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MULTIVARIATE ESTIMATES OF THE PERMANENT COMPONENTS OF GNP AND STOCK PRICES* John ... This paper brings multivariate evidence to bear in the following way: consider a pair of time series, with the property that the ratio of the two series is stationary. For example, it is possible that the consumption/GNP ratio is stationary, even if log consumption and 1ogGNP each have random walk components ...

256 J H Cochrane and A M Sbordone Permanent components of GNP and stock prices

twice the unconditional variance of the series If a series is a combination of a

stationary and a random walk component l k times the variance of k

differences tends to the variance of the random walk or permanent compo

It turns out that we can think interchangeably of series which contain a unit

root and series which are composed of a stationary and a random walk

component and that the variance of the random walk component can be

directly related to the effect of a univariate innovation on long horizon

These papers all provided interesting point estimates and unfortunately

large standard errors The problem is that there are few nonoverlapping long

runs of data available so unless strong restrictions are imposed on the form

of the estimated time series process the response of long run forecasts to a

unit innovation will necessarily be imprecisely measured

A natural response is to try to examine more series in the hope that

observations of several series over a short time horizon can in some sense

proxy for the observation of one series over a long horizon In this vein

Campbell and Mankiw 1987 examined GNP from many countries and

Poterba and Summers 1987 examined stock prices from many countries The

defect of this approach in its present implementation is that it is not clear how

much if any independent information is contained in estimates from several

countries at the same time

This paper brings multivariate evidence to bear in the following way

consider a pair of time series with the property that the ratio of the two series

