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Conditioning in models with logs
Journal
of Econometrics
38 (1988) 127-143.
North-Holland
CONDITIONING IN MODELS WITH LOGS* David A. BELSLEY Boston College, Chestnut Hdi, MA 02167, USA CCREMS, MIT, Cambridge, MA 02139, USA The conditioning diagnostics of Belsley, Kuh and Welsch (1980) are applicable to models that are linear in parameters and (structurally interpretable) variates. Different diagnostics are required when models contain different types of non-linearities. It is shown here, however, that the BKW diagnostics are also applicable to models containing logarithmic terms once the variates are first e-scaled, i.e., scaled to have a geometric mean equal to the base e of the natural logarithms,
1. Introduction The conditioning diagnostics of Belsley, Kuh and Welsch apply to the OLS estimator of the strictly linear, basic model y = /3,x,
+ . . . +/3,x,
+ E.
(1980) (BKW)
(1)
Strictly linear means it is linear in both parameters and variables, and basic means its variates X, E .CZJ?~ are structurally interpretable; that is, the model is parameterized so that its variates are in a form in which numerical changes in their values can be assessed as being unimportant or inconsequential relative to the real-life situation being modeled.’ In this case, a large scaled (for equal column length) condition number K(X) of the n X p data matrix X = [X1... XP] is meaningfully interpretable as the potential magnification factor by which small relative changes in the X, could be transformed into large relative changes in the OLS estimates b = (XTX)-lXTy, and thus K(X) becomes a meaningful measure of conditioning in this case. In the event, however, that the Xi are not themselves structurally interpretable, but rather are non-linear transformations such as X, = ln(Z,) or X, = 2,2, *Special thanks go to the late Edwin Kuh for comments and to Ana Aizcorbe and Sean Doyle for research assistance. All computation was done on the TROLL system at MIT. This research was funded in part by a Mellon Grant to Boston College and in part by the National Science Foundation under Grant No. IST-8420614. ‘This concept is defined in Belsley (1984,1986) and Belsley and Oldford (1986). Since the condition number in this context measures the potential sensitivity of the OLS estimator to small relative changes in the data, it is clear that this measure can only be meaningfully interpreted when the relative data change it is based on can itself be meaningfully interpreted; that is when the data are structurally interpretable.
0304-4076/88/$3.50~1988,
Elsevier Science Publishers
B.V. (North-Holland)
128
D.A. Belslq, Conditioning in models with logs
of structurally interpretable Zi, then the value K(X) will typically provide little meaningful diagnostic information about the true conditioning of the OLS estimator. These issues are developed in Belsley and Oldford (1986) where a general conditioning analysis is defined and the problem of assessing the conditioning of non-linear models is addressed. There we learn the following: (a)
(b)
(c)
There is no single diagnostic procedure for carrying out a general conditioning analysis; each form of non-linearity presents its own concerns. Unlike estimation, where ‘non-linearity’ often refers only to non-linearities in parameters, conditioning analyses are affected by non-linearities in both the parameters and the variates. As already indicated above, for any conditioning analysis to provide meaningful diagnostic numbers, it must be applied to data that are structurally interpretable.
There is, however, one form of non-linearity in the variates that lends itself to a reasonably straightforward conditioning analysis: the logarithmic transform - a fortunate case, since logarithmic transformations are frequently encountered in models used in the natural and the social sciences. In this paper we find that the procedures of BKW remain relevant once each structurally interpretable 2, entering in a logged term is first e-scaled, that is, scaled so that the geometric mean g,, of its elements is equal to e, the base of the natural logs. Where needed for distinction, the term L-scaled indicates scaling for equal Euclidean length, usually effected by scaling for unit length. In sections 2-5 we view the various issues involved and provide a solution. Section 6 provides several illustrations. 2. A view of the issues Assume for the moment that the X, in (1) are the basic, structurally interpretable variates. Models that differ from (1) only in scale (the units in which the X, are measured) are clearly structurally equivalent, but, as noted in BKW (p. 120), their respective data matrices can have very different condition numbers. Without a canonical scaling, then, the condition number K(X) provides little meaningful diagnostic information, since different values can apply to the same ‘real-life’ situation. In the linear-model context of BKW, this problem can be shown to be resolved through column equilibration, L-scaling each column of X to have the same, usually unit, Euclidean length. A similar problem arises when some of the X, in (1) are logs of structurally interpretable variates2 but the resolution is different. Consider the model Y =
PI&+ P21n(Z2) + &ln(-%>+ 6
(2)
‘The term ‘log’ is used here generally to mean logarithm. The discussion takes place in terms of natural logs, denoted ‘In’, but the results are readily generalized below to apply to logs of any base r.
D.A. Belsley, Conditioning in models with logs
129
I Qn(Z;)
= Qrl(c*)r +
,( =Qn(c,)l+ Qn(Z,) :I
:
:: : :
Fig. 1
where Z, and Z, are structurally Resealing the Z, as
z,* = cizi, merely renames (2) is structurally
interpretable
and
i= 1,2,
the ‘real-life’ phenomenon equivalent to
y = &L + &ln(Z,*)
+ &ln(Z,*)
1 is the vector
of ones.
(3)
represented
+ e,
by the Z,, and hence
(4)
where fir = /3r - ln(c,) - ln(c,). But the conditioning of the ln(Z,*) in (4) can differ greatly from that of the ln(Z,) in (2) even though the correlation between the Z, is invariant to such scaling. Hence a canonical scaling is required for meaningful diagnostics based on the ln(Z,). To see the effect of scaling on the conditioning of log variates, consider fig. 1. Here L represents the n-vector of ones (1. ..l)‘, Z, = (zl.. . z,,)~ the n values zi of a structurally interpretable variate Z,, and ln(Z,) the n-vector
130
D.A. Belsley, Conditioning
in models with logs
(ln(zi). . .ln(z,))r. Resealing 2, to Zi* = c,Z, adds to ln(Z,) the vector ln(c,)r and clearly has the effect of ‘sliding’ ln(Z,) along the affine subspace Hi containing ln(Z,) and parallel to L. Thus, any point along H, is structurally equivalent to ln(Z,), i.e., results simply from renaming Z,. Similar considerations apply to Z, and ln(Z,). Let us begin, then, as in fig. 1, with ln(Z,) and ln(Z,) at right angles, so these two variates are perfectly conditioned. Simply by renaming (scaling) the Z,, we can slide them both upward, reducing the angle (Ybetween them and thus arbitrarily worsening their conditioning. This result clearly holds whether or not the three vectors are coplanar as in fig. 1. The question thus arises as to the scaling for the Z, that produces ln(Z,*) whose conditioning provides meaningful diagnostics. 3. Measuring conditioning of models with logs An answer to this question depends on what we wish a conditioning diagnostic to measure. A general definition of ill conditioning is given in Belsley and Oldford (1986) and can be summarized as:
Dejinition.
A conditioned if understanding small can give
model, equation, or estimator w =f( u) [w E .B”“, u E .9fn] is ill relative changes 6~ in u that can be interpreted (from an of the phenomenon being modelled) to be inconsequentially rise to unacceptably large relative changes 6 w in w.
As noted, in the strictly linear, basic model (l), the L-scaled condition number K(X) of the matrix of structurally interpretable X, provides just such a measure for the OLS estimator. That is, a K(X) of 100 indicates that a 1 percent relative change in the data could produce up to a 100 percent relative change in the OLS estimates.3 If the X, are ln(Z,), the same considerations necessarily apply to the ln(Z,); that is, K(X) measures the potential relative sensitivity in b with respect to small relative changes in the ln(Z,), but not with respect to small relative changes in the Z,. And, if it is the Z, that are structurally interpretable, it is this latter measure that is desired. A solution is suggested by recalling that for a scalar w (and natural logs) din(w) PC ln(4
1
dw (5)
In(w)7
Here we see that relative changes in In(w) are the same as relative changes in w only if w scaled to the base e so that In(w) = 1. Of course, the X, and Z, are vectors and not scalars; so taking a hint from the preceding, it seems ‘Measured
as a vector norm, such as IlSbll/llbll.
See Belsley (1982) and BKW (1980).
