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. u, Lfiu: -1... t.b..lrl‘ {thriQO’nvv-b‘vrloiu- Littl.f.‘l):Lu.wEEL=‘b“t£-n. n:

 

LEADING INDICATORS IN STRUCTURAL ECONOMETRIC MODELS
WITH APPLICATIONS IN MULTIVARIATE TIME SERIES
ANALYSIS ABOUT THE COMMERCE DEPARTMENT LEADING

INDICATORS AND A PROPOSED MONETARY LEADING INDICATOR

By
Paul Koch

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1980

”NOW/’7

ABSTRACT

LEADING INDICATORS IN STRUCTURAL ECONOMETRIC MODELS
WITH APPLICATIONS IN MULTIVARIATE TIME SERIES
ANALYSIS ABOUT THE COMMERCE DEPARTMENT LEADING

INDICATORS AND A PROPOSED MONETARY LEADING INDICATOR

By
Paul Koch

The Commerce Department leading indicator approach has
been criticized as being void of economic theory.

In this study a leading indicator approach is formu-
lated which is firmly embedded in an economic theoretical
framework expressed as a dynamic, structural econometric model.
A time series model in which leading indicators play a
special role is derived directly from this structural model.
In this context forecasts of the objective variable can be
made with the current information provided by the leading
indicator. The variance of the forecast errors can also be
obtained in the analysis.

The current state of the art of forecasting with
econometric models uses the Final Form approach. The
forecasting ability of this approach is compared with that
of the proposed leading indicator approach.

In light of the prOposed approach, the Commerce

Paul Koch

Department's leading indicators are evaluated. Bivariate1finm
series models are built, describing the empirical relation—
ships between economic activity and certain economic time
series which the Commerce Department deems as useful leading
indicators (components of their Composite Index of Leading
Indicators). This examination reveals some possible flaws
with the Commerce Department approach. Most of the Commerce
Department leading indicators examined display no significant
lead over economic activity. Furthermore, one of the few
Commerce Department leading indicators which displays a
considerable lead, is seen to have a relationship with
economic activity that is contrary to the way it is employed
by the Commerce Department.

These flaws cast more doubt on the usefulness of the
Commerce Department leading indicator approach, and possibly
provide some insight as to why the approach has performed so
poorly in the past.

Finally, Money is considered as an alternative leading
indicator. A multivariate time series model is developed,
describing the empirical, dynamic relationship between Money
and economic activity. This model is expanded at length to
account for various problems with the sample period reviewed.
The empirical results are discussed with their implications

toward some considerations in Monetary Theory.

To my family

ACKNOWLEDGEMENTS

I must first express my sincere gratitude to my
thesis advisor, Robert H. Rasche, for his invaluable advice
and guidance throughout this study. The time and effort
which he generously gave has greatly promoted my under-
standing of Monetary Theory. His insight has been instru-
mental in solving the many problems which inevitably arise
during any such project.

I also wish to acknowledge the efforts of my guidance
committee. James M. Johannes provided many helpful comments
concerning exposition at all stages of the study. Peter
Schmidt initially stimulated my interest in econometrics,
and has provided helpful advice throughout my graduate
career. Mark Ladenson also generously gave of his time in
examining my work.

It is also important to recognize my fellow graduate
students at Michigan State University for their contributions
on my behalf. Mike Thomson deserves special attention in
this regard.

Finally I would like to thank Terie Snyder for typing

this tedious manuscript.

ii

TABLE OF CONTENTS

List of Tables

List of Figures.

CHAPTER

I

II

III

INTRODUCTION

LEADING INDICATORS IN STRUCTURAL
ECONOMETRIC MODELS

Introduction
The Framework - - - . .
Comparing the Relative Forecasting
Abilities of the Final Form and
the Proposed Leading Indicator
Approach. . . . . . . . .
The Final Form
The Proposed Leading
Indicator Approach
Example 1 - - - -
Example 2
FOOTNOTES. ..........
APPENDIX 1: Fitting the Transfer
Function in Equation (2.10)
APPENDIX 2: A Useful Convention of
Zellner and Palm - . . . - -

THE PROBLEM OF SPECIFICATION ERROR

Introduction . -

A Discussion of the Problem
FOOTNOTES . . .
APPENDIX 3: Proof that Ordinary Least

Squares Estimation Yields the
Appropriate Estimate- - -

iii

Page

vi

\IE

13
13

18
26
35
51
57

62

66
67
71

73

TABLE OF CONTENTS (cont'd)

CHAPTER

IV AN EVALUATION OF THE COMMERCE DEPARTMENT
LEADING INDICATORS. . . . . .

Introduction .

Critique of the Commerce Department
Approach. .

Empirical Evaluation

Comparison With Beta Coefficients.

Implications .

Conclusions.

V MONEY AS A LEADING INDICATOR

Introduction
Empirical Examination of the
Relationship Between Money and
Industrial Production .
The Money Stock Data
The Identification Stage
Estimation of the Single Input
Transfer Function . . .
Expanding the Model to Account for
the Energy Price Supply Shocks of
the Early 1970's - BCD92. . . .
The Identification Stage of
Building the Two Input
Transfer Function . .
Estimating the Two Input
Transfer Function .
Implications of the Final Model.
Expanding the Model to Account for the
Energy Price Supply Shocks of the
Early 1970's - Fuel Prices. .
The Identification Stage
The Estimation Stage .
Implications of the Final Model.
Expanding the Model to Account for
Supply Shocks Due to Strikes in
the Labor Force . . . . . . . .
FOOTNOTES. . . . . . . . . . . . .

VI CONCLUSION . . . . . . . . . . . . . . . .
APPENDIX H: Data Sources. . . . .

REFERENCES . . . . . . . . . . . . . . . .

iv

Page

75
75
77
81
103
10H
123
125
125
131
131
136

1U2

1H9

151
163
170

179
183
188
19k

198
20”

211
215

219

Table

IV-1

IV-2

LIST OF TABLES

Univariate Models for the Commerce Department
Leading Indicators

Bivariate Models for the Commerce Department
Leading Indicators . . . . . . . . . .

Money Stock Series — 1959

Cross Correlation Function: Prewhitened Money
Stock and Industrial Production. . . . . . .

Forecasting Industrial Production With Filtered
Money Stock: l973.9-75.2.

Cross Correlation Functions: Prewhitened Filtered
Money Stock and BCD92. . . . . . . .

Forecasting Industrial Production With Filtered
Money Stock and BCD92: .1973.9-75.2. . . .

Forecasting Industrial Production With Filtered
Money Stock and BCD92: 1975.3-79.7. .

Cross Correlation Functions: Prewhitened Filtered
Money Stock and Fuel Prices.

Forecasting Industrial Production With Filtered
Money Stock and Fuel Prices: 1973.9-75.2. .

Forecasting Industrial Production With Filtered
Money Stock and Fuel Prices: 1975.3—79.10

Page

83

88

133

138

150

159

166

168

189

190

192

LIST OF FIGURES

Figure

“.1 Impulse Response Function: BCDl Average
Workweek.

“.2 Impulse Response Function: BCD3 Layoff Rate.

”.3 Impulse Response Function: BCD8 New Orders

H.H Impulse Response Function: BCD19 Index of
Stock Prices . . . . . . . . . .

”.5 Impulse Response Function: BCD29 Index of
Housing Starts. . . . . . .

M.6 Impulse Response Function: BCD32 Vendor
Performance . . .

”.7 Impulse Response Function: BCD92 % Change in
PPI . . . . . . . . . . . . . .

”.8 Impulse Response Function: BCDIOS Real
Money Supply - Ml

u.9 Impulse Response Function [Beta Coefficients]:
BCDl. . . . . . . . . . . . .

H.10 Impulse Response Function [Beta Coefficients]:
BCD3. . . . . . . . . . . . . . .

A.ll Impulse Response Function [Beta Coefficients]:
BCD8. . . . . . . . . . . . .

H.12 Impulse Response Function [Beta Coefficients]:
BCD19 . . . . . . . . . . . . . . . .

H.13 Impulse Response Function [Beta Coefficients]:
BCD29 . . . . . . . . . . . . . . . . . .

n.1u Impulse Response Function [Beta Coefficients]:
BCD32 O C O O O O O O O O O O O O O O

4.15 Impulse Response Function [Beta Coefficients]:

BCD92 . . . . . . .

Page

95
96

97

98

99

100

101

102

105

107

109

117

LIST OF FIGURES (cont'd)

Figure Page

H.16 Impulse Response Function [Beta Coefficients]:

BCD105. . . . . . . . . . . . . . . . . . . 119
5.1 Time Paths of Nominal Income and Its

Components After a Monetary Shock , , , , , 129
5.2 Impulse Response Function: Prewhitened

Money Stock and Industrial Production

(Identification Stage). . . . . . . . . . . 190
5.3 Step Response Function: Prewhitened

Money Stock and Industrial Production

(Identification Stage). . . . . . . . . . . 193
5.9 Discontinuous Impulse Response Function

- Filtered Money Stock . . , , , , , , , , 195
5.5 Continuous Impulse Response Function

- Filtered Money Stock . . . , , , , . , , 196
5.6 Impulse and Step Response Functions

Implied by Equation (5.27) - Money Stock. , 172
5.7 Impulse and Step Response Functions

Implied by Equation(5.27) - BCD92 . . . . . 180

5.8 Impulse and Step Response Functions

Implied by Equation (5.27)‘ - Money Stock . 195
5.9 Impulse and Step Response Functions

Implied by Equation (5.27)‘ - Fuel Prices . 199

vii

CHAPTER I

INTRODUCTION

With the advent of the Great Depression in the 1930's
came the assigned task of the NBER of developing a leading
indicator approach to forecasting, in the hope of helping
to prevent another such catastrophe. Wesley Claire Mitchell
and Arthur Burns collected data on various economic time
series and set up criteria for choosing leading indicators
from among these. Over the years the Commerce Department
leading indicator approach has evolved into its present
state, the current Composite Index of Leading Indicators
(CLI).

This approach has been criticized as being void of
economic theory (Koopmans, 19u7). It is argued that theory
should be used in choosing leading indicators, or the under-
lying structural relationships are unknown, and thus the
leading indicators cannot be used for policy decisions.
Promoters of the Commerce Department approach have responded
to this criticism by claiming that a theoretical foundation
is present, since many of the series in the CLI reflect
either direct or indirect measures of demand for various
components of output, or reflect factors which have an impact

on demand. In any case, it is further argued that if a

method without underlying theory predicts better, then for
certain uses it should be preferred.

The record of the Commerce Department approach,
however, has been less than satisfactory. The CLI has
displayed two major faults: it has indicated many false
downturns, and has displayed a highly variable lead time at
true turns. These faults cast much doubt on the usefulness
of this approach in the role of forecasting.

This study develops a leading indicator approach
which is built upon an economic theoretical foundation, as
expressed in a dynamic, structural econometric model. Time
series models in which leading indicators play a special
role are derived directly from this theoretical framework.
In this context, current observations of the leading
indicators can be used to forecast the objective variable,
and the variance of the forecast errors can be obtained.

The current state of the art of forecasting uses the
Final Form of an econometric model. The forecasting ability
of this approach is compared with that of our leading
indicator approach. Two examples are presented illustrating
the approach, and this comparison of forecasting abilities.

In light of our proposed leading indicator approach,
we evaluate the Commerce Department leading indicators by
building bivariate time series models describing the empirical
relationships between the level of economic activity and
eight of the components of the CLI. Five of the "leading

indicators" examined display no significant lead over

economic activity. The other three components exhibit the
kind of relationships with economic activity that a good
leading indicator is expected to have. However, the
component which shows the greatest lead (the Producer Price
Index of Crude Materials) is seen to have a negative rela-
tionship with economic activity, while it is used in a
positive role in the CLI. This analysis sheds light on some
potential reasons for the poor record of the Commerce
Department approach.

Finally, Money is considered as an alternative
leading indicator. The dynamic, empirical relationship
between Money and real GNP is examined in the context of a
multivariate time series model. The model is expanded to
account for two kinds of supply shocks occurring in the
sample period: the energy price shocks of the early 1970's,
and strikes in the Labor Force. At all stages of its
development, the model indicates a stable relationship
between Money and real GNP, suggesting that Money may be

quite useful in the role of leading indicator.

CHAPTER II

LEADING INDICATORS IN STRUCTURAL ECONOMETRIC MODELS

Introduction

 

A leading indicator can be defined loosely as an
economic time series whose movements in some sense consist-
ently lead economic activity. More formally, a leading
indicator can be defined in the following context. We have
some presumed knowledge of the joint distribution of Ipt+k

and LIt’ f(IPt+k’LIt)’ where

IPt+k = some measure of economic activity in

period t+k (e.g. the index of industrial

production),
LIt = some leading indicator which we define, in
period t.
In period t we know the value of LIt' The leading

indicator approach to forecasting suggests that we can use

this knowledge of LIt to tell us more about the distribution

 

of IPt+k’ Thus we are interested in:
f(IP LI )
. _ t+k’ t

This is the context in which leading indicators can

be useful. We presumably know more about the distribution

of IPt+k given LIt’ than without that information. That is,

we can provide better forecasts of IP by using the condi-

t+k

tional distribution, f(IP' ILIt)’ than by using the

t+k

unconditional distribution, f<IPt+k)' Box and Jenkins
present an example showing the improvement in forecasting
using the conditional distribution over that using the
unconditional distribution.1 The amount of uncertainty in
forecasting a time series using their leading indicator
(measured as the standard error of the forecasts) is sub-
stantially less than the uncertainty present in the appro-
priate model without the information provided by the leading
indicator. This, as well as the record of the leading
indicator approach, establishes the usefulness of leading
indicators in forecasting.

In their pioneering work with leading indicators,
Mitchell and Burns originally analyzed #87 different time
series and finally selected 71 by the criteria listed below.2
This collection has been updated periodically, and now
contains 70 indicators: 30 leading, 15 coincident, and 7
lagging (and 28 of less importance). In general the series
considered must lead at no less than 2/3 of the reference
cycle turning points as defined by the NBER, to be considered
a leading indicator.3

Mitchell and Burns listed the following criteria to
select the better indicators (the NBER uses roughly the same

criteria).” A series is a better leading indicator:

l. The longer are its average leads at past revivals.

2. The more uniform are these leads in occurrence
and length.

3. The closer its specific cycles come to having a
one—to—one correspondence to the reference cycles.

u. The more clearly defined are its specific cycles.

5. The less intense are its erratic movements in
comparison with the amplitude of its specific
cycles.

6. The fewer are the changes in the direction of its
month-to—month movements.

7. The smaller and more regular are the seasonal
variations that have to be eliminated before the
specific cycles can be studied.

8. The larger is the number of past revivals covered
by the series.

9. The farther back in time any irregularities in
conformity to business cycle revivals have
occurred.

10. The broader is the range of activities represented
by the series.

11. The more stable is the economic significance of
the process represented.

Koopmans5 criticized the work of Burns and Mitchell
as choosing indicators to predict business cycle peaks and
troughs without any apparent economic theory behind their
methods. He argued that economic theory is useful in
choosing those indicators which will best predict, and if
theory is not used, the findings and results can not be used
for policy decisions or other useful tasks because the under-
lying structural relationships are unknown.

Vining6 later argued against Koopman‘s criticism.

Vining suggested that the usefulness of alternative methods

should be evaluated by the results achieved by each. If a
method without underlying theory (i.e. pure forecasting)
predicts better, then for certain uses it should be preferred.

Since this exchange, promoters of the leading
indicator approach have been concerned with its theoretical
background. Today there is general agreement that there is
in fact a theoretical framework underlying the leading
indicator approach.7 The series included in the Composite
Index of Leading Indicators reflect either direct or indirect
measures of demand for various components of output, or
reflect factors which have an impact on demand. Changes in
these components of demand tend to lead changes in output in
the near future.

To date, this appears to be the main argument of
proponents of the leading indicator approach in defending

their use of leading indicators.

The Framework

 

We wish to explicitly formulate an economic
theoretical background for the leading indicator approach to
forecasting.

Consider a dynamic simultaneous equation model incor-
porating leading indicators, in the context of a general
linear multiple time series process. As Zellner and Palm8

indicate, a multiple time series process can be represented

as:

(2.2) H(L) zt = F(L) et t = l, ... , T

pxp pxl PXP pxl

where zt = a vector of random variables measured as

deviations from their means:

H(L) and F(L) are matrices whose elements are poly—

r
1]
nomials in L, the lag operator (h.. = X h.. LQ and
qij 13 2:0 132'
i. = fi°£L£)5
3 2:0 3
et = a vector of disturbances with E(et) = 0 and

' :
E(etet ) Ip.

Given prior information suggesting that some elements

of 21: are endogenous and some are exogenous, the above

system can be rewritten:

r w r w r 1
H11(L) H12(L) yt P11(L) F12(L) e

(2.3) =

LH21(L) H22(L)J Xt LF21(L) F22(L) e

 

 

 

 

 

 

 

 

where yt and e are of dimension plxl,

1t

. o ' + : o
xt and e2t are of d1mens1on p2xl, w1th pl p2 p,

and the Hij and Fij submatrices are of the appropriate

dimensions.

If yt is endogenous and x is exogenous, these restrictions

t

are implied:

H21(L) = 0, F12(L) = 0, and F21(L) = 0.

Hence the above system becomes:

Hll(L) yt + H12(L) xt = F11(L) e1t

plxpl plxl plxp2 p2x1 plxpl plxl
(2.u)

H22(L) xt = F22(L) e2t

p2xp2 p2xl p2xp2 p2xl

This is in the form of a dynamic structural system of
simultaneous equations, with the exogenous variables, xt,
generated by an ARMA process.

In our model, both IPt and LIt will be endogenous to

 

 

 

 

2':
the system. Let LIt - ylt’ IPt - y2t, and yt = the (pl-2)xl
vector of remaining endogenous variables. Then [H11(L) yt]
in(2-W)can be rewritten:
r ] , l
h11 I ylt *
where H (L) is the last
(2 5) h21 | H *(L) {$3 p -1 colimns of H (L)
' I 11 1 « 11 °
C yt
h’ I L J
L P1l J
. h _ h k + **
Wr1te hi1 as 11 - il hi1 ,
k . . . -
where hi1 18 of order k (h1ghest exponent 1s Lk 1),
d h** _ Lk ***
a“ 11 ' hil ;
ii 2-
with hi** : Z hil£L k ’
11 £=k

Consider the first column of H (L):

11
r W fh * ** \
h11 11 + h11
3% **
(2.6) h21 == h21 + h21
h: h *:+ h **
L p11) L p11 p11 J

 

 

 

 

10

We can now rewrite our structural system in (2.9).

 

 

 

 

 

 

 

 

 

 

 

 

 

' 1
h11 I
I
h21 I
I 2':
° + :
. I H11(L) yt Hl2(L) x Fll(L) elt
|
hplll
[h i: I . r 1 '1'] 9:9: I .
11 I J’1’: 11 I ylt]
3': in" X
h21 : y2t 21 : t
(2.7) : *: Hll(L) Lyt J + I 9*: 12(L) = Flfld e1t
[hp 1| hp 1 I
l J t l J
* r ‘*+ 1 k
Ih11 : l h11 : [L ylt
* *fifi x
h21 : 21 : t
(2.8) . : Hun.) yt + : : l2(1.) mummyc
h 5%] *z’n‘: I

This expresses our structural system in terms of our
endogenous variables yt, and a vector of exogenous variables

x and a lagged endogenous (predetermined) variable Lk ylt’

t
k
L yl't - 3':
x - xt °
t

*
Solving for yt in terms of Xt’ we get a system of

equations which we could call the "partial" final form.

11

 

 

 

 

 

 

h11 : ‘1 h11 :
* *3???
h21 : 21 :
3': k
(2.9) y = - I l H (L) I | H (L) x
t . &I 11 . .,,I 12 t
h I h I
( p11 J ( p11 J
r k . 1
h11 : ‘
2':
h21 : .
+ . -
I : Hll(L) Fll(L) 811:
h *I
L P11 J

This expresses the current endogenous variables in terms of
the current and lagged exogenous and one lagged endogenous

(predetermined) variable. This is in the form of a transfer
function with input x*

t

We have now derived a set of pl transfer functions

and output yt.

from our economic theoretical foundation, expressed in our
dynamic structural system of simultaneous equations. Given
our assumptions regarding et, we can fit a transfer function
model about the specific time series we wish to examine.

The second equation in the set of pl equations in
(2.9) gives us the transfer function model for IPt implied

by our structural system:

11

 

 

 

 

h11 : '1 h11 :
* :‘n'n':
h21 : h21 I
(2 9) - ° I H *(L) ° l H (L) *
' yt ' : I 11 : I 12 Xt
h "I h "I
( p11 J ( p11 J
r 5" j 1
11 I ‘
* I
21 I *
+ O
I : H11(L) F11(L) e1t
h *I
\ p11 J

 

 

This expresses the current endogenous variables in terms of
the current and lagged exogenous and one lagged endogenous
(predetermined) variable. This is in the form of a transfer
function with input x*

t

We have now derived a set of pl transfer functions

and output yt.

from our economic theoretical foundation, expressed in our
dynamic structural system of simultaneous equations. Given
our assumptions regarding et, we can fit a transfer function
model about the specific time series we wish to examine.

The second equation in the set of pl equations in
(2.9) gives us the transfer function model for IPt implied

by our structural system:

l2

 

 

 

 

 

 

 

 

 

 

 

[ z: )_ f ::
h11 : 1 hll l
k z'::'::‘: I
h21 I h21 : *
(2.10) y : IP : 1 . 2': .
2t t : I H11(L) : |H12(L) Ixt
11 33¢: h :‘n‘n‘c I
I p11 p11 ' I
\ J t
J
lx( +1) 2'
p? (p +l)xl
fr :1: I W 2
hll I -l
2':
h21 :
2':
' I
+ I I | H11(L) Fll(L) e1t
*l
h I
l
L pl J
I J2.
lxpl plxl

We have thus derived a transfer function model
expressing the relationship between our leading indicator of
economic activity and our measure of economic activity. This
model is firmly embedded in economic theory, as it is derived
from a dynamic structural system of simultaneous equations
describing the world. It is also amenable to empirical
testing, using Box and Jenkins time series methods.10

After identifying and fitting the model, we can
obtain optimal (minimum mean square error) forecasts for

IP given LIt’ and the variance of the forecasts errors.

t+k’

That is, we can extract the first two moments describing the

conditional distribution of IP given LIt, f(IP ILIt).

t+k t+k

This is the object of our analysis of the leading indicator

approach.

13

Hence we have a theoretically sound and empirically
testable framework in which we can study and use leading

indicators.

Comparing the Forecasting Abilities of the Final Form and
the Proposed Leading Indicator Approach

The Final Form

 

The current state of the art of forecasting uses the

Final Form of the structural model expressed in equation

(2.9):
_ -1 -1
(2.15) yt - - H11(L) H12(L) xt + H11(L) F11(L) e1t

To forecast yt+k’ the appropriate ARIMA models for the xt

are fitted, and x is projected k periods into the future.

t

Then the forecast for yt+k is produced as follows:

_ -l -1
yt+k - H11(L)E52(La xtIk + H11(L) Fll(L) elt+k

_1 g _l *9:
- [H11(L) H12(L)] xt+k - [H11(L) H12(L)] Xt+k

(2.16) yt+k

+

-1
Hll(L) Fll(L) elt+k

_ k
where the (i,j)th element of [HII(L) H12(L)] is of order < k

k 1

(highest term is L — ); and the (i,j)th element of

- kt
[H11(L) H12(L)] is of order 3 k, and is equivalent

- ***
to the (i,j)th element of [HlI(L) H12(L)] °(Lk).
' '1 (L)J* - k'1 L2 d
k 2'0 ***
L [H‘1(L) H (L)]..
ll 12 13

-1 **
[Hll(L) HlZCLHij
2..
k 1] h.. Lk-k
13£
£=k

L

1a
Using this notation, we can work with (2.16):

_ -1 *
- — [H11(L) H12(L)] x

yt+k t+k

-1 :‘::‘::': k -1
[H11(L)H12(L)] 1.x +-Hll(L)Fll(L)e

t+k 1t+k

_ l :'::': z":
t+k — [H11(L)Hl2(L)] XI

-1 a
- [H11(L) H12(L)] x

+

1
H l(L) Fll(L) elt+k

Forecasts of xt for periods t+1 through t+k are used in the

first term of the above expression, and the past history of

xt is used in the second term, to get the forecast for yt+k:

(2.17) §t<k) - [HIi(L) H1H(L)] x (k)- U111(L)Hl2(L)I**)<

}

+

l A
H l(L) F11(L) {elt+k

-1 ( 1*.
- [H11(L) H12 L) xt(k)

[H11(L)Hl 2( f** fikk
L) Xt+ [H11(L)F11(L)] e1t

Here yt(k) is the vector of pl forecasts in period t,

of yt+k° Note that xt (k) refers to the vector of forecasts

for the p2 inputs, xj (k), j = l, ... , p2. It is understood

that in this context, [Lb xj t(k)] = x t(k-b) if b < k
xj(t+k-b) if b 3 k
Further note that in this forecast, Hl l(L) F11(L) elt+k
effectively reduces to [H1 l(l) F11(L)]***e1t since E(e1t+k)= 0

for any k = l, 2, ... Here the three asterisks imply the same
reconstruction of the matrix [H11(L) F11(L)] as is used

regarding other matrices throughout the paper.

15

After identifying and fitting the appropriate ARIMA

models for the p2 inputs in x , minimum mean square error

t
forecasts, Xt(i)’ are made for i = l, 2, ... , k.

The forecast error for the jth input is:

A

(2.18) 2X (1)

jt : xjt(l) ’ Xj(t+i)

i—l

‘ n50 wjn aj<t+i-n)

' = . + . . + . .
Slnce xj(t+i) at+i wjlat+1el w]? at+i-2 +
th(l) = Et(xj(t+i)lpa8t Xt]

: +

Wji at wj(i+1) at—l + Wj(i+2) at-2 +

where the W. are the weights of the ARIMA process for the

jn
jth input, written in pure moving average form;
and where ajt is white noise;
. _ ’ 2 _ 2 _
1.e. E(ajt) - O, E(ajt) - Oja’ and E(ajtajt_n)- 0,
for any input j, for any t, and for any n f 0.
A ii].
Note that E[e (i)] = E, W. a. .
xjt “:0 3n j(t+1-n)
iil
: W. E(a. . ) 0
+ _
n=0 jn j(t 1 n)

The variance of the forecast error is:

A . A . 2 1:1 2 2

(2.19) Var [eX (1)] = E[ex (1)] = 2 W.n O'a
jt jt n=0 3 3

Now consider the forecast error of the Final Form

model, from equations (2.16) and (2.17):

16

(2.20) E (k)
y

§ (k)
t t

’ yt+k

. J.

- _ l A _ l ‘4': 2'. n
— -[Hll(L)Hl2(LJ] Xt(k)+.[H11(L)F11(LJ] e1t

-1 -l
+ [Hll(L)H12(L)]xt+ -'H11(L)f11(LJ el

k t+k

A
0'0 0..

_ - 1 :': _ 1 :': .. ..
- -[H11(L)H12(L)] xtCkf-[H11(L)H12(L)] xt

-1 *k*
+ [H11(L) F11(L)] e1t

J. J.

_1 '2: _1 I :':.. ..
+ [Hll(L)H12(L)] XI, + [H11<L)L12(L)J x

k t

0'.

_. 1 :'::': ..
[H11(L)IHJ(L)] e

-1 a
- [Hll(L)Fll(L)] e lt

1t+k"

.. —1 '3 A
- —[H11(L)Hl2(L)] [xt(k)-xt+ ]

k

-1 *fi*
- [H11(L)H12(L)] (Xt_xt)

-1 a
- [H11(L)fil(LJ] elt+k

-e )

-1 ***
+ [Hll(L)Fll(L)] (e1t 1t

- -1 *A
- -[H11(L)H12(LJ] [ext(k)]
-1 *
[H11(L)F11(L)] elt+k

A - _l 3% A
Note that E[eyt(k)] — -[H11(L)1112(L)]IE[ext(k)]

__1 *
— [H11(L)F11(L)] E(elt+k)

= 0
since the parameters in our matrices are fixed;
since E[ex (k)] = 0 as shown in (2.18); and

t
S1nce E(elt+k) = 0 from our assumpt1ons.

It should be clear that in this context,

17

xt(k—b)—x if b < k

eX (k-b) t+k-b

b“ _ t
[L eX (k)] -

t .
Xt+k—b-Xt+k~b if b 3 k.

Observe that the forecast error of each endogenous
variable reduces to a function of the forecast errors of a11
inputs, and the noise associated with Ell pl endogenous
variables occurring over the forecast period.

Assuming that
(i) the input series are mutually independent,

(ii) each input is independent of each disturbance term,and
(iii) the parameters of our model are known with certainty;

consider the variance of the forecast errors.

(2.21) Var I; (k)] E[e (k)]2
yt yt

-1 k A
E{-[H11(L)H12(LJJ [ext(k)]

-1( ( k 2
- [Hll L)Fll L)] elt+k}

E{—[H-1(L)H (foté (km2
ll 12 x

+ '1( ) ( 1* 2
E[[Hll L Fll L) elt+k]

-1 * A 2
E{—[Hll(L) H12(L)1 [ext(k)]}

_1( s
+ [[Hll L) F11(L)]] Var (elt+k)

by (ii) and (iii) above and our assumptions as to elt'

- *
Here [HlI(L) Fll(L)]] is the transformation made by

- *
squaring each element of the matrix [HlI(L) F11(L)] . See

 

footnote(12)ftm'a convincing argument that this is the

appropriate transformation.

18

In this context, the variance of the forecast error
of each endogenous variable is a function of the variances
of the forecast errors of ill inputs, the various covariances
between forecast errors of different time horizons implied
by [H;%(L) H12(L)]* (see footnote (12)),and the variances of
the disturbances associated with £11 endogenous variables.

From our initial assumptions regarding e on page n,

It
we have Var(elt) = Var(elt+k) = 1. Thus equat1on (2.21) can
be rewritten:
(2.22) Var[e (k)] = E{—[H'1(L) H (L)]” [e (k)]}2 +
yt ll 12 xt

-1 k r 1
+ [[H11(L) F11(L)1] 1

 

 

The Proposed Leading Indicator Approach

 

Consider the forecast errors of our leading indicator
approach.
We can rewrite equation (2.9) from our previous work

on the leading indicator approach, in order to examine yt+k°

 

 

 

 

 

 

r *I “-1 r *** I 1
h11 h11
I , I 1
: : H11CL) : * *: H12(L) Xt+k +
- * *
(2'23) yt+k ‘ ‘ h ll h 1 I
L P1 J L p1 J
fhli‘. I ‘I‘v-l
‘ I * (L)
+ .1 *: H11(L) F11 elt+k
hp 1| J
L 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ka y
* _ lt+k lt
But xt+k - =
Xt+k Xt+k
r h ._l r *A* I
h11 ' 11 '
' * ' ylt I
Thus, yt+k : _ Z I Hll(L) : I H12(L) X I
* 2'*
h lI h 1‘ I t+k
L pl I J L pl I J
V .t. 1
r “ 1-1
h11 :
k
+ I I Hll(L) Fll(L)I elt+k
h "I
L p11] J
I J
Hence our forecasts are:
I .2 .-l , we: .
hll l hll l
(2 2a) A (k) ' i H * ° I ylt
° yt ‘ ’ : (I 11(L) : * : H12(L) .
. ':§:
h lI h 1: xt(k)
(p1 I . (p1 I .
' k -1 111*
' 1
h11 I
2.
+ I I I H11(L) Fll(L)I e1t
Ih *:
(I pllI . .

