Milton Friedman

English

Research Papers in Economics

This PDF is a selection from an out-of-print volume from the National Bureau
of Economic Research
Volume Title: The Interpolation of Time Series by Related Series
Volume Author/Editor: Milton Friedman
Volume Publisher: UMI
Volume ISBN: 0-87014-422-7
Volume URL: http://www.nber.org/books/frie62-1
Publication Date: 1962
Chapter Title: CONCLUSIONS
Chapter Author: Milton Friedman
Chapter URL: http://www.nber.org/chapters/c2067
Chapter pages in book: (p. 23 - 29)relatives are used to compute deviations from trend. Similarly, instead of
(43), one could use
=logx —log[(1 — (44)
which combines logarithmic and arithmetic transformations.
Condition (3)—facilitating the accurate estimation of the required param-
eters—becomes a somewhat independent condition when the parameters are
estimated from "test" series. It then dominates the transformations to be ap-
plied to these series. This problem is only touched on in the present paper and
needs much further attention.
V. CONCLUSIONS
This paper deals with the problem of estimating intermediate values of a
time series. This can be done by mathematical interpolation using only the
known values of the given time series or by using one or more related series
whose values are known for the desired time intervals and whose movements
are supposed to be correlated with the movements of the series to be inter-
polated. These notes are restricted to the simplest form of the problem: that
in which only the known values immediately preceding and following the value
to be interpolated are used explicitly in mathematical interpolation and in
which only one related series is used. This simple case probably covers the great
bulk of the interpolation performed in practice.
The major conclusions of our analysis can be summarized as follows:
1. Mathematical interpolation and interpolation by related series are not
substitute methods; rather they are complementary. In the words of our prac-
tical maxim I: First interpolate mathematically. In general, getting as good an
estimate by mathematical interpolation as possible will involve transforming
the data into a form in which straight-line interpolation can be expected to
give unbiased estimates of the unknown values or of seasonally adjusted un-
known values if interpolation is done for intervals that are fractions of a year.
In the words of our practical maxim II: Carry out interpolation by related series
with seasonally data. If the final series is desired in seasonally unad-
justed form, the seasonal should be estimated and combined with the values
obtained by straight-line interpolation, even if the seasonal is estimated from
the related series. This gives a first approximation to the unknown values. Call
it the "trend value."
2. A related series can then be used to improve this approximation by pro-
viding an estimate of the deviation of the unknown value from it.
3. For this purpose, trend values of the related series should be obtained by
mathematical interpolation as in point (1), including the seasonal component,
if any.
4. The related series and its trend values should be expressed in a form
(logarithmic, relative to trend, etc.) for which the deviations of the trans-
formed series from the similarly transformed trend values can be expected to
be homogeneous over time and linearly related to the deviations, correspond-
•ingly transformed, of the original series.
235. An estimate should be made from "test" series or otherwise of the stand-
ard deviation of these deviations for the related series (0-0)andalso for the series
to be interpolated and of the correlation coefficient of the two sets of devi-
ations (puv).
6.The size of the correlation coefficient determines the extent of improve-
ment that can be attained by use of the related series. Some improvement is
possible as long as it is not zero.
7. Compute an estimate of=puv0-u/from the estimates in point (5). Call
this estimate &Taketimes the deviation of the related series asan estimate
of the deviation of the series being interpolated. This is, method Mg.
8. Methods widely used transfer the deviation of the related series in full to
the series being interpolated. This is equivalent to using a value of1, hence
is designated method M1. Method M1 will worsen rather than improve matters
unless puv0-u/0-vgreater than 1/2 and will seldom yield smaller errors on the
average than It may nonetheless be worth using on occasions simply
because it requires less information and hence is
9. In practice, M1 has been used so widely and uncritically that it must have
often yielded poorer results than mathematical interpolation alone and may