D.A. Belsley, Conditioning
131
in models with logs
reasonable to scale the 2, so that, at some representative value z”* for the elements of the scaled vector Z,* = cIZ,, one has ln( z”*) = 1. This occurs if one chooses
(6)
c, = egzyl,
where
g,, is the geometric
g,, =
mean of Z, = (zlr . . . z,,)~, that is
(nz,;1l’Y
(7)
J
Then the geometric mean gz: of Z,* = eg, ‘Z, is e, and, at the representative unscaled value .? = e-‘g,,z”* = gz,, we value t”* = g,: = e and its corresponding have dln(Y*) ln(z*)
=
d,Y* t”* = 7’
(8)
The first equality holds since ln(gz:) = 1, and the second because scaling does not affect relative values. The preceding readily generalizes to accommodate logs to any base r. In this case, at the representative value 2*, (5) becomes dlog,(
2*)
log,( 5*)
log,(e) = log,@*)
d2* ,?* ’
(5’)
and the desired scaling is clearly that which sets log,(e) = log,(z”*) or, again, where z”* = g,: = e. For any base, then, the same canonical scaling is indicated as appropriate for assessing the conditioning of the logged variates Z,: e-scaling, or that which makes the geometric mean g zY of the scaled data series Z,* equal to the base e of the natural logarithms. For this scaling, small relative changes in the variates ln(Z,*) are approximately the same as those in the Z,* and hence as those in the structurally interpretable variates Z,.
4. A canonical
scaling for logs
Assume, then, some or all of indicates that, if are first e-scaled shall denote X*,
it is desired to assess the conditioning whose columns are in logged form. those structurally interpretable variates (the others left unchanged), resulting in the conditioning diagnostics of BKW
of a data matrix X, The preceding result entering in log form a data matrix that we apply directly to X*.
132
D.A. Belsley, Conditioning in models with logs
That is, it is suggested using the L-scaled condition number of X* to measure the potential relative change in the OLS estimator b that could result from a small relative change in the structurally interpretable 2, that enter the model in logged form. Various cases can arise depending upon whether X contains all logged variates or is a mixture of logged and non-logged variates. X=[X,... Case 1: Inthedatamatrix X,], some of the X, may be In(Z), but for each variate it is Xi and not Zi that is structurally interpretable. In this case, no e-scaling is necessary; the techniques of BKW can be applied directly to the X,. That is, all Xi should be L-scaled (which occurs automatically as the first step of the BKW procedure), and it is the L-scaled condition number K(X) and its variance-decomposition proportions that are the desired diagnostic measures for the conditioning analysis. Case 2: The Xi are log transforms ln(Zj) of structurally interpretable Z,. First, e-scale the Z, to Z,* having geometric mean g,: = e, and then use the techniques of BKW to examine the L-scaled condition number K(X*) of X* = [ln( Z,* ) . . . ln(Z,* )] and its variance-decomposition proportions. Thus, the Zj are first e-scaled to Zi* and then the ln(Zi*) are L-scaled for equal (unit) length. This later transformation does not alter the relative magnitudes of the d ln(Z,* )/ln( Z,* ) but is needed for the basic equilibration reasons given in appendix 3A of BKW. Case 3 (the general case): Some X, (which may be logged variates) are themselves structurally interpretable while others are ln(Z,) for which it is the Z, that are structurally interpretable. Here we have a mixture of the previous two cases. First, e-scale only the structurally interpretable Zi variates to Z,* as in case 1, then examine the L-scaled condition number of the matrix X* whose columns are the union of the structurally interpretable X, and the ln( Z,*). This resulting matrix is the proper input to the collinearity diagnostics of BKW, and its condition indexes and associated variance-decomposition proportions are the desired conditioning diagnostics. This diagnostic procedure is illustrated in section 7, but, in order to assess its performance, it is first necessary to examine the magnitude of the condition number that can be anticipated from its use. 5. The magnitude of K(X*)
Consider the two alternative specifications: (hereafter known simply as the linear model),
the model linear in the x’s
and the model linear in the logged x’s (hereafter known simply as the logged model)
(10)
133
D.A. Belsley Conditioning in models with logs
These models are clearly quite similar and often are considered as alternative specifications for the same real-life economic phenomenon. Furthermore, since with e-scaling a given relative shift in the x,, in (9) produces roughly the same relative shift in the ln(xit) in (lo), we should be very surprised if the condition number of the e-scaled logged data from (10) were not similar to that of the non-logged data from (9). This prospect can be informally motivated as follows. The Bi in the logged model (10) can be interpreted as the elasticities of y, with respect to the xir, while the /3, in the linear model (9) can be interpreted as the partial derivatives 8yJaxi,. Let ii be the OLS estimates of the elasticities 6, and bi be those of the partials /I;. Then we know that the condition number of the logged data bounds the relative change aii/b, in 6, with respect to a given relative change ~3ln(x,,)/ln(xi,) in ln(x,,), while the condition number of the basic data bounds the change dbi/b, in b, with respect to a given relative shift axiJxi, in xii. Of course, with e-scaling 8 ln(x,,)/ln(x,) = 8xi,/xir, so it is needed only to compare dbi/gi and db,/b, for a given axi,/xi,. This can be done roughly by converting bj into an * elasticity estimate like ii, i.e., by considering b, = b,(x,,/y,). Now, for a given relative change in xit (y, held constant), we would expect the potential sensitivity of the two elasticity estimates to be similar; that is, we would expect aAi/hi and a&,/6, to have a similar bound. But,
ah ab;(xit/Yt) (Xitabi+ b,ax,,) x=
bi(xit/yt)
=
bixit
r3bi
=b+-’
I
ax,, xit
(11)
so, when the data are ill conditioned and dbj/bi x=-axiJxi, by definition, a&,/h; = abi/bi. Under e-scaling, then, we expect 86,/g, and abi/b, to have similar bounds, and hence we anticipate that the condition number of the logged e-scaled data to be similar in magnitude to that of the basic data. It remains for the illustrations of section 7 to show that this indeed occurs in practice. 6. A special case of interest The treatment of one special case deserves attention: that of ln(2,) and ln(Z,‘). With a little thought beforehand, these two variates are not likely to occur together in the same model since ln( 2,‘) = 2 ln( Z,), and they are exactly collinear. The situation could arise, however, if one logged variate happened to be very closely the square of another, or, as could all too easily happen, if a given equation containing both Z, and Z,2 were mechanically ‘logged’ in order to test whether the logged or non-logged specification seemed better. The issues involved in assessing this situation are most easily made clear geometrically.
134
D.A. Belsley, Conditioning in models with logs
I
QnE’)=!Qn(cZ)
Q
I?iii(Z)Qn(2) I
I
0
-
man(z)l
Fig. 2
Consider first the geometric effect of e-scaling. In fig. 2, the 1 vector is oriented vertically, and In(Z) = (ln(z,). . . ln(z,))r is any n-vector of logs. As is well known, the arithmetic mean of any vector is determined by its orthogonal projection into the space of I. The point A, then, is rnlnCZl~, where m mean of the elements of In(Z). And, fr~.q=e~;a~~j~~~~Am th e arithmetic In(z) = ln(JJz;) ‘in = ln(g,), we see A is also ln(gZ)L. Thus, for any vector ln( W) E H$,, the affine space orthogonal to 1, ln( g,) = 1, and the elements of W have geometric mean g, = e. Now, recalling that, as we scale Z we slide In(Z) vertically along PC,, we see that the scaling c that produces a vector Z* = cZ with geometric mean e is that which places ln(Z*) at the intersection of H, and Nz.4 Consider next, then, applying the preceding to the model y = &L + &ln(Z)
+ &ln(Z2)
+ E,
(12)
containing both In(Z) and ln(Z’) = (ln(zf), . . ln(zi))r= 2ln(Z). These two vectors are plotted along with 1 in fig. 3. It is clear that In(Z) and ln(Z*) are perfectly collinear and that such a specification would make little sense. But note that, after Z is e-scaled to Z*, the resulting ln(Z*) and ln(Z**) are no longer collinear. It would seem that such scaling would remove our ability correctly to diagnose collinearity in the original data. This is not so, however, 4Computational note: the scaled vector Z* is most easily calculated by noting that Z* = iii( Z ) + L, where iii(Z) -In(Z) -m ,n(Zj~, That is, In(Z) is first put into deviation-from-mean form G(Z), and then 1 is added to each element. In the event that logs to base r are being used, one would instead add log,(e) to each element to give log,(Z) + log,( e)c.