is stationary For example it is possible that the consumption GNP ratio is

stationary even if log consumption and 1ogGNP each have random walk

components or that the dividend price ratio is stationary even if log

dividends and log stock prices each have random walk components

If the ratio of two series is stationary the random walk component of the

two series must be exactly the same we can express each series as a sum of a

common random walk component and separate stationary components If we

couldn t do this the ratio of the two series would contain a random walk

More precisely if log X and log W must be differenced to obtain

stationary series if they contain random walk components yet log X W is

stationary then there must be a representation

log X z c log W z C 1

where z is a random walk and c and c are stationary If the Z entered

with different coefficients or if there was a random walk component in one

series not present in the other then log X W would contain a random walk

Now if the two series can be expressed as the sum of a common random

walk component and distinct stationary components then l k times the

J H Cochrane and A M Sbordone Permunent components of GNP und stock prices 251

variance of k di erences of each series as well as l k times the covariance of

k diflerences of the two series must tend to exactly the same number the

innovation variance of the common random walk component In the representa

tion 1 all these quantities are l k var z z var z z t for large

Then we can estimate the variance of the permanent or random walk

component of one series GNP stock prices from l k times the variance of

k differences of the other series consumption dividends or l k times the

covariance of k differences of the two series

How do we decide what k is large enough or that the limit has been

adequately reached The choice of k is exactly the choice of a window width

of a spectral density estimator larger k smaller window gives less bias but

more uncertain estimates while smaller k larger window gives more precise

but more biased estimates Operationally we stop at a k large enough that

business cycle fluctuations are ironed out and only the long run remains on

the order of 20 or 30 years We also stop at k half the sample size which

amounts to taking a variance based on two data points Since we have no

reason to prefer k 20 or k 30 etc we present results for a variety of k

and hope that the results are robust in a range of k

Figs 1 and 2 present l k times the variance and covariance of k differences

for stock prices and dividends and GNP and consumption respectively Once

we impose the assumption that the consumption GNP ratio or the

dividend price ratio is stationary the three lines in each figure are each

estimates of the variance of one underlying random walk component

Which of the three estimates or which combination of the three is the best

estimate of the underlying random walk component As long as the difference

k is finite each variance or covariance is a biased estimate of the variance of

the underlying random walk component because some stationary components

are still included in the k differenced series On the other hand though each

variance or covariance contains no independent information about the vari

ance of the one common random walk component the stationary components

are not perfectly correlated so there is some independent movement in each

variance or the covariance in finite samples The optimal combination of the

variance and covariance is thus a tradeoff between the extra bias of series with

stronger stationary components against the reduction in variance that occurs

when you combine estimates with some independent information

Below we will argue that for pairs of series like GNP and consumption or

stock prices and dividends in which one series consumption dividends is

nearly a random walk has a flat graph of l k times the variance of

k differences the reduction in standard error is not worth the increase in bias

so the best combined estimate is provided by just looking at l k times the

variance of k differences of the series which is closest to a pure random walk

consumption and dividends

258 J H Cochrane und A M Sbordone Permanent components of GNP and stock prices

Hence the best multivariate estimate of the variance of the random walk

component of GNP is provided by the variance of k differences of consump

tion the best estimate of the variance of the random walk component of stock

prices is provided by the variance of k differences of dividends These

estimates are about half previous univariate estimates See figs 1 and 2 or

tables 4 and 5 the consumption and dividends lines are about half of the price

and GNP lines

1 1 Some comments on the estimation strategy

The major advantage of using several time series in this way is that it

reduces the bias associated with finite k differencing of the time series of

interest GNP stock prices The standard errors are mostly associated with

the standard error of measuring the common random walk component and not

the individual stationary components hence these bivariate estimates do not

significantly reduce the standard errors associated with univariate estimates

Hence it does not seem useful to generalize the procedures of this paper to

multiple time series that are all cointegrated for example to include many

other components of GNP The other components of GNP have more sta

tionary components than consumption their graph of l k times the variance

of k differences starts higher and slopes down more than the consumption

graph winding up slightly higher than l k times the variance of k differences

of GNP so including them will only bias the estimate of the common random

walk component while adding little independent information

In order to use multivariate information to reduce the standard errors we

would have to find other series that are not cointegrated with GNP stock

prices but have the same variance of a random walk component or variance

ratio Then observations of several series will add some independent observa

tions on the variance of a random walk component For example if the GNP

or stock prices of several countries are not cointegrated if the ratios of their

prices are not stationary or if the squared covariance of their k differences is

not equal to the product of the variances of their k differences and yet each

follows the same process has the same variance of k differences at all k then

