Multivariate Estimates Of The Permanent Components Of Gnp -Books Download

MULTIVARIATE ESTIMATES OF THE PERMANENT COMPONENTS OF GNP

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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


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