D.A. Belsley, Conditioning in models with logs
Qtl(2’)
135
Qn(Z*')
Oi Fig. 3
since the three vectors 1, ln(Z*) and ln(Z * * ) remain perfectly collinear (coplanar), and the perfect ill conditioning of X = [ 1, ln(Z ), ln(Z *)I also necessarily exists in the scaled matrix X* = [I, ln( Z*), ln(Z * * )] but in a different form. The preceding argument depends on the presence of the constant vector L in the model (12) (i.e., pi # 0) and seems therefore to lack generality, but this is not the case. It is an occasionally unappreciated fact about models containing logged terms that they must always include an intercept. This is because there is no natural unit for measuring any variable - I am always free to ‘deci’ any units you prefer. Thus, if one postulates the homogeneous model y = fi,ln(Z> a simple change non-homogeneous
+ P@(w)
+ e,
of units Z* = CZ (c # 1) produces model
y = pit + &ln(Z*)
+ &ln( W) + E,
(13) the structurally
equivalent
(14)
with pi = - ln( c) # 0. Thus, the homogeneous form (13) requires an untenable assumption of unique natural units for the variates. A model with log variates, then, must always include an intercept, and, in assessing the conditioning of the data relevant to such a model, the data matrix being assessed must always include 1, the constant column of ones. Because of this, the method of e-scaling proposed above will never fail to detect collinearity between In(Z) and ln(Z ‘), since it will always materialize as collinearity among L, ln(Z*) and ln(Z2*).
136
D.A. Belsley, Conditioning in models with logs
7. Illustrations
The e-scaling introduced above is chosen to provide conditioning diagnostics for models with logs whose magnitudes are able to support an interpretation similar to those of BKW for strictly linear models, namely, a measure of the relative sensitivity of the resulting OLS estimator b to a given relative change in the structurally interpretable variates Z, [not the ln(Z,)]. The efficacy of e-scaling in achieving this goal is illustrated through analyses of three different economic data sets corresponding to a consumption function, a commercial and industrial loans function, and a labor-force participation function. The conditioning of the data is assessed and reported first (case I) as if they came from the strictly linear model y=p11+p2x*+
...
+&Xp+&,
(15)
and second (case II) as if they came from a model linear in the logs ln(y)=&b+&1n(X2)+
... +/?Jn(X,)+e.
(16)
In this latter instance, the results are reported both with e-scaling (case IIa) and without e-scaling (case IIb) in order to demonstrate that the e-scaling is necessary to obtain stable and comparable diagnostic results. Furthermore, assessment is made and reported (case III) assuming the model to possess various mixtures of logged and non-logged terms. In each instance, it is the X, and not the ln(X;) that are assumed structurally interpretable. The structurally interpretable consumption-function data derive from the model [written in its basic linear form (15)] C(T) = pi1 + &C( T- 1) + &DPZ( T) + &r(T) +&ADPZ(T)
+ E,
(17)
where I = constant vector, C = total consumption, 1958 dollars, DPZ = disposable personal income, 1958 dollars, r = interest rate (Moody’s Aaa). These data are essentially those described in chapter 3 of BKW, differing only in the deletion of the last year which had a negative value of ADPZ. All series are annual, 1948-1973. The commercial- and industrial-loans data are derived from MCL(T)=&c+&MCL(T-l)+P,YGPPZ(T) +&[RCL(T)
-RTB(T)]
+E,
08)
137
D.A. Belsley, Conditioning in models with logs
where constant vector,
1
=
MCL
= commercial and industrial loans at all commercial banks,
YGZW = gross private product, = average bank rate on short-term commercial and industrial loans, = average yield on three-month U.S. Treasury bills.
RCL RTB
These data, from Kuh and Schmalensee (1972), are quarterly 1959-11 to 1969-IV.
series from
The secondary labor-force participation data derive from LCSLSPOP(T)
=&r+&ER(T) +&TIME(T)
+&LCSLSPOP(T-1) + E,
(19)
where
LCSLSPOP TIME
= constant vector, = ratio of civilian employment to total population of working age, = ratio of secondary civilian labor force to secondary population, = a time dummy.
These data, also from Kuh and Schmalensee, are annual series, 1948-1962. No effort is made here to explain the meaning of the basic, logged, or mixed-logged regression models behind these data sets. The purpose of this exercise is only to examine the efficacy of the proposed conditioning diagnostics in correctly assessing the presence of near dependencies in the data. The condition indexes for cases I, II (a and b), and III are summarized for each data set in table 1. Most of the comparative information needed for this study can be seen from these results alone. The detailed variance-decomposition proportions matrices are provided in an appendix for those interested in a more complete analysis of how the different transformations and scalings affect diagnosed variate involvement. The first important observation comes from comparing columns II(a) and II(b). It is clear that e-scaling has a major impact on the magnitudes of the condition indexes of the logged variates. It is further seen that e-scaling can cause these magnitudes either to decrease (as for the first two data sets) or increase (as for the last). That this can happen is readily predictable from fig. 1. The second important observation comes from comparing columns I and II(a). Here we see that e-scaling, consonant with section 5, always brings the condition indexes of the logged data into line with those of the basic data and
138
D.A. Belsley, Condrtioning in models with logs Table 1 Comparative
condition
indexes.
Case II
Case I
Basic data
(a) With e-scaling
1 6 10 40 361
1 4 9 40 339
Case III (b) Without e-scaling
Consumption-function
Representative mixtures data
1 9 14 232 1900
1 4 9 32 113
1 6 10 46 381
1 5 15 101
1 5 15 141
1 9 194 331
1 12 146 351
Commercial- and industrial-loans data 1 5 16 146
1 5 19 164
1 4 43 852 Secondary labor-force participation
1 9 194 333
1 12 154 346
1 21 156 238
data
so achieves the desired end of providing a stable diagnostic measure that meaningfully measures the potential sensitivity of a relative change in the OLS estimate to a relative shift in the basic, structurally interpretable data as opposed to the logged data. We also see that failure to e-scale logged data typically results in grossly distorted diagnostic magnitudes which can be either too high or too low. Although examples of both cases occur here, our experience indicates the behavior of the first two data sets to be most representative; that is, failure to e-scale logged data most often produces highly inflated condition indexes. Comparison of the mixture cases with column I indicates that e-scaling of logged variates, even when intermixed with non-logged variates, also tends strongly to produce diagnostic measures with magnitudes very comparable to those of the basic data. The first mixture in the consumption-function data deserves special mention here since these magnitudes seem to be quite different. Examination of table A.III(a) in the appendix, however, reveals in this case that, of the three variates C, DPI and ADPI involved in the strongest basic near dependency (the one corresponding to 367 in table A.I), only the last two have been logged. This differential treatment of the variates involved in a linear dependency will, of course, have a tendency to alter it - in this case weaken it so
D.A. Belsley, Conditioning
139
in models with logs
that its condition index falls from 367 to 113. In the second mixture among the consumption-function data, seen in table A.III(b), only the variate r was logged, leaving the strongest near dependency among the other variates untouched, and here we see the e-scaled diagnostics are telling a virtually identical story as in the basic case. E-scaling in the mixture case, then, seems to be working in accord with intuition and expectations. As to the effect of e-scaling on the diagnostic of variate involvement, it is readily seen from the variance-decomposition proportions matrices in the appendix that the e-scaled logged data show similar patterns of variancedecomposition proportions as those for the basic data. Failure to e-scale can alter these patterns to produce misleading results as can be seen from comparing the role of the constant term in the consumption-function data (tables A.11 and AI), the role of ln(RCL-RTB) in the commercialand industrial-loans data (tables B.11 and B.I), and the role of TIME in the secondary labor-force participation data (tables C.11 and C.1). 8. Conclusion Our goal was to find a canonical scaling for logged data able to produce stable conditioning diagnostics having the same interpretative basis and meaning as those of BKW applied to strictly linear, basic models. E-scaling - scaling structurally interpretable variates that appear in logged terms so that their geometric mean is e, the base of natural logarithms - is seen precisely to serve this end. The e-scaled logged variates along with the non-logged variates in the equation provide the proper input to the conditioning diagnostics of BKW, the first step of which is to L-scale (for unit length) all variates. E-scaling should be applied only to structurally interpretable variables Z, that enter the equation as ln(Z,). If the variate X, = ln(Z,) is itself the structurally interpretable variate, it should be treated normally in the BKW diagnostics. Appendix: Variance-decomposition
proportions matrices
In this appendix, the variance-decomposition proportions matrices for each case for each data set. The table numbers are keyed by: A. B. C.