 

 

Here again, xt(k) refers to the vector of forecasts
for the p2 exogenous inputs, xjt(k), j = 1, 2, ... ,p2.
Further, the simplification of the disturbance structure in
this forecast is analagous to that in the Final Form
forecast in equation (2.17)

After identifying and fitting the appropriate ARIMA

20

models for the p2 inputs in Xt’ minimum mean square error
A

forecasts, Xt(i)’ are made for i = l, 2, ... ,k. The

related forecast errors and their variances are expressed

in equations (2.18) and (2.19). From equations (2.23) and

(2.2”) we get the forecast error of our model:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2.25) eyt(k) = yt(k) - yt+k
I, 3’: ._1, :'::'::': \‘
hll l hll I
. I 5‘: I ylt
= -I : I H (L). : I H (L) I
. l 11 . I 12 x
* 1.1 xt(k)
h 1 l h 1 l
L P1 I J L P1 I .I
L J
frh * I 1"]. ‘0":
11 I
+I : 'H *(L) r (L)» e
' a I 11 11 1t
h I
IL p11 I I
J
(I a I_1I *2: I‘
hll l hll I
. I * . ' ylt
+ I : I HllCL) I IH12CL) I
h *l h 11* l xt+k
LP 1 LP 1
I 1 I I 1 I J,
V it -1 ‘
Ih I I
11 I * I
- 4 I I H11(L) F11(L) elt+k
*
h I
1
IL pl I J J

 

 

To work with this, we will again use our conventional
notation (*, **, ***) to break up these matrices into those

containing parameters with lags < k, and those containing

21

Rewriting (2.25):

parameters with lags > k.

 

 

 

 

 

 

t k
l (t
yAx
{ll-Ill

.«u

‘ )

L
(
2
1
H
[Mall

.fl .3

nun onul

* l
1 .0 p

m h

l

.

W )

L
(
*1
l
H
.21
.R1
1 co.
_
.-
)
k
(
t
y
Ae

 

 

 

 

)
L
(
2
l
H
.3
.8 .n»
.«n can 1
."n l
l .p
urn h
l
_
)
L
(
.fl 1
l
H
.3 l
.8 l
1 .0 p
m h

 

 

 

 

 

 

 

 

 

 

 

\'|ll|l
b.“
t
l t
y X
{I'll
.3
.3
.8
‘ ) J
L
(
2
1
H
.8
k *.l
.81 .8 1
.31 on. p
MW. mw
l
_
‘ ) J
L
.(
.81
l
H
.81 .81
1 0.0 p
mm .W
....

 

 

 

 

t
1
e
.3
.3
...»
‘l ) J
L
(
l
1
F
l
.
‘ ) J
L
(
.31
l
H
*1
t 1 1
1 on. p
mm h-
....

 

....

.K

+

t

l

e

\I

L

(

1

l

P.
ill.)

)

L

.(

*1

l

H
.rl
l l
l .p
h h
i

 

 

 

 

 

 

 

 

 

 

22

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lIIIIIJ IIKJIIJ
t _
1 ) R +
Y k y t i
_ (\ ...K X \J
t t _ _ k
l X +L .T. \l (
y A e l X ..L O +L
{Ili y l X
.n .. (IIIIIIK e A e
‘ L J 0.1 - r L
t w J .m k t ...»
l ) J4 L + l V I}
e L ‘1 JJ t e W
(\ \l l (\ \.I
2 ”w e .x L
l 2 .«u I...
H l .3 .fl 2
L J llllll H W i J ‘ L J l
) llllll ) ) H
L .x .3 L L llllll
(x .u .u l .n l (x ( .x
l .8 l on“ 1 .3 l on“ 1 l 1 .fl .3 1
1 .a l . P .x l . P 1 l .x l .x 1
P h h h h F F .3 1 . P
r L f L h h
l l l l 1 r it
. _ . _ _ l
‘ J \. J \I J N J W J —
\J \l \l \l \I l
L L L L L \l
I.\ I.\ ...(\ ( I.\ L
.r l .. l ... .1 .w l . l (
l l 1 1 l .3 l
H H H H H l
llllll H
.n. 1 ... l ...: l .3 l .u. l .3 l
l l .. l l . l l .3 ..l. l .3 1 l .8 l l
l o o P o. l o p 1 p l o o p l p l o o P
h hL ave rh hL an hL ,h hL rh hL h h It
1‘ t O r J L r J L r 11 IL r J L r {i
..D
a _ _ . + _
— e Z __
h \J
t k
{\
g t
n y
.l A e
y
f
.1 )
l 6
D. 2
m .
Cl 2
S (

 

 

f :9: "l ‘ v
f \
hll l
I * L
— f . IH11(L) F11(L) elt+k
,I
I
1
LL pl | J J

 

 

Observe that in this model, the forecast error of
each endogenous variable is a function of the forecast
errors of all inputs, and the noise associated with all pl
endogenous variables occurring over the forecast period.
Compare equation (2.26) with equation (2.20).

Again, we wish to consider the variance of the
forecast errors, given these assumptions:
(i) the exogenous input series are mutually independent,
(ii) each exogenous input is independent of each disturbance,
and

(iii) the model parameters are known with certainty.

A A 2
Var[ey (k)] E[ey (k)]

 

 

 

 

 

 

 

 

 

t t
'"'. * w-lr *** a‘*
h11 : h11 : 0
*
'-' E “i : :H11(L) . :H12(L) ‘ ji—
* *** ex k)
L p1 I J p1 I J,
rrh * w-l 1* _12
ll
*
-< . l H11(L) Fll(L)L e1t+k
*
h I
LL p111 J J _‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I. r 3': ‘-1r :'::'::‘: , 2
Ihll I hll I
| g; . I 0
= B _, I IH11(L) I |H12(L) } ;*————-
. *I h *v€*I ex (k)
h I I t
b J
rrrh 3‘: | 1"]. .0. ’I2
11
' ° I * (L)
+ B ‘ : {H11(L) F11 I elt+k
a!
I
1
LLL pl I J J
J

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r I, a ,_1, *** ,‘a q
I I
h11 I h11 , 0
' I ' I ______
= E _‘ . lHun.) : |H12(L) I A
* *** e (k)
h 1' h 1 I xt
P P
LLL ll J I 1 I ‘JI J
fr! *| W-l \wi‘
hlll
+ I ' IH *(L) F (L) II E(e )2
I : . 11 11 lt+k
*
h I
1
LLLplI J J‘

 

 

 

 

[by (iii) above and our assumptions as to elt]
Note that as before, double brackets around a matrix,
{{ })*, refer to the transformation of the original

matrix with single brackets, in which each element is

 

squared. Again refer to footnote (12).
Thus our forecast error variance finally reduces to

the following:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I 2': , _ 1 :': :‘: :': 2
I F; 1
h11 ' H11 '
I a . I 0
= E -I : 'H11(L) : 'H12(L) I A
h *I ***| ext(k)
I I
II P11: . I P11 I .J
b I
If * - :1: .12
rh I 1 1
11
I I * (L)I
+ .
E I : :H11(L) F11 e1t+k
*
I
I pll| J
LI 1 J
[by (ii) above]
I Ifh * I 1-1rhfnka' I 1“" 'I
11 I * 11 I 0
' I ’ I ______
= E -I : IH11(L) : IH12(L) I A
* *fi* 3 (k)
h 1' h 1 I xt
P1 I P I
fr * I -l H"
hll I I
+ I ° IH *(L) F (L) II H(e )2
I : I 11 11 1t+k
*
h I
II plll J I;
[by (iii) above and our assumptions as to elt]

Note that as before, double brackets around a matrix,

{{

}}*, refer to the transformation of the original

matrix with single brackets, in which each element is

squared.

 

Again refer to footnote (12).

Thus our forecast error variance finally reduces to

the following:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I. r :‘t , _1 :'::'::‘: , ,‘c ‘2
Ih11 I Ih11 I , 0
A :‘z
(2.27) V[e (k)] = IE -I : I (L) . III (L)I A
O I 1
yt *I ll :‘cs‘n'cI ex (k)
hp 1: hp 1 I t
I. LL 1 I J I l I JJ .I
p1x(p2+1) (P2+1)X1
(If k ‘_1 “«
h11 :
k
+ II : :H11(L) F11(L)II V(e1t+k)
*
h I
l
plxpl plxl

In this context, the variance of the forecast error
of aaah endogenous variable is a function of the variances
of the forecast errors of all exogenous inputs, the
appropriate covariances between forecast errors of different

time horizons implied by

 

 

 

 

 

', * I‘lr *éé ,‘g
h11 I , h11 I I
I I IH11(L) : IH12(L)
h *I h°***1
1| 1 I
L I pl J I p1 J I

 

(again see footnote (12)), and the variances of the
disturbances associated with all endogenous variables.

Compare equation (2.27) with equation (2.21).

 

26

Example 1
Consider an IS-LM model.
Commodity Market
' : + + +
(1) Ct a blYt—l b2rt—l elt
: _ + +
(2) It It bBPt—l e2t
: +
(3) Yt Ct It
Money Market
M?
_ : +
(u) Pt buYt-l bsrt-l + eat
5 -
(5) Mt - MO + b6rt + eHt
s _ d
(6) Mt - Mt
(7) Pt = Pt
where Ct = consumption, It = investment, Yt = output,
rt = "the" interest rate, P1: = commodity price
level
a = autonomous consumption, ft = autonomous
investment
and the e. are disturbances with E(e. ) = 0,
1t it
B (e. e. = O and E(e. )2 = O. for i = l 2 3 u.
1t 3t 3 1t 1 9 3 3 ’ 9
i f j.

The model consists of seven equations and seven unknowns:

d

Yt’ rt, Pt’ Mt’

M C and It'

s
t’ t’

Note thatcnu~dynamic formulation simply says each

right hand side

endogenous variable affects the left hand

27

side variable with a lag of one period, with one exception.
The exception is the Money Supply equation. This equation
reflects the likelihood that,banks will react quickly and
efficiently in adjusting their excess reserve positions, in
response to changes in the interest rate.

I wish to assume that a Keynesian aggregate supply
curve corresponds to this world. That is, assume:
(i) whatever output is demanded can be produced, and
(ii) Pt = Ft’ as expressed in equation (7).

Note that we can express the system as two equations in two

unknowns. These are the IS and LM relationships.

IS: Yt = Ct + It
= a + blthl + bZPt—l + e1t + It + bBPt-l
+ e21:
.. : — +
(7) (l blB) Yt a + It + (b2 b3)B rt + (e1t+e2t)
where B = the backshift operator, or lag operator
(previously specified as L).
d - 5
LM Mt - Mt
_ + _ _ : +
bup Yt—l bsp rt—l + P eSt M0 be rt + eat
(8) (—b6+b5PB) rt = M0 - buPB Yt + (eat-PeBt)
Let elt+e2t - eSt’ and eat—Fe3t = eSt’
noting that E(e5t) = H(efit) = 0, and E(e5t-e6t) = 0.

We now have two equations [(7) and (8)] in two

unknowns: Yt and rt. This is the classic IS—LM problem, in

28

a linear structural model framework.

The candidate for a leading indicator of Yt in this

context is rt. Observe that rt affects Yt+l through its

effect on Ct+l and It+ 1n the commodity market, and

l

affects Y through its effect on Yt+l’ which is involved

t+2

1n the IS equation for Yt+2. Further, rt affects Yt+l

through its effect on the LM relationship in period t+l. In
short, rt is a factor in the determination of the locations
of both the IS and LM relationships in period t+l.

Consider equations (7) and (8) in matrix form.

-(b2+b3)B (l—blB) rt a l 1 e51:

II
+

(9)

_ + _ — —
( b6 bSPB) buPB Yt M0 0 1t eBt

This is our structural model with endogenous

variables rt and Yt’-

ourtransformation,vmflll.separate the polynomials in B

and exogenous variable, It. Applying

multiplying r into a component with lags, k < l, and a

t,

component with lags, k 3 l.

O (l-blB) r (b2+b3)B a l r e

t t 5t

_ = _ l +
-b6 buPB Yt -b5PB M0 0 ‘ft e6t

or:

0 (l-blB) rt (b2+b3) a 1 Brt 'e5t

_ = _ 1 +
-b6 buPB Yt -b5P IQ) 0 I; IeSt

P I
Now we can solve for our endogenous variables, Y: ,

 

29

 

 

Br
1:
in terms of our predetermined variables, 1
.—1 It
rt 0 (l—blB) (b2+b3) a 1 Brt
: l
Yt —b6 buPB J —b5P M0 0 It
.-1
0 (l-blB) e5t
+
-b6 buPB , e6t
-1 _
0 (l-b B) b PB
Substitute' l ' l u
' _ b6(l-blB7
--b6 buPB b6
rt 1 buPB -(l-bfn (b2+b3) a l
= b l-bB
6 l —
Yt b6 0 -b5P M0 0
1 buPB -(l-blB) e5t
+
b6 l-blB
b6 0 e61:

Multiplying through the matrices:

I”t
Yt
[buPB(b2+b3)+b5P(l-blB)J [abuPB-MOU-blBfl
1
-
b6(l 131—713
L b6(b2+b3) ab6

 

Br

HI

bPB‘

 

H

 

 

30

b PBe - (l-blB)e6t)

This is the transfer function model implied by our
dynamic structural system of simultaneous equations.

Explicitly, the two transfer functions are:

 

[b BB(b +b )+b F(l—b 8)]
r1: = (const)r + u 2b (l bsB) l (Brt)
t 6 ' 1
buBB _
+ (I )
b6 l—blB t
+ 1 [b‘FBe - (l—b B)e J
b6(l-blB) u 5t 1 6t
b6(b2+b3) b6 _
Y : (CODSt) + fl (Br ) + j—T—y (I )
t Yt b6 1-bl t b6 1 blB t
1
+ [b e J
_
b6(l blB) 6 5t

From the second transfer function, it is clear that
Yt is a function of lagged values of our leading indicator,
rt. Thus we can fit this transfer function for Yt’ and
come up with the estimated mean and variance of Yt given
past rt. And more importantly, we can come up with an

estimate of Yt+1 given r That is, we can estimate the

t.

mean and variance of the conditional distribution,f(Y II‘

t+l t)°
This is the object of our analysis of the leading indicator
approach to forecasting. Consider the relative forecasting

abilities of the Final Form (FF) approach and our leading

indicator (LI) approach, in the context of example 1.

31

In the form of equation (2.”), we have:

Hll(B) yt + H12(B) xt = Fll(B) e1t
-(b2+b3)B (l—blB)- rt —a —l l e5t
+ :
(-b6+b5PB) buPB Yt —M0 0 It e6t

The Final Form:

-1

.. + -
rt (b2 b3)B (1 blB) a l l
Yt (—b6+b5PB) buPB M0 0 It

-(b +b )B (l-b B) ‘1 e

2 3 1 5t
+
(-b6+b5PB) buPB e6t

Call the matrix to be inverted, A.

- — 2 —
det A — -buP(b2+b3)B + (l-blB)(b6—b5PB)
= b - b FE — b b B + b b BB2 — b F(b +b )62
6 5 1 6 1 5 u 2 3
_ — — 2
— b6 — (b5P+blb6)B + F(ble-bq(b2+b3))B
buPB -(l-blB)
adjoint A =
(be-bSPB) -(b2+b3)B
A‘1 = _
buPB —(l—blB)

1
— — 2
b6-<b5P+blb6)B+P(blb5_bu(b2+b3))B (bB-bSBB) -(b2+b3)B

32

Substituting this into the FF and moving the determinant to

the left hand side, we get the following.

r
t
—- — 2
[be-(b5P+blb6)B+P(blb5~bu(b2+b3))B ] :
Y
t
buPB -(l-blB) a l l
bS-bSPB —(b2+b3)B M0 0 It
+ buPB -(l-blB) eSt
bB—bSPB -(b2+b3)B 861:
Pt [abuPB-M0(l-blB)] buPB l
[det A] = _ _ _
Yt [a(b6-bSPB)-M0(b2+b3)B] bB—bSPB It
+ bMPBGSt - (l-blB)e6t
(be—bSPB)e5t - (b2+b3)Be6t

Thus we have the Final Form transfer functions:

[det A] rt = (const)rt + buPBEIt] + buPBe5t
- (l-blB)e6t
(1) [det A] Yt = (const)Yt + (be-bSPB)[It]

+ (bs—bSPB)e5t — (b2+b3)Be6t

These are to be compared with the transfer functions of our

leading indicator approach which we derived in the example:

33

(ii) b6(l-blB)rt (const)1n +'[buPB(b2+b3)+b5P(l-blB)[Brt]

t

+ '— _ + _
buPBEIt] b PBe

u 5t ’ (l-blB)eBt

+b3)[Brt]+-b6[lg +b6e

(111) b6(l-blB)Yt (const)Y + b6(b 5t

t 2

We are interested in the models for Yt' Consider the
forecasts of Yt+l implied by each of the approaches above,
in turn.

The Final Form:

From equation (i), we get the model for period t+l:

— — 2
[bB-(b5P+blb6)B+P(blb5-bu(b2+b3))B ]Yt+l

t + _ ’ _ - _
(cons )Y (b6 bSPB)[It+l]+-(b6 bSPB)e

t+1 5t+l

- +
(b2 b3)Be

6t+l
Yt+l = (const)Y + bi (6515+blb6) Yt
t+l 6
_ 3; F(b b —b (b +b ))Y’ +-34 (b -b BB)[I ]
b6 1 5 u 2 3 t-l. b6 6 5 t+l
+ l. (b _b fiB) e - —1—(b +b ) Be
b6 6 5 5t+l b6 2 3 6t+l

Now expand. the remaining Final Form coefficients on
the right hand side into the values corresponding to the
various lags of the right hand side variables and distur-

bances.

3H

 

 

- j; —
Yt+l - (const) + b6 (b5P+blb6) Yt
l -— _
- —— - +
b6 P (blb5 bu(b2+b3))Yt_l [It+1]
b F b _ b +b
5 — 5 2 3
— ——— [I 1 + e - ___ e - e
6 t 6t+l 6 5t b6 6t
From this, the Final Form forecast is made:
A — l _
Yt(l) - (const) + b; (bSP + blb6) Yt
1_ _ 1
- -—— — + +
be P (blb5 bu(b2 b3)) Yt-l [It(l)]
_ _ i +
‘ b5? [it] ’ :5? e51: " 1326b3 e6t
6 6 6

From the above two expressions we get the Final Form

forecast error:

A A

(iv) eYt(l)= ‘yt(1) - Yt+l

[It(l) - I

t+l]

= [£- (1)] -
It

"85t+1 e5t+1

The variance of the Final Form forecast error follows:

(1)] E[eY (1)]2
t t

E{[$- (1)]
It

Var[eY

- e }2
5t+l

2

Btéf (1)]2 + F(e )

t

(given the assumption that It and 35t are uncorrelated)

5t+l

The Leading Indicator Approach:

From equation (iii) we get the model for period t+l:

35

(l-blB) Yt+l = (const) + (b2+b3)[Brt+l] + [Tt+l]
+ eSt+l
Yt+l = (const) + blYt + (b2+b3)[rt] + [Tt+l]
+ e5t+1

From this model our forecast is made:

Yt(l) '= (const) + blYt + (b2+b3)[rt] + [It(l)]

And from these two expressions we get the forecast

€I‘I‘OI’Z

)

(v) eY (1) §t(1)- Y

t+l

= [It(l) - I ]

t+l : [ef (1)] ’ e5t+1

— e

5t+l t
Comparing equations (iv) and (v), we see that our

leading indicator approach and the Final Form approach yield

the same forecast errors in this example. Our leading

 

indicator model, however, appears in a much simplier form

in this case.

Example 2

 

Consider a stochastic model which is an extension of
Samuelson's Multiplier-Accelerator Model combined with

Metzler's Inventory Model.

(1) Ct = a + bYt—l + e1t

(a short run consumption function)

36

(2) It = It + C(Ct-Ct-l) + e2t

(an investment model with c

the accelerator
coefficient, and

It = autonomous investment)
(3) Vt = d(Ct—l—Ut-l) + e3t
(Vt = production for inventory purposes)
Identities:
(u) Ut = Ct-l
(Ut = production to meet anticipated sales in periodtfl
(5) Y = U + v + I

t t t t

(income accounting identity)
: Ct-l + Vt + It
(el,e2, and e3 are disturbances with E(ei) = 0 and

E(eiej) = 0 for i # j.)
Equation (3) says that inventories are rebuilt

according to the difference between previous consumption and

sales.
Substituting for Yt in (l), and for Ut in (3); the

system becomes:

Ct = a + b(Ct-2 + Vt-l + It—l) + e1t
(6) It = It + c(ct - Ct-l) + e2t
Vt = d(Ct-l - Ct-Z) + e3t

Rewriting this system, with B = the backshift

operator:

37

2 _
(l-bB ) Ct - a + bB Vt + bB It + e1t
(7 : _ + _
) It It C(l B) Ct + e2t
V = d(B-B2) C + e
t t 3t
In matrix form:
r_ _ 2 _ 1 I 1 r 1 1 r 1
b8 (1 bB ) bB vt a 0 I 1 elt
(8) O -c(l-B) 1 Ct = 0 1. It + e2t
2 .
l -d(B—B ) 0 I O 0 e
L J k t] k ]( J ( 3t]

 

 

 

 

 

 

 

 

 

 

Note that this system consists of three equations and

three endogenous variables. It is exogenous.

Production for inventory purposes, Vt’ is the
candidate for a leading indicator. Increases in Vt affect
increases in Y1: directly, as production for inventory
purposes is included in Yt' Increases in Vt also affect

increases in C and thus affect increases in Yt+l through

t+l;

the effect of Ct+l on I and affect 1ncreases 1n Yt+2

t+l;

through the effect of Ct+l on Ut+2'

should be consistently followed by movements in Ct+l’ It+l’

Hence movements in Vt

and Yt+l°
We can apply OUPtransformation to the system in (8)

by separating the polynomials in B which multiply Vt’ into a

component with lags k < l, and a component with lags k 1 1.

Note that since the period under consideration in this model

is one year, a lead of one period is relatively substantial.

 

 

 

 

 

 

 

 

 

 

 

2 1 r 1 r 1 r 1 r 1
f0 (l-bB ) -bB vt bB a o vt elt
(9) O —c(l-B) 1 Ct : 0 0 l l. +_ e2t
2 ...
.1 -d(B—B ) o I .ItJ .0 0 OJ (ItJ .e3tJ
or equivalently:
2 1 r 1 r 1 r 1 r 1
r0 (l-bB ) —bB vt b a o B v,C elt
o -c(l-B) 1 ct = 0 0 1 1 + e2t
2 _
1 -d(B-B ) 0 I o 0 0 I e
. J ( tJ ( J . t J . 3U

 

 

 

 

 

 

 

 

 

Solving the system for the endogenous variables in

terms of the predetermined variable, [B Vt]’ and the exogenous

variable, It:

 

 

 

 

 

 

 

 

Ivt‘ F0 (l-bB2) -bB"1fb a 0‘ [B vt‘
(10) ct = 0 -c(l-B) 1 o 0 1 1
2 _.
I 1 -d(B—B ) o o 0 0 I
I tJ L J . L J I t J
r 2 1-1 1
o (l-bB ) -bB Ielt
+ 0 -c(1-B) l e2t
2
l -d(B—B ) o e
I J . 3’CJ

 

 

 

 

Call the matrix we wish to invert, A.

det A (l-bBZ) - bc(B-B2) = l — ch + b(C--l)B2

‘

'd(B-B2) 66(62-63) [l-ch+b(c-l)B2]

1 b8 0

[adjoint A]
c(l-B) (l-bB2) o

 

 

I

39

 

 

 

 

Thus,
2 2 3 2
d(B—B ) bd(B —B ) [l—ch+b(c—1)B ]
A'1 = l 2 1 b8 0
_ + _
With this substitution, (10) becomes:
Vt
Ct =
LItJ
r 2 2 3 2 1
d(B-B ) bd(B -B ) [l-bCB+b(C-1)B ] b a th
l 2 1 b8 0 0 0 l l
l‘bCB+b(C'1)B c(1-B) (l-bB2) 0 o o It
L J

 

d(B-B2) bd(B2-B3) [l—ch+b(c-1)B2] e

 

 

1 lt

+ 1 b8 0 e
1-ch+b(c-1)B2 2 2t
c(l-B) (l-bB ) 0 e3tJ

Multiplying out the coefficient matrices, and moving the

determinant to the left hand side:

 

 

t
(11) [l-ch+b(c—1)B2] ct =
LItJ
rbd(B-B2) ad(B-B2) bd(Bz-Bs) th
b a b8 1 +
Ibc(l-B) ac(l-B) (l-bBZ) It

 

39

 

 

 

 

 

 

 

 

 

 

Thus,
2 2 3 2
d(B-B ) bd(B -B ) [l-ch+b(c-1)B J
A“1 = l 2 1 b8 0
_ + _.
1 bCB b<c l>B c(l-B) (l-bBZ) 0
With this substitution, (10) becomes:
' 1
Vt
ct =
(ItJ
r 2 2 3 2 1 r 1
d(B-B ) bd(B -B ) [l-ch+b(c-1)B J (6 a 0 th
1 2 1 b8 0 o 0 1 1
I I J I J
d(B-B2) bd(Bz-B3) [l-ch+b(c-1)B2] elt‘
+ 1 1 DB 0 e
1-ch+b(c—1)B2 2 2t
c(l-B) (l-bB ) o e3t

Multiplying out the coefficient matrices, and moving the

determinant to the left hand side:

 

 

Vt
(11) [l-ch+b(c-1)B2] ct =
IItJ
rbd(B-62) ad(B-B2) bd(B2-B3)‘ 'th)
b a b8 1 +
Lbc(l-B) ac(l-B) (1-662) , LI

 

 

 

 

NO

(11) (cont'd.)

r 1

 

_ + _ + — -
delt-l delt-Z bde2t_2 bde2t_3 e3t bce3t_l+b(c l)83t-2
elt + be2t-1
_ + _
I celt ce1t-1 e2t be2t—2 I

 

This is the transfer function model implied by our
dynamic structural system of simultaneous equations. Our

inputs are the leading indicator, V and the exogenous

.t,

variable, I Explicitly, the three transfer functions are:

t.

[l-ch+b(c—l)B2] Vt = (const)V + bd(B-B2)(BVt)
t

2 3 —
+ bd(B -B )(It) + [delt-l - delt-2 + bde2t_2

- bde + e - bce + b(c-l) e ]

2t-3 3t 3t-l 3t—2

(12)
[l-ch+b(c-1)B2] c = (const) + b(BV ) + 66(I )
t Ct t t

4.
+ [elt beZt-l]

[l-ch+b(c-1)B2] It (const)I + bc(l-B)(BVt)

t

2 _
+ (l-bB )(It) + [ce1t - celt—l

]

+ e2t ' be2t-2

Finally, we can aggregate our transfer function models

to yield the time series model for Y implied by our dynamic

t,

structural system of simultaneous equations.

Lil

From equation (5) we have:

4.
Vt + It

From this identity and the transfer function models for the
components in (12), it is clear that Yt is a function of
lagged values of our leading indicator, Vt'

Hence, using time series methods, we can fit these
transfer functions and come up with the estimated mean and

variance of”Y given values of our leading indicator in

t+l
previous periods. That is, we can estimate the first two
moments of the conditional distribution of Yt+1 given Vt’
f(Yt+l|Vt). This is the object of our analysis of the
leading indicator approach to forecasting.

With this knowledge we can produce optimal forecasts
of Yt’ which are presumably better than forecasts produced
without the incorporation of our knowledge of the structural

relationships between our leading indicator and the other

variables in our model.

Consider the relative forecasting abilities of the
Final Form approach and our leading indicator approach, in

the context of example 2.

