcapu– starts at zero by assumption that – interpret as Fed raising FFR if CAPU rises
44Conclusions• We come to the same model interpretation
and policy conclusions no matter what the ordering, i.e. no matter which assumption we use, or
• So, accept the analysis
45Understanding through Simulation
• We can not get back to the primitive fron the standard VAR, so we might as well simplify notation
• y(t) = (/(1- ) +[ (+ )/(1- )] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (edcapu(t) + edffr(t))/(1- )
• becomes y(t) = a1 + b11 y(t-1) + c11 w(t-1) + d1 x(t) + e1(t)
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• And w(t) = (/(1- ) +[(+ )/(1- )] y(t-1) + [ (+ )/(1- )] w(t-1) + [(+ (1- )] x(t) + (edcapu(t) + edffr(t))/(1- )
• becomes w(t) = a2 + b21 y(t-1) + c21 w(t-1) + d2 x(t) + e2(t)
•
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Numerical Example
y(t) = 0.7 y(t-1) + 0.2 w(t-1)+ e1(t)w(t) = 0.2 y(t-1) + 0.7 w(t-1) + e2(t)
where e1(t) = ey(t) + 0.8 ew(t)
e2(t) = ew(t)
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• Generate ey(t) and ew(t) as white noise processes using nrnd and where ey(t) and ew(t) are independent. Scale ey(t) so that the variances of e1(t) and e2(t) are equal
– ey(t) = 0.6 *nrnd and
– ew(t) = nrnd (different nrnd)
• Note the correlation of e1(t) and e2(t) is 0.8
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Analytical Solution Is Possible
• These numerical equations for y(t) and w(t) could be solved for y(t) as a distributed lag of e1(t) and a distributed lag of e2(t), or, equivalently, as a distributed lag of ey(t) and a distributed lag of ew(t)
• However, this is an example where simulation is easier
50Simulated Errors e1(t) and e2(t)
51Simulated Errors e1(t) and e2(t)
52Estimated Model
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Y to shock in w
Calculated
0.8
0.76
0.70
Impact of a Shock in w on the Variable y: Impulse Response Function
Period
Imp
act
Mult
iplier
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4 5 6 7 8 9
Calculated
Simulated
Impact of shock in w on variable y
Impact of a Shock in y on the Variable y: Impulse Response Function
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Period
Impac
t M
ultip
lier
Calculated
Simulated