Consumption-function data, Commercialand industrial-loans Labor-force participation data.
I. II. III.
Basic data, Logged data, (a) with e-scaling and (b) without Mixtures, logged and non-logged.
In the various mixture the column headings.
are given
data,
cases, the logged and non-logged
e-scaling, variates
are clear from
140
A.I.
D.A. Belsley, Conditioning in models with logs
The consumption-function data Table A.1 Basic consumption-function
data; variance-decomposition
v=(h)
Condition index
CONST
1 6 10 40 361
W b2) C(T-
0.001 0.072 0.206 0.330 0.391
1)
0.000 0.000 0.000 0.006 0.994
proportions,
W 4)
W b.,1
DPI( T)
r(T)
0.000 O.OOfl 0.009 0.006 0.994
W bs) ADPI
OMKI 0.000 0.033 0.964 0.003
0.001 0.174 0.110 0.150 0.565
Table A.11 Logged
consumption-function
data; variance-decomposition
proportions.
(a) With e-scaling Condition index 1 4 9 40 339
van b, ) CONST 0.001 0.023 0.331 0.265 0.379
W b3)
W b, )
ln[C(T - l)]
ln[ DPI( T)]
O.OOG 0.000 O.OtXl o.OO5 0.995
0.000 0.000 0.000 0.007 0.993
W 41
W 41
WON
ln[ADPI(T)]
0.000 0.000
0.003 0.282 0.098 0.045 0.572
0.020 0.887 0.092
(b) Without e-scaling Condition index 1 9 14 232 1900
van&) CONST 0.000 0.000 0.001 0.944 0.054
van b, ) ln[C(T- l)]
var( b, ) ln[ DPZ( T)]
O.OOG 0.000 0.000 0.005 0.995
0.000 O.ooO O.OOG 0.007 0.993
van b4) ln[r(T)l
W 6,) ln[ADPI(T)]
0.000 0.001 0.073 0.835 0.091
O.cill 0.212 0.150 0.064 0.573
Table A.111 Mixed consumption-function
data; variance-decomposition
proportions.
(a) Condition index 1 4 9 32 113
Mb,) CONST
van b, ) C(T- 1)
0.002 0.033 0.429 0.361 0.175
0.000 0.000 0.001 0.015 0.985
vanhI) CONST
var(bz) C(T- 1)
0.001 0.061 0.200 0.243 0.495
0.000 0.000 0.000 0.006 0.994
van b, ) ln[ DPI(T)]
van b4) r(T)
O.OOG 0.000 0.001 0.074 0.925
O.ooO 0.000 0.027 0.645 0.328
var( b, ) DPI(T)
Mr(T)l
van b, ) ln[ADPI(T)] 0.007 0.606 0.187 0.134 0.067
(b) Condition index 1 6 10 46 381
O.OilO 0.000 O.ooO 0.008 0.992
W b4)
0.000 O.OQO 0.022 0.902 0.076
-~a(b, ) ADPI 0.002 0.185 0.118 0.181 0.515
141
D.A. Belsley, Conditioning in models with logs
A.2.
Commercial- and industrial-loans data Table B.1 Basic loans data variance-decomposition
Condition index 1 5 16 146
proportions
var(&) CONST
var( b, ) MCL(T1)
var( b, ) YGPPZ( T)
0.000 0.000 0.057 0.942
0.000 0.001 0.009 0.989
0.000 0.000 0.000 1.000
var( bz,) RCL(T) -RTR(T) 0.003 0.156 0.539 0.302
Table B.11 Logged loans data; variance-decomposition
proportions.
(a) With e-scaling Condition index 1 5 19 164
var( b, ) CONST
var( b, ) ln[MCL(T-
var( b, ) ln[ YGPPZ( T)]
l)]
0.000 0.001 0.007 0.992
0.000 0.000 0.186 0.814
0.000 0.000 0.000 1.000
(b) Without Condition index 1 4 43 852
var( b, ) CONST 0.000 0.000 0.005 0.995
var( b, ) ln[MCL(T-
0.004 0.273 0.599 0.124
e-scaling var( b, ) ln[ YGPPZ( T)]
l)]
var( b4 ) ln[RCL(T) -RTB(T)]
0.000 0.000 0.013 0.987
var( b4 ) In[RCL(T) -RTB(T)] 0.010 0.544 0.315 0.131
0.000 0.000 0.000 1.000 Table B.111
Mixed loans data; variance-decomposition
proportions
(4 Condition index 1 5 15 101
Condition index 1 5 15 141
var( b,) CONST
var( b, ) MCL(T1)
var( b, ) ln[ YGPPZ( T )]
var( b4 ) RCL(T) - RTB(T)
0.000 0.000 0.060 0.940
0.000 0.004 0.033 0.963
0.000 0.000 0.001 0.999
0.003 0.160 0.507 0.331
var( b,) CONST
var( b, ) MCL(T1)
var( b, ) YGPPZ( T)
var( b4 ) ln[RCL(T) - RTB(T)]
0.000 0.000 0.059 0.941
0.000 0.001 0.009 0.989
0.000 0.000 0.000 1.000
0.003 0.183 0.557 0.257
D.A. Belsley, Conditioning in models with logs
142
A.3. Secondary labor-force participation data Table C.1 Basic labor-force Condition index 1 9 194 333
participation
data; variance-decomposition
proportions.
Mb,) CONST
vaQ 6, ) ER(T)
var( b, ) LCSLSPOP( T - 1)
var( k, ) TIME
0.000 0.000 0.489 0.511
0.000 0.000 0.001 0.999
0.000 0.000 0.502 0.498
0.001 0.118 0.823 0.058
Table C.11 Logged
labor-force
participation
data; variance-decomposition
proportions
(a) With e-scaling Condition index 1 12 1.54 346
var( b, ) CONST 0.000 0.001 0.399 0.600
var( bz ) HER(T)1
W b3 1 ln[LCSLSPOP(T-
0.000 0.000 0.003 0.997 (b) Without
Condition index 1 21 156 238
l)]
W b., 1 In[ TIME]
0.000 0.000 0.319 0.681
0.000 0.181 0.654 0.165
e-scaling
var( b, ) CONST
var( b, ) ln[ER(T)I
var( b3 ) ln[LCSLSPOZ’( T - l)]
var( b4 ) ln[ TIME]
0.000 0.000 0.542 0.458
0.000 0.002 0.431 0.567
0.000 0.003 0.005 0.992
0.000 0.081 0.201 0.718
Table C.111 Mixed labor-force
participation
data; variance-decomposition
proportions
(4 Condition index 1 9 194 331
var( b, ) CONST
var( b, )
MERV)I
0.000 0.000 0.487 0.513
0.000 0.000 0.001 0.999
var( b, ) LCSLSZ’OP(T-
1)
var( b4 ) TIME
0.000 0.000 0.511 0.489
0.001 0.118 0.826 0.056
var( b, ) ER(T)
var( b3) LCSLSPOF( T - 1)
var( b4 ) In[TZME]
0.000
0.000
0.001 0.331 0.668
0.000 0.001 0.998
0.000 0.000 0.323 0.677
0.000 0.189 0.661 0.149
@I Condition index 1
12 146 351
var(b,) CONST
D.A. Belsley, Conditioning
in models with logs
143
References Belsley, D.A., 1982, Assessing the presence of harmful collinearity and other forms of weak data through a test for signal-to-noise, Journal of Econometrics 20, 211-253. Belsley, D.A., 1984, Demeaning conditioning diagnostics through centering, The American Statistician 38, 73-82. Belsley, D.A., 1986, Centering, the constant, first-differencing, and assessing conditioning, in: D.A. Belsley and K. Kuh, eds., Model reliability (MIT Press, Cambridge, MA). Belsley, D.A. and R.W. Oldford, 1986, The general problem of ill-conditioning and its role in statistical analysis, Computational Statistics and Data Analysis 4, 103-120. Belsley, D.A., E. Kuh and R.E. Welsch, 1980, Regression diagnostics: Identifying influential observations and sources of collinearity (Wiley, New York). Kuh, E. and R. Schmalensee, 1972, An introduction to applied macroeconomics (North-Holland, Amsterdam).
of Econometrics
38 (1988) 127-143.