a pooled estimate can reduce the standard errors We leave this as a suggestion

for future research

1 2 Plan of the paper

The rest of this paper formalizes these arguments and presents our results

for stock prices and dividends and for consumption and GNP In section 2

we discuss the decomposition of first difference stationary series into sta

tionary and random walk components and relate that decomposition to the

J H Cochrane and A M Sbordone Permanent components of GNP and stock prices 259

property that ratios of the series may be stationary In particular we derive

the representation 1 above

In section 3 we discuss estimation and present some of the properties of the

l k times the variance of k differences technique In particular this section

argues that just looking at the variance of the series closest to a random walk

is the best combined estimate in cases like ours

Section 4 presents our results First we prove that the present value relation

implies that the dividend price and consumption GNP ratios should be

stationary We also apply univariate variance ratios to these series to check

this stationarity assumption Then we impose the assumption that the divi

dend price and consumption GNP ratios are stationary to measure the

random walk components in GNP and stock prices from l k times the

variance of k differences of consumption and dividends

2 Representation of time series with unit roots

2 1 Decomposition into stationary and random walk components

Let y be an N dimensional vector time series Throughout we will assume

that yt is stationary in first differences in particular we will assume that

1 L y has a moving average representation

1 L y A L 2 1

L is the lag operator Ly Y 1 is an N dimensional vector of means

A L is an N x N matrix of lag polynomials with A 0 IN the N X N

identity matrix i e A L E ZNe IAje j and Edis an N dimensional

vector of innovations E E 0 E E E 2 Ir is a positive definite matrix

and E E E 0 for j f 0

We can construct a decomposition of yt into a stationary component and a

random walk component with a multivariate generalization of the Beveridge

and Nelson 1981 decomposition

1 L z l A l Et 2 2

c A L q where AT E A and A 1 E A

260 J H Cochrune and A M Sbordone Permaneni components of GNP and stock prices

In this decomposition z lim mEty k kp so z can be interpreted as

a stochastic trend or permanent component of y Also A l gives the

limiting response of E y to an E innovation at time t Stock and Watson

1986 derive this and some other representations

From 2 1 the spectral density of 1 L y at frequency 0 is

so the variance covariance matrix of changes in the permanent components

1 L z is the same as the spectral density of 1 L y at frequency 0 and is

the same as the spectral density matrix of 1 L z as well To avoid

repeating these three interpretations we will denote this matrix by k

9 A l ZA l S l var l L z S i jZ e iO

We ll denote the elements of k by so that is the variance of the

permanent component in element i of y and is the covariance of the ith

and jth permanent components

Although it is derived in the context of one of many possible decomposi

tions the matrix 9 gives a complete characterization of the unit root or

cointegration properties of a series in a finite sample Given the spectral

density of 1 L y at frequencies other than 0 we can always construct a

trend stationary series by changing the value of the spectral density to be 0 at

frequency 0 Cochrane 1987 discusses this point in detail

2 2 Cointegration

In this system the time series y are said to be cointegrated if there is an

N x M matrix Y rank 44 such that e yr is stationary These concepts and

terminology are due to Engle and Granger 1987 The columns of Y are

called the cointegrating vectors Now y a z Y c LY C A L E

and since linear combinations of stationary series are stationary this term

imposes no restrictions on A L However Y z z i a a A l e For

this term to be stationary we require

a 0 and LY A 1 0 2 3

In turn A l 0 implies cw A l Z A 1 OL P 0 so both A 1 and k must

be of rank N M for the system to be cointegrated Since we will use logs of

series cointegration amounts to the statement that ratios of the series are

stationary

Granger and Engle 1987 show that if y is cointegrated there is an

equivalent error correction representation

J H Cochrane and A M Sbordone Permanent components of GNP and stock prices 261

where B is a matrix of constants and y is a N X M matrix See Granger and

Engle for the construction of this representation from 2 1 This representa

tion shows how changes in y depend on how far away a y r is from its

equilibrium value

Fama and French s 1987 regressions of future returns on dividend price

ratios are estimates of this error correction form our variance based estimates

properties of the original representation Since either representation can be

derived from the other we are after the same phenomenon just as Fama and

French s 1986 regressions of future returns on past returns are measures of

the same phenomenon as Poterba and Summer s univariate variance ratios

2 3 A representation of cointegrated series that measures the importance of

cointegration by the size of random walk components

Cochrane 1986 emphasized that the univariate spectral density at 0

variance of random walk component and Ca 2u 2 is a useful as well as a

complete characterization of the unit root properties of a series because it

allows us to measure the importance of a unit root on a continuous scale from

0 to cc rather than just ask is there or isn t there a unit root For a vector of

time series y we can ask the further question how many unit roots common

trends in the Stock Watson language are there This section derives a

rewriting of the Beveridge Nelson representation that allows us to quantify

the importance of the N potential unit roots rather than ask simply what is

their number

Express the spectral density matrix at frequency 0 as

where A is a diagonal matrix of eigenvalues of the spectral density matrix

organized from highest to lowest

and Q is a corresponding orthogonal matrix of eigenvectors Since k is

symmetric it has a full set of linearly independent eigenvectors so this

r With this representation multivariate variance covariance ratio estimates or other multi

variate spectral density estimates can play the same role with regard to Stock and Watson s tests

for the number of random walk components that the univariate variance ratio tests do to the