In the form of equation (2.u), we have:

 

 

 

 

U2

 

 

 

 

 

 

 

 

H11(B) yt + H12(B) xt = Fll(B) elt
r-bB (l-bBZ) —bB‘ 'vt‘ ‘-a 0‘
0 -c(l—B) 1 ct + o -1
2
1 —d(B—B ) 0 I 0 0
L I tJ \ J
The Final Form:
(Vt‘r-bB (l-bBQ) —bB"lfa 0‘ 1
ct: o —c(l-B) 1 o 1. It
2
I 1 —d(B-B ) 0 0 0
I tJ. J . J
-bB (l-bB2) -bB ‘1 e
lt
+ 0 - -
c(l B) l e2t
2
1 —d -
(B B ) 0 e3t

Call the matrix to be inverted, A.

det A

adjoint A

A-l

l

—_

det

 

 

1 e11:
It : e2t
.831“)

(l-bB2) - bc(B—BZ) - bd(BQ-Bs)

2

2 + has

2 - ch + bCB

l - b8

1 - ch + (be-b—bd)B2 + 666

 

rd(B-Bz) bd(Bz-Ba)
: 1 13B
c(l-B) (1-b62)-bd(62-B3

[adjoint A]

+ de3

3

‘

(l-sz)-bc ( 6-62)

bB

 

) bc(B~BZ)

H3

Substituting this into the Final Form and moving the

determinant to the left hand side, we get the following:

Vt
[det A] Ct =
It
d(B-Bz) bd(B2-B3) l—bB2-bc(B-BQ) a 0
1
= 1 bB b8 0 1 [— I
I
2 2 3 2 t
c(l-B) l—bB -bd(B -B ) bc(B-B ) 0 0
d(B-Bz) bd(BZ—B3) l-bBZ-bc(B-B2) e1t
+ 1 b8 bB e2t
c(l-B) l-sz-bd(Bz-B3) bc(B-Bz) e3t

Multiplying through the matrices:

 

2 2 3 1
Vt ad(B-B ) bd(B —B ) l
[det A] Ct = a bB I
2 2 3 t
It ac(l-B) l-bB -bd(B -B ),
r 2 2 3 2 2 1
d(B-B )elt+bd(B -B )e2t+[l-bB -bc(B-B )]e3t
+ elt + bBe2t + bBe3t

2

 

 

2 3 2
c(l-B)elt+[l-bB -bd(B -B )]e2t+bc(B-B )e3t

I J

Consider each of the three Final Form transfer functions in

turn. First;

[l-ch+(bc-b-bd)B2+bd33] vt = (const)v
t
2 3 — 2 2 3
+ [bd(B -B ) [It] + d(B—B )e1t + bd(B -B )e2t
2

2
+ [l-bB -bc(B-B )]e3t

H3

Substituting this into the Final Form and moving the

determinant to the left hand side, we get the following:

 

 

 

 

 

 

,
(Vt
[det A] Ct =
LItJ
'd(B-62) bd(BZ-B3) l—bBQ-bc(B—BQ)‘ a o
1
= 1 b8 bB o 1 [f I
2 2 3 2 t
Lc(1-B) l-bB -bd(B -B ) bc(B-B ) J o o
d(B-B2) bd(BZ-B3) l-bB2—bc(B-B2)‘ elt‘
+ 1 bB bB e2t
C(l-B) l-bBZ-bd(BZ-B3) bc(B-B2) 1 eat}

Multiplying through the matrices:

 

 

 

 

’v ‘ 'ad(B-B2) bd(BZ-Bs> ‘
t l
[det A] Ct 1' a 138 T
2 2 3 t
LIt, Lac(1-B) l-bB -bd(B -B ),
r 2 2 3 2 2 1
d(B-B )elt+bd(B -B )e2t+[l-bB -bc(B-B )Je3t
+ e1t + bBe2t + bBe3t
2

 

2 3 2
—bd(B -B )]e2t+bc(B-B )e3t

 

Lc(l-B)elt+[l—bB

J

Consider each of the three Final Form transfer functions in

turn. First;

[l-ch+(bc-b—bd)82+bd83] vt = (const)v
t
2 3 — 2 2 3
+ [bd(B -B ) [It] + d(B—B )e1t + bd(B -B )e2t
+ [l-sz-bc(B-B2)]e3t

U3

Substituting this into the Final Form and moving the

determinant to the left hand side, we get the following:

[det A]

 

 

tJ
'd(B-B2)

l

 

LC(l-B)

(d(B-BQ)

l

 

{C(l-B)

Multiplying through the matrices:

Vt

C

[det A] t

 

 

 

LItJ

 

_ + _
Lc(l B)e1t [1 b8

2

 

 

 

 

bd(Bz-B3) l—bBQ-bc(B-B2)‘ a 0‘
bB bB
l-bBZ-bd(B2-B3) bc(B-B2) J

bd(BZ—B3) l-bB2-bc(B—B2)‘
bB bB
l-sz-bd(B2-B3) bc(B-B2) 1
‘ (ad(B-B2) bd(B2-B3) ‘ 1
a bB [TI]
2 2 3 t
Lac(l-B) l-bB -bd(B -B )J
'd(B-Bz)elt+bd(B2-BB)e2t+[l-sz-bc(B-B2)Je3t
e1t + bBe2t + bBe3t

2 3 2
—bd(B —B )]e2t+bc(B-B )e3t

[

 

 

‘

 

J

I

l

1

t

Consider each of the three Final Form transfer functions in

turn. First;

[l-ch+(bc-b—bd)Bz+bd83] vt

+

2

+ [l-bB

2
-bc(B-B )]e3t

(const)v

2 3 — 2
[bd(B -B ) [It] + d(B-B )e1t + bd(B

t 2

3
-B )e2t

an

(i) Vt = (const)vt + cht_l+(b+bd-bc)Vt_2--bdvt_3
+ deIt_2] - bd[It_3]'t[delt_l-delt_2-+bde2t_2
- bde2t__3 + e3t - bce3,t_l + b(c—l)e3t_2]
Second;
[l-ch-t(bc-b-bd)B2+de3]Ct == (const)Ct+ bB[It]
+ elt + bBe2t + bBe3t
(ii) ct = (const)Ct+-bcCt_14-(b+bd-bc)Ct_2-det_3-+b[I$€l]
+ [elt + be2t-l + best-1]
Third;
[I-ch+(bc-b-bd)62+de3]It==(const)It
+ [I-bB2 - bd(B2—B3)][I J + c(l-B)e
t lt
+ [l—sz-bd(B2-B3)]e2t + bc(B—B2)e3t
(iii) It == (const)It4-bclt_l+-(b+bd-bc)It_2-bdlt_3+'[It]

- b(l+d)[It_2]+bd[It_3]+[celt—celt_l+e2t

- 1 _
b(l+d,e2t_2+bde2t_3+bce3t_l bce3t_2]

Note here that It is endogenous and If is exogenous.

The aggregation of equations (i), (ii), and (iii)

yields the Final Form transfer function of Yt Vt + Ut + It

vt + ct_1-+It.

This is done on the following page.

H5

Yt = (const)Vt + cht_l + (b+bd—bc)Vt_2 - det—B
+ deft-Z] - bd[I£_3] + [delt-l-delt-Z
+ bde2t_2-bde2t_3 + e3t - bce3t_l + b(c-l)e3t_2]
+ (conSt)Ct-l + bcCt_2 + (b+bd-bc)Ct_3 — det-u
+ Mic-2J + [elt-l + be2t-2 + best-2J
+ (const)It + bCIt-l + (b+bd-bc)It_2 - det-B
+ [It] - b(l+d)[It_2] + deIt_3J
+ [celt—celt_l+e2t-b(l+d)e2t_2+bde2t_3+bce3t_l
- bce3t_2]

: (const) + bCEVt_1+Ct—2+It-l]

+ (b+bd—bc)[Vt_ +C +1 3

2 t_3 t_2]-bd[Vt_ +C +1

3 t-N t-3

+ [It] + [celt+(l+d-c)elt_l-delt_2+e2t+e3t]

(iv) Y (const) + bc[Yt_l] + (b+bd-bc)[Yt_2] - deYt-3]

...

- + + +
[It] [cel e e

t 2t + (l+d-c)elt_ -de ]

3t 1 lt—2

Note that (const) = (const) + (const) + (const) .
Vt Ct-l It
Our initial assumptions as to these disturbances were:

_ 2 _ 2
E(eit) - 0, E(eit) - Oi , H(eiteit-k

and E(eitejt) = 0 for 1, 3=l,2,3 and 1 i 3.

We see that the disturbance structure of our aggregation of'

) = 0 V k i 0,

Yt is a second order autoregressive model about white noise.

1+6

From equation (iv) we get the Final Form forecast.

Yt+l = (const) + bc[Yt] + (b+bd—bc)[Yt_l] - bd[Yt_2]
+ [Tt+l] + [celt+1+92t+1+e3t+l+(1+d-C)elt_delt-I]
(v) 2t(l) = (const) tbCEYt]'t(b+bd-bc)[Yt_l]-bd[Yt_2]
+ [It(l)]+ (l+d-c)e1t - delt-l
2Yt(1) = §t(1) - Yt+l

A

’ [It(l) ’ It+lJ ' ce1t+1 ' e2t+1 ‘ e3t+1

[eft(l)] — celJc+1 - e2t+1 - e3t+l

A

A 2
Var[e (1)] E[e (1)]
Y1: Y‘t

A 2
Var[eft(l)]+ c Var(elt+l)‘+Var(e2t+l)

)

+

Var(e3t+1

Now consider our leading indicator approach. Our

leading indicator is V and we have our three transfer

t3
functions from equation (12) in the example. Consider each
in turn.
First:

[l-ch+b(c-1)B2]\Q: = (const)Vt + bd(B-BZ)[Vt_l]

2 3 —
+ bd(B -B )[It] + [delt_l-delt_2+bde2t_2-bde2t_3

+ e3t - beeBt-l + b(c-l) €3t-2]

U7

(V1) Vt = (const)Vt + bCVt-l -b(c-1)Vt_2 + bd(Vt_2]

- bd[Vt_3] + bd[It_2] - bd[It_3] + [deit—1*“it-2

+bde2t_2-bde2t_3+e3t-bce3t_l+b(c-l)e3t_2]
Second;
[l-ch+b(c-1)B2] Ct = (const)Ct + bEVt-l] + bETt-l]
+ [elt+beZt-l]
(vii) Ct = (const)C + bCCt-l — b(c—1)Ct_2 + bEVt-l]

t

+ bEIt-l] + [elt+be2t_l]

Third;
[l-ch+b(c—l)82] It = (const)It + bc[Vt_l] - bc[Vt_2]
+ [If] - b[I£_2] + [celt-Celt-1+e2t-be2t-2]
(viii) It = (const)I + bCIt-l - b(c-1)I,C__2 + bc[Vt_l]

t

- bc[Vt_2] + [It] -b[It_2] + [ce1t_celt—1+e2t

- be ]

2t-2

The aggregation of equations (vi), (vii), and (viii)
yields the transfer function for Yt implied by our leading

indicator approach.

(ix)

H8

(const)Vt + bCVt-l - b(c-1)Vt_2-+bd[Vt_2]

- bd[Vt_3] + bd[1t_2]-bd[lt_33'*[delt_1‘delt-2

+ bde2t_2 - bde2t_3 + e3t - bce3t-1

+ b(c—l) e3t_2] +
(const)Ct-l'tbcCt_2-b(c—1)Ct_3'tb[Vt_2]'*b[T¥_fl
+ [elt-l + be2t-2] +

(conSt)ItI+bCIt-l-b(C-1)It-24-bC[Vt-l]-bC[“F23

]

-ce +e -be

lt-l 2t 2t-2

+ [It] - tht_21 + [celt

(const)+bc[Vt_l+Ct_2+It_l]--b(c-1)[Vt_2+Ct_3

+ I ] + bc[Vt_l]+'b(l+d—c)[Vt_2]-bd[Vt_3]

t-2

+ [If] + bd[I£_ J - deIt_3]+-[celt

2

+ (1+d-C)elt- -de 2+e

+ _
1 lt- 2t bde2t-2 bde2t-3

J

+ e -bce + b(c-l)e

3t 3t-l 3t-2

(const) + bCYt- -b(c-1)Yt_ 'tbc[Vt_ ]

l 2 l

+ b(l+d-c) [v ]-bd[Vt_3]+-[It]+-bd[It_ J

t-2 2

- bd[1t_3] + [celt+82t+e3t]

+ [(l+d-c)elt_ -b ]

1 ce3t-1

+ [-delt_2+bde2t_,&xc—1)e3t_2J+-[—bde2t_3]

that (const) == (const)v + (const)C -+(const)I.

t t-l t

(ix) Y =

Note that (const) =

H8

- b(c—1)Vt_ -+bd[Vt_ J

2

(const)V + cht_ 2

t l

— deVt_3] + bd[1t_2]-bd[1t_33'*[deit_1’de1t-2

+ bde - bde + e - bce

2t-2 2t-3 3t 3t-l
+ b(c-l) e3t_2] +
(const)ct_l-+bcct_2-b(c—1)ct_3-+b[vt_2]-+b[II;g
+ [elt-l + be2t-2] +

(const)It'tbcIt_l--b(c-1)It_2

—be ]

+
ezt

‘CeIt-I 2t-2

+ [It] - b[It_2] + [ce1t

(const)'tbCEVt_1+Ct_2+It_l]-b(C-1)[Vt_2+ct_3

+ I

t_,] + bcEVt_l]4-b(1+d—c)[Vt_2]-bd[Vt_3]

+ [If] + deII_ J - bdliI't_3]-+[celt

2

-bde

'de 2t-2 2t-3

+ + -
(1 d Ckfitel 1t_2+e2t+bde

]

+ e3t‘bce3t-1 + b(C‘l)e3t-2

1

(const) + bCYt- -—b(c-1)Yt_ 4-bc[Vt_

l 2 l

+ b(1+d-c) [v ]-bd[Vt_3]+-[It]+-bd[It_ J

t-2 2

+
e3t]

1

+e

- bd[1t_3] + [celt 2t

-b

+ [(l+d-c)elt_l ce3t_l

+ [-de +bde +b(c-1ka ]'*[-bde J

lt-2 3t-2 2t-3

+ (const)C
t t-l

2t-2
(const)V

+ bc[Vt_l] - bc[Vt_2]

‘+(const)I.

t

(ix) Y =

Note that (const) == (const)V + (const)C +(const)I.

H8

(const)Vt + bCVt-l - b(c-1)Vt_2+-bd[Vt_2]

- deVt_3] + bd[1t_23-'bd[1t-3]'*[981t-1‘delt-2

+ bde2t-2 ‘ bde2t-3 + e3t ’ bce3t-1

+ b(c-l) e3t_2] +
(conSt)ct_l'+bCCt-2"b(c_l)Ct-3'+b[Vt-2]'+b[jt49
+ [elt-l + be2t—2] +

(const)I 'tbcIt_l-b(c-l)It_2'tbc[Vt_l]-bC[fl}2]

t
1

-ce +e —be

+ [It] ' b[It-2] + [Ge lt—l 2t 2t-2

lt

(const)'tbc[Vt_l+Ct_2+It_l]-b(C-1)[Vt_2+ct_3

+ It-2] + bc[Vt_l]4'b(l+d—c)[Vt_2]-bd[Vt_3]

+ [If] + deI£_2] - deI’t_3]-+[ce1t

+ + - - + -
(l d exit-l delt-2+e2t bde2t_2 bde2t_3

+ e -bce + b(c-l)e

3t 3t-l 3t-2J

]

(const) + bCYt— -b(c-1)Yt_ 4-bc[Vt_

l 2 l

+ b(l+d-c) [vt_2]-bd[vt_3]+-[It]+-bd[It_2]

- deIt_3] + [celt+82t+83t]

+ [(l+d-c)elt_ -b J

1 ce3t-1

+ [-de +bde +b(c-lk3 ]'t[-bde ]

lt-2 2t-2 3t-2 2t-3

t t-l

t

H9

Also, given our assumptions regarding the disturbances, eit’

the noise structure of Yt is seen to be a third order auto-

regressive scheme.

From equation (ix) we get the forecast of our

leading indicator approach.

(x)

t+l

Yt(l)

)

(l

Varfe

)

Y

+-bde

t

(1)]

(const) + cht - b(c-l) Y + bCEVt]

t-l

b(l+d-c)[Vt_ J — devt_2] + [Ith + deIt_ I

l l

bd[It_2] + [celt+l+e2t+l+e3t+l]

[(l+d-c)elt-bce3t] + [—delt-l+bde2t-l

b(C—l)e3t-l] + [-bde ]

2t-2

(const) + cht-b(c-1)Y -+bc[Vt]

t-l

b(l+d-c)[Vt_ J - bd[Vt_2] + [It(1X1+ deIt_ ]

l l

deIt_2] + [(l+d-c)elt-bce3t] + [-delt?1

+b(c—l)e3t_1] + [-bde ]

2t-l 2t—2

Yt(l) - Yt+l

[It(l) - It+l]'+[celt+l+e2t+1+e3t+1]

[ef£(l)] + [celt+1+e2t+l+e3t+l]

HEY (1)]2

t

A 2
Var[eft(1)]‘I c Var[e1t+l]'+Var[e2t+l]

Varfe3t+l]

...

Also, given our assumptions regarding the disturbances,

us

e 0
1t’

the noise structure of Yt is seen to be a third order auto-

regressive scheme.

From equation (ix) we get the forecast of our

leading indicator approach.

(x)

t+l

Yt(1)

>

Var[e

(1)

Y

+bde2t_l +b(C-l)e3t-l] + [-bde

t

(1)]

(const) + cht - b(c—l) Y + bc[Vt]

t-l

b(l+d-c)[Vt_l] - deVt_2] + [It+g + deIt_ 3

l

bd[It_2] + [celt+l+e2t+l+63t+l]

[(1+d-c)elt—bce3t] + [-delt-l+bde2t—l

b(C-l)e3t-l] + [-bde ]

2t-2

(const) + cht-b(c-1)Y ~+bc[Vt]

t-l

A

b(l+d-c)[Vt_ J - deVt_2] + [It(lfl + deIt_ I

l l

deIt_2] + [(l+d—c)elt-bce3t] + [-deltvl

2t-2:I

Yt(1) - Yt+l

[It(l) ' It+l“[ce1t+1+e2t+1+e3t+1J

[2f (1)] + [ce ]

t lt+l+82t+l+e3t+l

E[e

2
(1)]
Yt

VarEQT (1)]4-c2VarEe ]'+Var[e ]

t
+ Var[e3t+l]

lt+l 2t+l

H9

Also, given our assumptions regarding the disturbances, eit’

the noise structure of Y is seen to be a third order auto-

t

regressive scheme.

From equation (ix) we get the forecast of our

leading indicator approach.

(x)

t+l

Yt(l)

)

(l

Var[e

)

Y

-+bde

t

(1)]

(const) + cht — b(c—l) Y + bc[Vt]

t-l

b(l+d-c)[Vt_ J - .bd[vt_23 + [I£+g + deIt_ 3

l 1

deIt_2] + [ce ]

' +
lt+l+e2t+l e3t+1

[(1+d—c)elt-bce3t] + [-delt_l+bde2t_l

b(c-l)e3t_l] + [—bde J

2t-2

(const) + cht-b(c-1)Y -*bc[Vt]

t-l

b(l+d-c)[Vt_

l] - bd[Vt_2] + [It(1xl+ deIt_l]

deIt_2] + [(l+d—c)elt-bce3t] + [-deltvl

+b(C-l)eBt-l] + [-bde ]

2t-l 2t-2

Yt(l) - Yt+l

[It(l) - It+l].+[celt+l+92t+l+83t+l]

[ef£(1)] + [celt+l+e2t+1+83t+1]

BE;Y (1)]2

t
J

]'+Var[e

VarEeT (l)]+'c2Var[e
t

Var[e3t+l]

lt+l 2t+l

....

50

Note that the one step ahead forecast error for the leading
indicator approach is identical with that of the Final Form
approach.

Hence we have outlined an approach with an explicit
theoretical background in which leading indicators can be
studied and used, and which performs as well as the Final
Form approach.

It is not surprising that the two approaches yield
the same forecast errors for forecasts within the horizon
of our leading indicators' lead. They are obtained from
essentially the same information set. They are just

different algebraic manipulations of the same model.

51

FOOTNOTES

CHAPTER II

Box and Jenkins, Time Series Analysis, Holden Day, Inc.,
1977, pp. HOH-HlO.

 

Evans, M.K., Macroeconomic Activity, New York: Harper and
Row, 1969, Chapter 16.

 

The Handbook of Cyclical Indicators, A Supplement to the
BCD, May,l977, pp. 170-185.

 

Evans, op. cit.

Koopmans, T.C., "Measurement Without Theory," Review of
Economics and Statistics, August919H7.

 

Vining, R., "KOOpmans on the Choice of Variables to be
Studied and of Methods of Measurement," Review of
Economics and Statistics, May,19H9.

 

Harris,I%JV.and Jamroz, D., "Evaluating the Leading
Indicators," Monthly Review of the Federal Reserve

Bank of New York, June,1976.

 

Zellner and Palm, "Time Series Analysis and Simultaneous
Equation Econometric Systems," Journal of Econometrics,
2, 197H, pp. l7-5H.

Theil and Boot, "The Final Form of Econometric Equation
Systems," Review of the International Statistical
Institute, Volume 30:2; 1962, pp. 136-152.

 

0Box and Jenkins, op. cit.
Box and Jenkins, op. cit., p. HOH.

Well known fact:
If Zt is a vector of mutually independent random

variables with E(zt) = O and E(zt zt-i) = 0, i = 1,2, ... ,

and if H is a matrix with known coefficients,

52

then
Var{[H]zt} = E{[H]zt}2 = [[311 Var(zt);

where [[H]] is the transformation of [H] made by squaring

each element in [H].

 

A simple example:

Let 1 -2 e

 

 

llt
[H] : ; zt :
3 0 e21t
with the e's having properties identical to those in examplel.
1 -2 ellt
Var{[H]zt} = Var 1 I
I 3 0 e211;
First, with our Second, the straightforward
transformation; development;
1 u e11t e11t'2821t
= Var = Var
9 0 e21t 3e11t
+ ..
Var(ellt) HVar(e21t) Var [ellt 2e21t]
9Var(ellt) Var[3e11t]
Var(ellt)+HVar(e21t)
9Var(ellt)

It should be noted that in our model, each element
in [H] is a polynomial in L, the lag operator. In this
context, to transform [H] into [[H]] we need to square the

coefficient of each different power of L appearing in every

polynomial comprising an element in [H].

53

A less simple example:

Let (B-3B2) 2B2 ellt
[H] = 3B3 0 ; zt = e
21t
I B—3B2 2B2 e ‘
llt
Var{[H]zt} = Var 1 3 I
I 3B O e2lt J

 

 

First, with our transformation;

2 2 2 2 2
(1) B+(-3) B (2) B ellt
: V
2 3
(3) B 0 ezlt
2 2
(B+QB ) HB ellt
= V
3
9B 0 eth
' BV(e )+9B2V(e )+HB2V(e )
llt llt 21t
3
I 9B V(ellt)
' +
V<ellt—l) 9 V(ellt_2)+H V<e2lt-2)
I 9 V<ellt-3)

 

Second, the straightforward development;

 

r 2 2
(B-3B )ellt + 28 ealt
: V 3
( 38 ellt
’ 2 2
- +
Be11t 3B ellt 23 e21t
= V
I 3B3e

llt

5H

3 2

eilt—f'e11t-2+ e21t-2

= V

3e1112-3

V(e ~3e

llt-l 11t-2+2

e21t—2
V(3ellt_3)

)+9V(e )+HV(e )

V< 11t-2 2lt-2

e11t-1

9V(ellt-3)

Clearly, for this to be a valid transformation, not
only must our disturbances be mutually independent, but

they must not be autocorrelated as well. i.e. E(zt Zt-i): 0,

i = 1,2,
This assumption holds concerning our error structure,

elt’ but it does not hold in general concerning our inputs,

Xt' In particular, consider equation (2.21). We cannot

. - *

apply the same transformation to the matrix [H1I(L)H12(L)]
- k

as we apply to [H11(L) F11(L)] because the elements in the

vector of forecast errors, eX (k), are autocorrelated.

A A t

e.g. E[ex (k-1)~ex (k-2)]
1t 1t

k-2 k-3
= 3 DEC wlnat+(k—1)-n ngo wlnat+(k-2)-n

(by substituting from equation (2.18))
k-3

‘ 2

I1

2

wlnwl(n-l)ola ( 0

1

This point becomes obvious in our 1ess simple example,

e ex (k)
if we replace 2 = llt with z ' = “ 1t(k) .
t eth

t e
21t

55

We will find in taking the variance that this transformation

of [H] will be inappropriate since e (k) is autocorrelated
It
with past forecast errors.
Let 6—3132 2132 e (k)
xlt
[H] = ; zt' =
BB3 0 eX (k)
2t

with the forecast errors having properties like
those in equation (2.18).

B—3B2 2B2 e (k)

Var{[H]zt'} = v 1t
363 0 eX (k)
2t
First, with our transformation;
B+gB2 us? ex (k)
= v 1t
983 0 ex (k)
A 2t
V[e (k-l)+9V[eX (k—2)]+HV[eX (k-2)]
xlt 1t 2t

9V[; (k-3)]
x1t

(inappropriate)
Second, the straightforward development;

vim—362);x (k) + 232; (k)]
lt x2t

Var{[H]zt'}
V[3Bae (k)]
xlt

V[e (k-1)-3ex (k-2)+2ex (k-2)]
xlt 1t 2t

V[3; (k-3)]
xlt

56

F—

v[; (k-1)] + 9vtéx (k-2)

 

Xlt 1t
= + uvt; (k-2)] - 66E; (k—1)-éX (k—2)]
X2t Xlt 1t
9V[; (k-3)]
_ Xlt J

 

The difficulty is due to more than one power of B
appearing in any polynomial which is a single element in
[H]. In the variance, the presence of these different
powers of B requires consideration of the given variable's
autocorrelation.

Although we cannot make a transformation of
[H1%(L) H12(L)] , we can still calculate the variance of
the forecast errors as expressed in equation (2.21) in a
relatively straightforward manner. See the examples

for applications.

APPENDIX 1

FITTING THE TRANSFER FUNCTION IN EQUATION (2.10)

APPENDIX 1

FITTING THE TRANSFER FUNCTION IN EQUATION (2.10)

How do we fit the transfer function in equation (2.10)?

 

X

ka
We can prewhiten the input, xt* = I 1t]
1:

To do this, first consider the bottom p2 elements of xt*;

namely Xt' From (2.H) we have:
H22(L)xt = F22(L) e2t
or -1
Xt = H22 (L) F22(L) e2t

This is the appropriate ARIMA model for Xt' We can substitute

this into x * for the bottom p2 elements, and these inputs

t
will be "prewhitened." Then the transfer function will be

in terms of the predetermined variable input, Lk ylt’ and the

. . _ -1 .
p2 prewhltened ifiputs, xt - H22 (L) F22(L) e2t. That 18,
x ... = L ylt

t -1
H22 (L) F22(L) e

 

2t

We still need to prewhiten ylt’ our leading indicator,
in order to work with the system in (2.10) in terms of all
prewhitened inputs, We must therefore fit y1t with its

appropriate ARIMA model.

The first equation in the set of pl equations in

58

(2.9) gives us the time series model for y1t implied by

our dynamic structural system of simultaneous equations:

 

 

 

 

 

 

 

 

 

 

 

* ‘-l [ fififi ‘
Ihll I h11 I k
h * I h *A* I L ylt
(2.11) y = —J 21 | . 21 | x
1t I Hll“(L) 'H12(L) r t
I I I
II 1 A
h h ..a’:..
I
( lx(p2+l) ‘ 1-
F 3 1‘1 (p +1)xl
h11 I 2
h * |
21
+ *
J : H11 (L) F11(L)I e1t
I l
h 3 I I
It p11 I J1.
1x
pl p1X1

This is a transfer function for y1t with input xt*. This
equation is interesting in that lagged y1t is an input to
the transfer function model for ylt'

The first element of the 1x(p2+l) vector,
1

 

 

 

 

 

 

,
r * 1-1r 31* ‘
h11 I h11 I

* ***
I h21 I h21 I
a
I l Hll’(L) I H21(L) )
' l I
h a h ***|

multiplies Lk y1t in (2.11). Call this first element of the
vector, g(L). The first term in the sum on the right hand
side of (2.11) can thus be written: [g(L) Lk ylt]' This
can be moved to the left hand side of (2.11) in order to

collect all the terms in ylt'

59

 

 

 

 

 

 

 

 

k _ k
(2.12) ylt + g(L) L ylt - (1 + g(L) L ) y1t
If 1"].r 1‘
5': :‘::':‘:
h11 I h11 . I
3': :‘n‘n":
J h21 I h21 I
: .. 3':
: l Hll (L) : l H12(L) > xt
o I o I
h a" h :‘c:'::':l
. P 1 J . P 1 J
I l 1 J10-
I 9 1"]. ‘
Ih11 I
I 1121:: :
+ 3': '
Z lHll (L) Fll(L)) e1t
' l
h 3': I
1
I. p, J J1.

 

 

Note that this vector multiplying xt is the same

vector as in (2.11), without the first element, g(L). This

can now be solved for ylt:

 

 

 

 

 

 

 

r 2’: ‘-l[ :‘:':‘: “
h11 I h11 "I
2': :‘n’n‘:
h21 I h21 I
(2.13) ylt = -(1+g(L)Lk)‘1< : .Hll='=(L) : IH12‘L’ )xt
. I . I
h :‘al h :‘fil
\L p11 J I p11 J11“-
r 1
f 1-1
h11 I
k -1 h21* '
+ (1 + g(L)L ) J . Hll*(L) F11(L)) e1t
1 l
h. "

 

 

 

This gives us the transfer function for our leading indicator,

ylt’ implied by our structural model, in terms of exogenous

60

variables only. This can be fitted by prewhitening the

input, substituting [H22_1(L) F22(L) €2t] for xt:

(2.1a) y1t = —(1+g(L)Lk)‘1I I [H22‘1(L> F22(L)e2t]
10—

+ (1+g(L)Lk)'l I I elt
1.

- - Lk ylt

We can now substitute this model for y into x *== ——————
lt t xt
Hence we have X * entirely prewhitened, and can now work

t

with our model for the measure of economic activity in

 

 

 

 

 

 

 

 

 

 

 

 

 

(2.10):
- k k -1 -1 I
[‘L (1+8(L)L ) {}1._H22 (L)F22(L)e2t
* + Lk(1+g(L)Lk)‘lI} e I
Xt = 1. 1t
-1
H22(L) F22(L) e2t
rrh ":l l-lfh '.'.J. I I‘
11 I 11 I
a fi‘*
I h21 I h21 I I .
(2.10) y2t - - : IHll«(L) : [H12(L) [Xt"I
' I ' I
h *I h "* I
1 1
LL Pl I J L pl I 1J2.
r! .3. "'l W
hlldb :
I h21 I *
+ . I H11 (L) F11(L)+ e1t
' I
*l J J2.
I

61

Note that equation (2.13) is the same equation as is implied
in the structural system in equation (2.”).

The Final Form for the system is:

_ -1 -l
(2.15) yt - -H11(L) H12(L) x + H11(L) F11(L) e

t 11:

The first equation of the Final Form is the same as equation

(2.13).

APPENDIX 2

A USEFUL CONVENTION OF ZELLNER AND PALM

APPENDIX 2

A USEFUL CONVENTION OF ZELLNER AND PALM

At this point let me interject a useful convention
which Zellner and Palm point out, when working with this
kind of model transformation (see footnote 8).

In both the Final Form and leading indicator
approaches, our model is expressed as a vector of endogenous
variables in terms of a set of linear combinations of
predetermined variables and disturbances, in a dynamic
framework. In matrix form, our coefficient matrix is the
product of two known matrices (say A and B), one of which
is in inverse form. That is, our model is of the form;
yt = [A'1131xt + [A—lCJet.

From our presumed knowledge of A, B, and C, we can

compute A-1 = HE%—A [adjoint A], and thus we know [A—lB]
and [A—lC]. Note that in our context, each element of

[A-lB] will be the ratio of two polynomials in L, with the
denominator being the determinant of A; i.e. [A-lB]

= 35%f3 [adj A][B]. A distributed lag which is the ratio
of two polynomials in L implies a lag of infinite order.

Hence we have a quite complicated system.

We can simplify this system by multiplying both sides

62

63
of the equation of our model by [det A];

i.e. [det Aly,C = [adj A][B]Xt + [adj A][C]et

Here our system is in the form of a transfer function
with current and lagged yt's in terms of current and lagged
xt's and disturbances. An interesting aspect of this system
is that the order and parameters of the autoregressive part
of each equation will be the same. This is true because
the determinant multiplying the vector, yt, is a single
polynomial in L.

Note that with this manipulation of our models,
equations (2.16), (2.17), (2.23), and (2.2”) will be changed

as follows.

Final Form approach;

(2.16)‘ det[Hll(L)] yt+k —[adj H11(L)][H12(L)] x

t+k

+[adj H11(L)][F11(L)] elt+k

-[adj H11(L)][Hl2(L)] xt(k)

(2.17)' det[Hll(L)]yt(k)
+ {[adj H11(L)][F11(L)J}***e1t

Leading Indicator approach;

h11* l
' ' I :': :
(2.23) det : I Hll (L) yt+k
h *,
p11

6H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f f WI
:': ‘ r ‘
hll J . 11 I y
= _J . . . lt
adj : I H11 (L) I :H12(L) I x
* . t+k
h I J h '*I J
L l k
f (h :1 W ‘
11 I
+4 ' : I 2':
adj . I H11 (L) F11(L)I elt+k
Rh ' I
hllz: I
' ° :': _
(2.2M) det I I H11 (L) yt(k)..
h '
p11 '
f {h ' I ‘ Ih u I 1‘
11 I 11 l y
:_ J . . * . 1t
adj : I H11 (L) : [H12(L) I A
h .I h ...I xt(k)
L L p11 J L p11 J J
I Ih . ‘
11 I
+ ' 2':
Jad] : J Hll (L) F11(L)> e1t
h *I
L L P11 J

In this form the forecasts of yt(k) in equations
(2.17)' and (2.2M)‘ will not only be in terms of the history
and forecast profile of Xt’ but will also depend on the past
history and forecast profile of yt itself. This is due to
the determinant multiplying the vector of endogenous
variables. Furthermore, the autoregressive part of all
of the pl forecasts in yt(k) will be identical.