North-Holland
CONDITIONING IN MODELS WITH LOGS* David A. BELSLEY Boston College, Chestnut Hdi, MA 02167, USA CCREMS, MIT, Cambridge, MA 02139, USA The conditioning diagnostics of Belsley, Kuh and Welsch (1980) are applicable to models that are linear in parameters and (structurally interpretable) variates. Different diagnostics are required when models contain different types of non-linearities. It is shown here, however, that the BKW diagnostics are also applicable to models containing logarithmic terms once the variates are first e-scaled, i.e., scaled to have a geometric mean equal to the base e of the natural logarithms,
1. Introduction The conditioning diagnostics of Belsley, Kuh and Welsch apply to the OLS estimator of the strictly linear, basic model y = /3,x,
+ . . . +/3,x,
+ E.
(1980) (BKW)
(1)
Strictly linear means it is linear in both parameters and variables, and basic means its variates X, E .CZJ?~ are structurally interpretable; that is, the model is parameterized so that its variates are in a form in which numerical changes in their values can be assessed as being unimportant or inconsequential relative to the real-life situation being modeled.’ In this case, a large scaled (for equal column length) condition number K(X) of the n X p data matrix X = [X1... XP] is meaningfully interpretable as the potential magnification factor by which small relative changes in the X, could be transformed into large relative changes in the OLS estimates b = (XTX)-lXTy, and thus K(X) becomes a meaningful measure of conditioning in this case. In the event, however, that the Xi are not themselves structurally interpretable, but rather are non-linear transformations such as X, = ln(Z,) or X, = 2,2, *Special thanks go to the late Edwin Kuh for comments and to Ana Aizcorbe and Sean Doyle for research assistance. All computation was done on the TROLL system at MIT. This research was funded in part by a Mellon Grant to Boston College and in part by the National Science Foundation under Grant No. IST-8420614. ‘This concept is defined in Belsley (1984,1986) and Belsley and Oldford (1986). Since the condition number in this context measures the potential sensitivity of the OLS estimator to small relative changes in the data, it is clear that this measure can only be meaningfully interpreted when the relative data change it is based on can itself be meaningfully interpreted; that is when the data are structurally interpretable.
0304-4076/88/$3.50~1988,
Elsevier Science Publishers
B.V. (North-Holland)
128
D.A. Belslq, Conditioning in models with logs
of structurally interpretable Zi, then the value K(X) will typically provide little meaningful diagnostic information about the true conditioning of the OLS estimator. These issues are developed in Belsley and Oldford (1986) where a general conditioning analysis is defined and the problem of assessing the conditioning of non-linear models is addressed. There we learn the following: (a)
(b)
(c)
There is no single diagnostic procedure for carrying out a general conditioning analysis; each form of non-linearity presents its own concerns. Unlike estimation, where ‘non-linearity’ often refers only to non-linearities in parameters, conditioning analyses are affected by non-linearities in both the parameters and the variates. As already indicated above, for any conditioning analysis to provide meaningful diagnostic numbers, it must be applied to data that are structurally interpretable.
There is, however, one form of non-linearity in the variates that lends itself to a reasonably straightforward conditioning analysis: the logarithmic transform - a fortunate case, since logarithmic transformations are frequently encountered in models used in the natural and the social sciences. In this paper we find that the procedures of BKW remain relevant once each structurally interpretable 2, entering in a logged term is first e-scaled, that is, scaled so that the geometric mean g,, of its elements is equal to e, the base of the natural logs. Where needed for distinction, the term L-scaled indicates scaling for equal Euclidean length, usually effected by scaling for unit length. In sections 2-5 we view the various issues involved and provide a solution. Section 6 provides several illustrations. 2. A view of the issues Assume for the moment that the X, in (1) are the basic, structurally interpretable variates. Models that differ from (1) only in scale (the units in which the X, are measured) are clearly structurally equivalent, but, as noted in BKW (p. 120), their respective data matrices can have very different condition numbers. Without a canonical scaling, then, the condition number K(X) provides little meaningful diagnostic information, since different values can apply to the same ‘real-life’ situation. In the linear-model context of BKW, this problem can be shown to be resolved through column equilibration, L-scaling each column of X to have the same, usually unit, Euclidean length. A similar problem arises when some of the X, in (1) are logs of structurally interpretable variates2 but the resolution is different. Consider the model Y =
PI&+ P21n(Z2) + &ln(-%>+ 6
(2)
‘The term ‘log’ is used here generally to mean logarithm. The discussion takes place in terms of natural logs, denoted ‘In’, but the results are readily generalized below to apply to logs of any base r.
D.A. Belsley, Conditioning in models with logs
129
I Qn(Z;)
= Qrl(c*)r +
,( =Qn(c,)l+ Qn(Z,) :I
:
:: : :
Fig. 1
where Z, and Z, are structurally Resealing the Z, as
z,* = cizi, merely renames (2) is structurally
interpretable
and
i= 1,2,
the ‘real-life’ phenomenon equivalent to
y = &L + &ln(Z,*)
+ &ln(Z,*)
1 is the vector
of ones.
(3)
represented
+ e,
by the Z,, and hence
(4)
where fir = /3r - ln(c,) - ln(c,). But the conditioning of the ln(Z,*) in (4) can differ greatly from that of the ln(Z,) in (2) even though the correlation between the Z, is invariant to such scaling. Hence a canonical scaling is required for meaningful diagnostics based on the ln(Z,). To see the effect of scaling on the conditioning of log variates, consider fig. 1. Here L represents the n-vector of ones (1. ..l)‘, Z, = (zl.. . z,,)~ the n values zi of a structurally interpretable variate Z,, and ln(Z,) the n-vector
130
D.A. Belsley, Conditioning
in models with logs
(ln(zi). . .ln(z,))r. Resealing 2, to Zi* = c,Z, adds to ln(Z,) the vector ln(c,)r and clearly has the effect of ‘sliding’ ln(Z,) along the affine subspace Hi containing ln(Z,) and parallel to L. Thus, any point along H, is structurally equivalent to ln(Z,), i.e., results simply from renaming Z,. Similar considerations apply to Z, and ln(Z,). Let us begin, then, as in fig. 1, with ln(Z,) and ln(Z,) at right angles, so these two variates are perfectly conditioned. Simply by renaming (scaling) the Z,, we can slide them both upward, reducing the angle (Ybetween them and thus arbitrarily worsening their conditioning. This result clearly holds whether or not the three vectors are coplanar as in fig. 1. The question thus arises as to the scaling for the Z, that produces ln(Z,*) whose conditioning provides meaningful diagnostics. 3. Measuring conditioning of models with logs An answer to this question depends on what we wish a conditioning diagnostic to measure. A general definition of ill conditioning is given in Belsley and Oldford (1986) and can be summarized as:
Dejinition.
A conditioned if understanding small can give
model, equation, or estimator w =f( u) [w E .B”“, u E .9fn] is ill relative changes 6~ in u that can be interpreted (from an of the phenomenon being modelled) to be inconsequentially rise to unacceptably large relative changes 6 w in w.
As noted, in the strictly linear, basic model (l), the L-scaled condition number K(X) of the matrix of structurally interpretable X, provides just such a measure for the OLS estimator. That is, a K(X) of 100 indicates that a 1 percent relative change in the data could produce up to a 100 percent relative change in the OLS estimates.3 If the X, are ln(Z,), the same considerations necessarily apply to the ln(Z,); that is, K(X) measures the potential relative sensitivity in b with respect to small relative changes in the ln(Z,), but not with respect to small relative changes in the Z,. And, if it is the Z, that are structurally interpretable, it is this latter measure that is desired. A solution is suggested by recalling that for a scalar w (and natural logs) din(w) PC ln(4
1
dw (5)
In(w)7
Here we see that relative changes in In(w) are the same as relative changes in w only if w scaled to the base e so that In(w) = 1. Of course, the X, and Z, are vectors and not scalars; so taking a hint from the preceding, it seems ‘Measured
as a vector norm, such as IlSbll/llbll.