well have done so more frequently than it has yielded better results. One reason
for such an outcome is that the relevant criterion in choosing related series and
in judging the extent of improvement possible through their use. is the size of
Puv• This correlation may be small even though the correlation between the
original and the related series afr known dates is high, yet the latter is often the•
criterion implicitly or explicitly used to chooseand judge interpolators.
10. A point that is important for practical work arises when only one com-
ponent of a broader total is unknown for the desired dates. In such a case, in
the words of our practical maxim III: Perform interpolation only on the part of
a series that is unknown for the dates for which interpolation is to be done; never
on a broader total, part of which is known for those dates:
APPENDIX NOTES
1. Relation between parameters of the distributions of the original and
formed variables. Letandrepresent the mean and standard deviation of the
variable to be designated by a subscript and p the correlation coefficient be-
tween the two variables designated by subscripts. Further, consider the more
general transformation
=— [(1— + WjX2J (i)
=— [(1—w2)yo+ w2y2], (ii)
whereis the relative weight attached to the terminal value in computing the
straight-line trend. This transformation covers explicitly not only the case in
the text but also both nonequally spaced intervals and more than one inter-
mediate value to be interpolated. The relation between the parameters of the
(ui,universe and those of the (x0, x2, Yo,Y2) universe is then as follows:
= — (1— — (iii)
242 2 22 22
=+ (1 — + —2(1—
(iv)
— + 2w2(1 —
withandgiven by the same formulas except that y replaces x in all sub-
scripts;
2 2
+ (1 + Wipx2y20z20y2
— (1—w2) [p +
(v)
— +
+ — [Px0y20100y2 +
If we suppose on grounds of symmetry that
= (vi)
— = (vii)
Px0y0 =Pxaj1Px2y2 = (viii)
then (iv) and (v) reduce to:
= —w2+ u —(1— — + w1(1 — (ix)
and
= —+ —(1— +
• (x)
— —I— + — px2yo) 1.
Theseformulas may suggest the simplicity of exposition gained by the simple
transformation (7).
2. Straight-line interpolation in the light of statistical considerations. The use
of straight-line interpolation in the transformation (i) and (ii) is not at all a
self-evident step .requiring no justification, once the problem is viewed sta-
tistically. From that point of view, the question is how to form the best estimate
offrom x0 and x2 alone. The answer depends very much on what is assumed
known about the parameters of the multivariate distribution of the x's. For
simplicity, accept the symmetry assumption (vi), return to the case in the text
of three equally spaced values of X, and for this case adopt the additional sym-
metry assumption
= (xi)
The least squares estimate of x1 from x0 and x2 is then given by
* Pzozi Pxoxi =+ (Xo—+ (x2— (xii)
1 + Px0x2 1 + Pxoz2
Ifwe suppose that theare unknown but equal, then the best estimate of all
25thefrom x0 and x2 alone is the mean of x0 and x2. Inserting these estimates
for thein (xii) gives
*xo+x2 = , (xiii)
or the straight-line estimate. Similarly if the means are supposed unequal but
linearly related to time, the same result follows.
Suppose, however, that the x's have already been adjusted for any longer
period trend or cycle movements so that all the means can be taken as zero.
Equation (xii) would then reduce to (xiii) only if
1
= (1+ pzox2). (xiv)
It is perhaps clear intuitively why, under these highly special assumptions'
straight-line interpolation may not be justified. For example, suppose x0, xi'
and x2 are independent, uncorrelated observations with zero means. Then x0
and x2 provide no information relevant to estimating x1 and the best estimate
of x1 is zero.
3. Relation betweenand multiple regression method of estimation. William
Kruskal has derived the necessary and sufficient conditions for the estimate of
x1 obtained from the multiple regression of x1 on x0, x2, yo, yi, and Y2tobe equal
to that given by when all parameters are assumed known. Letstand for
an arbitrary one of the five independent variables and coy () forthe covari-
ance of the variables in the brackets. Then the necessary and sufficient con di-
tion derived by Kruskal is that there exist two numbers A and B not both
zero for which
A coyu)=Bcoy v)
for E equal successively to x0, x2, Yo, y',andY2.Iam greatly indebted to Kruskal
for this proof.
A particular set of conditions satisfying this requirement, and which are
therefore sufficient but more stringent than necesary, is the following:
1
= +0x2px0x2) (xv)
1
= +oxopxox2) (xvi)
= (ox2px2y2+ oxopxoy2) (xvii)
1= (0x0pz0y0+ 0z2px2y0), (xviii)
plus four additional conditions obtained from these by replacing the x's