Dickey Fuller etc tests for the presence or absence of a single unit root they allow one to

estimate the same quantities while imposing less additional structure

262 J H Cochrane and A M Sbordone Permanent components of GNP and stock prices

representation exists When the spectral density matrix 9 is of rank N M

less than N M of the hj are 0

Define a new N dimensional series of innovations by Y Q A l Then

we can rewrite the random walks z in terms of these innovations as

I c z J where E v O E y v A 2 6

Since the variances of the last N M elements of Y are 0 this representation

expresses yt in terms of N M random walks or common trends whose

innovation variances are the eigenvalues of k

Consider the problem of finding M L nearly cointegrating vectors when

there are really M In a large sample we want to find M L vectors aj

j 1 2 M L that minimize

fin var cy I L z Ol aj aj Q A Q aj 2 7

subject to an arbitrary normalization for the vector ai The answer is pick

for aj the eigenvectors corresponding to the M L lowest eigenvalues of the

spectral density matrix The minimized variance of the permanent compo

nent of the corresponding linear combinations of y are then

var 1 L a z Xj 2 8

The representation 2 6 and its interpretation 2 7 2 8 provide a multi

variate extension of the quantity a l u used for univariate time series If

there are M cointegrating vectors M of the Xj elements of A are 0 there are

M linear combinations of y that are stationary and so the N dimensional

series yt can be expressed as a sum of only N M random walk components

plus stationary components As the series yt becomes closer and closer to

being cointegrated with M 1 cointegrating vectors the M 1 th hj will

approach 0 and the N M 1 th random walk vN M 1 1will contribute less

and less to the variance of the long term forecasts z In applications where it

is more interesting to measure the size of a univariate random walk component

rather than test for its presence or absence of a unit root this representation

suggests we estimate the variance of the N potential vif rather than test for

their number

Take the normalization as ICY

1 Then we want to minimize nJQAQ u ol a Any linear

algebra textbook e g Wang 1976 p 253 under Rayleigh s Quotient shows that this quotient

is minimized by taking ai as one of the eigenvectors or columns of Q and the value of the

minimized quotient is the corresponding eigenvalue or element of A

3Phillips and Ouharis 1986 derive the asymptotic distribution of eigenvalues of the spectral

density matrix at frequency 0 under the null that the matrix is in fact of full rank

J H Cochrane and A M Sbordone Permanent components of GNP and stock prices 263

2 4 EfSect of cointegration on

Later on we will impose the restriction that the consumption income ratio

and the dividend price ratios are stationary This assumption implies that the

logs of these series are cointegrated with cointegrating vector l 11 In this

section we ask what effect does this assumption have on the matrix 9

Consider a pair of time series y x w cointegrated with cointegrating

vector Y l YJ x Y w is stationary Think of x as GNP and w as

consumption or x as stock prices and w as dividends Cointegration implies

a A 1 0 and hence a A l EA l a k 0 This requirement means for

the elements of q

J L c llh 2 11

2 9 shows that cxl can be found from an OLS regression of the permanent

components 2 10 shows that the relative variance of the random walk

components is determined by the cointegrating vector a and 2 11 reminds us

that k is singular so that there is effectively only one random walk compo

nent We can rewrite 2 9 2 11 as

constant 2 12

or in a common trends representation

w z cw 2 13

z z i nl nt i i d

If Y l l then x w is stationary or with x log X and

wt log w log X W is stationary Otherwise log X v l is stationary

When Y l 11 and log X W is stationary then 2 13 reduces to

wf z cw 2 14

z z r l nt 9 i i d

264 J H Cochrane and A M Sbordone Permanent components of GNP and stock prices

This is eq 1 in the introduction In words x and wt can be expressed as the

sum of a common random walk component and a stationary component In

1 1 I constant 2 15

We will use l k times the variance of k differences of x and w to estimate

the diagonal elements of k and l k times their covariance to estimate the

off diagonal elements From 2 15 if the log X W is stationary l k times

the variance of k differences of x l k times the variance of k diflerences of w

and l k times the covariance of their k differences all approach the same number

as k grows We can read the variance of the random walk component of

income for example from the variance of the k differences of consumption

and from the covariance of the k differences of consumption and income as

well as from the variance of k differences of income

3 Estimation

3 1 l k times variance of k diflerences as a spectral density estimator

Most tests for cointegration like tests for unit roots are based on parsi

monious VAR representations For examples see Stock and Watson 1986 or

Granger and Engle 1987 In retrospect this is surprising because the

cointegration and unit root properties of a series are entirely a function of k

the spectral density at 0 and spectral density is usually not estimated from

parsimonious VAR or VARMA representations

These techniques impose restrictions across frequencies to estimate the

spectral density at 0 from high frequency information in any finite sample

Most commonly one has to impose that the errors are described by a short

AR or MA process This assumption essentially bounds the slope of the

spectral density near zero Direct estimation of the spectral density at frequency