See Belsley (1982) and BKW (1980).
D.A. Belsley, Conditioning
131
in models with logs
reasonable to scale the 2, so that, at some representative value z”* for the elements of the scaled vector Z,* = cIZ,, one has ln( z”*) = 1. This occurs if one chooses
(6)
c, = egzyl,
where
g,, is the geometric
g,, =
mean of Z, = (zlr . . . z,,)~, that is
(nz,;1l’Y
(7)
J
Then the geometric mean gz: of Z,* = eg, ‘Z, is e, and, at the representative unscaled value .? = e-‘g,,z”* = gz,, we value t”* = g,: = e and its corresponding have dln(Y*) ln(z*)
=
d,Y* t”* = 7’
(8)
The first equality holds since ln(gz:) = 1, and the second because scaling does not affect relative values. The preceding readily generalizes to accommodate logs to any base r. In this case, at the representative value 2*, (5) becomes dlog,(
2*)
log,( 5*)
log,(e) = log,@*)
d2* ,?* ’
(5’)
and the desired scaling is clearly that which sets log,(e) = log,(z”*) or, again, where z”* = g,: = e. For any base, then, the same canonical scaling is indicated as appropriate for assessing the conditioning of the logged variates Z,: e-scaling, or that which makes the geometric mean g zY of the scaled data series Z,* equal to the base e of the natural logarithms. For this scaling, small relative changes in the variates ln(Z,*) are approximately the same as those in the Z,* and hence as those in the structurally interpretable variates Z,.
4. A canonical
scaling for logs
Assume, then, some or all of indicates that, if are first e-scaled shall denote X*,
it is desired to assess the conditioning whose columns are in logged form. those structurally interpretable variates (the others left unchanged), resulting in the conditioning diagnostics of BKW
of a data matrix X, The preceding result entering in log form a data matrix that we apply directly to X*.
132
D.A. Belsley, Conditioning in models with logs
That is, it is suggested using the L-scaled condition number of X* to measure the potential relative change in the OLS estimator b that could result from a small relative change in the structurally interpretable 2, that enter the model in logged form. Various cases can arise depending upon whether X contains all logged variates or is a mixture of logged and non-logged variates. X=[X,... Case 1: Inthedatamatrix X,], some of the X, may be In(Z), but for each variate it is Xi and not Zi that is structurally interpretable. In this case, no e-scaling is necessary; the techniques of BKW can be applied directly to the X,. That is, all Xi should be L-scaled (which occurs automatically as the first step of the BKW procedure), and it is the L-scaled condition number K(X) and its variance-decomposition proportions that are the desired diagnostic measures for the conditioning analysis. Case 2: The Xi are log transforms ln(Zj) of structurally interpretable Z,. First, e-scale the Z, to Z,* having geometric mean g,: = e, and then use the techniques of BKW to examine the L-scaled condition number K(X*) of X* = [ln( Z,* ) . . . ln(Z,* )] and its variance-decomposition proportions. Thus, the Zj are first e-scaled to Zi* and then the ln(Zi*) are L-scaled for equal (unit) length. This later transformation does not alter the relative magnitudes of the d ln(Z,* )/ln( Z,* ) but is needed for the basic equilibration reasons given in appendix 3A of BKW. Case 3 (the general case): Some X, (which may be logged variates) are themselves structurally interpretable while others are ln(Z,) for which it is the Z, that are structurally interpretable. Here we have a mixture of the previous two cases. First, e-scale only the structurally interpretable Zi variates to Z,* as in case 1, then examine the L-scaled condition number of the matrix X* whose columns are the union of the structurally interpretable X, and the ln( Z,*). This resulting matrix is the proper input to the collinearity diagnostics of BKW, and its condition indexes and associated variance-decomposition proportions are the desired conditioning diagnostics. This diagnostic procedure is illustrated in section 7, but, in order to assess its performance, it is first necessary to examine the magnitude of the condition number that can be anticipated from its use. 5. The magnitude of K(X*)
Consider the two alternative specifications: (hereafter known simply as the linear model),
the model linear in the x’s
and the model linear in the logged x’s (hereafter known simply as the logged model)
(10)
133
D.A. Belsley Conditioning in models with logs
These models are clearly quite similar and often are considered as alternative specifications for the same real-life economic phenomenon. Furthermore, since with e-scaling a given relative shift in the x,, in (9) produces roughly the same relative shift in the ln(xit) in (lo), we should be very surprised if the condition number of the e-scaled logged data from (10) were not similar to that of the non-logged data from (9). This prospect can be informally motivated as follows. The Bi in the logged model (10) can be interpreted as the elasticities of y, with respect to the xir, while the /3, in the linear model (9) can be interpreted as the partial derivatives 8yJaxi,. Let ii be the OLS estimates of the elasticities 6, and bi be those of the partials /I;. Then we know that the condition number of the logged data bounds the relative change aii/b, in 6, with respect to a given relative change ~3ln(x,,)/ln(xi,) in ln(x,,), while the condition number of the basic data bounds the change dbi/b, in b, with respect to a given relative shift axiJxi, in xii. Of course, with e-scaling 8 ln(x,,)/ln(x,) = 8xi,/xir, so it is needed only to compare dbi/gi and db,/b, for a given axi,/xi,. This can be done roughly by converting bj into an * elasticity estimate like ii, i.e., by considering b, = b,(x,,/y,). Now, for a given relative change in xit (y, held constant), we would expect the potential sensitivity of the two elasticity estimates to be similar; that is, we would expect aAi/hi and a&,/6, to have a similar bound. But,
ah ab;(xit/Yt) (Xitabi+ b,ax,,) x=
bi(xit/yt)
=
bixit
r3bi
=b+-’
I
ax,, xit
(11)
so, when the data are ill conditioned and dbj/bi x=-axiJxi, by definition, a&,/h; = abi/bi. Under e-scaling, then, we expect 86,/g, and abi/b, to have similar bounds, and hence we anticipate that the condition number of the logged e-scaled data to be similar in magnitude to that of the basic data. It remains for the illustrations of section 7 to show that this indeed occurs in practice. 6. A special case of interest The treatment of one special case deserves attention: that of ln(2,) and ln(Z,‘). With a little thought beforehand, these two variates are not likely to occur together in the same model since ln( 2,‘) = 2 ln( Z,), and they are exactly collinear. The situation could arise, however, if one logged variate happened to be very closely the square of another, or, as could all too easily happen, if a given equation containing both Z, and Z,2 were mechanically ‘logged’ in order to test whether the logged or non-logged specification seemed better. The issues involved in assessing this situation are most easily made clear geometrically.
134
D.A. Belsley, Conditioning in models with logs
I
QnE’)=!Qn(cZ)
Q
I?iii(Z)Qn(2) I
I
0
-
man(z)l
Fig. 2
Consider first the geometric effect of e-scaling. In fig. 2, the 1 vector is oriented vertically, and In(Z) = (ln(z,). . . ln(z,))r is any n-vector of logs. As is well known, the arithmetic mean of any vector is determined by its orthogonal projection into the space of I. The point A, then, is rnlnCZl~, where m mean of the elements of In(Z). And, fr~.q=e~;a~~j~~~~Am th e arithmetic In(z) = ln(JJz;) ‘in = ln(g,), we see A is also ln(gZ)L. Thus, for any vector ln( W) E H$,, the affine space orthogonal to 1, ln( g,) = 1, and the elements of W have geometric mean g, = e. Now, recalling that, as we scale Z we slide In(Z) vertically along PC,, we see that the scaling c that produces a vector Z* = cZ with geometric mean e is that which places ln(Z*) at the intersection of H, and Nz.4 Consider next, then, applying the preceding to the model y = &L + &ln(Z)
+ &ln(Z2)
+ E,
(12)
containing both In(Z) and ln(Z’) = (ln(zf), . . ln(zi))r= 2ln(Z). These two vectors are plotted along with 1 in fig. 3. It is clear that In(Z) and ln(Z*) are perfectly collinear and that such a specification would make little sense. But note that, after Z is e-scaled to Z*, the resulting ln(Z*) and ln(Z**) are no longer collinear. It would seem that such scaling would remove our ability correctly to diagnose collinearity in the original data. This is not so, however, 4Computational note: the scaled vector Z* is most easily calculated by noting that Z* = iii( Z ) + L, where iii(Z) -In(Z) -m ,n(Zj~, That is, In(Z) is first put into deviation-from-mean form G(Z), and then 1 is added to each element. In the event that logs to base r are being used, one would instead add log,(e) to each element to give log,(Z) + log,( e)c.
