econ 240c
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Econ 240C. Lecture 16. Part I. VAR. Does the Federal Funds Rate Affect Capacity Utilization?. 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 for total industry?. - PowerPoint PPT PresentationTRANSCRIPT
Econ 240C
Lecture 16
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 for total industry?
4
5
6
Preliminary Analysis
• The Time Series, Monthly, January 1967 through April 2008
7
70
75
80
85
90
70 75 80 85 90 95 00 05
TCU
0
5
10
15
20
70 75 80 85 90 95 00 05
FFR
8Capacity Utilization Total Industry:
Jan. 1967- April 2008
9
0
10
20
30
40
72 74 76 78 80 82 84 86 88
Series: TCUSample 1967:01 2008:04Observations 496
Mean 81.41028Median 81.25000Maximum 89.40000Minimum 70.90000Std. Dev. 3.695047Skewness -0.263366Kurtosis 2.814450
Jarque-Bera 6.445440Probability 0.039847
10
11
12Identification of TCU
• Trace
• Histogram
• Correlogram
• Unit root test
• Conclusion: probably evolutionary
13
14
0
20
40
60
80
2 4 6 8 10 12 14 16 18
Series: FFRSample 1967:01 2008:04Observations 496
Mean 6.499980Median 5.760000Maximum 19.10000Minimum 0.980000Std. Dev. 3.326970Skewness 1.112618Kurtosis 4.931202
Jarque-Bera 179.4119Probability 0.000000
15
16
17Identification of FFR
• Trace
• Histogram
• Correlogram
• Unit root test
• Conclusion: unit root
18Pre-whiten both
19
Changes in FFR & Capacity Utilization
-8
-6
-4
-2
0
2
4
70 75 80 85 90 95 00 05
DFFR
-4
-3
-2
-1
0
1
2
70 75 80 85 90 95 00 05
DTCU
20Contemporaneous Correlation
-4
-3
-2
-1
0
1
2
-8 -6 -4 -2 0 2 4
DFFR
DT
CU
21Dynamics: Cross-correlation
Two-Way Causality?
22In Levels
Too much structure in each
hides the relationship between
them
23In differences
24Granger Causality: Four Lags
25Granger Causality: two lags
26Granger Causality: Twelve lags
27Estimate VAR
28Estimation of VAR
29
30
31
32
33
34
35Specification
• Same number of lags in both equations
• Use liklihood ratio tests to compare 12 lags versus 24 lags for example
36
37
Estimation Results
• OLS Estimation
• each series is positively autocorrelated– lags 1, 18 and 24 for dtcu– lags 1, 2, 4, 7, 8, 9, 13, 16 for dffr
• each series depends on the other– dtcu on dffr: negatively at lags 10, 12, 17, 21– dffr on dtcu: positively at lags 1, 2, 9, 24 and
negatively at lag 12
38We Have Mutual Causality, But
We Already Knew That
DTCU
DFFR
39Correlogram of DFFR
40Correlogram of DTCU
41
Interpretation
• We need help
• Rely on assumptions
42
What If
• What if there were a pure shock to dtcu– as in the primitive VAR, a shock that only
affects dtcu immediately
43Primitive VAR (tcu Notation)
dtcu(t) =1 + 1 dffr(t) + 11 dtcu(t-1) + 12 dffr(t-1) + 1 x(t) + edtcu (t)
(2) dffr(t) = 2 + 2 dtcu(t) + 21 dtcu(t-1)
+ 22 dffr(t-1) + 2 x(t) + edffr (t)
Primitive VAR (capu notation)
(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)
45The 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
46
DTCU
DFFR
shock
47Standard 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) +
48
• 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))
49
DTCU
DFFR
shock
edtcu(t)
edffr(t)
Reality
50
DTCU
DFFR
shock
edtcu(t)
edffr(t)
What If
51EVIEWS
52Economy affects Fed, not vice versa
53Interpretations• Response of dtcu to a shock in dtcu
– immediate and positive: autoregressive nature
• Response of dffr to a shock in dffr– immediate and positive: autoregressive nature
• Response of dtcu to a shock in dffr– starts at zero by assumption that – interpret as Fed having no impact on TCU
• Response of dffr to a shock in dtcu– positive and then damps out– interpret as Fed raising FFR if TCU rises
54Change the Assumption Around
55
DTCU
DFFR
shock
edtcu(t)
edffr(t)
What If
56Standard 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))
57
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7 8 9 10
Response of DTCU to DTCU
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
1 2 3 4 5 6 7 8 9 10
Response of DTCU to DFFR
-0.2
0.0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 9 10
Response of DFFR to DTCU
-0.2
0.0
0.2
0.4
0.6
1 2 3 4 5 6 7 8 9 10
Response of DFFR to DFFR
Response to One S.D. Innovations ± 2 S.E.
58Interpretations• Response of dtcu to a shock in dtcu
– immediate and positive: autoregressive nature
• Response of dffr to a shock in dffr– immediate and positive: autoregressive nature
• Response of dtcu to a shock in dffr– is positive (not - ) initially but then damps to zero– interpret as Fed having no or little control of TCU
• Response of dffr to a shock in dtcu– starts at zero by assumption that – interpret as Fed raising FFR if CAPU rises
59Conclusions• 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
60Understanding 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)
61
• 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)
•
62
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)
63
• 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
64
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
65Simulated Errors e1(t) and e2(t)
66Simulated Errors e1(t) and e2(t)
67Estimated Model
68
69
70
71
72
73
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