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

4 Preliminary Analysis

5 The Time Series, Monthly, January 1967through May 2003

6 Federal Funds Rate: July 1954-April 2006

7 Capacity Utilization Manufacturing: Jan April 2006

8 Changes in FFR & Capacity Utilization

9 Contemporaneous Correlation

10 Dynamics: Cross-correlation

11 Granger Causality

12 Granger Causality

13 Granger Causality

14 Estimation of VAR

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23 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

24 Correlogram of DFFR

25 Correlogram of DCAPU

26 We Have Mutual Causality, But We Already Knew That DCAPU DFFR

27 Interpretation We need help Rely on assumptions

28 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

30 The Logic of What If A shock, e dffr, 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, e dcapu, 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

31 DCAPU DFFR shock

32 Standard VAR dcapu(t) = (        /(1-     ) +[ (    +     )/(1-     )] dcapu(t-1) + [ (    +     )/(1-     )] dffr(t-1) + [(    +      (1-     )] x(t) + (e dcapu  (t) +   e dffr  (t))/(1-     ) But if we assume    then  dcapu(t) =    +    dcapu(t-1) +   dffr(t-1) +    x(t) + e dcapu  (t) + 

33 Note that dffr still depends on both shocks dffr(t) = (        /(1-     ) +[(      +   )/(1-     )] dcapu(t-1) + [ (      +   )/(1-     )] dffr(t- 1) + [(      +    (1-     )] x(t) + (    e dcapu  (t) + e dffr  (t))/(1-     ) dffr(t) = (        +[(      +   ) dcapu(t-1) + (      +   ) dffr(t-1) + (      +    x(t) + (    e dcapu (t) + e dffr  (t))

34 DCAPU DFFR shock e dcapu  (t) e dffr  (t) Reality

35 DCAPU DFFR shock e dcapu  (t) e dffr  (t) What If

36 EVIEWS

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38 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    –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

39 Change the Assumption Around

40 DCAPU DFFR shock e dcapu  (t) e dffr  (t) What If

41 Standard VAR dffr(t) = (        /(1-     ) +[(      +   )/(1-     )] dcapu(t-1) + [ (      +   )/(1-     )] dffr(t- 1) + [(      +    (1-     )] x(t) + (    e dcapu  (t) + e dffr  (t))/(1-     ) if    then, dffr(t) =      dcapu(t-1) +   dffr(t-1) +    x(t) + e dffr  (t)) but, dcapu(t) = (        + (    +     ) dcapu(t- 1) + [ (    +     ) dffr(t-1) + [(    +      x(t) + (e dcapu  (t) +   e dffr  (t))

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43 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    –interpret as Fed raising FFR if CAPU rises

44 Conclusions 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

45 Understanding 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) + (e dcapu  (t) +   e dffr  (t))/(1-     ) becomes y(t) = a 1 + b 11 y(t-1) + c 11 w(t-1) + d 1 x(t) + e 1 (t)

46 And w(t) = (        /(1-     ) +[(     +   )/(1-     )] y(t-1) + [ (      +   )/(1-     )] w(t-1) + [(      +    (1-     )] x(t) + (    e dcapu  (t) + e dffr  (t))/(1-     ) becomes w(t) = a 2 + b 21 y(t-1) + c 21 w(t-1) + d 2 x(t) + e 2 (t)

47 Numerical Example y(t) = 0.7 y(t-1) w(t-1)+ e 1 (t) w(t) = 0.2 y(t-1) w(t-1) + e 2 (t) where e 1 (t) = e y  (t) e w  (t) e 2 (t) = e w  (t)

48 Generate e y (t) and e w (t) as white noise processes using nrnd and where e y (t) and e w (t) are independent. Scale e y (t) so that the variances of e 1 (t) and e 2 (t) are equal –e y (t) = 0.6 *nrnd and –e w (t) = nrnd (different nrnd) Note the correlation of e 1 (t) and e 2 (t) is 0.8

49 Analytical Solution Is Possible These numerical equations for y(t) and w(t) could be solved for y(t) as a distributed lag of e 1 (t) and a distributed lag of e 2 (t), or, equivalently, as a distributed lag of e y (t) and a distributed lag of e w (t) However, this is an example where simulation is easier

50 Simulated Errors e 1 (t) and e 2 (t)

51 Simulated Errors e 1 (t) and e 2 (t)

52 Estimated Model

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58 Y to shock in w Calculated

Impact of shock in w on variable y