Econ 240C
Lecture 17
2
Part I. VAR
• Does the Federal Funds Rate Affect
Capacity Utilization?
3
• 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
4
Preliminary Analysis
The Time Series, Monthly,
Jan uary 1967through May 2003
5
6
Changes in FFR & Capacity Utilization
Contemporaneous Correlation
7
Dynamics: Cross-correlation
8
Granger Causality
9
Granger Causality
10
Granger Causality
11
12
Estimation of VAR
13
14
15
16
17
18
19
20
21
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
Correlogram of DFFR
22
Correlogram of DCAPU
23
We Have Mutual Causality, But
We Already Knew That
DCAPU
DFFR
24
25
Interpretation
• We need help
• Rely on assumptions
26
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)
The Logic of What If
28
• 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
29
DCAPU
shock
DFFR
Standard VAR
• dcapu(t) = (  )/(1-  ) +[ ( +
 )/(1-  )] dcapu(t-1) + [ ( + 
)/(1-  )] dffr(t-1) + [( +   )/(1 )] x(t) + (edcapu (t) +  edffr (t))/(1- 
)
• But if we assume =0,
• then dcapu(t) =  + dcapu(t-1) + 
dffr(t-1) +  x(t) + edcapu (t) +

30
31
• Note that dffr still depends on both shocks
• dffr(t) = (  )/(1-  ) +[(  + )/(1 )] dcapu(t-1) + [ (  + )/(1-  )] dffr(t1) + [(  +  )/(1-  )] x(t) + ( edcapu (t) +
edffr (t))/(1-  )
• dffr(t) = (  )+[(  + ) dcapu(t-1) +
(  + ) dffr(t-1) + (  +  ) x(t) + ( edcapu
(t) + edffr (t))
Reality
edcapu (t)
DCAPU
shock
DFFR
edffr (t)
32
What If
edcapu (t)
DCAPU
shock
DFFR
edffr (t)
33
EVIEWS
34
35
Interpretations
36
• 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 =0,
– 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
37
Change the Assumption Around
What If
edcapu (t)
DCAPU
shock
DFFR
edffr (t)
38
Standard VAR
39
• dffr(t) = (  )/(1-  ) +[(  + )/(1 )] dcapu(t-1) + [ (  + )/(1-  )] dffr(t1) + [(  +  )/(1-  )] x(t) + ( edcapu (t) +
edffr (t))/(1-  )
• if =0
• then, dffr(t) =    dcapu(t-1) +  dffr(t-1) +
 x(t) + edffr (t))
• but, dcapu(t) = (  ) + ( +  ) dcapu(t1) + [ ( +  ) dffr(t-1) + [( +   ) x(t) +
(edcapu (t) +  edffr (t))
40
Interpretations
41
• 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 =0,
– interpret as Fed raising FFR if CAPU rises
Conclusions
• We come to the same model interpretation
and policy conclusions no matter what the
ordering, i.e. no matter which assumption
we use, =0,or =0.
• So, accept the analysis
42
Understanding through Simulation
43
• 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)
44
• 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)
•
45
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)
46
• 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
47
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
Simulated Errors e1(t) and e2(t)
48
Simulated Errors e1(t) and e2(t)
49
Estimated Model
50
51
52
53
54
55
56
Y to shock in w
Calculated
0.8
0.76
0.70
Impact of shock in w on variable y
Impact of a Shock in w on the Variable y: Impulse Response Function
0.9
0.8
0.7
Impact Multiplier
0.6
0.5
Calculated
0.4
Simulated
0.3
0.2
0.1
0
0
1
2
3
4
5
Period
6
7
8
9
Impact of a Shock in y on the Variable y: Impulse Response
Function
1.2
Impact Multiplier
1
Calculated
Simulated
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
Period
6
7
8
9
10