0 k often imposes fewer such restrictions As in the univariate case it can

capture classes of time series behavior such as long horizon mean reversion

that are precluded by the restrictions imposed by estimating parsimonious

VAR representations

This section shows how l k times the variance of k differences is a

conventional spectral density estimate In particular it is a member of a class

of spectral density estimates that we call variance of filtered data estimates

Estimates in this class are equivalent implementations of the usual weighted

covariance and smoothed periodogram estimates in the sense that for any

member of one class there is an equivalent member of the other two available

by Fourier transformation

J H Cochrane and A M Sbordone Permanent components of GNP and stock prices 265

Start with a stationary N dimensional time series Ay Define its finite

Fourier transform x o as4

X W T l t e Ay VTSWlT 3 1

Its periodogram is defined for T frequencies between r and rr

Oj x W x W aj 2rj T j T 1 2

where means complex conjugation and transposition

The periodogram is an unbiased but inconsistent estimate of the spectral

density matrix at w because its variance does not decline to 0 as T CO

Therefore the spectral density is conventionally estimated as a weighted sum

of nearby periodogram ordinates

S emiw C WbJ 4qQ 3 3

3 3 is the smoothed periodogrum estimate of the spectral density If we

promise that the weighting function will approach a delta function as T co

but at a slower rate than T 3 3 is a consistent estimate of the spectral

If we Fourier transform the weighting function

w k 1 eik Wjv w dv 3 4

then the Fourier transform of 3 3 is

eCiw J eCik w k f k 3 5

where f k is a consistent estimate of the kth autocovariance E Ay Ay

This is the weighted covariance estimator

The variance of long differences is an instance of a third equivalent class of

estimators Define a two sided lag polynomial F ee such that

lF e I W Y a 3 6

4 We chose to put the 2 in the inverse Fourier transform rather than include a 2 in

both the forward and inverse transform so that the identity between the spectral density at 0 and

the variance of 1 L z is preserved without an intervening 2n

266 J H Cochrune and A M Sbordone Permaneni components of GNP and stock prices

Then form a filtered version of the original series

r t Wd AA 3 7

The variance of this filtered series is

var y t 2a IF e i 12S e i dv 3 8

Therefore the sample variance of filtered data is asymptotically equivalent to

the weighted covariance and smoothed periodogram estimators of the spectral

density 3 3 and 3 5

We can express the variance of long differences estimate as the variance of

filtered data

b g eCio var 3 9

The corresponding smoothed periodogram estimate is from 3 3

sin2 k l 2 0

and the weighted covariance estimate is

e iO i k k lry 3 11

which is the Bartlett estimate of the spectral density at frequency 0

The usual procedure is to pick a k or window width and then calculate one

of the above estimates If k is too small the estimate will be biased from

including too many far away periodogram ordinates If k is too large it will

have a large variance because few periodogram ordinates are included A plot

like figs 1 and 2 represents the result of experimentation with different k or

window widths We hope to find a region in which the results are insensitive to

the choice of the window width

The smoothed periodogram weighted covariance and variance of filtered

data estimates are asymptotically equivalent In a finite sample there are

differences between the three estimates For example the variance of k

differences estimate corresponds to a calculation of the autocovariances that

underweights observations k away from the beginning and end of the data set

compared to the conventional estimate of the autocovariance These differ

J H Cochrane and A M Sbordone Permanent components of GNP and stock prices 261

covariance

d 0 6 12 18 24 3

difference years

Fig 1 l k x variance and covariance of k differences for stock price and dividends Units