throughout by the corresponding y's and the y's by the corresponding x's.
The meaning of these conditions is that they assume that u and v are each
Uncorrelated with x0, x2, yo, and Y2,hencethat the terminal values contain no
information relevant to the prediction of u from v.
26If we accept the symmetry assumptions (vi), (vii), and (viii), then these
conditions reduce to the following conditions on the correlation coefficients:
1
Pxox4 =Pxixz = (1 + pxox2) (xix)
1
=py11j2 = (1+ py0,,2) (xx)
.1
pxoyi= = + pxoy2) (xxi)
1
PxujoPz2yi = (Pxu+ pxwo). (xxii)
Condition (xix) is, of course, the same as conditions (xi) and (xiv) since the
problem in Appendix Note 2 is a component of the problem considered in this
note.
4. Preliminary observations on the problem of distribution. Perhaps the sim-
plest form of the problem of distribution that still preserves its essential fea-
tures is the conversion of annual data to semi-annual data. To render this
analogous to the problem of interpolation, we must consider three years for
which we have annual totals for the series to be interpolated and semi-annual
figures for the related series. The problem is to distribute the annual total of
X for the middle year on the basis of the other data. We must use three years
to avoid arbitrariness; if data for the end year contain information for the
distribution of the total for the middle year, then data for the initial year must
contain the same kind of information.
Let the (unknown) semi-annual values of X be given by x1, x2, x3, X4,x5,x6,
the (known) semi-annual values of Y by y1, Y2,y3, y6; the annual values of
the two series by
=x1+ x2,
'71Y1+Y2,
fl3Y3+Y4,
'15= Us+
The analogue to the correlation method for interpolation is first to get esti-
mates of x3 and x4 from andand then to correct these estimates on the
basis of the deviations of y3 and Y4fromsimilar estimates based on
Inorder to assure satisfaction of the constraint
=x3+ x4, (xxiii)
it will be simplest to work with
X4 — x3, (xxiv)
27rather than with x4 and x3 themselves. Given an estimate of the simulta-
neous solution of (xxiii) and (xxiv) will give the desired estimates.
The transformation of variables analogous to (7) is then
1
u = — — —Ei)
1
(xxv)
V = — — —
Therest of the interpolation analysis then carries over directly to these
transformed variables. In particular, of course, the three practical maxims
apply in full.16
As for interpolation, the justification for using as an estimate of
is not self-evident. The relevant analysis is similar to that in Appendix
Note 2. For simplicity, let us again make symmetry assumptions that the x's
all have the same standard deviation and that the serial correlations of x
depend only on the interval between the items correlated and not on which par-
ticular items are correlated. Letequalthe correlation between, values of x
separated by i time units (i.e., by i half-years). All the information provided
by the annual totals on is clearly contained in
= — Ei
(xxvi)
= —
Themultiple regression of on and reduces, under our assump-
tions, to
* P1P3
= + — •(xxvii)
2 + 2Pi— Pa — 2P4 — P5
If,by analogy with the interpolation problem, we suppose that the mean dif-
ferences between successive semi-annual observations are unknown but equal,
then the best estimate from the annual observations alone of = and
of = SO (xxvii) reduces to
* 1
= — —Ei), (xxviii)
which is identical with the negative of the second term on the right-hand side
of (xxv).
If we suppose that the annual totals have already been adjusted for any
n sub—periods, with n >2, the situation though similar is more complex. Given the constraint equivalent
to (Xxiii), there are only n—i independent variables but there is no particular way of transforming the n variables
into n—i that has the direct appeal of (xxiv). However, it does not much matter which way it is done,long as
both X and Y are treated alike. Perhaps the 8implest, and the moat like (xxiv), is to use the n —1 first differences
between contiguous observations.
28longer-period trend or cycle movements so that all the means and differences
can be taken as zero, then (xxvii) would reduce to (xxviii) only if
—7p3 + = 2. (xxvix)
Interestingly enough, if the pattern of the p'sisthe natural extension of
(xiv), i.e., each correlation coefficient is the arithmetic average of the preced-
ing and following one,'7 the coefficient of in (xxvii) turns out to beinde-
pendently of the value of
Thispattern could not of course continue indefinitely, since it would ultimately lead to correlation coefficients
less than —1.
29

The Interpolation of Time Series by Related Series

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