Econ 240C

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