variance of one year change in price

covariance

consumption

d 0 3 6 9 12 15

difference years

Fig 2 l k X variance and covariance of k differences of GNP and nondurable services

consumption Units variance of one year change in GNP

268 J H Cochrane and A M Sbordone Permanent components of GNP and stock prices

ences give rise to different small sample properties In particular the small

sample bias can differ

The variance of k differences gives a clear picture of exactly what feature of

the data drives the estimate The essential characteristic that any estimate of

the spectral density at 0 picks out is whether there is a great deal of variability

left at long horizons This is in fact the only distinguishing feature of a

random walk component By taking long differences of the levels of a series

which is the same as taking a long moving average of its differences we are

filtering out the high frequency variation in the series and leaving in only the

long horizons

In principle one might get better more efficient results by using other

filters than a simple unweighted moving average because the unweighted

moving average lets in a certain amount of high frequency information

through side lobes in its Fourier transform Other filters amount to other

window shapes of smoothed periodogram estimates In experiments with

these data sets however we have found very little difference in either results

or standard errors from using other filters

3 2 Standard errors

The asymptotic distribution of spectral density estimates is also standard

Koopmans 1974 gives the asymptotic variance of the Bartlett estimator at

frequency 0 as

S e fp 1 5 3 12

where 4 is l k times the sample variance of k differences our estimate of

the spectral density at frequency 0

To assess the accuracy of the Bartlett formula in samples of our typical size

we ran a few Monte Carlo experiments reported in table 1 The mean value

of the variance of k differences was close to 1 00 in all these experiments In

the first row of table 1 we simulated 100 observations of a pure random walk

with no drift and we report standard errors of l k times the variance of k

differences We calculated the variance of k differences using formula 3 19

That formula includes some corrections for small sample bias discussed

In the second row we extend the sample to T 200 observations Note that

the standard errors are very close to the same for equal values of k T This

behavior is typical of all Monte Carlo experiments we ran

The third row gives the results when the process includes and we estimate a

drift Here the process is a random walk with a drift term of 1 and the

variance of k differences is calculated by 3 18 below

J H Cochrane and A M Sbordone Permanent components of GNP and stock prices 269

Monte Carlo standard errors for variance of k differences 500 trials a

Model when no mean removed y y E 0 1

Model when mean removed y 1 y e 0 1

100 k T 1 2 3 4 5 10 20 30 40 50

T lOO 0 137 0 160 0 200 0 231 0 263 0 409 0 607 0 772 0 888 0 896

T 200 no mean 0 105 0 167 0 203 0 232 0 256 0 366 0 561 0 733 0 913 1 093

T 100 no mean 0 139 0 178 0 210 0 246 0 271 0 379 0 563 0 710 0 853 0 992

Bartlett 0 115 0 163 0 200 0 231 0 258 0 365 0 516 0 632 0 730 0 816

Sampling error in 500 trials is about 0 02

The fourth row gives the asymptotic standard error from the Bartlett

formula 3 12 These are close to the Monte Carlo values and best for

When applying the standard errors we could choose LOuse the 4 at each k

to scale the standard errors or to choose one qk the Lkat the largest k as

the estimate of J for all k We followed the latter choice in the applications

Since we will not use the covariance of k differences or estimates which are

combinations of variances of k differences we do not report the correspond

ing Monte Carlo results

3 3 Optimal combinations Bias

We have three estimates of the same quantity l k times the variance of k

differences of each series and l k times the covariance of k differences

Which one or which combination should we use

The asymptotic distribution theory is of no help here because asymptoti

cally all three estimates are identical Eq 2 14 shows that when T and k are

large l k times the sample variance of k differences of each series and l k

times their sample covariance all approach l k times the sample variance of k

differences of z Hence the asymptotic variance covariance matrix of the

three estimates

Vcovk var w wlek 3 13

co Lk W Wl k

is a constant var 1 L z times a 3 X 3 matrix of ones

For finite differences k and a finite sample T the stationary components c

in 2 14 still enter the variance of k differences so the variance covariance

matrix of the sample variance and covariance of k differences is not singular