D.A. Belsley, Conditioning in models with logs
Qtl(2’)
135
Qn(Z*')
Oi Fig. 3
since the three vectors 1, ln(Z*) and ln(Z * * ) remain perfectly collinear (coplanar), and the perfect ill conditioning of X = [ 1, ln(Z ), ln(Z *)I also necessarily exists in the scaled matrix X* = [I, ln( Z*), ln(Z * * )] but in a different form. The preceding argument depends on the presence of the constant vector L in the model (12) (i.e., pi # 0) and seems therefore to lack generality, but this is not the case. It is an occasionally unappreciated fact about models containing logged terms that they must always include an intercept. This is because there is no natural unit for measuring any variable - I am always free to ‘deci’ any units you prefer. Thus, if one postulates the homogeneous model y = fi,ln(Z> a simple change non-homogeneous
+ P@(w)
+ e,
of units Z* = CZ (c # 1) produces model
y = pit + &ln(Z*)
+ &ln( W) + E,
(13) the structurally
equivalent
(14)
with pi = - ln( c) # 0. Thus, the homogeneous form (13) requires an untenable assumption of unique natural units for the variates. A model with log variates, then, must always include an intercept, and, in assessing the conditioning of the data relevant to such a model, the data matrix being assessed must always include 1, the constant column of ones. Because of this, the method of e-scaling proposed above will never fail to detect collinearity between In(Z) and ln(Z ‘), since it will always materialize as collinearity among L, ln(Z*) and ln(Z2*).
136
D.A. Belsley, Conditioning in models with logs
7. Illustrations
The e-scaling introduced above is chosen to provide conditioning diagnostics for models with logs whose magnitudes are able to support an interpretation similar to those of BKW for strictly linear models, namely, a measure of the relative sensitivity of the resulting OLS estimator b to a given relative change in the structurally interpretable variates Z, [not the ln(Z,)]. The efficacy of e-scaling in achieving this goal is illustrated through analyses of three different economic data sets corresponding to a consumption function, a commercial and industrial loans function, and a labor-force participation function. The conditioning of the data is assessed and reported first (case I) as if they came from the strictly linear model y=p11+p2x*+
...
+&Xp+&,
(15)
and second (case II) as if they came from a model linear in the logs ln(y)=&b+&1n(X2)+
... +/?Jn(X,)+e.
(16)
In this latter instance, the results are reported both with e-scaling (case IIa) and without e-scaling (case IIb) in order to demonstrate that the e-scaling is necessary to obtain stable and comparable diagnostic results. Furthermore, assessment is made and reported (case III) assuming the model to possess various mixtures of logged and non-logged terms. In each instance, it is the X, and not the ln(X;) that are assumed structurally interpretable. The structurally interpretable consumption-function data derive from the model [written in its basic linear form (15)] C(T) = pi1 + &C( T- 1) + &DPZ( T) + &r(T) +&ADPZ(T)
+ E,
(17)
where I = constant vector, C = total consumption, 1958 dollars, DPZ = disposable personal income, 1958 dollars, r = interest rate (Moody’s Aaa). These data are essentially those described in chapter 3 of BKW, differing only in the deletion of the last year which had a negative value of ADPZ. All series are annual, 1948-1973. The commercial- and industrial-loans data are derived from MCL(T)=&c+&MCL(T-l)+P,YGPPZ(T) +&[RCL(T)
-RTB(T)]
+E,
08)
137
D.A. Belsley, Conditioning in models with logs
where constant vector,
1
=
MCL
= commercial and industrial loans at all commercial banks,
YGZW = gross private product, = average bank rate on short-term commercial and industrial loans, = average yield on three-month U.S. Treasury bills.
RCL RTB
These data, from Kuh and Schmalensee (1972), are quarterly 1959-11 to 1969-IV.
series from
The secondary labor-force participation data derive from LCSLSPOP(T)
=&r+&ER(T) +&TIME(T)
+&LCSLSPOP(T-1) + E,
(19)
where
LCSLSPOP TIME
= constant vector, = ratio of civilian employment to total population of working age, = ratio of secondary civilian labor force to secondary population, = a time dummy.
These data, also from Kuh and Schmalensee, are annual series, 1948-1962. No effort is made here to explain the meaning of the basic, logged, or mixed-logged regression models behind these data sets. The purpose of this exercise is only to examine the efficacy of the proposed conditioning diagnostics in correctly assessing the presence of near dependencies in the data. The condition indexes for cases I, II (a and b), and III are summarized for each data set in table 1. Most of the comparative information needed for this study can be seen from these results alone. The detailed variance-decomposition proportions matrices are provided in an appendix for those interested in a more complete analysis of how the different transformations and scalings affect diagnosed variate involvement. The first important observation comes from comparing columns II(a) and II(b). It is clear that e-scaling has a major impact on the magnitudes of the condition indexes of the logged variates. It is further seen that e-scaling can cause these magnitudes either to decrease (as for the first two data sets) or increase (as for the last). That this can happen is readily predictable from fig. 1. The second important observation comes from comparing columns I and II(a). Here we see that e-scaling, consonant with section 5, always brings the condition indexes of the logged data into line with those of the basic data and
138
D.A. Belsley, Condrtioning in models with logs Table 1 Comparative
condition
indexes.
Case II
Case I
Basic data
(a) With e-scaling
1 6 10 40 361
1 4 9 40 339
Case III (b) Without e-scaling
Consumption-function
Representative mixtures data
1 9 14 232 1900
1 4 9 32 113
1 6 10 46 381
1 5 15 101
1 5 15 141
1 9 194 331
1 12 146 351
Commercial- and industrial-loans data 1 5 16 146
1 5 19 164
1 4 43 852 Secondary labor-force participation
1 9 194 333
1 12 154 346
1 21 156 238
data
so achieves the desired end of providing a stable diagnostic measure that meaningfully measures the potential sensitivity of a relative change in the OLS estimate to a relative shift in the basic, structurally interpretable data as opposed to the logged data. We also see that failure to e-scale logged data typically results in grossly distorted diagnostic magnitudes which can be either too high or too low. Although examples of both cases occur here, our experience indicates the behavior of the first two data sets to be most representative; that is, failure to e-scale logged data most often produces highly inflated condition indexes. Comparison of the mixture cases with column I indicates that e-scaling of logged variates, even when intermixed with non-logged variates, also tends strongly to produce diagnostic measures with magnitudes very comparable to those of the basic data. The first mixture in the consumption-function data deserves special mention here since these magnitudes seem to be quite different. Examination of table A.III(a) in the appendix, however, reveals in this case that, of the three variates C, DPI and ADPI involved in the strongest basic near dependency (the one corresponding to 367 in table A.I), only the last two have been logged. This differential treatment of the variates involved in a linear dependency will, of course, have a tendency to alter it - in this case weaken it so
D.A. Belsley, Conditioning
139
in models with logs
that its condition index falls from 367 to 113. In the second mixture among the consumption-function data, seen in table A.III(b), only the variate r was logged, leaving the strongest near dependency among the other variates untouched, and here we see the e-scaled diagnostics are telling a virtually identical story as in the basic case. E-scaling in the mixture case, then, seems to be working in accord with intuition and expectations. As to the effect of e-scaling on the diagnostic of variate involvement, it is readily seen from the variance-decomposition proportions matrices in the appendix that the e-scaled logged data show similar patterns of variancedecomposition proportions as those for the basic data. Failure to e-scale can alter these patterns to produce misleading results as can be seen from comparing the role of the constant term in the consumption-function data (tables A.11 and AI), the role of ln(RCL-RTB) in the commercialand industrial-loans data (tables B.11 and B.I), and the role of TIME in the secondary labor-force participation data (tables C.11 and C.1). 8. Conclusion Our goal was to find a canonical scaling for logged data able to produce stable conditioning diagnostics having the same interpretative basis and meaning as those of BKW applied to strictly linear, basic models. E-scaling - scaling structurally interpretable variates that appear in logged terms so that their geometric mean is e, the base of natural logarithms - is seen precisely to serve this end. The e-scaled logged variates along with the non-logged variates in the equation provide the proper input to the conditioning diagnostics of BKW, the first step of which is to L-scale (for unit length) all variates. E-scaling should be applied only to structurally interpretable variables Z, that enter the equation as ln(Z,). If the variate X, = ln(Z,) is itself the structurally interpretable variate, it should be treated normally in the BKW diagnostics. Appendix: Variance-decomposition
proportions matrices
In this appendix, the variance-decomposition proportions matrices for each case for each data set. The table numbers are keyed by: A. B. C.