The presence of these "lagged" forecasts of yt will

65

cause complications when we consider the variance of the

forecast errors, since forecasts of yt(l), yt(2), ... ,
yt(k-l) will be correlated with each other, with forecasts
of xt(l), Xt(2)’ ... , Xt(k)’ and w1th elt'

This is obvious in the examples, in which this

convention is used.

CHAPTER III

THE PROBLEM OF SPECIFICATION ERROR

Introduction

 

In Chapter II we proposed a framework for studying and
using leading indicators. We outlined a procedure for
building transfer function models by setting up a structural
model, and deriving the time series models directly from
this explicit theoretical background.

In Chapters IV and V we will consider multivariate
time series models which describe various empirical relation-
ships between "established" leading indicators and economic
activity. We will build the transfer functions empirically
by following the procedures outlined in Box and Jenkins'

Time Series Analysis.1 This procedure is chosen over the

 

framework formulated in Chapter II.

The transfer functions we examine in Chapter IV
consist of economic activity (Industrial Production) as the
output, and as a single input, the leading indicator under
consideration.. In light of Chapter II, it may be argued that
there is an econometric problem of omitted variables in this
approach. At the outset, our single—input transfer functions
will appear to reflect a belief that the level of economic

activity is adequately "explained" by the use of just one

RR

67

input. The framework in Chapter II shows that the number of
inputs in the transfer function implied by any given econo-
metric model will be equal to the number of exogenous vari-
ables plus the number of leading indicators in that model.
Even the simplest structural econometric model will imply a
transfer function with more than one input.

It is important to emphasize that our work is not
done with the presumption that a single input is sufficient
to explain the movements in economic activity. We follow
the Box and Jenkins procedure because we are interested in
the dynamic relationships which exist empirically between
economic activity and each of the leading indicators under
consideration.

Furthermore we argue in this chapter that the bias
introduced in the parameter estimates of our single input
transfer functions through the omission of variables, does
not present a serious problem.:tfthe following conditions
characterize the model being studied.

(i) The main objective of building the modeliésforecasting.

 

(ii) The variables in the underlying econometric model are

drawn from a joint distribution which is covariance

 

stationary.

 

A Discussion of the Problem

Suppose that the true model describing the world we

are examining is

: +...+ 1’
(3'1) yt lelt + B2X2t BKth et

68

or
(3.2) Y = X8 + e

where Y is a Txl vector of observations on the endogenous
variable, expressed as deviations from the mean; X is
a TxK matrix of explanatory variables expressed as
deviations from their respective means, which are
drawn from a multivariate Normal distribution2;
B iSéinl vector of true model parameters; and

2

8 iSéiTxl vector of disturbances, with €%N[0,0 IT].

We use only Xlt' Thus
(3.3) E(yt|xlt) = lelt + B2E(x2t|xlt) + ... + BKE(th|xlt)
Under our assumptions,
0
- 12 -
E("2tlxlt) ' "E Xlt ' C2Xlt
o
l
o
_ l3 _
H(XBtIXlt) ‘ 2 Xlt ” C3Xlt
o
l
(3.”)
o
_ 1K -
E<XKtlxlt) ‘ 2 Xlt " Clet
0
l
where Olj = Cov (xlt’xjt) for j = 2, 3, . . , K
2 _ 3
and 01 - Var (xlt)'

Note that the cj are constants, for j = 2, 3, ... , K.

69

Thus
(3'5) B<ytixlt) z lelt + 82°2x1t + "' + BKClet
= (81 + 82c2 + ... + BKCK) xlt

= 8" Xlt

Hence we can write the model as

) = 0.

(3.6) yt = 8* x + v where E(vt|xlt

1t t

Ordinary Least Squares will yield a consistent estimate
of 8*. Furthermore, given stationarity, vt has other nice
properties like homoskedasticity (since the diagonal of
the covariance matrix of the conditional density given in
footnote (3) does not depend on t).

It is obvious that bias is present since 8* ¢ 81
unless x1t is uncorrelated with other xjt (in which case the

0 would be zero for j = 2,3, ... ,K). However, if condi—

13'
tion (i) on page 2 characterizes our study, we do not care
about the economic interpretation of 81. In this case, we

are concerned with forecasting, and 8* is appropriate for

 

that. Indeed, [8* t] is a much better forecast of yt than

x1
[81 xlt], since it is exactly E(ytlxlt).

The second condition of page 2 is important because
it implies that the cj; j = 2, 3, ..., K, will remain stable
over time. If all the variables in the underlying econo-
metric are drawn from a joint distribution which is

covariance stationary,u then the correlation between the

69

Thus
: +

(3.5) E(ytlxlt) lelt 82c2x1t + ... + BKCKXlt
= (81 + 82c2 + ... + BKCK) x1,t

: ‘1"
8 x11:

Hence we can write the model as

(3.6) yt = 8* x + v where E(vtl ) = 0.

1t t xlt

Ordinary Least Squares will yield a consistent estimate
of 8*. Furthermore, given stationarity, vt has other nice
properties like homoskedasticity (since the diagonal of
the covariance matrix of the conditional density given in
footnote (3) does not depend on t).

It is obvious that bias is present since 8* ¢ 81
unless x1t is uncorrelated with other xjt (in which case the

0 would be zero for j = 2,3, ... ,K). However, if condi—

lj
tion (i) on page 2 characterizes our study, we do not care
about the economic interpretation of 81. In this case, we

are concerned with forecasting, and 8* is appropriate for

 

that. Indeed, [8* t] is a much better forecast of yt than

).

x1

[81 1, Since it 18 exactly E(yt|x1t

The second condition of page 2 is important because

xlt

it implies that the cjs j = 2, 3, ..., K, will remain stable
over time. If all the variables in the underlying econo-
metric are drawn from a joint distribution which is

covariance stationary,” then the correlation between the

70

the single input included in our model and each variable
which is omitted, will remain stable in the future. That

is, the variances and covariances in the constants, cj, will
not vary over time. Hence, in E(xjtlxlt) (j = 2, ..., K) we
have incorporated the way that the xjt (j = 2, ..., K) change

on average when xlt changes. We still miss the information

 

in [xjt - E(xjtl

serious problem over time.

xlt)]’ but given condition (ii), this is not

FOOTNOTES

CHAPTER III

lBox, George and Jenkins, Gwilym, Time Series Analysis,
Holden Day, 1976.

2This assumption that the Xit5 i = l, ... ,K are drawn
from a multivariate Normal distribution, is consistent with
the notion expressed in ChapterII that the explanatory
variables follow ARIMA processes. That is, each explanatory
variable can be expressed as an infinite order Moving
Average process about a white noise series which is assumed
to be Normally distributed.

3Dhrymes, Phoebus J., Introductory Econometrics,
Springer-Verlag, New York Inc., 1978, pp. 364-66.

 

Consider his Propositions 7 and 8, restated: Given
the set of K random variables, X (le), expressed as devia—

tions from their means.
X1 N [0} 2:11 212
’

Let X m N[0,E], or equivalently,
X 0 X X

2 21 22
. . _ z 2 .
where X1 is (lxl), X2 18 (K l)xl, Ell 01 18 (lxl),
. _ I .
£22 18 (K-l)x(K-l), and £12 - 221 IS lx(K-l).
Note that 212 = 221' is the vector of (K-l) covariances:
Clj = Cov(xlt,xjt); j = 2, 3, ... ,K.

Under these conditions,

X m NEG, 2 l;

l 1; X2 m NEG, 2

ll 22

and the conditional density of X2 given X1 is

-1 —1
N[221211 X1’ 2:22 ’ 221211 212] ‘

71

72

The first moment of this conditional distribution is the
set of K—l conditional expectations:

C12
H(thlxlt) : O 2 Xlt = C2X11-
1
C
_ l3 -
E<x3tlxlt) ’ 2 Xlt ' C3Xlt
1
C
_ lK _
E<XKt|Xlt) ’ O 2 Xlt ‘ Clet
1

uIf the joint distribution is Normal, then covariance
stationarity implies strict stationarity.

APPENDIX 3

PROOF THAT ORDINARY LEAST SQUARES ESTIMATION

YIELDS THE APPROPRIATE ESTIMATE

APPENDIX 3

PROOF THAT ORDINARY LEAST SQUARES ESTIMATION
YIELDS THE APPROPRIATE ESTIMATE

Here we will show that Ordinary Least Squares
estimation of the misspecified system yields a parameter
estimate with an expected value identical to 8* in equations
(3.5) and (3.6).

Let X and 8 be partitioned as follows.

8

TxK Txl Tx(K-l) le B2 (K'1)X1

In this case, equation (3.2) becomes

Y = x3 + e
B
_ 1
- [X1 X2] [-52]+ E
: X181 + X282 + 5

We estimate the following misspecified system.

: 5': 3'; =
Y X18l + e where 6 X282 + 8

Ordinary Least Squares estimation yields the following.

A -1
' '

-1 _
' t + +
(Xle) Xl[xlBl X282 5]

73

7H

-1 — -1
: + ' ' + ' '
81 (Xle) XlX282 (Xle) Xle
E[8 J = a + (x'x )‘lx'x E
l l l l l 2 2
T
2 _ 1 2
Note that Cl — T 2 x11:
t-l
_ l '
- T(X X1)

Thus (x'x )'1 = [To 21'1 = _l.

l l l T 2

o
l
T
Furthermore o . = Cov(x x ) = l X x x.
’ lj lt’ jt T t=l lt jt
_ l '
- T [X1X2]lj

I = -

Thus XlX2 T[ol2 013 ... 01K]

Substituting these into the expected value of our least

squares estimate, we get the following.

A
-1 _
: I v
E[Bl] 81 + (xlxl) x1x282
_ 012 °13 O11<
‘ 81 + 2 2 ' 2 [82]
C U U
1 1 1
: 81 + C2B2 + C383 + + CKBK

CHAPTER IV

AN EVALUATION OF THE COMMERCE DEPARTMENT
LEADING INDICATORS

Introduction

 

The huge amount of effort exerted toward developing
the Commerce Department leading indicators has resulted in
the current Composite Index of Leading Indicators (CLI).
This index represents the fifth complete reworking of the
indicator selection. The first was compiled by Burns and
Mitchell in 1938, the second and third by G.H. Moore in
1956 and 1960, the fourth by Moore and Shiskin in 1967, and
the fifth by the Commerce Department in 1975 (see references
[17], [H7], and [75].

The CLI is constructed from the following twelve
series which have been evaluated as the "most useful"
leading indicators, given the Commerce Department's scoring
procedure.

Business Conditions Digest Series 1 (BCDl): Average

Workweek of Production Workers, Manufacturing

BCD3: Layoff Rate, Manufacturing

BCD8: Value of Manufacturers' New Orders for
Consumer Goods and Materials

BCD12: Index of Net Business Formation

75

76

BCD19: Index of Stock Prices

BCD20: Contracts and Orders for Plant and Equipment
BCD29: Index of Housing Starts

BCD32: Vendor Performance

BCD36: Change in Inventories on Hand and on Order

BCD92: Percent Change in Sensitive Prices (PPI of
Crude Materials)

BCDlOH: Percentage Change in Total Liquid Assets (M7)

BCD105: Real Money Supply, Ml

First the monthly data is standardized so that each
series displays the same average absolute monthly change.
This makes it possible to combine series with different
units of measurement.

Next these standardized series are combined into a
weighted average, with the weights reflecting the overall
performance scores of each component series as a cyclical
indicator. The score for a given series depends on its
economic significance, statistical adequacy, cyclical timing,
conformity to business cycles, smoothness, and currency
(how quickly the data are available). The weight given to
each series is the ratio of the performance score of that
series to the average of the scores of all series in the
CLI ([5], [33]).

Finally this weighted average is subjected to a
"reverse trend adjustment." Here the trend of the weighted
average is made to equal the trend of the Composite Index

of Coincident Indicators (CCI). This trend can be viewed

77

as a linear approximation to the secular movement in
economic activity. The rationale for this reverse trend
adjustment is as follows. Many of the twelve series listed

above relate to changes or percent changes in output, prices,

 

or monetary aggregates rather than levels, and thus display
no significant trend. Hence the weighted average of leading
indicators constructed in the first two steps displays many
small declines which are not indicative of a coming drop in
the CCI. Reverse trend adjustment is intended to overcome
this difficulty ([33], [3M], [37], [68]).

The series resulting from these three steps is the
CLI.

Critique of the Commerce Department Approach to Leading
Indicators

 

In Chapter II we developed a theoretical framework
for leading indicators. It is important to examine whether
the leading indicator approach of the Commerce Department
outlined above,is appropriate in the context of our work
in Chapter II.

Is there a theoretical framework justifying the
above construction of the CLI?

This point was examined empirically as early as
1957 by John E. Maher [H0] and again in 1973 by Saul Hymans
[37]. They each ran a multiple regression of the CCI on
the components of the CLI, to see if the resulting coeffia

cients resembled the weights imposed on the twelve series

78

in the Commerce Department construction. Their results
showed three leading indicator series with insignificant
coefficients, and five more with coefficients of the wrong
sign.

In all fairness to the Commerce Department approach,
it should be noted that this regression equation does not
represent a behavioral relation. Hymans clearly states
that it is "merely an exercise in curvevfitting, or — at
best - some kind of pseudo reduced form equation." Hence,
this exercise is not an appropriate test for the existence
of a theoretical framework underlying the Commerce Depart—
ment approach.

A further criticism which has been used in defense
of the Commerce Department approach, is that such a
regression equation examines the relationship between leading
indicators and the measure of economic activity at all

points intime. It is claimed that the Commerce Department

 

is only concerned about the lead of an indicator just prior
to true turning points in the economy, and does not worry
about the interim periods ([30], [37]).

Hence, "the CLI is alleged to be constructed so as to
maximize the use of the turning point information contained
in the component leading indicator series. The statement
that a turning point in economic activity is typically
preceeded by a turning point in many of the component series
of the CLI, is not meant to imply direct causality. If it

did, one would attempt to estimate a behavioral or

79

technological relation that could be expected to hold outside
the sample, that would have directly interpretable coeffi—
cients, and so on. Rather, the statement implies something
about the process through which those forces that do lead

to turning points Operate within the structure of the U.S.
economy" ([37)].

If these statements constitute the best defense of
the Commerce Department leading indicator approach, then it
seems that there is no explicit, quantifiable theoretical
framework underlying the approach. Instead, promoters of
the approach rely on stories about the trade cycle which
suggest reasons why many of the component leading indicator
series might turn down before economic activity.

These arguments would pp: justify the Commerce
Department approach in the light of our Chapter II. There
must be an explicit theoretical framework, expressable in
terms of a system of structural dynamic equations, for our
approach to be applicable.

Furthermore, the common argument that these series
should only be expected to consistently lead economic
activity at true turning points, suggests a significant
weakness in their value as forecasters.‘ If we had an
indicator or an index of indicators which was successful at
predicting £323 turning points and of no predictive value
at any other time, it would be useful. However, it is
difficult to imagine relying on such an indicator to predict

a downturn which has not yet occurred! It seems that at

80

best, this indicator would be useful in an expost role,
verifying that a downturn has already occurred. In fact,
Julius Shiskin and G.H. Moore have admitted that the
Commerce Department approach has only been useful in such
an ex post role ([17], [2H], [37], [H8], [H9], [50], [52],
[65], [68]).

In any case the existing CLI has not met with great
success, nor has any of its components alone, nor has any
of the other eighteen series classified as leaders by the
Commerce Department ([12], [17], [18], [21], [22], [2H],
[28], [29], [30], [31], [33], [3H], [37], [39], [H3], [59],
[67], [72]). The leading indicators have displayed small
average leads (only two of the thirty leading indicators
have an average lead at true turns of more than five months
[22]), and their lead times have varied greatly from cycle
to cycle. In addition, they have often signalled false
downturns. Understandably, the CLI has exhibited the same
problems. Though it has not failed to signal a true down-
turn since World War II, it has displayed lead times which
have varied greatly, and it has signalled many false downv
turns ([3], [22], [3H], [37], [HH], [70]). These problems
make the Commerce Department leading indicator approach
quite unreliable as a useful predictor of business cycle

turns.

81

Empirical Evaluation

 

Given the above arguments in the ongoing debate
concerning the Commerce Department leading indicators, a
more extensive examination of the relationships between
these indicators and the level of economic activity is in
order.

Here we consider eight single—input transfer function
models with the Industrial Production Index (I.P.) as output,
and with one of the leading indicator series used in the CLI
as input. This technique allows the data to inform us about
the extent and the form of the relationship between each
leading indicator and I.P., as well as the average lead
involved in the relationship at each point in time.

The first step in this analysis is to identify and
fit the univariate [UKDMX model appropriate for each leading
indicator. These models are used to prewhiten the time
series in each transfer function. This prewhitening
procedure transforms each input into a series of innovations
which presumably contains the information relevant to that
leading indicator that is not "explained" by previous
observations of the indicator. This series of innovations
is compared with the corresponding series of innovations
created by applying the same prewhitening transformation to
the output, I.P. The cross-correlation between these two
series provides us with information describing the relationv

ship between the input and the output, and hence, the form

82

of the transfer function.

In working with this technique, data which is not
seasonally adjusted is required on all the time series
involved. The time series methods applied, effectively do
theor own seasonal adjustment in building the transfer
function models.

This has presented a problem in collecting our data
on the leading indicators. Though the numbers are readily
available in the Business Conditions Digest, all but two of
the twelve series we need appear in seasonally adjusted form.
The search for the numbers in unadjusted form has produced
just eight of the twelve leading indicators. Consequently,
our analysis centers on these eight leading indicators.

The univariate models for these eight leading
indicators are reproduced in Table IVPl.

To examine the stability of these models, we split
the sample of each model into two relatively equal sub—
samples, and reeestimate. In most cases the new parameter
estimates remain within one standard deviation of the
original estimate from the model with the entire sample.

The new x2 statistics and Residual Standard Errors (RSE's)
also generally remain close to the corresponding figures
from the model with the entire sample. Furthermore, the

sum of squared residuals (SSE) for the estimated models in
each subsample, generally amount to approximately 50 percent
of the SSE of the model with the entire sample. All these

observations suggest a high degree of stability!

83

TABLE IV—l

UNIVARIATE MODELS FOR THE COMMERCE
DEPARTMENT LEADING INDICATORS

[The number below each parameter estimate is its standard
error.]

 

BCDl: Average Workweek

Sample: l9H7 - September, 1979: n = 393.

 

(u.1> (l-B12)(l-B)log(zt) = (l-.21188)(1v.82089812)at
(.05) (.033)
X28 = 3u.7 RSE = .007u
—2 (RSE)2 ( 007H)2
R = 1 "' 7-2—— = 1 " ‘ 2 = .1457”
Goutput (.0100H6)
where o is the standard error of

output
(1-812)(l-B)log(zt).

BCD3: Layoff Rate

Sample: 19H7 — August, 1979: n = 392.
12

 

(u.2) (l-Bl2)(l-B)log(zt) = <1-.710953 ) at
(.037)
2 _ _
x29 - 33.5 RSE — .167H
—2 ( 167H)2
R = l n '° .3157

(.20236)2

8H

TABLE IV—l (cont'd.)

 

BCD8: Value of Manufacturers‘ New Orders

Sample: 1958 ~ May, 1979: n = 257

 

(0.3) (l-Blz)(l-B)log(zt) = (1..09590312)at
(.0H8)
2 _ _
x29 — 26.3 RSE - .0355
—2 < 0350)2
R = l - ' 2 = .3107

(.0H2878)

BCD19: Index of Stock Prices

Sample: 19H? - October, 1979: n = 39H.

 

 

(0.0) (l-.2186B)(l-Bl2)(l-B) log(zt) = (l—.8216Bl2)at
(.051) (.030)
X28 2 30.7 RSE = .0353
—2 ( 0353)2
R = 1 — ' 2 = .0113
(.0H6007)
BCD29: Index of Housing Starts
Sample: 1959 — August, 1979: n = 2H8.
(0.5) (1-B12)(1-B)log(zt) = (l-.2H912B+.30H7882
(.005) (.007)
+ .19372135 — .5952512)at
(.0H5) (.0H8)
2 - _
X26 _ 07.1 RSE - .0895
—2 ( 0895)2
R = 1 - ‘ = .3529

(.11213)2

85

TABLE IV-l (cont'd.)

 

BCD32: Vendor Performance

Sample: 19H7 — October, 1979: n = 39H.

 

(0.6) (l-.2992B + .069287 + .1559B1u)(l-B12Xer)log(z )

(.050) (.050) (.008) t

= (1—.6600812)at
(.0H0)

2
x = z

26 39.2 RSE .125
—2 ( 125)2
R = l — ° = .H725

(.17211)2

BCD92: Percent Change in PPI of Crude Materials

Sample: February, l9H7 - August, 1979: n = 391.

(0.7) (l-.3756B — .2H0583)(l-812)(l-B) log(zt)
(.0H9) (.0H6)
= (l-.1H68B2 — .7331812 + .099Blu)at
(.056) (.037) (.055)

X2 = 38.5 RSE = .0157

2
R2 = l _ (.0157) 2 : .H732
(.021631)

 

BCD105: Real Money Supply (Ml)
Sample: 19H? - September, 1979: n = 393.

(0.8) (l-.20358)(l-812)(1-B) log(zt)
(.052)
= (1 + .1905883 _ .56297812)at
(.003) (.000)
2
x = =
27 33.0 RSE .0059
2
R2 = 1 _ (.0059) = .2936

 

(.00702)2

86

Two mild exceptions to the stability described above
appear in the models for Housing Starts (BCD29) and
Producer Prices (BCD92). Observe that these models are
more complicated than the surprisingly simple models which
describe the other leading indicators. The problem with
both of these series is that the model parameter estimates
vary somewhat more in the subsamples. Though some of the
parameter estimates remain within one standard deviation
of the estimate for the entire sample, others move outside
this band of one standard deviation, and a few move outside
the (23) confidence band. However, both of these models
have the redeeming qualities that the X2 statistics and the
RSE's are quite robust, and the SSE‘s for the subsamples
constitute close to 50% of the SSE for the entire sample.
Furthermore, the £23m of each of these models remains
appropriate in all subsamples considered.

It is not surprising that these two models are the
least stable. Housing starts have been subject to the whims
of Regulation Q enforcement; and the PPI since 1973 has
been subject to some degree to the whims of OPEC pricing.
In light of these facts, it is remarkable that these series
behave as well as they do.

The extent of the stability of these eight models
is fairly amazing, given their simplicity, the length of
the sample period, and the volatility of many of the leading
indicator series.

With these univariate models established, we can

87

examine the cross-correlation function between the pre—
whitened input and the prewhitened output of each of our
eight transfer functions. In so doing we obtain information
about the form of the transfer functions under considera-
tion. With this information, we proceed to the estimation
stage and finish building our eight models, which are
displayed as equations (H.9) through (H.16) in Table IVw2.

To examine the stability of these models, we split
the sample of each into two relatively equal subsamples and
re-estimate. The resulting parameter estimates appear
directly beneath those for the entire sample, for each of
our eight models in Table IV—2. Examination of the parav
meter estimates of these subsamples indicates that our
models are quite stable. A study of the diagnostic checks
for each model supports this finding.

We conclude that the models adequately represent the
bivariate relationships between each of the eight leading
indicators under consideration and Industrial Production.

We are interested in the impulse response function
implied by each of our models. These eight functions are
listed and plotted in Figures (H.l) through (H.8). We
would like to compare these eight functions in order to
make some evaluation about the strengths and weaknesses of
each in the role of leading indicator. However since the
inputs are not all measured in the same units, and since
each input behaves differently (in particular, since each

input has a different standard error), this comparison

88

TABLE IV—2
BIVARIATE MODELS FOR THE COMMERCE
DEPARTMENT LEADING INDICATORS

 

 

 

 

 

yt = BCDH7 Index of Industrial Production
(i) xt = BCDI: Average Workweek
Sample: January, l9H7 — September, 1979 (n: 393)
12 mo 12
(H.9) (l-B )(l-B)log(yt) I:EI§ (l-B )(l—B)log(xt)
12
+ (1-812B )at
w = 1.0260 61 = .6777 012 = .7H62
O (.079) (.036) (.035)
2 _ -
x”, — 60.7 R58 _ .0117
r 2 1
R2 = l _ AéRSE)
0output
2
= l _ (.0117) 2 : .652H
L (.0198H6) ,
Sample: January, 19H7 — April, 1963 (n = 196)
w = 1.2599 61 = .6098 012 = .7670
(.133) (.056) (.053)
2 - _
x“, — 00.0 RSE — .0136
—2 ( 0136)2
R = l — ° = .530H

(.0198H6)2

89

TABLE IV—2 (cont'd.)

 

 

Sample: May, 1963 — September, 1979 (n = 197)
mo = .7093 51 = .7099 812 = .7012
(.092) (.008) (.055)
x37 = 57.8 R58 = .0096
—2 ( 0096)2
R : l — ° 2 3 .7660
(.019805)
(ii) xt = BCD3 Layoff Rate
Sample: January, l9H7 - August, 1979 (n = 392)
w -w B
(0.10) (l-Blz)(l-B)log(y ) = —O—l—(l-B12)(l-B)log(x )
t 2 t
1-62B
12
+ (1-012B )at
w = -.0300 ml = 0315 52 = .5927 912 = .7565
O (.003) ( 003) (.005) (.035)
x37 - 56.0 RSE = .0116
R2 = .6580
Sample: January, 19H? - April, 1963 (n = 196)
w = -.0381 w = .0331 5 = .5569 e = .7609
O (.005) l (.005) 2 (.065) 12 (.053)
X07 = 08.5 R58 = .0139
R2 = .5090
Sample: May, 1963 - August, 1979 (n = 196)
w =-.0299 ml = .0261 52 = .6597 912 = .7095
(.00H) (.00H) (.062) (.058)
X07 2 02.9 R58 = .0096
R2 = .7660

90

TABLE IV-2 (cont'd.)

 

(iii)

(0.11)

(iv)

(H.12)

xt = BCD8 Value of Manufacturers' New Orders
Sample: January, 1958 - May, 1979 (n = 257)
w -w B
(1-812)(l-B)log(y ) = —9—4l— (1-812)(l-B)log(x )
‘t 2 ‘t
1-62B
12
+ (1-8128 )at
w = .2396 w = —.0865 6 = .3H71 0 = .695H
0 (.019) l (.018) 2 (.062) 12 (.009)
x39 = 39.1 R58 = .0103
R2 = .7306
Sample: January, 1958 - December, 1969 (n = lHH)
w = .2306 ml 3 -.0731 62 = .3175 012 = .6H9H
(.027) (.026) (.102) (.072)
x39 = 02.2 R58 = .0116
R2 = .5580
Sample: January, 1970 - May, 1979 (n 113)
w = .2616 ml = -.102H 62 = .3852 012 = .7353
(.026) (.025) (.076) (.079)
x39 = 25.2 R58 = .0091
R2 = .7897
Xt = BCD19 Index of Stock Prices
Sample: January,l9H7 - October, 1979 (n = 39H)
(1-812)(1-B)log(y ) = ——9——82(1—812)<1-8)1og(x )
t 1-618 t
1-0 B12
+ 12 a
l-¢lB t
w = .0670 51 = .8155 812 = .7902 *1 = .2739
(.015) (.059) (.032) (.051)
x35 = ”5.6 R58 = .0132
R2 = .0020

91

TABLE IV-2 (cont'd.)

 

(v)

(0.13)

Sample January, 1907 - May, 1963 (n = 197)
6 = .0961 31 = .6903 812 = .8007 $1 = .2931
(.031) (.101) (.006) (.070)
2 _ _
X06 35.8 R58 — .0159
R2 = .3581
Sample: June, 1963 - October, 1979 (n = 197)
w = .0537 5 = .8919 8 = .7830 g = .1831
O (.012) l (.036) 12 (.051) l (.076)
2 _ _
X06 38.5 R58 — .0103
R2 = .7306
xt = BCD29 Index of Housing Starts
Sample: January, 1959 - August, 1979 (n = 208)

(l-Blz)(l—B)log(yt)

2
wOB9(1-Bl )(l—B)log(xt)

 

1-8 812
+ 12
at
l—¢lB
w = .0172 812 = .5908 *1 = .3797
(.007) (.056) (.050)
X06 = 37.2 R58 = .0120
I R2 - .6096
Sample: January, 1959 - April 1969 (n = 120)
w = .0278 012 = .0052 61 = .2569
(.012) (.09H) (.100)
2 _ _
XL,6 — 32.7 RSE - .0130
R2 = .5001

92

TABLE IV-2 (cont'd.)

 

 

(vi)

(0.10)

Sample: May, 1969 - August,l979 (n = 120)

A A A

w = .0125 612 : .6539 $1 = .5031
(.096) (.082) (.090)
2 _ _
X06 — 39.0 R58 - .0121
R2 = .6283
xt = BCD32 Vendor Performance
Sample: January, 1907 - October,l979 (n = 390)
(L)
(1-812)(l-B)log(y ) = ~—9—— 82(1—812)(1—B)log(x )
t 1-618 t
1—0 812
+ _11121__
1-¢ B at
1
w = .0180 61 = .7551 912 = .7811 ¢1 = .2770
(.000) (.078) (.033) (.052)
x36 = 05.0 R58 2 .0133
R2 = .5509
Sample: January, 1907 - May, 1963 (n = 197)
w = .0163 61 = .8625 912 = .8061 ¢3 = .2800
(.005) (.072) (.050) (.070)
2 _ -
x£46 - 30.9 R58 - .0160
R2 = .3500
Sample: June, 1963 - October, 1979 (n = 197)
0) = .0279 §_= .0870 812 = .7775 01 = .2138
(.008) (.176) (.051) (.079)
2 _ -
X06 - 03.2 R58 - .0105
R‘ = .7107

93

TABLE IV—2 (cont'd.)

 

 

(vii) xt = BCD92 % Change in PPI of Crude Materials
Sample: January, 19H? - August,l979 (n = 392) 12
1-9 B
12 _ mo 10 12 12
(14.15) (l-B )(l-B)log(yt) - ITO? B (1-B )[Xt]+ 1—¢1B at
w =‘.1062 51 = .7792 912 = .77H7 $1 = .2886
(.033) (.083) (.030) ( 051)
2 _ _
X06 — 37.7 RSE - .0136
R2 = .5300
Sample: January, 1907 - April, 1963 (n = 196)
w = -.0700 61 = .8596 612 = .8051 91 = .3101
(.000) (.096) (.052) (.070)
2 _ _
X05 - 30.5 RSE - .0168
R2 = .2830

Sample: May, 1963 - August, 1979 (n = 196)

A A A

w = -.1965 6 = .5052 8 = .6689 0

= .1595
O (.009) 1 (.105) 12 (.060) l (.079)
2 _ - .
X06 - 00.7 R58 — .0107
R2 = .7093
(viii) xt = BCD105 Real Money Supply - M1
Sample: January, 1907 - September, 1979 (n = 393)
(L)
(0.16) (1-812)(1-B)log(y ) = —°— 85(1-812)(1—B)log(x )
t 1-618 t
1-812812
+ ———————— a
1-018 t
Q = .3267 31 = .6511 612 = .7960 ¢1 = .3029
(.109) (.153) (.033) (.050)
2 _ _
X05 - 06.8 R58 - .0136

R = .530H

90

TABLE IV-2 (cont'd.)

