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

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

6Federal Funds Rate: July 1954-April 2007

7Capacity Utilization Manufacturing:

Jan. 1972- April 2007

8

65

70

75

80

85

90

75 80 85 90 95 00 05

CAPUMFG

0

5

10

15

20

75 80 85 90 95 00 05

FFR

January 1972 - April 2007

9Changes in FFR & Capacity Utilization

10Contemporaneous Correlation

11Dynamics: Cross-correlation

12In Levels

13In differences

14Granger Causality

15Granger Causality

16Granger Causality

17Granger Causality

18

Estimation of VAR

19

20

21

22

23

24

25

26

27

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

28Correlogram of DFFR

29Correlogram of DCAPU

30We Have Mutual Causality, But

We Already Knew That

DCAPU

DFFR

31

Interpretation

• We need help

• Rely on assumptions

32

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)

34The 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

35

DCAPU

DFFR

shock

36Standard 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) +

37

• 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))

38

DCAPU

DFFR

shock

edcapu(t)

edffr(t)

Reality

39

DCAPU

DFFR

shock

edcapu(t)

edffr(t)

What If

40EVIEWS

41

42Interpretations• 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

43

Change the Assumption Around

44

DCAPU

DFFR

shock

edcapu(t)

edffr(t)

What If

45Standard 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))

46

47Interpretations• 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

48Conclusions• 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

49Understanding 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)

50

• 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)

•

51

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)

52

• 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

53

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

54Simulated Errors e1(t) and e2(t)

55Simulated Errors e1(t) and e2(t)

56Estimated Model

57

58

59

60

61

62

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