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Econ 240C Lecture 16
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2 Part I. VAR Does the Federal Funds Rate Affect Capacity Utilization?
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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?
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6 Preliminary Analysis The Time Series, Monthly, January 1967 through April 2008
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8 Capacity Utilization Total Industry: Jan. 1967- April 2008
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12 Identification of TCU Trace Histogram Correlogram Unit root test Conclusion: probably evolutionary
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17 Identification of FFR Trace Histogram Correlogram Unit root test Conclusion: unit root
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18 Pre-whiten both
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19 Changes in FFR & Capacity Utilization
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20 Contemporaneous Correlation
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21 Dynamics: Cross-correlation Two-Way Causality?
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22 In Levels Too much structure in each hides the relationship between them
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23 In differences
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24 Granger Causality: Four Lags
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25 Granger Causality: two lags
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26 Granger Causality: Twelve lags
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27 Estimate VAR
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28 Estimation of VAR
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35 Specification Same number of lags in both equations Use liklihood ratio tests to compare 12 lags versus 24 lags for example
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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
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38 We Have Mutual Causality, But We Already Knew That DTCU DFFR
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39 Correlogram of DFFR
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40 Correlogram of DTCU
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41 Interpretation We need help Rely on assumptions
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42 What If What if there were a pure shock to dtcu –as in the primitive VAR, a shock that only affects dtcu immediately
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43 Primitive VAR (tcu Notation) dtcu(t) = 1 + 1 dffr(t) + 11 dtcu(t-1) + 12 dffr(t-1) + 1 x(t) + e dtcu (t) (2) dffr(t) = 2 + 2 dtcu(t) + 21 dtcu(t-1) + 22 dffr(t-1) + 2 x(t) + e dffr (t)
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Primitive VAR (capu notation)
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45 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
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46 DTCU DFFR shock
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47 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) +
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48 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))
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49 DTCU DFFR shock e dtcu (t) e dffr (t) Reality
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50 DTCU DFFR shock e dtcu (t) e dffr (t) What If
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51 EVIEWS
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52 Economy affects Fed, not vice versa
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53 Interpretations 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
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54 Change the Assumption Around
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55 DTCU DFFR shock e dtcu (t) e dffr (t) What If
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56 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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58 Interpretations 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
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59 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
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60 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)
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61 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)
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62 Numerical Example y(t) = 0.7 y(t-1) + 0.2 w(t-1)+ e 1 (t) w(t) = 0.2 y(t-1) + 0.7 w(t-1) + e 2 (t) where e 1 (t) = e y (t) + 0.8 e w (t) e 2 (t) = e w (t)
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63 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
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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 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
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65 Simulated Errors e 1 (t) and e 2 (t)
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66 Simulated Errors e 1 (t) and e 2 (t)
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67 Estimated Model
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73 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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