 

Sample: January 1907 - May, 1963 (n = 197)

w = .3055 01 = .5018 012 = .8055 01 = .3717
(.177) (.333) (.008) (.072)
X36 = 36.8 R58 = .0160
R2 = .3171

Sample: June, 1963 — September, 1979 (n = 196)

w = .3378 6 = .7123 0 = .6881 ¢l = .2578

O (.103) l (.151) 12 (.059) (.076)
2 _ - _
X06 - 35.5 R58 - .0111
—2

R = .6872

 

xt

95

FIGURE 0.1
GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]
= BCDl Average Workweek

0. .25 .5 .75 l.

(k) +++++++++.+++++++++.+++++++++,+++++++++,+++++++++

(OCDNCDUW-KwMHCD
><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><xgéégggggggggé

VALUES [vk]

.102605E+01
.695305E+00
.H71231E+00
.319351E+00
.216H22E+00
.1H6668E+00
.99396lE-01
.673602E-01
.HS6H96E-01
.309365E-01
.209655E-01
.lH2082E-01
.962880E-02
.652538E-02
.HH2222E—02
.29969lE-02
.203099E-02
.137639E-02
.932771E-03
.63213HE-03
.H28393E-03
.290320E-03
.l967H8E-03
.133335E-03
.903600E—00
.612367E-0H
.H1H998E-0H
.2812H2E-OH
.190596E-0H
.129166E-0H
.875308E-05
.593215E-05
.H02021E-05
.272HH7E-05
.18H636E-05
.l25127E-05
.807977E—06
.57H669E-06
.389H50E-06
.263928E-06
.178862E-06
.12121HE-06
.821H61E-07
.556699E-07
.377272E-07
.255675E-07
.173269E—07
.11707‘ ‘7
.795

96

FIGURE 0.2
GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]
xi = BCD3 Layoff Rate
. -.5 -.25 0. .25

(k)+++++++++,+++++++++,+++++++++.+++++++++,+++++++++ VALufis[vk]
0 xxx -.303560£~01
1 xxx -.315098E—01
2 xx -.2035288—01
3 xx -.186759E—01
” x -.120691E—01
5 x -.110692E-01
5 x -.715335E—02
7 x -.656075E—02
8 x —.023980E—02
9 x —.388856E—02
10 x -.251290E—02
11 x —.2300768—02
12 x —.1089028—02
13 x -.136603E-02
15 x —.8827818—03
16 x -.8095098—03
17 x —.523226E-03
18 x —.079880E—03
19 x —.3101158—03
20 x -.280025£—03
21 x -.183806E-03
22 x -.1685798—03
23 x —.108902E—03
2” x -.999172E—00
25 x .-.505702E—00
26 x -.5922108-00
27 x —.382709E—00
28 x —.3510008-00
29 x —.226832E-00
30 x -.208000E—00
31 x -.1300008—00
32 x -.1233068—00
33 x -.796809E—05
3” x -.7308358-05
35 x -.072293E—05
36 x -.0331678-05
37 x -.279929E-05
38 x -.256739E—05
39 x -.1659108—05
00 -x -.152169E-05
”1 x -.983375E-06
”2 x -.901909E-06
”3 x -.5828088-06
00 x -.530563E-06
”5 x -.3050558-05
06 x -.316836E—06
07 x —.2OH751E-06
”8 x —.187789E-06
x -.l21356E—06

X

97

FIGURE H.3

GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]
= BCD8 Value of Manufacturers' New Orders

t -.25 0. .25 .5

00 +++++++++.+++++++++.+++++++++,+++++++++,+++++++++VALUES [Vk]

0 xxxxxxxxxx .2395828+00

l xxxxx .8608588-01

2 xxxxx .8315508-01

3 xxx .300180E-01

” xxx .2885218-01

5 x .1001888-01

5 x .100176E—01

7 x .361623E-02

8 x .3075978—02

9 x .1255108-02
10 x .120681E-02
11 x .035602E-03
12 x .0188668-03
13 x .151205E—03
1” x .1053838-03
15 x .520812E-00
15 x .5006028—00
17 x .1821558-00
18 x .1751008—00
19 x .5322338—05
20 x .607877E-05
21 x .219039E-05
2? x .210989E-05
23 x .7515038-06
2” x .732313E-06
25 x .2503558-06
25 x .250176E-06
27 x .9175008-07
28 x .882208E-07
29 x .318065E-07
30 x .306202E-07
31 x .110535E-07
32 x .1062788-07
33 x .383650E—08
3” x .368876E-08
35 X .1331598-08
35 x .l28031E-08
37 x .H62175E-09
33 x .0003778—09
39 x .1600108—09
”0 x .150237E-09
”1 x .5567738-10
”2 x .5353328-10
”3 x .193208E—10
”” x .185806E-10
”5 x .6707308-11
”5 x .6009008-11
”7 x .2328028-11
”8 x .223837E-ll

98

FIGURE 0.0
GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]
xt = BCD19 Index of Stock Prices
-.25 0. .25 .5

(k)+++++++++,+++++++++,+++++++++,+++++++++,+++++++++VALUES [Vk]

0 X 0.

l x 0.

2 xxxxx .5738018-01

3 xxxx .5095008-01

H xxxx ' .0081368—01

5 .xxx - .365068E-01

6 xxx .2980098-01

7 xx .2030688—01

8 xx ‘ .198229E—01
-9 xx .161661E-01
10 x .1318398-01
11 x .1075198-01
12 x .8768008-02
13 x .715091E-02
1” x .583177E-02
15 x .0755988-02
15 x .3878538-02
17 x .316313E-02
18 x .257963E-02
19 x .2103758-02
20 x .l71567E-02
21 x .1399188-02
2? x .1101078-02
23 x .930576E-03
2” x .758911E-03
25 x .618910E-03
26 x .5007028-03
27 x .011631E-03
28 x .3356978-03
29 x .273770E-03
30 x .223267E-03
31 x .1820818-03
32 x .1080928-03
33 x .121100E-03
3” x .987601E-00
35 X .8050178-00
35 x .5568008-00
37 x .535672E-00
38 x .0368558—00
39 x .356268E-OH
”0 x .2905078-00
”1 x .2369098-00
”2 x .1932398-00
”3 x .157592E-00
”” x .1285208—00
“5 x .1008128-00
”5 x .8507728-05
”7 x .697091E-05
Ll8 x .568097E-05

99

FIGURE 0.5
GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]
xt = BCD29 Index of Housing Starts
-.25 0. .25 .5

(k) +++++++++.+++++++++.+++++++++,+++++++++,+++++++++VALUES[vk]

(DmQCDUW-FCDNF-‘O
000000000
.0 I. O I O O

:172268E-01

N
.5.-
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXEEX><><><><><><><><

DOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
O O O I I O O O O I O O O O O O O C O O C O O O O O O O O O O

100

FIGURE 0.6
GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]
xt = BCD32 Vendor Performance
-.25 0. .25 .5

(k)+++++++++.+++++++++.+++++++++,+++++++++.+++++++++ VALUESEVk]

0 X 0.

1 X 0.

2 XX .1800808-01

3 X .137775E—01

0 X .1050058-01

5 X .8060088-02

5 x .5169078-02

7 X .0719988-02

8 X .361105E-02

9 X .276265E-02
10 X .2113588-02
11 X .161701E-02
12 X .1237108-02
13 X .906059E-03
10 X .7200878-03
15 X .5539678-03
16 X .0238158-03
17 X .3202028-03
18 X .2080638—03
19 X .1897828—03
20 X .1051908-03
21 X .1110818—03
22 X .8098338-00
23 X .5501698-00
20 X .0970158-00
25 x .3805508-00
25 X .2911028-00
27 X .2227008-00
28 X .1700088-00
29 X .l30372E-OH
30 X .9970168-05
31 X .763079E-05
32 X .5837978-05
33 X .0056378-05
30 x .3017028-05
35 X .2510218-05
36 x .200002E-05
37 x .153012E-05
38 x .1170538-05
39 x .8955968-05
00 X .5851818-06
01 X .5202018-06
02 x .0010038-05
03 x .3058208-06
00 X .2307308-06
“5 x .179585E-06
05 X .137392E-06
07 X .1051138-05
08 x .800170E-07

X

101

FIGURE 0.7

GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]

t
—.5 -.25 0.

= BCD92 % Change in PPI of Crude Materials

25

(k)+++++++++,+++++++++,+++++++++.+++++++++,+++++++++

(DGDQCDUW-C'wNI—‘O

><><><><><><><><><><

XXXXX
XXXX
XXXX

XXX
XXX

XXX
XXX

><><><><><><XXXXXXXXXXXXXXXXXXXXX><><><><

IOOOOOOOOOO
000...... 0

VALUES[vk]

.106213E+00
.827565E-01
.6HH800E—01
.502397E-01
.391HHHE-01
.300995E-01
.237638E-01
.l85156E-01
.lHH265E-01
.112H05E-01
.875803E-02
.68238HE-02
.531682E-02
.HlH262E-02
.322773E-02
.251090E-02
.l959H9E-02
.15267HE-02
.118956E-02
.926853E-03
.722160E-03
.562673E-03
.H38HO9E-03
.3H1587E-03
.266109E-03
.207371E-03
.161573E-03
.125890E-03
.980879E-0H
.76025HE-OH
.59SH7lE—0H
.H63963E-0H

361H98E-0H
281662E-OH
219058E-0H
l70991E-0H
133228E-0H
103805E-OH
808802E-05

xt - BCD105 Real Money Supply - M1
—.25 0. .25 .5

(k)+++++++++.+++++++++.+++++++++.+++++++++.+++++++++ VALUES[vk]
0 X 0.

l X 0.

2 X 0.

3 X 0.

0 X 0.

5 XXXXXXXXXXXXXX .326655E+00
6 XXXXXXXXX .212692E+00
7 XXXXXX .138088E+00
8 XXXXX .901725E-01
9 XXXX .587132E-01
10 XXX .382290E-01
11 XX .208920E-01
12 XX .162077E-01
13 X .105532E-01
1“ X .687138E—02
15 X .0070108—02
15 X .291318E-02
17 X .189683E-02
18 X .123507E—02
19 X .8001778-03
20 X .523616E-03
21 X .300937E-03
22 x .221991E-03
23 X .1005038—03
2” X .9011518-00
25 X .6128038-00
25 x .3990098-00
27 X .259803E-OH
28 x .169163E-00
29 X .1101058-00
30 X .717l8lE-05
31 X .066971E-05
32 X .3000558—05
33 X .197976E-05
3” X .l28907E-05
35 X .839337E-06
35 X .5065108-05
37 X .3558008-06
38 X .2316978-06
39 x .150863E-06
”0 x .982301E—07
”1 x .639596E-07
”2 x .016055E-07
”3 x .2711628-07
”3 x .l76559E-07
”5 X .110961E-07
”5 X .708538E-08
“7 X .087389E-08
”8 X .317309E-08

102

FIGURE 0.8

GRAPH OF IMPULSE RESPONSE WEIGHTS [vk]

103

cannot be made with the functions listed in Figures (0.1)
through (0.8). We need to transform the impulse response
weights in each function into beta coefficients in order

to make this comparison.

Comparison With Beta Coefficients

 

To illuminate this requirement, consider an illustrae
tion. The impulse response weights in these figures are
analagous to regression coefficients, such as "6" in the
following model.

Suppose the true model is

Y = a + bX + 8

Then suppose the line minimizing the sum of squared errors

(the Ordinary Least Squares regression line) is

Y = a + bX
Then Y = a + bX + e

A

where e represents the residuals of the regression.
The regressed line will pass through the point, (X,Y).
i.e. Y = a + bX

From this it follows that

 

 

Y .. Y : b(X -- Y) + e
or O Y,: Y = bO XA- X + e

 

 

_ A0 _
or [Y-Y] : bl[XA'X]+X€3—
C

100

In this last equation the input and the output are

A O
standardized. [b :3] is the beta coefficient, which measures

0

the relationship btheen the standardized input and the
standardized output. Observe that it is obtained by
multiplying the regression coefficient, 6, by the ratio of
the standard deviation of the input to the standard
deviation of the output. It can readily be compared with
the beta coefficient Of any other similar regression in
which the input and output are standardized.

We can transform the impulse response weights in
each of our functions listed in Figures 0.1 through 0.8
into beta coefficients, by simply multiplying each

goefficient in each impulse response function by the ratio,

Q

75, relevant to that particular model. This is done in
O
y
Figures 0.9 through 0.16. In these eight figures the
, magnitude of the relationship between each leading indicator

and Industrial Production can be readily compared.

Implications

 

Upon examining these figures, we are immediately
struck by a distinct fault which characterizes five of the
relationships: the lack of any substantial lead. These
series are hailed by the Commerce Department as leaders,
which suggests that current movements in each indicator
Should be consistently followed by movements in economic
activity after some lag. They are developed with the

expressed purpose of providing information about future

105

FIGURE H.9
GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficients]

Xt = BCDl Average Workweek
The figures are obtained by multiplying the 09 weights in
Figure 0.1 by ~0100”5 .

.019806

The standard error of the series, (l—Blz)(l-B)log(x ), is
.0100H6. t

The standard error of the series, (1-812)(1-B)log(y ), is
.019806. . t

' .25 . 0. .25 .5
(k)+++++++++,+++++++++.+++++++++,+++++++++,+++++++++ VALUES
0 XXXXXXXXXXXXXXXXXXXXXX .519380E+00
l XXXXXXXXXXXXXXX .351982E+00
2 XXXXXXXXXX .238536E+00
3 XXXXXXX .161655E+00
H XXXXX .109522E+00
5 XXX .7H2H30E-01
6 XXX .503101E-01
7 XX .300976E-01
8 XX .231077E-01
9 XX .156600E-01
10 X .106127E-01
11 X .712916E-02
12 X .087008E-02
13 X .330263E-02
10 X .223852E-02
15 X .151703E-02
16 X .102808E-02
17 X .696725E—03
18 X .072167E-03
19 X .319985E-03
20 X .216852E-03
21 X .106959E-03
22 X .995930E-00
23 X .670939E-00
20 X .057002E-OH
25 X .309979E-OH
26 X .210071E-0H
27 X .1H236HE-0H
28 X .960793E-05
29 X .653835E-05
30 X .003099E—05
31 X .300280E-05
32 X .203502E-05
33 X .137912E—05

106

FIGURE H.9 (cont'd.)

-.25 0. .25 .5
(k) +++++++++ .+++++++++ ,+++++++++ , +++++++++ ,+++++++++ VALUES
30 X .9306238-06
35 X .5333908-06
36 X .0292008-06
37 X .290896E—06
38 X .1971398-05
39 X .1336008-06
00 X .9053958-07
01 X .613583E-07
02 X .015822E-07
03 X .281800E—07
00 x .190970E-07
05 x .1290228—07
05 X .877080E-08
07 X .5903988-08
08 x .002819E-08

107

FIGURE 0.10

GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficients]
x = BCD3

t

Layoff Rate

The figures are obtained by multiplying the 09 weights in
Figure 0.2 by -20235 .

.019806

The standard error of (1-812)(1—B)log(xt) is .20236, and
the standard error of (l-B12)(l-B)1og(yt) is .019806

-.5

-.25 0.

.25

(k)+++++++++,+++++++++,+++++++++.+++++++++,+++++++++

(DQOUW-L'WNHO

XXXXXXXXXXXXXXX
XXXXXXXXXXXXXX
XXXXXXXXX
XXXXXXXX

XXXXX

XXXXX

XXXX

XXXX

XXX

XXX

XX

XX

XX

XXXXXXXXXXXXXXXX><><><><><><><><><><

+ VALUES

.350311E+00
.321290E+00
.207630E+00
.190029E+00
.123063E+00
.112867E+00
.729392E-01
.668968E-01
.032312E-01
.396098E-01
.256232E-01
.235005E-01
.151869E-01
.139287E-01
.900129E—02
.825560E—02
.533508E-02

089310E-02

.316210E-02
.290010E-02
.187018E-02
.l71892E-02
.111083E-02
.101881E-02
.658391E-03
.603808E-03
.390230E-03
.357902E-03
.231290E-03
.212128E-03

137086E-03
125729E-03
812508E-00
705197E-00
08157HE-00
001679E—00
285030E-00
261780E—00
169170E-00

108

FIGURE 0.10 (cont'd.)

-.5 0.

(k)+++++++++,+++++++++,+++++++++,+++++++++,+++++++++ VALUES

39 x -.155159E-00
00 X -.1002708-00
01 x -.919633E-05
”2 x -.590302E-05
”3 X -.505068E—05
00 x -.352200E—05
”5 X -.323062E-05
”6 x -.208775E-05
07 X -.191079E-05
08 X -.1237018-05

108

FIGURE 0.10 (cont'd.)

-.5 0.

(k)+++++++++,+++++++++,+++++++++,+++++++++,+++++++++ VALUES

39 x —.155159E—00
00 x -.100270E-00
01 x -.919633E-05
02 X -.590302E-05
03 X -.505068E-05
00 X -.352200E-05
”5 x -.323062E-05
06 X -.208775E—05
07 X -.191079E-05
08 X -.l23701E-05

109

FIGURE 0.11

GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficients]

xt = BCD8 Value of Manufacturers' New Orders

These figures are obtained by multiplying the H9 weights in
Figure 0.3 by .

.019806

The standard error of (1-812)(1—B)log(xt) is .002878, and
the standard error of (l-Blz)(l-B)log(yt) is .019806

—.25 0. .25 .
QJ+++++++++,+++++++++.+++++++++,+++++++++,+++++++++ VALUES
6 XXXXXXXXXXXXXXXXXXXXX .5176258+00
l XXXXXXXX .1868568+00
2 XXXXXXXX .179660E+00
3 XXXX .608550E-01
0 XXX .623576E-01
5 XX .225102E-01
6 XX .216030E-01
7 X .7813008-02
6 X .751212E-02
9 X .271178E-02
10 X .260736E-02
11 X .9012208-03
12 X .900975E-03
13 X .326680E-03
1” X .3101058-03
15 X .1133888-03
16 X .10902lE-03
17 x .393552E-00
18 X .3783968-00
19 X .1365968-00
20 X .131330E-00
21 X .0701058—05
22 X .055809E-05
23 X .l60556E—05
2” X .1582198-05
25 X .5711098-06
26 X .5091568-06
27 x .1982388-06
23 X .190600E-06
29 X .6880558-07
30 X .6615608-07
31 X .2388158—07
32 X .229617E-07
33 X .8288908-08
3” X .7969708-08
35 x .287695E-08
36 X .2756168-08
37 X .998506E-09

38
39
00
01
02
03
00
05
06
07
08

-.25

FIGURE 0.11 (cont'd.)

0.
(k)+++++++++.+++++++++.+++++++++.+++++++++.+++++++++ VALUES

><><><><><><><><><><><

110

.25

.5

.960093E—09
.306580E—09
.333235E-09
.l20293E-09
.115660E-09
.017519E-10
.HOlHHlE-lO
.100915E-10
.13933HE-10
.502977E-11
.083608E-11

111

FIGURE 0.12
GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficients]
Xt = BCD19 Index of Stock Prices

These figures are ogtained by multiplying the 09 weights in

Figure 0. 0 by _______
.019806

The standard error of (l- 81:)(1- B)log(xt ) is .006007, and
the standard error of (1-812)(l— B)log(y:) is .019806

-.25 0. .25 .5

UO+++++++++.+++++++++.+++++++++.+++++++++.+++++++++ VALUES

LDCDQCDU'l-C'WMF—‘O

X

X
XXXXXXX
XXXXXX
XXXXX
XXXX
XXXX
XXX
XXX

XX

XX

XX

XX

XX

XX

XXXXXXXXXXXXXXXXXXXXXXX

OO
0

.156201E+00
.127386E+00
.103887E+00
.807228E—01
.690937E-01
.563080E—01
.05953HE-01
.307672E-01
.305629E-01
.209251E-01
.203269E-01
.165772E-01
.135192E-01
.110253E-01
.899lHHE-02
.733277E-02
.598010E-02
.087690E-02
.397727E-02
.320358E-02
.260523E-02
.215726E-02
.l75931E-02
.103077E-02
.117009E—02
.950203E-03
.778213E-03
.630650E-03
.517578E-03
.022100E-03
.30023HE-03
.280730E-03
.228906E—03
.186712E-03
.152269E-03
.l20179E-03

38
39
00
01
02
03
00
05
06
07
08

-.25

FIGURE 0.12 (cont'd.)

0.
UO+++++++++.+++++++++.+++++++++.+++++++++.+++++++++ VALUES

><><><><><><><><><><><

112

.25

.5

.101272E-03
.825901E—00
.673506E-00
.509295E-00
.007967E-00
.365330E—00
.297935E-00
.202975E—00
.l98153E—00
.161600E—00
.131789E-00

113

FIGURE 0.13
GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficients]

xt = BCD29 Index of Housing Starts

These figures are obta§ned by multiplying the 09 weights

in Figure 0. 5 by ________
.019806 12
The standard error of (1-B )(1—B)log(xt) is .11213, and

the standard error of (1-812)(l—B)log(yt) is .019806

-025 0. .25 .5
UO+++++++++.+++++++++.+++++++++.+++++++++.+++++++++ VALUES

(oooqmmszHo
000000000
0 0 0 0 0 0 0 0

XXXX .973315E-01

HFJPJHFAPHAPJH
QOU'I-ITCDMl—‘O

wwwwwwmmmmwwmmm
:wwl—‘OLOCDQGU‘I-COONHO
0 0 0 0

com
0501

H
(D
XXXXXXXXXXXXXXXXXX><><><><><><><><><><><><><><><><><><><><

OOOOOOOOOOOOOOOOOOOOOOOOOOOO
0 0 0 0 0 0 . 0 0 0 0 0 0 0 0 . 0 . 0 0 0 0

(.0
\1

110

FIGURE 0.13 (cont'd.)

-, 0. .25 .5
G<)+++++++++,?+§L+++++++,+++++++++ . +++++++++ , +++++++++ VALUES
38 X 0
39 X 0
00 X 0
01 X 0
02 X 0
03 X 0
00 X 0
05 X 0
06 X 0
07 X 0
08 X 0

115

FIGURE 0.10
GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficientS]
xt = BCD32 Vendor Performance
These figures are obtained by multiplying the 09 weights
in Figure 0.6 by ~17211 .
.019806
The standard error of (l-Blz)(l-B)log(xt) is .17211, and

the standard error of (l-B12)(l—B)log(yt) is .019806

0. .25 .5

UO+++++++++.+++++++++.+++++++++.+++++++++.+++++++++ VALUES

LOCDNIOTUWSCDNHO

X 0.

X 0.

XXXXXXX .156170E+00
XXXXXX .119082E+00
XXXXX .910101E-01
XXXX .699339E-01
XXX .535033E-01
XXX .009330E-01
XX .313160E-01
XX .239585E-01
XX .183296E-01
XX .100232E-01
X .107285E-01
X .820759E-02
X .627908E-02
X .080016E-02
X .367500E—02
X .281192E—02
X .215127E—02
X .l60580E-02
X .125916E-02
X .963325E-03
X .736999E-03
X .563805E-03
X .031312E-03
X .330023E—03
X .252086E-03
X .193166E—03
X .107783E-03
X .113062E-03
X .860987E-00
X .661763E-00
X .506285E-00
X .387336E-00
X .296333E-00
X .226712E-00
X .173007E-00
X .132696E-00

38
39
00
01
02
03
00
05
06
07
08

-.25

FIGURE 0.10 (cont'd)

0. .
QJ+++++++++,+++++++++,+++++++++,+++++++++,+++++++++ VALUES

><><><><><><><><><><><

116

.25

5

.101520E-00
.776686E-05
.590208E—05
.050602E-05
.307796E—05
.266083E—05
.203568E-05
.155701E-05
.ll9050E-05
.911569E—06
.693398E-06

117

FIGURE 0.15
GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta coefficients]

xt = BCD92 % Change in PPI of Crude Materials

These figures are obtained by multiplying the 09 weights in
Figure 0.7 by -021531 .
.019806

The standard error of (1-812)[xt] is .021631, and
the standard error of (l-Bl2)(l-B)log(yt) is .019806

-.5 -.25 0. .25
Q)+++++++++,+++++++++,+++++++++,+++++++++,+++++++++ VALUES

0 X 0

l X 0

2 X 0

3 X 0

0 X 0

5 X 0

6 X 0

7 X 0

8 X 0

9 X 0.
10 XXXXXX -.1157568+00
ll XXXXX -.901998E—01
12 XXXX -.7027958-01
13 XXX -.SH7580E-01
1” XXX -.026651E-01
15 XX -.332027E-01

.259012E-01
.201809E—01

HP4
\107
XX
XX
ll

18 XX -.1572018-01
19 X -.l22515E-01
20 X -.905575E-02
21 X -.703759E-02
22 X —.5795038-02
23 X -.051522E-02
20 X —.351800E—02
25 X -.270110E-02
26 X -.2l3573E-02
27 X -.l66006E-02
28 X -.129655E-02
29 X -.101022E—02
30 x —.787113E-03
31 x —.613281E-03
32 X -.H77801E-03
33 x -.372310E-03
30 X -.290087E-03
35 X -.2260228-03
36 x -.176105E-03
37 X -.l37213E-03

118
FIGURE 0.15 (cont'd)

-05 -o25 0., 025
Qj+++++++++,+++++++++,+++++++++,+++++++++,+++++++++

38
39
00
01
02
03
00
05
06
07
08

><><><><><><><><><><><
I

VALUES

.106910E-03
.832993E-00
.609029E—00
.505693E-00
.390012E-00
.306995E—00
.239197E—00
.186370E-00
.105211E-00
.113101E-00
.881508E-05

119

FIGURE 0.16

GRAPH OF IMPULSE RESPONSE WEIGHTS [Beta Coefficients]

xt = BCD105 Real Money Supply - Ml

These figures are obtained by multiplying the 09 weights in
Figure 0.8 by 2

.019806
The standard error of (l- 81:)(1-B)1og(xt ) is .00702, and
the standard error of (1-812)(1-B)log(y:) is .019806.
-.25 0. .25 .5

QJ+++++++++,+++++++++,+++++++++,+++++++++,+++++++++ VALUES

0 X 0.

1 X 0.

2 X 0.

3 X 0.

0 X 0-

5 XXXXXX .115506E+00
6 XXXX .752302E-01
7 XXX .089865E-01
8 XX .318961E-01
9 XX .207682E-01
10 XX .135226E-01
11 X .880089E-02
12 X .573305E-02
13 X .373292E-02
10 X .203057E202
15 X .158260E-02
16 X .103006E-02
17 X .670950E—03
18 X .H36873E-03
19 X .280056E—03
20 X .185215E—03
21 X .120597E-03
22 X .785235E-00
23 X .511283E—00
20 X .332907E-00
25 X .216763E-00
26 X .101138E-00
27 X .918935E—05
28 X .598370E-05
29 X .389612E-05
30 X .253680E-05
31 X .165179E-05
32 X .107551E-05
33 X .700288E-06
30 X .055975E-06
35 X .296893E-06
36 X .193310E-06
37 X .l25870E-06

120

FIGURE 0.16 (cont'd.)

-.25 0. .25 .5

&Q+++++++++,+++++++++,+++++++++,+++++++++,+++++++++ VALUES

38 X .819567E-07
39 X .533638E-07
00 X .307063E-07
01 X .226200E-07
“2 X .lH7310E-07
”3 X .959160E-08
00 x .620531E-08
”5 X .006600E-08
06 X .260776E-08
”7 X .l72001E-08
08 X .ll2250E-08

121

movements in economic activity. Furthermore, they are
deemed useful in forming and implementing monetary and
fiscal policy decisions aimed at stabilizing economic
activity.

To be truly useful in such a role, a leading
indicator should Show a consistently long lead over
Industrial Production. How long should this lead be? There
are various decision rules about evaluating when a turning
point occurs in any of these series. A common rule
"accepted" at this time states that a series has experienced
a turning point if a succession of increases (or decreases)
is followed by three successive decreases (or increases).
With this decision rule, a leading indicator must certainly

show a consistent lead of more than three months over economic

 

activity, if it is to be of any value in forecasting a
turning point in economic activity.

According to this rule, the recognition lag of a need
for stabilization policy action will be three months
(provided that the decision rule is followed). After this
lag there will be an action lag and an outside lag before
policy actions actually have a desired effect on economic
activity. The action lag could be extremely long itself
if fiscal policy is the desired tool, given the inertia of
the Congressional decision-making process. The action lag
could be relatively short if monetary policy is implemented.
There has been much debate about the length of the outside

lag in our economic system, though there is general

122

agreement that it extends at least over several months.

In this light we see the need for leading indicators
to display leads which are several months longer than the
three months necessary to recognize a turning point. From
Figures 0.9 througthlB we see that BCDl, BCD3, and BCD8
show no lead at all over Industrial Production. BCD19 and
BCD32 display leads of just two months each. BCD105, the
Real money supply, has a lead of five months, which may not
be long enough to be very useful in the role desired, though
it is much better than zero or two months. BCD29, the Index
of Housing Starts, has a lead of nine months, which
suggests that it may supply useful information as a leading
indicator.

Finally, the leading indicator with the longest lead
of the eight considered is BCD92, the % change in the PPI of
Crude Materials. Its impulse responSe function implies that
a sustained 1% increase in (the seasonal difference of) the
growth rate of the PPI will be followed by a decrease of
about one tenth of 1% in (the seasonal difference of) the
growth rate of Industrial Production after ten months, and
further decreases in the following months. This negative
relationship could reflect a movement along some demand curve,
and thus follows our economic intuition. It is remarkable
to note, however, that the Commerce Department uses this
leading indicator in a positive role in the CLI [See the
Handbook of Cyclical Indicators, pages 2, 3, 61.]. That is,

BCD92 is used by the Commerce Department as if an increase

123

in the PPI were consistently followed by increases in
Industrial Production. Our model in equation (0.15) and
Figures 0.7 and 0.15 suggests that this is an inappropriate
use of BCD92!

It is important to note that this same negative
pattern is implied by the estimated coefficients for the
sample period ending in 1963 [See Table IV-2, (vii)]. This
indicates that the negative relationship is quite stable,
rather than just a phenomenon of the "supply shocks" in the
early 1970's.

Given all these considerations, we are left with just
two of these eight leading indicators whicfilshould contribute
positively to the CLI, as constructed by the Commerce
Department: BCD29, the Index of Housing Starts, and BCD105,

the Real money supply - M1.

Conclusions

 

The Commerce Department construction of the CLI is not
backed by an appropriate theoretical framework, as outlined
in Chapter II.

