Econ 240C

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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
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The Time Series, Monthly, January 1967through
May 2003
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Federal Funds Rate July 1954-April 2006
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Capacity Utilization Manufacturing Jan. 1972-
April 2006
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Changes in FFR Capacity Utilization
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Contemporaneous Correlation
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Dynamics Cross-correlation
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Granger Causality
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Granger Causality
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Granger 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

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Correlogram of DFFR
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Correlogram of DCAPU
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We 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

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Primitive VAR
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The 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 b1 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
shock
DFFR
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Standard VAR

• dcapu(t) (a1 b1 a2)/(1- b1 b2) (g11 b1
  g21)/(1- b1 b2) dcapu(t-1) (g12 b1
  g22)/(1- b1 b2) dffr(t-1) (d1 b1 d2 )/(1-
  b1 b2) x(t) (edcapu (t) b1 edffr (t))/(1-
  b1 b2)
• But if we assume b1 0,
• then dcapu(t) a1 g11 dcapu(t-1) g12
  dffr(t-1) d1 x(t) edcapu (t)

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• Note that dffr still depends on both shocks
• dffr(t) (b2 a1 a2)/(1- b1 b2) (b2 g11
  g21)/(1- b1 b2) dcapu(t-1) (b2 g12
  g22)/(1- b1 b2) dffr(t-1) (b2 d1 d2 )/(1-
  b1 b2) x(t) (b2 edcapu (t) edffr (t))/(1- b1
  b2)
• dffr(t) (b2 a1 a2)(b2 g11 g21)
  dcapu(t-1) (b2 g12 g22) dffr(t-1) (b2 d1
  d2 ) x(t) (b2 edcapu (t) edffr (t))

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Reality
edcapu (t)
DCAPU
shock
DFFR
edffr (t)
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What If
edcapu (t)
DCAPU
shock
DFFR
edffr (t)
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EVIEWS
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Interpretations

• 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 b1 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

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Change the Assumption Around
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What If
edcapu (t)
DCAPU
shock
DFFR
edffr (t)
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Standard VAR

• dffr(t) (b2 a1 a2)/(1- b1 b2) (b2 g11
  g21)/(1- b1 b2) dcapu(t-1) (b2 g12
  g22)/(1- b1 b2) dffr(t-1) (b2 d1 d2 )/(1-
  b1 b2) x(t) (b2 edcapu (t) edffr (t))/(1- b1
  b2)
• if b2 0
• then, dffr(t) a2 g21 dcapu(t-1) g22
  dffr(t-1) d2 x(t) edffr (t))
• but, dcapu(t) (a1 b1 a2) (g11 b1 g21)
  dcapu(t-1) (g12 b1 g22) dffr(t-1) (d1
  b1 d2 ) x(t) (edcapu (t) b1 edffr (t))

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Interpretations

• 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 b2 0,
• interpret as Fed raising FFR if CAPU rises

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Conclusions

• We come to the same model interpretation and
  policy conclusions no matter what the ordering,
  i.e. no matter which assumption we use, b1 0, or
  b2 0.
• So, accept the analysis

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Understanding through Simulation

• We can not get back to the primitive fron the
  standard VAR, so we might as well simplify
  notation
• y(t) (a1 b1 a2)/(1- b1 b2) (g11 b1
  g21)/(1- b1 b2) y(t-1) (g12 b1 g22)/(1- b1
  b2) w(t-1) (d1 b1 d2 )/(1- b1 b2) x(t)
  (edcapu (t) b1 edffr (t))/(1- b1 b2)
• becomes y(t) a1 b11 y(t-1) c11 w(t-1) d1
  x(t) e1(t)

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• And w(t) (b2 a1 a2)/(1- b1 b2) (b2 g11
  g21)/(1- b1 b2) y(t-1) (b2 g12 g22)/(1-
  b1 b2) w(t-1) (b2 d1 d2 )/(1- b1 b2) x(t)
  (b2 edcapu (t) edffr (t))/(1- b1 b2)
• 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

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Simulated Errors e1(t) and e2(t)
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Simulated Errors e1(t) and e2(t)
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Estimated Model
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Y to shock in w Calculated 0.8 0.76 0.70
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Impact of shock in w on variable y
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