Furthermore, our study indicates that five of the
eight leading indicators considered display empirical
relationships with Industrial Production which do not reflect
the characteristics of a good leading indicator. They Show
no significant lead time. This suggests that these five
series may not merit the status given them by the Commerce

Department. Their qualifications for the role of leading

12H

indicator appear to be lacking.

Three of the eight series considered display empirical
relationships with Industrial Production which do reflect the
characteristics of a good leading indicator: BCD29, BCD92,
and BCD105. However, BCD92, the % change in the PPI of Crude
Materials, proves to have a negative empirical relationship
with Industrial Production, while the Commerce Department
uses it in a positive role in the CLI. This leaves BCD29 and
BCD105 which appear to be the only leading indicators of the
eight considered, which might contribute positively to the
Commerce Department's CLI.

Given these observations, it is not surprising that
the Commerce Department's leading indicator approach has been
so unreliable. The question that remains is why so much
effort has been, and continues to be, spent on its development
and use in a predictive role. It seems clear that it will
continue to be relied upon in its ex post role of verifying
that turning points in economic activity have already taken
place. Our study suggests that at best, it should be limited

to this role.

CHAPTER V

MONEY AS A LEADING INDICATOR

Introduction

 

A huge literature exists on the role of Money in an
economy, and its relationship to real GNP. Under the current
state of thought toward Monetary Theory, what kind of
relationship might we expect to see between Money and real
GNP?

Friedman describes the adjustment of nominal income
to Monetary shocks in the context of a system of simultan-
eous differential equations.1 This system is an attempt
to explain (a) the short run adjustment of nominal income
to a change in autonomous variables; (b) the short run
division of a change in nominal income between prices and
real output; and (c) the transition between this short run
situation and long run equilibrium.

In this framework it is suggested that anything
which produces a discrepancy between the nominal quantity
of Money demanded and the quantity supplied, or between
their rates of change, will cause the rate of change in
nominal income to depart from its anticipated (permanent)

value. In general form,

125

126

 

* s d
dY _ dY dM dM s d
(a) aft- - f[(a{:‘) , —(—1_'E— , dt , M , M]
where Y = Py = nominal income,
P = the price level, y = real GNP,
M8 = Money Supply,
M(:1 = nominal amount of Money demanded,

and a * denotes the anticipated (or "permanent")
value of that variable.

A linearized version of (a) might be:

* s d
dlogY _ dlogY dlogM dlogM
' —_—_— _
(‘3) dt ' ( dt ) + W dt dt ]

 

+

¢(logMS - logMd)

Next, the division of a change in nominal income
between prices and output depends on two major factors:
anticipations about the behavior of prices, and the current
level of output compared with its full employment (permanent)

level. We can express this in general form as:

,7, it
dP _ dY dP dy ,
at ‘ gE'd—F’(EF)’(EF)’Y’YJ
(b)
dy dY dP * dy *
at = “a? (at) 3 (a) ’Y’ V”

where the form of g and h must be consistent with the
identity, Y = Py.

A linearized version (If (b) might be:

127

 

 

 

difigP : (difigP) + “Edifigy _ (dlwogY) ]

+ Y [logy - logy*]

(b)'
A

 

 

dlogy : dlogy _ dlogY _ dldoth
dt (—?fiT—) + (1 a)[ dt ( ) *1

Y [logy - logy*]

In their general form, the equations in (b) do not by
themselves specify the path of prices or output beginning
with any initial position. In addition we need to know how
anticipated values are formed. Presumably these are
affected by the course of events so that, in response to
a disturbance which produces a discrepancy between actual
and anticipated values of the variables, there is a feedback
effect that brings the actual and anticipated values

together again. To put this in general terms, we must have:

 

[Mm] = jtdldtogpmn ,
[91,12,61hn* = kEMWH ,
(c)
y*(t) = m[y(T)],
P*(t) = n[P(T)],
where t stands for a particular point in time, and T for

a vector of all dates prior to t.

A disturbance of long run equilibrium introduces

discrepancies in the two final terms in parentheses on the

128

right hand side of equation (a)'. This will cause the rate
of change in nominal income to deviate from its permanent
value, which through the equations in (b)‘, produces
deviations in the rate of price change and output change
from their permanent values. These will, through the
equations in (c), produce changes in the anticipated values
that will eventually eliminate the discrepancies between
measured and permanent values.

In the context of the above system, consider as such

a disturbance of long run equilibrium, a permanent increase

dlogMS
dt

frame in Figure 5.1 shows the time path of the money stock

in , the growth rate of the Money Supply. The first
before and after such a shock. The second frame shows the
equilibrium path of nominal income.

The slopes of the time paths in these two frames
must be equal, since in equilibrium nominal income will grow
at the same rate as the money stock, given the framework of
Friedman's model. However, the equilibrium path of Y after
this shock will be at a higher level than that of the Money

Stock. This is because part of the increase in dtfiEY w111

 

consist of an increase in §%%§E. With this increase in

inflation fully anticipated in equilibrium, it is now more
costly to hold money. As a result there will be a decline
in the real quantity of money demanded relative to income;
i.e. a rise in desired velocity. This rise will be
achieved by a rise in nominal income over and above that

required to match the rise in the nominal quantity of money.

129

FIGURE 5.1

TIME PATHS OF NOMINAL INCOME AND ITS COMPONENTS,
AFTER A MONETARY SHOCK

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

log M log Y
//
t t

t t

o o

d log Y
dt
6?
\
(Si—13%!) :1, _ _ _
(d log Y)
dt
t
o
d log y d log P
dt dt
(SQEELEH- _____ L _______
dt
(Ci—19.8.1) ________ __ - _ (910813) ________________
dt 0 dt
t t t t

130

The equilibrium path of nominal income will be like the
solid line in the second frame of Figure 5.1 rather than the
dashed line.

We are interested in the adjustment process involved
in the above scenario. It is apparent that in order to
produce the shiftimmthe equilibrium path of nominal income
from the dashed line to the solid line, nominal income must
rise over some period at a faster rate than the final
equilibrium rate. That is, there must be an overshooting,
or a cyclical reaction in the rate of change in nominal
income. The third frame in Figure 5.1 summarizes the

various possible adjustment paths of difigY consistent with

 

the theory presented above. The one common feature of all

possibilities is that the area above the (difing line must

 

exceed the area below.

This chapter is concerned with the composition of the

 

path of difigY describing the adjustment to a changeixx
s
(Ql285_) . We want to know how this time path is broken up
dt

into the time paths of 9%figx-and difigp, as expressed in the

 

equations in (b)'. The time path of g%%§X-will reflect the

usefulness of the nominal quantity of Money as an indicator
of real GNP.

One such possible set of time paths consistent with
the equations in (b)' is diSplayed in the last two frames
of Figure 5.1. Note that the vertical sum of these two time

paths is the resulting path of difigY. Further note that in

 

this picture, the time path of gAggy-initially rises,

131

following an increase in dififm, but eventually this rise is

 

crowded out so that in the long run there is no rise in
9%figl. This reflects a situation in which money is neutral

in the long run.

The remainder of this chapter will examine the

empirical relationship existing between Q%%§l and 9%figfl.

Empirical Examination of the Relationship Between Money and
Industrial ProductIon

The Money Stock Data

In this role, we are concerned with the ability of
money to promote spending in the economy. Hence, an
appropriate definition of money to consider is:

MlB = Currency + Demand Deposits at commercial
banks + Other Checkable Deposits at all
depository institutions including NOW accounts,
ATS, Credit Union draft shares, and Demand
Deposits at Mutual Savings banks.

It is worth noting that this is not the definition which
Friedman would choose, since it excludes most Time Deposits.
However, we feel it is appropriate for the work in this
chapter.

Data on M1 (= Currency + Demand Deposits) is
available beginning with January, 1907. In 1960 and again
in 1962 the Fed changed the definition of Demand Deposits

at commercial banks. In 1960 the data were altered to

include Demand Deposits due to mutual savings banks and

132

foreign banks, and to exclude float as well as CIPC.2

These numbers were published from 1907 to 1962, when the
data were further amended to include foreign Demand Deposits
with Federal Reserve Banks and Demand Deposits that banks

in U.S. territories and possessions have at U.S. commercial
banks.3 The data on this definition of M1 were then
published beginning with 1907, and continued to be published
until 1980, when the definition was changed once again.

The data on this latest definition of M1 (namely MlB) have
been published beginning with January, 1959.

There is obviously a discrepancy between the old and
new definitions, since they measure different things. Table
V-l displays the components of the old Ml series as published
in the 1960 definition, and the new MlB series as published
in the 1980 definition. The last two columns show the
discrepancy for the twelve monthly observations in 1959.

We are interested in comparing phese two definitions since
the definition of Demand Deposits in 1960 is closer to the
definition of Demand Deposits in 1980, than is the 1962
definition. Note that the currency component is identical
in the two definitions:h1Table V-l. The discrepancy

arises in the Demand Deposit component (noting that Other
Checkable Deposits are zero for the observations in 1959).
The Demand Deposit component in the 1960 definition exceeds
the Demand Deposit component in the 1980 definition by the

amount of Demand Deposits due to foreign official institu-

tions.

133

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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mmmH I mMHmmm MUOHm Mmzoz

HI> Mdm<B

130

We can construct a "complete" time series on the
money supply by using the 1960 definition of M1 from January,
1907 through December, 1958, and the 1980 definition of M18
from January, 1959 to the present. Call this series M1B*.
MlB* still contains the discrepancy shown in Table
V-l, due to the change in the definition of the Demand Deposit
component of M1. This will appear as a jump of approximately
1% in M1B*, between December, 1958 and January, 1959. We

can correct this fault by building an intervention model for

MlB*. The intervention term will be defined as follows.

1.0 for January, 1907 - December, 1958

0.0 for January, 1959 - December, 1979

This will operate as a dummy variable which should account for
the change in the definition of the Demand Deposit component
beginning in January, 1959. We anticipate a coefficient of
about (.01) for this intervention term, reflecting the 1%
jump in the series shown in Table V-l.

We proceed by first building the model for M1B* for

the sample, January, 1907 - December, 1979 (n = 396).

3 12 13

12 * _

A 2
9 = -.1688 9 = .0697 0 = .1078 R ==l— 2
3 (.000) 12 (.006) 13 (.006) _ CONN”)

— .1887
X2 = 32.0 RSE = .0003

135

From the form of this model, we build the interven-

tion model for MlB*.

12 _ 3 2 13
(5.2) (l-B )(l-B)[log(MlB*t)-pOIt] - (l—%B -81281—0138 )at

E = .0110 93 =—.1709 912 = .0633 913 = .1099
(.0036) ( 000) ( 006) (.006)
2 -
X06 — 51.3 RSE = .0002
,2 _ 1 (.0002)2
' ' ——_‘—_“2
(.00077)
= .2207

It is interesting to note that the coefficient of our
intervention term follows the information in Table V-l in
suggesting a Shift of about 1% in MlB*, in January, 1959.

We can now utilize the information provided by this
intervention model to splice our data and obtain a
"continuous" series from January, 1907 through December,

1979. The transformation of M1B* indicated in the interven-

tion model is following.

log(Mt) = log(MlB«t) - poIt

We are interested in levels of the money supply. That is,
we are really interested in Mt’ which we obtain by

exponentiating the above expression.
0 -
elog(Mt) : e[log(MlB t) + ( po)It]

e[logOfiB*t)]e(-po)lt

*
(5.3) Mt (MlB t)e o t

136

A

where po = .0110
and I = 1.0 for January, 1907 - December, 1958
t 0.0 for January, 1959 - December, 1979

Note that
{(M1B*t)e-'0110 for January, 1907 - December, 1958
M :
t

(M1B*t) for January, 1959 - December, 1979

This spliced series simply shifts the first 12 year segment

by 1.1%, to eliminate the jump in the series.

The Identification Stage

We can now proceed to examine the relationship between

A

the growth rate of Money, Mt’ and the growth rate of GNP,

yt, discussed earlier. We wish to build a transfer function

of the following form.

(5 0) (l-Bl2)(l-B)log(y ) = (01(B) (1-812)(1-8)1og(M ) + N
t 61(B) t It

Note: (l-B12)(l-B)1og(yt) = (1—B12)yt is stationary, and
(1-B12)(l-B)log(Mt) = (l-B12)Mt is stationary,

where yt = the Index of Industrial production,

and M is as defined in equation (5.3).

t

This relationship will give us the impulse and step response
functions describing the reaction of §t to changes in Mt.
The first step in building this transfer function
model is to establish the univariate model for the input,
Mt' This model is given in equation (5.2).
We now proceed to the second step, and move to

identify the form of the impulse response function,

137

vl(B) = wl(B)/5l(B), by calculating the cross-correlation
function between the prewhitened input and the prewhitened
output in the transfer function, (5.0). This function is
listed and plotted in Table V-2and Figure (5.2). Note that
the prewhitening model used is that in equation (5.2).

Note that at the identification stage, the impulse
response weights are calculated directly from the cross-
correlation coefficients as follows.

(k) s

 

 

I"
A as
vk = S B k = 0, 1, 2, ...
0L
e.g.
2 _ <.109)(.015823) _
vo ‘ .0001893 ‘ “5628

Note the remarkably smooth pattern of this unrestricted

cross-correlation function. This suggests an impulse

response function between Mt and yt which might follow either

of the following patterns:

(i) a damped cosine wave with a period of 08 months.

[——‘\\\\\\\H 4” lag in months

I 1W6
(ii) a damped "V" pattern which reverses according

to a similar time period.

1...———- lag in months

What does vl(B) suggest about the monetarist

proposition outlined at the beginning of the chapter? To

Sample:

138

TABLE V-2

Cross Correlations [raB(B)]

Series 1 - Prewhitened Money Stock{at}

Series 2 - Prehitened

Mean of Series 1
St. Dev.
Mean of Series 2
St. Dev.

Number of Lags
On Series 1

(k)

 

(OCDQCDU'i-ITOJNF-‘O

of Series 1

of Series 2

January, 1907 - November, 1979; n =

Industrial Production{8t}

 

 

BCDH7 Index of
= .l7720E-03
= .01893E-02 = SO
= -.72950E-00
= .15823E-01 = SB
Cross Number of Lags
Correlation On Series 2
ra8(k) (k)
.109 0
.119 l
.082 2
.150 3
.052 0
.120 5
.093 6
.021 7
.096 8
.008 9
.020 10
.003 11
.023 12
—.009 13
-.018 10
-.086 15
-.076 16
-.090 17
-.036 18
-.072 19
.019 20
-.083 21
-.065 22
-.091 23
-.086 20
—.058 25
-.102 26
-.100 27
—.126 28
.039 29
-.073 30
.079 31
-.008 32
-.020 33
-.088 30
-.013 35
—.001 36

395

Cross

Correlation

r8a(k)

 

.109
.190
.052
.051
—.062
—.020
-.080
.017
.062
.007
.010
-.090
-.007
-.098
.003
-.085
-.000
-.065
.022
-.020
.057
.089
.003
-.018
.025
-.060
.000
.001
.006
.070
.008
-.051
-.057
.010
-.037
.007
-.100

139

TABLE V-2 (cont'd.)

 

 

 

 

Number of Lags Cross Number of Lags Cross

on Series 1 Correlation on Series 2 Correlation
(k) raB(k) (k) r (k)

Ba

37 .097 37 .108
38 .108 38 -.038
39 .050 39 .028
00 -.003 00 -.010
01 .003 01 .008
02 .057 02 -.068
03 _ -.069 03 .035
00 -.082 00 -.071
05 -.000 05 -.037
06 .018 06 .008
07 .032 07 .023

08 .032 08 .005

moowmmstP-Io 75‘

V

100

FIGURE 5.2
GRAPH OF IMPULSE RESPONSE WEIGHTS (v1(B)

-105 -075 0. .75 1.5
,+++++++++.+++++++++,+++++++++,+++++++++.

XXXXXXXX
XXXXXXX
XXXXX
XXXXXXXXX
XXXX
XXXXXXX
XXXXXX
XX
XXXXXX
XXX
XX
X
XX
XXX
XX
XXXXX
XXXXX
XXXXX
XXX
XXXXX
XX
XXXXX
XXXX
XXXXXX
XXXXX
XXXX
XXXXXX
XXXXXX
XXXXXXX
XXX
XXXXX
XXXXX
.XXX
XX
XXXXX
XX
X
XXXXXX
XXXXXX
XXXX
X
X
XXXX
XXXX
XXXXX

VALUES (vk)

.56397E+00
.00906E+00
.30800E+00
.58296E+00
.19059E+00
.05006E+00
.30995E+00
.79710E-01
.36108E+00
.17989E+00
.76150E-01
.l2550E-01
.88038E-01
—.18333E+00
-.67603E-01
-.32399E+00
-.28506E+00
—.33901E+00
-.13598E+00
9.27075E+00
~.71382E-01
—.31007E+00
.20658E+00
.30273E+00
.32583E+00
.21701E+00
.38682E+00
.39283E+00
.07086E+00
.10836E+00
-.27006E+00
.29997E+00
-.18310E+00
—.91287E-01
—.33277E+00
—.07375E-01
—.53809E-02
.36520E+00
.00658E+00
.19006E+00
v.10152E—01
.12309E-01
.21611E+00
—.26086E+00
—.30900E+00

I

11111

101

FIGURE 5.2 (cont‘d.)

-105 -075 00 Q75 1.5
(k) .+++++++++.+++++++++.+++++++++.+++++++++. VALUES (vk)

05 XXX -.15067E+00
06, XX .67507E-01
07 XXX .12172E+00

08 XXX .12231E+00

102

discuss this question, we must consider the Step Response
Function: Vl(B), implied by this impulse response function,
since the monetarist proposition illustrated in Figure 5.1

A

shows the presumed reaction of yt to a sustained (step)
change in Mt'
We obtain this Step Response Function by summing

over the impulse response function:

V = kth element of V(B)

.th

k
= E v. where v. = 1 element of V(B).
i=0 1 1

V1(B) is listed and plotted in Figure (5.3).
Note the strong resemblance between Figure 5.3 and
the bottom left frame of Figure 5.1. We emphasize that
Vl(B) is obtained from the identification stage of our time
series model building, with no restrictions imposed.
Immediately the question arises as to what V1(B)
.might converge to, if allowed to follow the pattern shown.
In particular, will V1(B) converge to zero, as suggested
by the monetarist proposition? To answer this question, we
must proceed to the estimation stage and finish building this

transfer function model.
Estimation of the Single Input Transfer Function

Consider equation (5.0) and Figure (5.2), describing

the transfer function between Mt and yt. This presents a

problem in that a damped cosine wave with such a long period

103

FIGURE 5.3
GRAPH OF STEP RESPONSE WEIGHTS Vl(B)

(-2.0) 0. (2.0)

(k).+++++++++.+++++++++.+++++++++.+++++++++.

(DCDQOVU'l-C'le-‘O

.cirxwptr:(ouawcuuuwoawcuoaMFOK>M60KJMFQKJNFHPJHFHFJHFHPJHFH
UhruJNFJCDOCD\Jm(flI?wWOFJOLDGDQCDULCQJNFJCDQCD\JOCDifwrokJO

XXXX
XXXXXX
XXXXXXXX
XXXXXXXXXXX
XXXXXXXXXXX
XXXXXXXXXXXXXX
XXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXX
XXXXXXXXXXXXXX
XXXXXXXXXXXXX
XXXXXXXXXXX
XXXXXXXXXXXX
XXXXXXXXXX
XXXXXXXXX
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XXX
X
XXXX
XXX
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X
X
X
XX
XX
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VALUES [Vk]

.56397
1.01303
1.32183
1.90079
2.09938
2.55300
2.90339
2.98310
3.30018
.52007
.60022
.61277
.70121
.51788
.05020
.12625
.80079

2.50138

2.36500

2.09065

2.16603

1.85196
1.60538
1.26265
.93682
.71901
.33259
-.06020
-.53510
-.38670
-.66120
-.36123
-.50037
-.63566
—.96803
-1.01581
-1.02119
—.65595
-.20937
-.05891
-.06906
-.05671
.15900
-.10106
-.01090
-.56157

wwwwwwww

100

FIGURE 5.3 (cont'd.)

(—2.0) 0. (2.0)
(k).+++++++++.+++++++++.+++++++++.+++++++++. VALUES [Vk]
06 XXX -.09006
07 XXX -.37230

08 XX -.25003

105

cannot be represented parsimoniously as a ratio of two

polynomials in B.

Furthermore, we must operate within an

upperbound of eight or nine total parameters in our model,

due to the limitations of our time series computer program.

We have overcome these difficulties by filtering

the data on Mt prior to estimation, in a 20-month moving

average which follows one half of the period of the cosine

function. This is done as follows.
1 period = 08 months = 2Tr radians
- 1L ‘
1 month - 2” radians
. 23 in i 12
Filtered M = FM = Z cos(——)B (l-B )(l-B)log(M )
t t 1:0 20 t

We then use FMt as our input series in the following

transfer function.

(5.5) (l-Bl2)(l-B)log(y£) =

The 20th order polynomial in the

O
.________ +
1-6 B,” [FMt] N

0.)

1t
20

denominator will work with

our 20-month moving average of Mt to make an impulse

response function, vl(B), which resembles the damped cosine

wave we desire. If -1 < 6

20

function will appear as follows.

FIGURE 5.0

 

(520

/——\

= -.5)

< 0, then this impulse response

lag in
months

106

This picture is still not ideal, as it implies an
impulse response function which is discontinuous at all
lags which are multiples of 20. We can eliminate these

discontinuities by altering our filter slightly:

23 . .
(5.6) Filtered M = FM = [ X [A]cos(££)Bl](l—B12)(l-B)log(Mf)
t t 1:0 20
1.0 for i=0, 1, ... ,12
where A =
‘520 for 1:13, ... ,23

Note that this filter will change the appearance of Figure

 

5.0 as follows.

r\\\\\\ ,,/"'_—-‘“‘~\\ lag in

12 20 36 08 60 72

 

FIGURE 5.5
(520 C -.5) .
We can arrive at a model of this form by first

choosing a value of 62”, and filtering Mt according to (5.6)
with this value. Then we can estimate the transfer function,
(5.5), and check the value of 62”, to see if it differs
substantially from our initial choice used to filter Mt'

'If it does, we can use the new value of 62” to filter Mt
again with (5.6), and then re-estimate (5.5). We can

continue this iterative procedure until the estimate of

6 does not vary appreciably from the value used to filter

20

M Using this procedure will give us a transfer function

t0
between Mt and y1: with a "smooth" impulse response function,

as in Figure 5.5.

107

In the following work we iterate on 52” until the
estimate remains within a band of (:.005) from the value
used to filter Mt' Since the estimate of mo is in all
cases less than (.2), this will result in a discontinuity
at lag 20 in Figure 5.5 of less than (.001)[=(.2)*(.005)].

We are now ready to estimate this model. But we

must first consider some problems with our sample period:

January, 1907 - December, 1979. Does the transfer function,

(5.5), adequately describe a stable relationship throughout
this entire period? We suspect that the oil crunch of 1973
represents an episode for which (5.5) is inadequate. There
is a substantial literature on this topic, concerning the

supply shock to the economy resulting from the increase in

energy prices.5 This literature dwells on the change in the

structure of the world economy after this shock, and the
presumed impact on real and potential output. Tatom states
that "the large increase in the cost of energy resources
from 1972 to 1977 has had profound effects on productivity,
investment, and the long term growth path of the U.S.
economy." The study of Rasche and Tatom produced empirical

results which "support the argument that the new energy

regime imposed in 1970 permanently reduced potential output,

and suggests that "failure to account for energy prior to
1973 is not critical, but that serious inconsistencies
arise when the sample period is extended to include recent

years."

It is apparent that the supply shock of 1973 changed

I

108

the world we are studying. This phenomenon enters our
model as outlined in the beginning of the chapter in equation
(b), as a sudden change in anticipated potential output, y*.
For example in equation (b)', (log y*) will have changed
drastically during this episode. This will result in
alterations in the time paths of prices and real output.

These considerations move us to believe that the
presumed stable relationship between Mt and §t should pp: be
expected to hold during this oil crunch of 1973, and should
pp: be expected to account for the change in the world since
then. The transfer function in equation (5.5) would likely
overpredict yt during the oil crunch, and subsequently prove
to be inadequate.

We can examine this possibility by first estimating
the single-input transfer function in (5.5) over the sample
period May, 1950 - September, 1973, the period prior to

the oil crunch. The following model is the result.6

BCDH7 Index of Industrial Production

yt =
xt = FMt = Filtered Mt as in equation (5.6) with
6 = -.5207
20
(5 7) (1-812)(l-B)lo ( ) - ~———m£——— [x 1 + 1-612312 a
’ g yt ' 1 5 B20 t 1-618 t
7 20
6 = .1877 3 = -.5251 6 = .8005 0 = .2776
0 (.059) 2” (.252) 12 (.039) l (.052)
2 _ _
Xus - 37.7 RSE — .0135
—2

109

We are interested in how this model will forecast
over the next 18 months outside the sample, during the oil
crunch. Table V-3 shows the 18 one-step—ahead forecasts
obtained from this model by including an additional
observation on FMt and yt at each step. The forecast errors
show that the model drastically overpredicts yt in late
1970, though it performs fairly well for most of the other
18 months considered. Thus our suspicions as to the
adequacy of this single-input transfer function during the
oil crunch are possibly well-founded.

Expanding the Model to Account for the Energy Price Supply
Shocks of the Early 1970's - BCD92

We can correct this situation by considering a second
input which might capture the effect of the oil crunch in
1973, and thus enhance our model's ability to predict yt
during this period.

One such possible input is BCD92: the percent change
in the PPI of Crude Materials. Recall that this series
represents one of the three Commerce Department leading
indicators examined in Chapter IV, which displays the
characteristics a good leading indicator is expected to have.
The second to last column of Table V—3 shows the monthly
observations of BCD92. This series displays a noticeable
increase in late 1973 and early 1970. Recall from Chapter
IV that this leading indicator has a lead of approximately

ten months over Industrial Production. This suggests that

150

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ml> mqm<B

151

BCD92 may be successful in capturing the effect of the oil
crunch, and thus improve the poor forecasting performance
of our model in equation (5.7) during late 1970.
The Identification Stage of Building the Two Input Transfer
Funct1on

In building this two-input transfer function, it is
interesting to consider the cross-correlation function
between the two inputs, FMt and BCD92. In particular, we
are interested in the cross—correlation coefficients between
certain transformations of the two inputs. If these
coefficients are extremely small, then it will imply that the
cross-correlation function between each prewhitened input
and the similarly transformed output, can be used separately
to identify the respective impulse response function. This
implication is developed below.

After differencing to achieve stationarity, our two-

input transfer function can be written as:

. : + 4‘ ...
(5 8) yt V10Xlt v11X1t-1 + V12xlt-2 V13Xlt-3 + +

v20X2t + V21X2t-1 + v22X2t-2 + V23X2t-3 " + n

Then, on multiplying throughout in (5.8) by Xlt-k for k i 0,

we obtain

4.

(5'9) Xlt-kyt = VlOXlt-kxlt Vllxlt-kxlt-l + Vl2xlt-kxlt-2

+ "° + V20Xlt—kx2t + V21X1t-kx2t-l +

152

+
+ V22xlt-kx2t-2 + "° Xlt—knt

If we make the assumption that Xlt4<is uncorrelated with nt
for all k, taking expectations in (5.9) yields the set of

equations

(5.10) E[Xlt-kyt] = VlOEEXlt-kxlt] + VllBEXIt-kxlt-l] + ...+

V20E[xlt-kx2t] + V21EEX1t-kx2t-l] +"’

-or Y (k) = v Y (k) + v Y (k-l) + v Y (k-2)
xly 10 xlxl ll xlxl l2 xlxl
+ 000 +
v Y (k) + v Y (k-l) + v Y (k-2)
20 xlx2 21 xlx2 22 xlx2
+
for k = 0, 1, 2,
where Yab(k) is the cross-covariance at lag k between

series a and b.
This is the set of cross—covariances between our first input,
Xlt’ and the output, yt. Suppose that the weights, vlj and
v2., are effectively zero beyond some lag, k=K. Then the

first K+l equations in (5.10) can be written

 

_ F v
Yxly ' x1X1’X1X2
r 1
where Yxly(0)
Yxly(1)
Y :
le :
.(K)
(Yxly J

 

(K+l)xl

153

 

 

(2k+2)x1

F -
xlx1,xlx2

 

Y . (K) Y (K—l)". Y (0) (K _ ...
1 1 x1"1 X1X1 Yx1X2 ) YX1X2(K l) Y"1 2(0) J

(K+l)x(2K+2)

 

This system of equations can normally be used at the
identification stage of the Box and Jenkins modeling
procedure.7 If x1 and x2 are not significantly cross-

correlated, then the Yx x (k) terms drop out, and we can
1 2

substitute estimates of the autocorrelation function of

150

, Y (k), and the cross-correlations between x and y,

l xlxl l

Yx y(k), and solve the system for the top half of v, which
1

is the transfer function between x1 and y, vl(B).

X

However, it will be a rare case when the two inputs
are not significantly cross-correlated! Since the yxlx2(k)
are not generally zero, the system in (5.10) cannot be
solved. We have 1(+1 equations in 2K+2 unknowns in v, and the
system is not identified.

The problem is changed slightly when we first prewhiten

the series in our model. Suppose that the univariate models

for Xlt and x2t appear as follows.

-1
8 :
(5.11) ¢x (B) x (B)[Xlt] allt
l l
0’1 = ’
(5.12) 6X (B) X (B)[x2t] a22t
2 2
a a . . .
where llt and 22t are white n01se series,
with standard deviations s1 and s1 , respectively.
1 2

Further, define the following.

(5,13) ¢X1(B)e;:(8)[x2t] : (1th
(5.10)¢x1(B)0;:(B)[yt1 = Blt
(5.15) ¢x1(B)9;:(B)[nt] = 611:
(5.16)<px2(B)e;:(B)[xlt] = (:21,
(5.17)¢x2(B)e;:(B)[yt] = th

155

-1 _
(5.18) ¢X2(B)9X2(B)[nt] - €2t

Note that equations (5.13) through (5.15) are just
transformations of X2t’ yt, and nt where the transformation
is the prewhitening model for Xlt' Likewise equations (5.16)
through (5.18) are transformations of Xlt’ yt, and nt with
the prewhitening model for x2t°

In order to identify the impulse response function

between x and yt, vl(B), we apply the prewhitening

1t
transformation of our first input to the model in equation

(5.8). This yields the following model.

5-1 '1
(5.19) ¢Xl(B) Xl(B)[yt] vl(B)¢xl(B)0xl(B)[xlt]

-l
+ v2(B)¢X (B)0x (B)[x2t]

1 1
+ 0X (B)9;1(B)[nt]
1 1
OP Blt = vl(B)allt + V2<B>al2t + Elt

On multiplying through this by allt-k’ we obtain

(5°20) “11t-k81t = v1(B)°‘11t-k°‘11t + v2(B)°‘11t-k°‘12t
+ allt-kglt
If we make the assumption thatallt_k is uncorrelated with

alt for all k, taking expectations in equation (5.20)

yields the following.

156

l

(5.21) E[a 8 ] = vl(B)E[allt_kallt]'+v2(B)E[o

llt-k lt llt-Kal2t

(k) = v (B)Y (k) + v2(B)Y (k)

or Y
0‘11 B1 1 0110111 C111‘112

Since allt is white noise, the first term on the right hand

side of equation (5.21) reduces to [Vkoa 2]. Finally, if

11
(k) is zero for all k, then equation (5.21) reduces

Y%1%2

to the following.

(5 22) (k) - o 2 - (1118 (k)
. Ya B - Vk all 01" Vk - O 2
a 11

 

where Ya B (k) = E[0 llt- k Blt] is the cross- covariance at

 

 

0‘11 1
lag k between allt and Blt
Therefore,
pull 81(k)081 YO‘ll 81(k)
(5.23) v = since 0 (k) =
k Ga 0"1181 ° 08
11 . “11 1

Hence the cross-correlation function between the
prewhitened first input and the correspondingly transformed
output is directly proportional to the impulse response
function, vl(B). We can thus identify the form of the first

impulse response function by estimating the cross-correlation

 

function, r (k), and the standard errors of the pre-
0‘1161
.whitened first input and the similarly transformed output,

and then substituting into equation (5.23).
(5.20) v =

k 5a
ll

 

157

It is important to emphasize that this procedure

rests on the assumption that Ya a (k) is zero for all k.

11 12
We can estimate this cross-correlation function between the

 

prewhitened first input and the correspondingly transformed
second input, in order to evaluate the applicability of
this assumption. If the coefficients of r (k) are not
significantly different from zero, then we canigt reject
the hypothesis that the assumption holds.

Note that to identify v2(B), we will use the cross-
correlation function between the prewhitened second input
and the similarly transformed output, Ya B (k). The use

22 2
of this procedure will rest on the assumption that Ya a (k)

is zero for all k. The applicability of this assumpticzm21
is testable in the same way we test the assumption regarding
the identification of vl(B).

In summary, this analysis shows us that if r (k)

:912

and r (k) display coefficients which are not signifi-

“22“21

cantly different from zero, then the cross-correlation
function between each prewhitened input and the correspond-
ingly transformed output can be used separately to identify
the respective impulse response functions.

In continuing our empirical analysis, we now wish
to build the two-input transfer function, with FMt and
BCD92 as our two inputs. In order to be able to identify
the two impulse response functions separately, we are

interested in the two cross—correlation functions, ra a (k)
11 12

158

and rd 0 (k). These are presented in Table V-u.
2Thilstandard error of any given cross-correlation
coefficient, rab(k), is approximated by 1//fi:k.8 With
n = 352 in the sample of the transfer function we are
building, the standard error of ralla12(0) = the standard
error of ra22a21CO) = l//352 = .0533. To test the hypothesis
that the estimated coefficients are not significantly
different from zero, each coefficient (at lag zero) should

be compared with (1.96)*(.0533) = .10u5.

In the estimated cross-correlation function,

 

ra a (k), all the coefficients are less than .1095 except
11 12
the coefficient at lag 16; r (16) = -.111.
“11a12

The standard error of this coefficient = l

n-k

: ————l = .05146.
{352-16

and (1.96)*(.0596) = 1.07.

Therefore, r (16) is "significantly different from

“11312

zero" at the 95% confidence level. However, if we consider
the whole set of #9 coefficients in the cross-correlation
function, we would expect about 2 1/2 coefficients to be
"significantly different from zero" at the 95% confidence
level. Hence the cross-correlation function, ra a (k),

11 12
supports our assumption which implies that we can use the

cross-correlation function between prewhitened x1t (= Filtered
Money Supply) and the similarly transformed yt (= Industrial

Production Index), to identify v1(B).

159

TABLE V-u
Cross Correlations

Series all - Prewhitened Filtered Money Supply (62” = -.5600)
Series a12 - Prewhitened BCD92 Percent Change in PPI of

Crude Materials

 

 

 

 

n = 352
Mean of Series all = .u2260B-03
St. Dev. of Series all = .uuzosB-oz
Mean of Series a12 = .93668E-04
St. Dev. of Series a12 = .2182uE-01
Number of Lags Cross Number of Lags Cross
on Series a Correlation on Series a Correlation
11 12
(k) ra a (k) (k) ra a (k)
11 l2 12 11
0 -.023 0 -.023
l .018 l .023
2 -.0u2 2 .008
3 —.026 3 -.025
Q .090 H .008
5 .010 5 -.00N
6 -.06H 6 .0u2
7 -.005 7 -.055
8 -.027 8 -.06H
9 .077 9 ..060
10 -.063- 10 —.0Hl
11 —.oon 11 .059
12 -.051 12 .019
13 .057 13 -.005
1a .080 1” -.068
15 -.020 15 -.016
16 -.111 16 -.021
17 .062 17 .020
18 .030 18 -.0u7
19 -.022 19 .126
20 -.019 20 -.098
21 .021 21 —.060
22 .003 22 .032
23 .OHS 23 -.038
2H -.079 2” -.032
25 -.005 25 .060
26 .005 26 .068
27 -.037 27 —.030
28 -.002 28 —.051
29 -.035 29 .021
30 -.003 30 -.016
31 .030 31 -.056
32 .015 32 .095
33 -.035 33 .015

160

TABLE V-H (cont'd.)

 

 

 

 

 

Number of Lags Cross Number of Lags Cross
on Series all Correlations on Series al2 Correlations

(k) ra a (k) (k) ra a (k)

11 12 12 11

3H .006 3H —.0H8

35 -.012 35 -.027

36 .077 36 .052

37 -.039 37 -.018

38 -.0H6 38 -.031

39 -.032 39 .030

H0 .070 H0 .093

Hl .00H H1 -.105

H2 -.025 H2 .073

H3 -.033 H3 ‘ —.029

HH .025 HH -.020

H5 .012 H5 -.007

H6 -.010 H6 .0H8

H7 -.037 H7 .031

H8 —.016 H8 -.055

Prewhitening:

(l-.8673B)(l-.2197B-.3052B3)(l+.6109812)[FMt] =
10 13

(l+.1173B-.13H3B -.1819B )at

" same model on (l-B12)[BCD92].

161

TABLE V-H (cont'd.)

Cross Correlations:
Series a22 - Prewhitened BCD92 Percent Change in PPI of Crude

 

 

 

 

Materials
Series a21 - Prewhitened Filtered Money Supply (62u=-.5600)
n = 352
Mean of Series a22 = .80105E—03
St. Dev. of Series a22 = .lH037E-01.
Mean of Series a21 = .393HOB-02
St. Dev. of Series a21 = .9307HE-02
Number of Lags Cross Number of Lags Cross ,
on Series a22 Correlation on Series a2l Correlation
(k) r (k) (k) r a (k)
“22“21 0‘21 22
0 -.015 0 —.015
1 .0H9 1 .029
2 .061 2 .02H
3 .026 3 .081
H .021 H .0H6
5 -.002 5 .06H
6 .00H 6 .061
7 -.0H3 7 .037
8 -.055 8 .025
9 .001 9 .018
10 —.069 10 .008
11 -.050 11 .006
12 -.107 12 .006
13 -.l22 13 .053
lH -.139 1H .030
15 ~.13l 15 .058
16 -.117 16 -.021
17 -.078 17 .012
18 -.031 18 .011
19 -.015 19 -.002
20 -.036 20 -.015
21 -.003 21 -.023
22 -.029 22 -.007
23 -.029 23 -.015
2H -.011 2H -.05H
25 .0H1 25 -.00H
26 .080 26 -.019
27 .029 27 .0H3
28 -.005 28 -.002
29 .007 29 .017
30 .029 30 .03H
31 .002 31 .021
32 .033 32 .008

162

TABLE V—H (cont'd.)

 

 

 

 

 

Number of Lags Cross Number of Lags Cross

on Series a22 Correlation on Series a21 Correlation
(k) Pa a (k) (k) ra a (k)

22 21 21 22

33 .083 33 -.002
3H .02H 3H .0H2
35 -.006 35 .009
36 .019 36 .027
37 -.002 37 -.005
38 .030 38 -.038
39 .03H 39 .010
H0 .053 H0 -.029
H1 .016 H1 -.035
H2 ' .069 H2 -.009
H3 .027 H3 -.027
HH .011 HH -.0H2
H5 .056 H5 —.0H5
H6 .0H8 H6 -.025
H7 .00H H7 -.0H5
H8 .011 H8 -.03H

Prewhitening:

(1-.38508-.265883)(1-B12)[BCD92] = (1-.15H2B2-.8120B12

+ .197581u) at

" same model on the levels of FMt'

163

The estimated cross-correlation function, ra a (k)
displays five coefficients which are "significantly2different
from zero." Furthermore the tendency of positive coefficients
to be followed by positive coefficients and of negative
coefficients to be followed by negative coefficients, seems
to indicate that there is some correlation inherent in the
relationship between these two series which is not
effectively eliminated by the prewhitening model for BCD92.
This is somewhat disturbing. However we are reassured by
the fact that H3 of the H8 coefficients are not significantly
different from zero. This is the key characteristic in our
analysis of the applicability of the assumption that all
coefficients in Ya22a21(k) are zero. Hence we anticipate

that the cross-correlation function between the prewhitened

input, x (= BCD92), and the correspondingly transformed

2t
output, yt(=Industrial Production Index), will be instrumental

in identifying v2(B).
Estimating_the Two Input Transfer Function

We have already identified and estimated separately,
the two transfer functions with each of these inputs as the
single input. Thus we know what to expect as the form of
the two impulse response functions, vl(B) and v2(B), in our

two-input transfer function.

With this in mind, we build the following model.9

yt = BCDH7 Index of Industrial Production

16H

x FMt = Filtered Mt as 1n (5.6), w1th 62H::-°5231

1t
BCD92 % change in PPI of Crude Materials

X2t
Sample: May, 1950 - September, 1973 (n = 281)

.0)
(5.25) (1—B12)(l-B)log(yt) = ———°—2¢ [xlt]
1-5 8
2n
(.0,
o 10 12
+ l—6iB B (l-B )[x2t]
1-812812
+ 1-¢1B at
w = .1835 52a = -.5208 w' = -.061H 51 = .7805
(.058) (.260) O (.0u5) (.176)
912 = .7926 ¢1 = .25u0
(.ouo> (.051)
‘2 _ _ —2 _
XHB _ uo.u RSE - .013u R - .SHHl

A comparison of the coefficients in this equation with
those of equation (5.7) on page 1H8 shows that the
addition of the second input, BCD92, does not change the
appearance of the first transfer function much. In
particular, the coefficients 00, 62”, 812, and $1 are quite
insensitive to the addition of this second input. It is also
interesting to compare the coefficients 8; and 6i in equation
(5.25) with the coefficients 00 and 61 of the single-input
transfer function with BCD92 as the input, which appears
in Table IV-2 in Chapter IV. These coefficients are also

quite insensitive to the addition of the second variable,

FMt'

165

We are now interested in how this model will
forecast over the next 18 periods. Table V-5 shows the
18 resulting one—step-ahead forecasts. We see improvement
in reducing the large forecast errors in late 197H that
appear in Table V-3. Furthermore, the Theil Decomposition
statistics show marked improvement. In particular, the
RMSE is reduced to 1.758H6 from 2.053285. These character-
istics suggest that our second input, BCD92, is useful in
the role desired.

Our third step in building this model is to re-
estimate the two-input transfer function in equation (5.25)

over the sample period May, 1950 - March, 1975, the period

through the oil crunch. The resulting model follows.lo
yt = BCDH7 Index of Industrial Production
: : ° ° ° 0 : ..
x1t FMt Filtered Mt as 1n (5.6),w1th 2H .5520
x2t = BCD92 % change in PPI of Crude Materials
Sample: May, 1950 - March, 1975 (n = 299)
m
(5.26) (l-B12)(l-B)1og(y ) = ————9——— [x J
t 1- B29 1t
529
w.
0 10 12
+ W B (l-B )[X2t]
1‘812312
+ ———————— a
1—¢1B t
w = .1772 5 = -.5568 3' = -.0993 3' = .7707
O (.060) 2” (.290) O (.093) 1 (.123)
6 = .7699 3 = .2827
12 (.092) 1 (.061)
2 = 39.6 RSE = .0139 82 = .5991

x96

166

 

 

 

 

 

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167

Note that the coefficient estimates are extremely
stable as we increase the sample size from n = 281 to
n = 299, in moving from equation (5.25) to (5.26), even
though these eighteen additional periods represent a shock
to the economy.

We are now interested in how ihig model forecasts
over the next H 1/2 years. Table V—6 displays 53 one—step-
ahead forecasts for the H 1/2 years after the sample used
to estimate equation (5.26).

Examination of the Theil Decomposition statistics
in Table V-6 indicates that our model in equation (5.26)
performs remarkably well over this extremely long forecast
horizon. The RMSE is reduced to 1.33H221 from 1.758H6 in
Table V-5, and from 2.053285 in Table V—3. We conclude
that our model is appropriate, and proceed to the last step
11

of estimating over the entire sample period.

BCDH7 Index of Industrial Production

yt 3
x11: = FMt = Filtered Mt as in (5.6), W1th 62u=«-.5600
x2t = BCD92 % change in PPI of Crude Materials

Sample: May, 1950 - August, 1979 (n = 352)

(L)

 

O
2H[Xlt]

(5.27) (l-Blz)(l-B)1og(yt)
l-quB

w '
+ i:91§-B10(1-B12)[x2t]

1-912B

l-¢lB t

12

+
DJ

 

168

 

 

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w = .1726 52” = -.5583 wé = -.1036 0i 3 .68H2
(.052) (.196) (.0H2) (.1H8)
8 = .7776 0 = .2777
12 (.037) 1 (.056)
2 _ _ —2 _
XH6 - HH.1 RSE - .0129 R - .5775

Again, note that the coefficient estimates are
extremely stable as we increase the sample size from n = 299

to n = 352, from equation (5.26) to (5.27).
Implications of the Final Model

This is our model for the entire sample period,
describing the relationships between Mt and yt and between
BCD92 and yt. We are especially interested in the impulse
and step response functions between Mt and yt, in order
to examine in more detail the empirical evidence regarding
the monetarist proposition outlined at the beginning of
this chapter. This impulse response function is developed
below.

Let mt = (1-B12)(l-B)log(Mt)

Then the impulse response function is:

01(B) 00
(5.28) vl(B)[mt] : W [mt] 3 W [FMt]
‘ 29

where FMt is defined as in equation (5.6), as follows:

23 i“ i 1.0 for i = 0, ... ,12
FMt = [ Z (A)cos(§E)B )mt; with A =

i=0 -6 for i = 13, ...,23

2H

171

PMt = [1.0 + .99B + .96682 + .92983 + .8668” + .79385
+ .70796 + .60987 + .588 + .38389 + .259810 + .13311
- (—62u) .13813 - (-62u).259Blu — (—62u) .383815
- ('529) .5816 - (-52u).609817 - ("529) .707818
- (”529’ .793819 — (~52u).866B20 - (’529’ .92H821
- (-62u) .966822 - {-62u).99B23] mt

Thus 01(8), the polynomial in B comprising the numerator of

vl(B), is simply mo multiplying the above twenty-third order
polynomial in B.

A 23 in i
(5.29) 01(8) = 00 .Z (A)cos (§H)B

1-0

When combined with 61(8), the denominator of vl(B), the
resulting impulse response function is of the following form.
(5.30) vlCB) ”o 2 (A)cos(L P)B -+(62u) Z0 (A)cos(L BEB‘fi

i=0 i=

+

N70 O[: X (A)cos(L M1][8::J

A A 23 o o
3 in 1 72
(02”) Swo[i:0(A)COS(:fi)B ][B J +

+

A

From equation (5.27) we have 00 = .1726, -62H = .5600,
and 62” = -.5583. Using these estimates in equation (5.30)
yields the infinite order polynomial in B comprising vl(B).
The first H8 coefficients of this impulse response function
are listed and plotted in Figure 5.6, as well as those of

the associated step response function.

172

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177

We know that the impulse response function will
converge to zero (see Figure 5.5), and thus that the step
response function will also converge. In light of the
Monetarist proposition under consideration, we are interested
in what this step response function will converge to. We
can pinpoint this number as follows.

Beginning with the sum of the first twelve impulse
response weights, we can sum over the next 2” weights to get

a single figure which will be subtracted in the step response

 

function during the next 2H periods. We can then multiply
this same figure by (-52u) to get another figure describing
the total amount which will be added to the step response
function in the following 2H months. Multiplying this figure

2 will then yield the total amount to be subtracted

(\
by (-62u)
again in the following 2H months.

Continuing this procedure indefinitely would give us
the exact number to which Vl(B) converges. Continuing for

a few iterations will closely approximate this number.

The sum of the first l2 impulse response weights

= V12 = 1.u027.

The sum of the next 2” impulse response weights
= 1.H719.

Thus, V36 = l.H027 - 1.u719 = -.0692.

The sum of the following 2H impulse response weights
= (l.H719)(.5583)

= .8218

Thus, V = -.0692 + .8218 = .7526

60

178

The sum of the following 2H impulse response weights
= (1.9719)(.5583)2

= .9588.

Thus, v8” = .7527 - .9588
Continuing:

<1.9719)(.5583>3 = .2561
v108 = .2938 + .2551 =
(1.9719)(.5583)” = .1930
v132 = .5999 - .1930 =
(1.9719)(.5583)5 = .0798
v156 = .9059 + .0798 =
<1.9719>(.5583)6 = .0998
v180 = .9857 - .0995 =
(1.9719)(.5583)7 = .0299
v20” = .9921 + .0299 =
(1.9719)(.5583)8 = .0139'
v228 = .9570 - .0139 =
(1.9719)(.5583)9 = .0078
v252 = .9531 + .0078 =

Hence we see that

to approximately

.H570.

.5999
.M069
.H867
.MM21
.H870
.H531

.U609

.2938

the step response function converges

This suggests that money is not

neutral, but that a sustained increase in the growth rate of

the money stock will produce an increase in the growth rate

of Real GNP in the long run.

Finally we are interested in the impulse and step

response functions for the second input, BCD92, implied by

179

our model in equation (5.27). These are listed and plotted

in Figure (5.7).

Expanding the Model to Account for the Energy Price Supply
Shocks of the Early 1970's - Fuel Prices

In equations (5.25), (5.26), and (5.27), we included
a second input to account for the effect of the oil crunch
in 1973, and the subsequent change in the world. We used
as our second input, BCD92, the percent change in the PPI
of Crude Materials. This is interesting, since BCD92 is one
of the leading indicators which constitute the subject of
discussion of Chapter IV. However, much of the supply shock
literature listed in footnote 5 uses Fuel prices in this role.
Thus we now re—examine the relationships discussed in the
last section, with x2t=Fuel prices.

We must first consider the relationship between Fuel
prices and Industrial Production. The bivariate model
describing this relationship is presented below.12

BCDH7 Index of Industrial Production

yt

x PPI of Fuel, Power, and Related Products

t
Sample: May, 1950 - November, 1979 (n = 355)

(A)
12 ° _ O 8 12

(l-B )(l-B)log(yt) - Fé—i—B- B (l-B )(l-B)log(xt)

1-0 812
+ 12 a
1-¢1B t
O (.055) 1 (.169) 12 (.033) 1 (.053)
2 —2 _
= ”1.1 RSE = .0129 R - .5775

(k).+++++++++.+++++++++.+++++++++.+++++++++.

(DmflmU‘l-Fle—‘O

for x2t =

-.l O.

><><><><><><><><><Z><

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180

FIGURE 5.7

GRAPH OF IMPULSE RESPONSE WEIGHTS [v (B)]

BCD92 in equation (5.27)

.1

VALUES [vk]

IOOOOOOOOOO
O O .0 O O O O C O

.103629E+00
.709052E—Ol
-.H85147E-01
-.3319H7E-01
-.227lZSB—01
—.155u03E-01
-.106330E-01
-.727529E-02
—.u97789E-02
-.390597E—02
-.233093£—02
-.159u53E-02
-.109101E-02
-.7H6H87E-03
-.510761£-03
-.399H72B-03
-.239116E-03
-.163608E-03
-.1119”ME-03
-.765939E-0U
-.52H070E-0H
-.358579E-OH
-.245397E-0u
-.16787lE—0M
-.llHBBlE-0M
-.785898E-05
-.537727E-05
-.367923E-05
-.2517HOE-05
-.1722u5E-05
-.1l785”E-05
-.806377E-06
-.551739E-06
-.377510E-06
-.258300E-06
-.l7673hB-06
-.128925B-06

181

FIGURE 5.7 (cont'd.)
"ol 0. cl
(k),+++++++++,+++++++++,+++++++++,+++++++++, VALUES [vk]

U7 X -.82739OE-07
#8 X -.566116B—07

182

FIGURE 5.7 (cont'd.)

GRAPH OF STEP RESPONSE WEIGHTS [V2(B)] for x2t = BCD92

-.5

-.25 0.

.25

(k).+++++++++.+++++++++.+++++++++.+++++++++.+++++++++.VALUES

LOCDQCDU‘L'OOMF-‘O

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.323119
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.328163
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.328167
.328168
.328168
.328169
.328169
.328169
.328169
.328170

-.5

182

FIGURE 5.7 (cont'd.)
GRAPH OP STEP RESPONSE WEIGHTS [V2(B)] for x2t =

-.25 0.

.25

BCD92

(k),+++++++++.+++++++++,+++++++++,+++++++++,+++++++++.VALUES

LOCDQCWWICDNHO

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-.312U05
-.317383
-.320788
-.323119
-.32H713
-.SZSBOH
-.326551
-.327062
-.327Hll
-.327650
-.327BIH
-.327926
-.328002
-.328055
—.328091
-.328115
-.328132
-.3281M3
-.328151
-.328157
-.328161
-.328163
-.328165
—.328166
-.328167
—.328168
-.328168
-.328169
-.328169
-.328169
-.328169
-.328170

183

This model shows that the Fuel PPI displays a
substantial lead over Industrial Production. The last
column of Table V-3 shows the monthly observations of the
first log difference of the Fuel PPI, during the oil crunch
of 1973. This series displays a large increase in late
1973 and early 1979. Since the Fuel PPI has a lead of eight
months over Industrial Production, it may be successful in
capturing the effect of the oil crunch, and thus improving
the poor forecasting performance of our model in equation

(5.7) during late 1979.
The Identification Stage

As before, with our two inputs, FMt and BCD92, we are
now interested in the cross-correlation functions between the
two inputs, FMt and Fuel Prices; first transformed by the
prewhitening model for FMt, and second, transformed by the
prewhitening model for Fuel prices. These two cross-
correlation functions, r (k) and r (k), are listed

o‘11"‘21 “22021
in Table V-7.

Examination of r (k) shows that two coefficients
“11921
are "significantly different from zero." Since we expect
about 2 1/2 coefficients to vary from zero at the 95%
confidence level, this cross-correlation function supports
the assumption that these cross-correlations are zero. Hence
the cross-correlation function between prewhitened FMt and

the similarly transformed output series can be used to

identify the first impulse response function in this

189

 

TABLE V-7
Cross Correlations:
Series a11 - Prewhitened Filtered Money Supply (<521+ = -.5520)
Series a12 - Prewhitened PPI of Fuel Power and Related
n = 355 Products
Mean of Series all = .38159E-03
St. Dev. of Series all = .99258E-02
Mean of Series a12 = .78262E-09
St. Dev. of Series a12 = .13782E-01
Number of Lags Cross Number of Lags Cross
on Series all Correlation of Series a12 Correlation
(k) Pa a (k) (k) ra a (k)
11 12 ‘12 22
0 .096 0 .096
l .031 l .059
2 -.053 2 -.067
3 -.037 3 -.016
9 .003 9 -.061
5 .013 5 .057
6 —.095 6 .099
7 .069 7 -.055
8 —.059 8 .036
9 .090 9 -.197
10 -.058 10 .096
11 .192 11 . -.009
12 -.035 12 -.003
13 .023 13 -.020
19 —.008 19 -.092
15 .021 15 -.102
16 -.059 16 .168
17 .056 17 -.061
18 .005 18 .021
19 -.066 19 .091
20 .109 20 -.091
21 -.072 21 .023
22 .090 22 -.021
23 -.059 23 .018
29 .099 29 -.098
25 -.096 25 .010
26 .002 26 .098
27 .010 27 -.005
28 .018 28 -.027
29 -.012 29 .107
30 .009 30 -.080
31 .062 31 -.023
32 -.025 32 -.009
33 .026 33 .058

39 -.099 39 .002

185

TABLE V-7 (cont'd.)

 

 

Number of Lags Cross Number of Lags
on Series all Correlation on Series al2 Correlation
(k) Pa d (k) (k) ra a (k)
., ..,. 11 12 ., . ,. 12 11
35 .033 35 —.053
36 -.060 36 .059
37 .071 37 -.002
38 -.013 38 -.020
39 -.021 39 .085
90 .009 90 -.026
91 .039 91 -.079
92 —.019 92 .055
93 -.068 93 -.023
99 ' .015 99 .017
95 .021 95 -.062
96 .018 96 .082
97 -.051 97 -.039
98 -.079 98 .018
Prewhitening:

(1-.8681B)(1—.2l90B-.3091B3)(1+.6101B12)[FMt] =

(1+.1169B-.1397810—.181081”)a

t
-. same model on (1-B12)(1—B)log[Fue1 PPI].

186

TABLE V-7 (cont'd.)

Cross Correlations:

 

Series a22 - Prewhitened PPI of Fuel Power and Related
Products
Series a21 - Prewhitened Filtered Money Supply (629:'-’5520)
Mean of Series a22 = .77038E-03
St. Dev. of Series a22 = .96629E-02
Mean of Series a21 = .56007E-02
St. Dev. of Series a21 = .78979E-02
Number of Lags Cross Number of Lags Cross
of Series a22 Correlation on Series a21 Correlation
(k) r . (k) (k) r (k)
“22921 o‘21"‘22
0 -.000 0 -.000
l .095 1 -.029
2 -.005 2 -.092
3 -.078 3 -.069
9 .091 9 -.018
5 .005 5 -.021
6 -.091 6 -.031
7 -.032 7 .009
8 .001 8 .070
9 -.088 9 .005
10 -.020 10 .092
11 ' -.098 11 .166
12 -.038 12 .057
13 .039 13 .086
19 -.098 19 .086
15 -.070 15 .098
16 .111 16 .069
17 -.029 17 .096
18 .002 18 .055
19 .022 19 .057
20 .019 20 .191
21 -.009 21 .000
22 .018 22 .119
23 -.017 23 .079
29 .029 29 .069
25 .066 25 .031
26 .070 26 .061
27 -.007 27 .039
28 .099 28 .029
29 .050 29 .038
30 .003 30 .059
31 .003 31 .069
32 .063 32 .053
33 .026 33 .022

.066

00
4‘:

39 .003

187

TABLE V-7 (cont'd.)

 

 

Number of Lags Cross Number of Lags Cross
of Seriesa22 Correlation of Series a21 Correlation
(k) ra a (k) (k) Pa a (k)
22 21 21 22
35 —.009 35 .099
36 .069 36 .027
37 .098 37 .075
38 -.006 38 .090
39 .029 39 .013
90 .105 90 .008
91 -.095 91 .003
92 .033 92 .003
93 .018 93 -.039
99 .021 99 -.003
95 —.000 95 —.039
96 .011 96 .035
97 -.039 97 -.036
98 .098 98 -.098
Prewhitening:
(1-.6606B)(l-Bl2)(1-B)1og[Fuel PPI] = (1-.8923812)at

- same model on levels of FMt'

188

two-input transfer function.
An examination of r (k) also shows two
“22921
coefficients which vary from zero. Analagously, we can use
the cross-correlations between prewhitened Fuel prices and
the similarly transformed output series to identify the

second impulse response function in this two-input transfer

function.

The Estimation Stage

We are now ready to redevelop equations (5.25), (5.26),

and (5.27), with Fuel prices as our second input.13

yt = BCD97 Index of Industrial Production
x1t = FMt = Filtered Mt as in (5.6),w1th 62u=-.9920
x2t = PPI of Fuel, Power, and Related Products

Sample: May, 1950 - September, 1973 (n = 281)

(0

(5.25)' (1-B12)(l-B)log(yt) ° 2” [xlt]
1—6 B
29
w' 8 1
O
+ 1-618 B (l-B 3(1—8)1og[x2t]
1-0 812
+ 12 a
‘1:$IS“ t .
w = .1928 52l+ = -.9917 w; = -.0890 5i = .8882
(.051) (.260) (.073) (.131)
e = .8059 ¢ = .2537
12 (.090) l (.063)
X36 = 39.2 RSE = .0135 82 = .5373

A comparison of these parameter estimates with those

in equations (5.25) through (5.27) shows that the parameters,

189

A

and 01 are quite stable as we change from

A, A
“o ’ 529’ 12’
BCD92 as our second input, to Fuel prices. A comparison of

6
the parameters 05 and Si with the coefficients in the single-
input transfer function with input, Fuel prices (on page
179Lshows that the form of this impulse response function is
also quite stable when the second input, FMt’ is added.

We are interested in how this model will forecast
over the next 18 months, through the oil crunch. Table V-8
shows these 18 one-step-ahead forecasts. Comparison with
Table V—3 shows much improvement in reducing the forecast
errors appearing in late 1979 in our single-input model.
Comparison with Table V-5 shows that Fuel prices are more
successful in reducing these forecast errors in late 1979
than is BCD92. The Theil Decomposition statistics support
this finding.

We now proceed to re-estimate equation (5.26) with
Fuel prices as our second input.lu

BCD97 Index of Industrial Production

 

 

yt :
x1t = FMt = Filtered Mt as in (5.6), With 62u=«-.9600
x2t = PPI of Fuel, Power, and Related Products
Sample: May, 1950 - March, 1975 (n = 299)
, 12 “o
(5.26) (l-B )(1—B)log(y ) [x 1
t 29 1t
1-5 B
29
w' 8 12
o
+ 1_61B B (l-B )(l-B)log[x2t]
1—0 812
+ 12 a
t

1-¢1B

190

 

 

 

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191

mo 3 .1819 (52,4 = -.9582 w; = -.1132 6i = .8332
(.060) (.280) (.060) (.191)
0 = .7993 ¢ = .2729
12 (.090) 1 (.060)
2 _ _ —2 _
X96 - 37.9 RSE - .0133 R - .5509

Note again that the parameter estimates are extremely
stable as we increase the sample size from equation (5.25)'
to (5.26)‘.

We want to examine this model's ability to forecast
over the next 9 1/2 years. These 56 one-step-ahead forecasts
appear in Table V-9. The forecast errors indicate that
this model performs quite well over this long forecast
horizon, and the Theil Decomposition statistics Show marked
improvement over previous models. Hence we proceed to the

last step and re-estimate equation (5.27) with our new

second input, Fuel prices.15
yt = BCD97 Index of Industrial Production
x1t = FMt = Eiltered Mt as in (5.6), With
- -.5520
29
x21: = PPI of Fuel, Power, and Related Products

Sample: May, 1950 - November, 1979 (n = 355)

(A)
(5.27)' (l-B12)(l-B)log(yt) = O 29 [
l-62uB

J

 

xlt

“5 8 12
I:XI§ B (l-B )(1-B)1og[x2t1

+

12
1-612B

1-¢1B t

 

192

 

 

 

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193

 

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199

w = .1659 629 = -.5510 wé = -.1160 5i 3 .7195
(.051) (.209) (.056) (.155)
8 = .7989 ¢ = .2809
12 (.035) 1 (.055)
2 _ _ —2 _
X96 - 91.5 RSE - .0128 R - .5890

Again note the stability of the parameter estimates as we
increase the sample size,moving from equation (5.26)' to
equation (5.27)'. This is our model for the entire sample
period with these inputs. We are again interested in the

A

implied impulse and step response functions between Mt and
yt.

Implications of the Final Model

 

Again equations (5.28), (5.29), and (5.30) are
relevant in the development of these functions. From
equation (5.27)' we have mo = .1659, “629 = .552, and

A

629 = —.551. Using these estimates in equation (5.30) gives
us the impulse response function of this model, vl(B).
Figure 5.8 shows the impulse and step response functions.

It is not surprising that this figure bears much
resemblance to Figure 5.6.

To examine the convergence of this step response

function we follow the same procedure as before.

The sum of the first 12 impulse response weights

= Vl2 = 1.3983.

The sum of the next 29 impulse response weights
= 1.3959.

Thus, V = 1.3983 - 1.3959 = -.0975.

36

195

FIGURE 5.8
GRAPH OF IMPULSE RESPONSE WEIGHTS [vl(B)1
-.5 0. .5 1.

(k) . +++++++++ , +++++++++ ,+++++++++ . +++++++++ . VALUES [vk]
0 xxxx .1559
1 xxxx .1592
2 xxxx .1503
3 xxxx .1533
9 xxxx .1937
5 xxxx .1315
5 xxx .1173
7 xxx .1010
8 xxx .0830
9 xx .0535

10 xx .0930‘

11 x .0215

12 X 0.

13 x -.0119

19 x -.0237

15 xx -.0351

15 xx -.0958

17 xx -.0558

18 xx -.0698

19 xx -.0725

20 xxx -.0793

21 xxx -.0895

22 xxx -.0885

23 xxx -.0907

29 xxx -.0919

25 xxx -.0905

25 xxx -.0883

27 xxx -.0895

28 xxx -.0792

29 xx -.0725

30 XX -.0597

31 xx -.0557

32 xx -.0957

33 xx -.0350

39 x -.0237

35 x -.0119

35 x 0.

37 x .0055

38 x .0131

39 X .0193

90 xx .0252

91 XX .0307

92 xx .0357

93 XX .0900

99 xx .0937

95 xx .0955

95 xx .0987

97 xx .0500

98 xx .0505

196

FIGURE 5.8 (cont'd.)
GRAPH OF STEP RESPONSE WEIGHTS [Vl(B)]
0. .5 1.0 1.5

(k).+++++++++,+++++++++,+++++++++,+++++++++,

LOCDQOUW-PwNI—‘O

::Kpg-firz-FwwwwwwwwwwNNNNNMMNMMl—‘HHHI—‘I—‘l—‘I—‘l—‘H
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XXXX
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XXX
XX
X
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XX

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XX

XX

XX
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VALUES [Vk]

.1659
.3301
.9908
.6937
.7879
.9190
1.0363
1.1373
1.2203
1.2838
1.3268
1.3989
1.3989
1.3365
1.3128
1.2777
1.2319
1.1761
1.1113
1.0387
.9599
.8798
.7863
.6956
.6092
.5137
.9259
.3909
.2617
.1892
.1295
.0688
.0231
-.0119
-.0356
-.0975
-.0975
-.O909
-.0278
-.0085
.0157
.0979
.0831
.1231
.1668
.2139
.2621
.3121
.3626

197

The sum of the following 29 impulse response weights
= (1.3959)(.551)

= .7691.

Thus, V60 = -.0975 + .7691 = .7216.

The sum of the following 29 impulse response weights
= (1.3959)(.551)2

= .9238 .

Thus, v8” = .7215 — .9238 = .2978
Continuing:

(1.3959)(.551)3 = .2335
v108 = .2978u+ .2335 = .5313
(1.3959)(.551) = .1287
v132 = .53135- .1287 = .9025
(1.3959)(.551) = .0709
V155 = .90256+ .0709 = .9735
(1.3959)(.551) = .0391
v180 = .97357- .0391 = .9399
(1.3959)(.551) = .0215
v20” = .93998+ .0215 = .9559
(1.3959)(.551) = .0118
v228 = .95599- .0118 = .9991
(1.3959)(.551) = .0055
v252 = .9991 + .0055 = .9505

Hence the step response function will converge to
approximately .9980. Of course the implications are
analagous to those of our previous model in equation (5.27),
WIth x2t = BCD92.

Again, we are also interested in the impulse and

step response functions for the second input, Fuel prices,

implied by our model in equation (5.27)'. These are listed

198

and plotted in Figure 5.9.

Expanding the Model to Account for Supply Shocks Due to
Strikes in the Labor Force

 

At this point we need to consider another potential
fault with our model in equations (5.27) and (5.27)'. In
the 33 year sample period reviewed, the Index of Industrial
Production was substantially influenced at various times by
strikes in the Labor Force. We are concerned with the
performance of our models during these times. In this regard,
a list of specific strikes, their dates, and the sectors
of the economy affected, is presented below.

Late 1969: Steel Strike (116 days)

February, 1959: Coal

1970: Teamsters

General Motors (Fall, 139 days)

1979: Coal

1977: Longshoremen

An examination of the residuals of the models in
equations (5.27) and (5.27)' reveals very few outlyers in all
of the time periods listed above. Further, the few outlyers
which do appear near any of these time periods are not
extremely large. This suggests that strikes may not present
a serious problem in our models. However, we are interested
in the possibility of improving the models by including a
third input which accounts for these strike episodes. In

this role we use the number of hours of work stoppage due to

199

FIGURE 5.9
GRAPH OF IMPULSE RESPONSE WEIGHTS [V2(B)J
for x = Fuel PPI in equation (5.27)'

2t

4.1 0. .1
(k).+++++++++,+++++++++,+++++++++,+++++++++,

CDCDQOVU'I-L'CONHO

HPAPJH
(”NF-4C)

H
.1:

HFJFJHFJ
LOCDQCDUW

NNM
Ni-‘O

M
(A)

wwwwwwwwwwwwwmwm
(oooqmmzwwi—Iocoooqmm:

543-334:
(UNI-‘0

47-534:
03014":

><><><><><><><><

XXXXXXXXXXXXX
XXXXXXXXX
XXXXXXX

XXXXX

XXXX

XXX

XXX

XX

XX

XX

><><><><><XXXXXXXXXXXXXXXXXXXXXXXX

VALUES [vk]

lOOOOOOOO
O O. .0 O.

.116038E+00
.829025E-01
.592299E—01
.923152E-01
.302325E-01
.215996E-01
.159317E—01
.110251E-01
.787686E-02
.552759E-02
.902060E-02
.287250E-02
.205225E-02
.l96622E—02
.109753E—02
.798907E-03
.539595E-03
.382011E-03
.272925E-03
.199991E-03
.139311E-03
.995298E-09
.711087E—09
.508033E-09
.362962E-09
.259317E-09
.185268E-09
.132359E-09
.995555E-05
.575527E-05
.982599E-05
.399862E-05
-.295385E-05
-.l76029E-05
-.125753E-05
-.898510E-05
-.691936E-06
-.958629E-06
-.327555E-05

200

FIGURE 5.9 (cont'd.)

-.l 0. .1
(k),+++++++++,+++++++++,+++++++++,+++++++++, VALUES [Vk]

97 X -.239099E-06
98 X -.167251E-06

201

FIGURE 5.9 (cont'd.)

GRAPH OF STEP RESPONSE WEIGHTS [V2(B)] for x21: 2 Fuel PPI

(oooqmmzmwr—Io

-.25 0. .25

(k),+++++++++,+++++++++,+++++++++,++++++++++.+++++++++.VALUES

X 0.

X 0.

X 0.

X 0.

X 0.

X 0.

X 0.

X 0.
XXXXXX -.116038
XXXXXXXXX -.l9899l
XXXXXXXXXXX -.258170
XXXXXXXXXXXXX —.300986
XXXXXXXXXXXXXX -.3307l9
XXXXXXXXXXXXXXX -.352319
XXXXXXXXXXXXXXX -.367751
XXXXXXXXXXXXXXXX -.378776
XXXXXXXXXXXXXXXX -.386652
XXXXXXXXXXXXXXXXX -.392280
XXXXXXXXXXXXXXXXX -.396310
XXXXXXXXXXXXXXXXX —.399173
XXXXXXXXXXXXXXXXX —.901225
XXXXXXXXXXXXXXXXX -.902692
XXXXXXXXXXXXXXXXX —.903739
XXXXXXXXXXXXXXXXX -.909988
XXXXXXXXXXXXXXXXX —.905022
XXXXXXXXXXXXXXXXX —.905909
XXXXXXXXXXXXXXXXX —.905677
XXXXXXXXXXXXXXXXX -.905872
XXXXXXXXXXXXXXXXX -.906011
XXXXXXXXXXXXXXXXX -.906111
XXXXXXXXXXXXXXXXX -.906182
XXXXXXXXXXXXXXXXX -.906232
XXXXXXXXXXXXXXXXX —.906269
XXXXXXXXXXXXXXXXX —.906295
XXXXXXXXXXXXXXXXX -.906313
XXXXXXXXXXXXXXXXX -.906326
XXXXXXXXXXXXXXXXX —.906336
XXXXXXXXXXXXXXXXX -.906393
XXXXXXXXXXXXXXXXX —.906397
XXXXXXXXXXXXXXXXX —.906350
XXXXXXXXXXXXXXXXX -.906353
XXXXXXXXXXXXXXXXX -.906355
XXXXXXXXXXXXXXXXX -.906356
XXXXXXXXXXXXXXXXX -.906357
XXXXXXXXXXXXXXXXX —.906358
XXXXXXXXXXXXXXXXX —.306358
XXXXXXXXXXXXXXXXX -.906358
XXXXXXXXXXXXXXXXX -.906359
XXXXXXXXXXXXXXXXX -.906359

strikes.

x = BCD92,

2t
Yt :
Xlt ’
th '
X3t 3
Sample

and following, with x2t
BCD97

PM

BCD92

t

202

= Fuel prices.

The resulting models are shown below; first with

15,17

Index of Industrial Production

Filtered Mt as in (5.6), with

6

29

%

-.5360

change in PPI of Crude Materials

Work Stoppage due to Strikes

May.

1950 -

(5.31) (1-812)(1—8)1og(yt)

1t
2t

3t

BCD97

FMt

(.

July, 1979 (n =
wo
= [x 1
29 It
1'6298
Y
+ mo B10
1-6iB

+

1-612812
+———a
l-¢1B t
.5909 w' = -.0870
.171) (.038)
7578 51 = .3105
.039) (.055)
RSE = .0117

351)

12
(l-B )[xth

H 12
mo (l-B )(l—B)log[x3t1

'
1 (.195)
R2 = .5529

Index of Industrial Production

Filtered Mt as in (5.6), with 629: -.5231

PPI of Fuel, Power, and Related Products

Work Stoppage due to Strikes

Sample:

May,

1950 - July,

1979 (n =

351)

203

(1)

12 _ o
(5.317 (l-B )(l-B)log(yt) - —__—_—29 [xlt]

l-quB

!
+ mo B8(l-Bl2)(l-B)lo [x 1
1-5i8 3 2t

n 12
+ wo(l-B )(1—B)1og[x3t]

1-812B12
+-—————-—— a
1—¢18 t
w = .1881 529 = -.5189 35 = -.1319 51 = .5500
(.050) (.187) (.058) (.158)
w" = -.0122 0 = .7781 ¢ = .3098
O (.001) 12 ( 035) 1 (.059)
2 _ 2
59 2 RSE = .0117 R = .5529

X95

A comparison of equations (5.31) and (5.31)' with
equations (5.27) and (5.27)' reveals an apparently substantial
reduction in the RSE when X3t is included, although some
reduction is expected with the addition of any third
variable.

We are especially concerned with the performance of
our new models during the years in which strikes substan-
tially influenced GNP. The residuals of the models in
equations (5.31) and (5.31)‘ during these strike periods,
show no noteworthy improvement over those in equations
(5.27) and (5.27)'. This suggests that our models in
equations (5.27) and (5.27)' may be considered adequate

with regard to the problem of strikes.

FOOTNOTES

CHAPTER V

1 . . . .

Friedman, Milton, Milton Fr1edman's Monetary
Framework: A Debate with HIS Critics, ed. by Robert J.
Gordon, The UniversIty of Chicago Press, Chicago, 1979.

2Abbott, W.J., "A New Measure of the Money Supply,"
Federal Reserve Bulletin, October, 1960, p. 1105.

3Abbott, W.J., "Revision of the Money Supply Series,"
Federal Reserve Bulletin, August, 1962, p. 999.

uBox, G. and Jenkins, G., Time Series Analysis,
Holden Day, 1979, Chapter 11.

5The following seven papers represent a good portion
of the Supply Shock literature.

 

 

 

 

 

Denison, E.F., "Explanations of Declining Productivity
Growth," Survey of Current Business, August, 1979.

 

Hall, R.H. and Knut, A.M., "Energy Prices and the
U.S. Economy," Working Paper No. MIT-EL 79-093 WP, M.I.T.
Center for Energy Policy Research, August, 1979.

Hudson, E.A. and Jorgenson, D.W., "Energy Prices and
the U.S. Economy," Discussion Paper No. 637, Harvard
Institute of Economic Research, Harvard University, Cambridge,
Massachusetts, July, 1978.

Jorgenson, D.W., "The Role of Energy in the-U.S.
Economy," Discussion Paper No. 622, Harvard Institute of
Economic Research, Harvard University, Cambridge,
Massachusetts, May, 1978.

Rasche, R.H. and Tatom, J.A., "The Effects of the New
Energy Regime on Economic Capacity, Production, and Prices,"
Federal Reserve Bank of St. Louis Review, May, 1977.

Rasche, R.H. and Tatom, J.A., "Energy Resources and

Potential GNP," Federal Reserve Bank of St. Louis Review,
June, 1977.

209

205

Tatom, J.A., "Energy Prices and Capital Formation:
1972—1977," Federal Reserve Bank Of St. Lonis ReView, May,
1979.

6Note that we must supply the univariate model for
our input, xt = FMt, as the prewhitening model for the
estimation of this transfer function in equation (5.7).
Since our input consists of a 29-month moving average of Mt,
it is not surprising that we find the complicated univariate
model below.

x = FM = Filtered M as in (5.6), with

t t t
629 = -.5207
Sample: May, 1950—September, 1973: n = 281
_ , _ _ 3 _ 12 _ 2 10 13

(1 ¢lB)(l ¢1B ¢3B )(1 ¢12B )[xtJ- (1-62B ~810B -0138 )at
¢i = .8618 ¢1 = .2633 03 = .3303 ¢12 = —.6333

(.190) (.216) (.197) (.055)

6 = -.1683 0 = .1952 8 = .1993
2 (.115) 10 (.057) 13 (.055)
x30 = 61.9 RSE = .0093

7

op. cit., Box, G. and Jenkins, G., Chapter 11.
80p. cit., Box, G. and Jenkins, G., Chapter 11,;n 382.
9For use in the estimation of equation (5.25), the

univariate models for the two inputs over the sample, May,
1950-September, 1973 (n = 281), are the following.

x1t = FMt = Filtered Mt as in (5.6), w1th
52” = -.5231
Same model as in footnote 6, with:
(5 A A A
¢i = .8669 ¢l = .2579 ¢3 = .3267 ¢12 = -.6338
(.199) (.220) (.199) (.055)
A A A
0 = -.1652 8 = .1951 8 = .1992
2 (.117) 10 (.055) 13 (.055)
2

X90 = 52.1 RSE = .0093

206

= BCD92 % change in PPI of Crude Materials

x2t
3 12 _ 2 10 13
(1-918-935 )(1-5 )[th1 - (1-62B ~6128 -61uB )at
91 = .9133 03 = .1855 52 = .0977 912 = .7580
(.053) (.055) (.055) (.091)
614 = —.1586 x35 = 19.1 RSE = .0125
(.055)
10

For use in the estimation of equation (5.26), the
univariate models for the two inputs over the sample,
May, 1950-March, 1975 (n = 299), are the following.

x1t = FMt = Filtered Mt as in (5.6), w1th
629 = -.552
Same model as in footnote 6, with:
A. A A A
¢i = .8563 ¢1 = .2752 ¢3 = .3303 ¢12 = -.6080
(.223) (.298) (.162) (.051)
0 = -.1219 8 = .1919 0 = .1638
2 (.128) 10 (.052) 13 (.051)
2 _ -
X90 - 69.2 RSE — .0093
x21: = BCD92 % change in PPI of Crude Materials
Same model as in footnote 9, with:
A A A A A
¢ = .9290 ¢ = .2181 6 = .0739 e = .7739 9 = -1317
1 (.050) 3 (.055) 2 (.055) 12 (.092) 1” (.057)
x35 = 25.1 RSE = .0130
11

For use in the estimation of equation (5.27), the
univariate models for the two inputs over the sample,
May, 1950 — August, 1979 (n = 352), are the following.

xlt = FMt = Filtered Mt

Same model as in footnote 6, with:

as in (5.6), with 629:‘-'5600

1i = .8573 51 = .2197 ¢3 = .3052 ¢12 = -.5109
(.118) (.191) (.105) (.095)
5 = -.1173 5 = .1392 5 = .1819
2 (.085) 10 (.058) 13 (.057)
2 = 57.5 RSE = .0095

X90

207

x2t = BCD92 % change in PPI of Crude Materials

Same model as in footnote 9, with:
A A A A A
¢ = .3850 ¢ = .2658 8 = .1592 8 = .8120 8 = -.1975
1 (.059) 3 (.099) 2 (.058) 12 (.039) 1” (.057)

2 _ ..
X25 - 92.6 RSE - .0192
12

The univariate model for the input, the Fuel PPI,
is needed in the estimation of this single—input transfer
function. This model is presented below.

12 _ 12
(l-¢lBXl~B )(1—B) log [xtJA - (1-8128 ) at
¢ = .6606 8 = .8923
1 (.092) 12 (.031)
x2 = 52 8 RSE = 0098
96 ' '
where xt = PPI of Fuel, Power, and Related Products, and the

Sample period is May, 1950-November, 1979 (n = 355).

13For use in the estimation of equation (5.25)‘, the
univariate models for the two inputs over the sample,
May, 1950-September, 1973 (n = 281), are the following.

x1t = FMt = Filtered Mt as in (5.6), With

62” = -.992

Same model as in footnote 6, with:

A

¢i = .8669 01 = .2599 $3 = .3230 ¢12 = -.6239
(.195) (.221) (.199) (.055)
A A A
8 = -.1633 0 = .1967 0 = .1950
2 (.118) 10 (.057) 13 (.055)
2 _ A
x“0 - 59.9 RSE — .0093
x2t = PPI of Fuel, Power, and Related Products
Same model as in footnote 12, with:
¢ ' .2377 6 = .8086
1 (.051) 12 (.090)
2 = 93.9 RSE = .0079

208

ll1For use in the estimation of equation (5.26)‘, the
univariate models for the two inputs over the sample,
May, 1950—March, 1975 (n = 299), are the following.

x1t = FMt = Filtered Mt as in (5.6), w1th
62” = -.9600
Same model as in footnote 6, with:
¢i = .8539 21 = .2833 03 = .3206 212 = —.5995
(.299) (.273) (.179) (.052)
8 = -.1181 0 = .1961 0 = .1537
2 (.137) 10 (.052) 13 (.051)
2 _ _
X90 — 62.0 RSE - .0092
x2t = PPI of Fuel, Power, and Related Products

Same model as in footnote 12, with:

¢ = .5089 5 = .8318
1 (.098) 12 (.037)
2 _ _

X95 - 55.3 RSE - .0097

15

For use in the estimation of equation (5.27)‘, the
univariate models for the two inputs over the sample:
May, 1950-November, 1979 (n = 355), are the following.

xlt = FMt = Filtered Mt as in (5.6), w1th
62” = -.5520
Same model as in footnote 6, with
5i = .8581 51 = .2190 53 = .3091 ¢12 = -.5101
(.119) (.193) (.106) (.096)
8 = -.1169 8 = .1397 8 = .1810
2 (.087) 10 (.055) 13 (.057)
2 _ _
X“0 — 66.6 RSE - .0095
x2t = PPI of Fuel, Power, and Related Products

Same model as in footnote l2.

209

16For use in the estimation of equation (5.31), the
univariate models for the three inputs over the sample,
May, 1950—July, 1979 (n = 351), are the following.

xlt = FMt = Filtered Mt as in (5.6), With
6 -
2” _ 3.5360
Same model as in footnote 6, with:
¢i = .8627 21 = .2238 03 = .3132 ¢12 = -.6151
(.107) (.131) (.101) (.096)
8 = -.1l97 8 = .1997 8 = .1806
2 (.083) 10 (.059) 13 (.057)
2 _ _
X90 - 60.5 RSE - .0099
x = BCD92 % change in PPI of Crude Materials

2t

Same model as in footnote 11.

x3t = Work Stoppage due to Strikes
(1-812)(1-8) log [xst] = (1—812812) at
59 1:332:
x37 = 51.5 RSE = .909
17

‘ For use in the estimation of equation (5.31)‘, the
univariate models for the three inputs over the sample,
May, 1950-July, 1979 (n = 351), are the following.

x1t = FMt = Filtered Mt as in (5.6), With
629 = -.5231
Same model as in footnote 6, with:
h A A A
¢i = .8637 ¢1 = .2227 ¢3 = .3116 212 = -.6l36
(.109) (.133) (.102) (.097)
A A A
82 = -.ll90 0 = .1503 0 = .1792

(.089) 10 (.059) 13 (.057)

2 - -
X90 - 59.2 RSE - .0099

210

x2t = PPI of Fuel, Power, and Related Products

Same model as in footnote 12.

x3t = Work Stoppage due to Strikes

Same model as in footnote 16.

CHAPTER VI

CONCLUSION

The proposed leading indicator approach is supported
by an appropriate theoretical framework in the form of a
dynamic, structural econometric model. In this context,
information is obtained about the first two moments of the
conditional distribution, f(yt+k|LIt). This information
is the object of our analysis.

A comparison of the forecasting abilities of the
proposed approach with the Final Form, reveals essentially
that both approaches forecast equally well. That is, both
approaches imply the same forecast error variance for
forecasts within the lead of the leading indicator.

These observations suggest that the proposed
leading indicator approach may deserve more attention as an
alternative to the Commerce Department approach.

As we move from this proposed theoretical approach to
the empirical evaluation of the Commerce Department leading
indicators, justification is needed for the consideration
of only one input in the transfer functions developed. It
is argued that the bias introduced in the parameter

estimates through the omission of relevant variables, is

211

212

not a problem when forecasting is the main objective. In
fact the biased estimates are truly appropriate for fore-
casting since they contribute to an exact representation
of the expectation of the objective variable, conditional
on the single input utilized.

With this justification established, bivariate time
series models are built describing the empirical relation-
ships between economic activity and eight of the component
series in the Commerce Department‘s CLI. Five of these
models show a lack of any significant lead time in the
relationship. The other three models suggest relationships
with a lead worthy of a “leading indicator". However, the
leading indicator which displays the most significant lead
over economic activity is the Producer Price Index of Crude
Materials, which is seen to have a negative relationship
with economic activity, while the Commerce Department uses
it in a positive role. These revelations present some
possible reasons for the poor performance record of the
Commerce Department approach.

Finally, Money is proposed to fill the role of
leading indicator. The cross-correlation functions between
the prewhitened Money series and the correspondingly
transformed Industrial Production series implies an impulse
response function which might follow a damped cosine wave
with a period of four years. The associated step response
at the identification stage bears much resemblance to that

implied by the framework suggested by Friedman (1979). This

213

step response function supports a proposition that Money is
neutral after 36 months. The step response function
approaches zero in that time frame, and the impulse response
weights after 36 months, appear to be quite erratic, breaking
from the previous appearance of a smooth cosine wave. This
may indicate that they reflect only random behavior after
that point.

In this light, a bivariate time series model is
built, describing the empirical, dynamic relationship
between economic activity and Money. An infinite lag model
is estimated about the pattern indicated at the identifica-
tion stage discussed above. This model is expanded to
account for two different kinds of supply shocks occurring
in the sample period reviewed: the energy price shocks of
the early 1970‘s, and strikes in the Labor Force. At each
stage of the expansion of this model, a stable dynamic
relationship is observed between Money and economic
activity.

After 36 months, the estimated step response
function is seen to approach zero, supporting the
observations regarding the relationship at the identification
stage, However, since an infinite lag relationship is
estimated, the impulse response weights after 36 months are
constrained to continue to follow the damped cosine wave,
even if they truly represent only random movements about
zero. These impulse response weights after 36 months must

be considered in finding the steady state gain, the

219

convergence of the step response function. These weights
are seen to converge to approximately (.95) in the various
stages of expansion of the model.

This suggests a nonneutral long run relationship
between Money and real GNP, which is somewhat troubling in
light of the concepts of current Monetary Theory. Never—
theless, the model provides useful insight into the
empirical relationship between Money and real GNP, which is
so important in economic theory. In particular, the stable
relationship found implies that Money may provide useful

information as a leading indicator of economic activity.

APPENDIX 9

DATA SOURCES

APPENDIX 9

DATA SOURCES

The economic indicator series are available in the

 

Business Conditions Digest (BCD), but most are seasonally
adjusted.
The Index of Industrial Production (BCD97) appears

not seasonally adjusted in The Survey of Current Business,

 

and is available from 1997 to the present.

The Composite Index of Leading Indicators consists of
12 components: BCD series 1, 3, 8, 12, 19, 20, 29, 32, 36,
92, 109, and 105. Series 19, the Index of Stock Prices, and
series 32, Vendor Performance, are available not seasonally

adjusted in the Business Conditions Digest from 1997 to the

 

present.
BCD series 1, the Average Workweek of Production

Workers, appears not seasonally adjusted in The Survey of

 

Current Business, and is available from 1997 to the present.

 

BCD series 3, the Layoff Rate, is also available not

seasonally adjusted in The Survey of Current Business from

 

1997 to the present.
BCD series 8, the Value of Manufacturers' New Orders

for Consumer Goods and Materials, consists of the aggregation

215

216

of three series: (1) new orders for durable goods
industries plus (2) new orders for nondurable goods
industries with unfilled orders minus (3) new orders for
capital goods and defense products. New orders for durable
goods industries is available not seasonally adjusted in

The Survey of Current Business from 1997 to the present.

 

The latter two series are both available not seasonally
adjusted in the Census Bureau publication, Current Industrial
Reports - Manufacturers' Shipments, Inventories, and Orders,
from 1958 to the present (see call number C3.158/M3-1.6).
These categories are unavailable in this publication for
earlier years. Thus our constructed series BCD8 is
available from 1958 to the present.

BCD series 29, the Index of New Private Housing Units
Authorized by Local Building Permits (not seasonally
adjusted), can be constructed from the month to month
changes in series HSBBR in the Citibank Data Base. This is
available from 1959 to the present.

BCD series 92, the % Change in Sensitive Prices (the
PPI of Crude Materials, not seasonally adjusted), can be
constructed from series PWCMPX in the Citibank Data Base.

BCD series 105, the Real Money Stock, Ml’ not
seasonally adjusted, can be constructed by dividing the
Nominal Money Stock, M1 (available as series FZMl in the
Citibank Data Base), by the CPI for All Items (available in

The Survey of Current Business). This constructed series is

 

available from 1997 to the present.

217

We have been unable to locate data for BCD series 12,
20, 36, and 109, on a not seasonally adjusted basis.

Data for the unadjusted Nominal Money Stock series
employed in Chapter V is obtained by splicing two different
definitions of the Money Stock. This splicing technique is
developed extensively in the Chapter. The two Money Stock
series used are:

Ml from 1997.1 - 1958.12: published in the Federal

Reserve Bulletin, October, 1960.

 

MlB from 1959.1 - present: published in "Redefined

Money Stock Measures, Liquid Assets, and Related

Measures," Federal Reserve Release, March 29, 1980,

 

with recent updates available in "Federal Reserve
Statistical Release H.6," June 20, 1980.
The Producer Price Index for Fuel, Power, and Related

Products is published in The Survey of Current Business, from

 

1997 to the present. However, the numbers appear under

three different base periods:

1997.1 - 58.12 with base, 1997-99 = 100;
1959.1 - 66.12 with base, 1957-59 = 100;
1967.1 - present with base, 1967 = 100.

These segments are spliced into a complete series with base,
1967 = 100, as follows. The mean of the observations for
1957.1 - 59.12 with base, 1997-99 = 100, is calculated

[mean = 119.21. Then the mean of the observations for
1967.1 - 67.12 with base, 1957-59 = 100, is calculated

[mean = 103.6]. Next, the observations for 1997.1 — 58.12

218

with base, 1997—99 = 100, are divided by (1.1192*1.036).
And finally, the observations for 1959.1 - 66.12 with base,
1957-59 = 100, are divided by (1.036). These two segments,
together with the remaining segment from 1967.1 to the
present, represent a complete series for Fuel Prices, with
base, 1967 = 100.

Finally, the Number of Hours of Work Stoppage Due to
Strikes in the Labor Force is obtained from the series,

LHSTOP, in the Citibank Data Base.

LIST OF REFERENCES

[l]

[2]

[31

[9]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

REFERENCES

Abbott, W.J., "A New Measure of the Money Supply,"

Federal Reserve Bulletin, October, 1960.

 

"Revision of Money Supply Series," Federal
Reserve Bulletin, August, 1962.

 

Alexander, 3.8., "Rate of Change Approaches to

Forecasting - Diffusion Indexes and First
Difference," Economic Journal, June, 1958.

 

Boschan and Ebanks, "A New Method of Measuring Economic

Growth," American Statistical Association, August,
1978.

 

, and Zarnowitz, V., "Cyclical Indicators: An
Evaluation and New Leading Indexes," Business
Conditions Digest, May, 1975.

 

, and Zarnowitz, V., "New Composite Indexes of
Coincident and Lagging Indicators," Business
Conditions Digest, November, 1975.

 

Box, G. and Jenkins, G., Time Series Analysis, San

 

Francisco: Holden-Day Inc., 1976,

Bronfenbrenner, Martin, ed., Is The Business Cycle

 

Obsolete?, New York: John Wiley and Sons, 1969.

 

Burns, A.F., "Business Cycles: General," International

 

Encyclopedia of the Social Sciences, New York:
MacMillan Co., 1968, pp. 226-95.

 

, and Mitchell, W.C., Measuring Business Cycles,
New York, NBER, 1996.

 

, and Mitchell, W.C., "Business Cycles,"
International EncyC1opedia of the Social Sciences,
1968, Vol. 2; reprinted as "The Nature and Causes
of Business Cycles," in Burns, The Business Cycle
in a Chapging World, New York: NBER, 1969.

 

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