Consumption-function data, Commercialand industrial-loans Labor-force participation data.
I. II. III.
Basic data, Logged data, (a) with e-scaling and (b) without Mixtures, logged and non-logged.
In the various mixture the column headings.
are given
data,
cases, the logged and non-logged
e-scaling, variates
are clear from
140
A.I.
D.A. Belsley, Conditioning in models with logs
The consumption-function data Table A.1 Basic consumption-function
data; variance-decomposition
v=(h)
Condition index
CONST
1 6 10 40 361
W b2) C(T-
0.001 0.072 0.206 0.330 0.391
1)
0.000 0.000 0.000 0.006 0.994
proportions,
W 4)
W b.,1
DPI( T)
r(T)
0.000 O.OOfl 0.009 0.006 0.994
W bs) ADPI
OMKI 0.000 0.033 0.964 0.003
0.001 0.174 0.110 0.150 0.565
Table A.11 Logged
consumption-function
data; variance-decomposition
proportions.
(a) With e-scaling Condition index 1 4 9 40 339
van b, ) CONST 0.001 0.023 0.331 0.265 0.379
W b3)
W b, )
ln[C(T - l)]
ln[ DPI( T)]
O.OOG 0.000 O.OtXl o.OO5 0.995
0.000 0.000 0.000 0.007 0.993
W 41
W 41
WON
ln[ADPI(T)]
0.000 0.000
0.003 0.282 0.098 0.045 0.572
0.020 0.887 0.092
(b) Without e-scaling Condition index 1 9 14 232 1900
van&) CONST 0.000 0.000 0.001 0.944 0.054
van b, ) ln[C(T- l)]
var( b, ) ln[ DPZ( T)]
O.OOG 0.000 0.000 0.005 0.995
0.000 O.ooO O.OOG 0.007 0.993
van b4) ln[r(T)l
W 6,) ln[ADPI(T)]
0.000 0.001 0.073 0.835 0.091
O.cill 0.212 0.150 0.064 0.573
Table A.111 Mixed consumption-function
data; variance-decomposition
proportions.
(a) Condition index 1 4 9 32 113
Mb,) CONST
van b, ) C(T- 1)
0.002 0.033 0.429 0.361 0.175
0.000 0.000 0.001 0.015 0.985
vanhI) CONST
var(bz) C(T- 1)
0.001 0.061 0.200 0.243 0.495
0.000 0.000 0.000 0.006 0.994
van b, ) ln[ DPI(T)]
van b4) r(T)
O.OOG 0.000 0.001 0.074 0.925
O.ooO 0.000 0.027 0.645 0.328
var( b, ) DPI(T)
Mr(T)l
van b, ) ln[ADPI(T)] 0.007 0.606 0.187 0.134 0.067
(b) Condition index 1 6 10 46 381
O.OilO 0.000 O.ooO 0.008 0.992
W b4)
0.000 O.OQO 0.022 0.902 0.076
-~a(b, ) ADPI 0.002 0.185 0.118 0.181 0.515
141
D.A. Belsley, Conditioning in models with logs
A.2.
Commercial- and industrial-loans data Table B.1 Basic loans data variance-decomposition
Condition index 1 5 16 146
proportions
var(&) CONST
var( b, ) MCL(T1)
var( b, ) YGPPZ( T)
0.000 0.000 0.057 0.942
0.000 0.001 0.009 0.989
0.000 0.000 0.000 1.000
var( bz,) RCL(T) -RTR(T) 0.003 0.156 0.539 0.302
Table B.11 Logged loans data; variance-decomposition
proportions.
(a) With e-scaling Condition index 1 5 19 164
var( b, ) CONST
var( b, ) ln[MCL(T-
var( b, ) ln[ YGPPZ( T)]
l)]
0.000 0.001 0.007 0.992
0.000 0.000 0.186 0.814
0.000 0.000 0.000 1.000
(b) Without Condition index 1 4 43 852
var( b, ) CONST 0.000 0.000 0.005 0.995
var( b, ) ln[MCL(T-
0.004 0.273 0.599 0.124
e-scaling var( b, ) ln[ YGPPZ( T)]
l)]
var( b4 ) ln[RCL(T) -RTB(T)]
0.000 0.000 0.013 0.987
var( b4 ) In[RCL(T) -RTB(T)] 0.010 0.544 0.315 0.131
0.000 0.000 0.000 1.000 Table B.111
Mixed loans data; variance-decomposition
proportions
(4 Condition index 1 5 15 101
Condition index 1 5 15 141
var( b,) CONST
var( b, ) MCL(T1)
var( b, ) ln[ YGPPZ( T )]
var( b4 ) RCL(T) - RTB(T)
0.000 0.000 0.060 0.940
0.000 0.004 0.033 0.963
0.000 0.000 0.001 0.999
0.003 0.160 0.507 0.331
var( b,) CONST
var( b, ) MCL(T1)
var( b, ) YGPPZ( T)
var( b4 ) ln[RCL(T) - RTB(T)]
0.000 0.000 0.059 0.941
0.000 0.001 0.009 0.989
0.000 0.000 0.000 1.000
0.003 0.183 0.557 0.257
D.A. Belsley, Conditioning in models with logs
142
A.3. Secondary labor-force participation data Table C.1 Basic labor-force Condition index 1 9 194 333
participation
data; variance-decomposition
proportions.
Mb,) CONST
vaQ 6, ) ER(T)
var( b, ) LCSLSPOP( T - 1)
var( k, ) TIME
0.000 0.000 0.489 0.511
0.000 0.000 0.001 0.999
0.000 0.000 0.502 0.498
0.001 0.118 0.823 0.058
Table C.11 Logged
labor-force
participation
data; variance-decomposition
proportions
(a) With e-scaling Condition index 1 12 1.54 346
var( b, ) CONST 0.000 0.001 0.399 0.600
var( bz ) HER(T)1
W b3 1 ln[LCSLSPOP(T-
0.000 0.000 0.003 0.997 (b) Without
Condition index 1 21 156 238
l)]
W b., 1 In[ TIME]
0.000 0.000 0.319 0.681
0.000 0.181 0.654 0.165
e-scaling
var( b, ) CONST
var( b, ) ln[ER(T)I
var( b3 ) ln[LCSLSPOZ’( T - l)]
var( b4 ) ln[ TIME]
0.000 0.000 0.542 0.458
0.000 0.002 0.431 0.567
0.000 0.003 0.005 0.992
0.000 0.081 0.201 0.718
Table C.111 Mixed labor-force
participation
data; variance-decomposition
proportions
(4 Condition index 1 9 194 331
var( b, ) CONST
var( b, )
MERV)I
0.000 0.000 0.487 0.513
0.000 0.000 0.001 0.999
var( b, ) LCSLSZ’OP(T-
1)
var( b4 ) TIME
0.000 0.000 0.511 0.489
0.001 0.118 0.826 0.056
var( b, ) ER(T)
var( b3) LCSLSPOF( T - 1)
var( b4 ) In[TZME]
0.000
0.000
0.001 0.331 0.668
0.000 0.001 0.998
0.000 0.000 0.323 0.677
0.000 0.189 0.661 0.149
@I Condition index 1
12 146 351
var(b,) CONST
D.A. Belsley, Conditioning
in models with logs
143
References Belsley, D.A., 1982, Assessing the presence of harmful collinearity and other forms of weak data through a test for signal-to-noise, Journal of Econometrics 20, 211-253. Belsley, D.A., 1984, Demeaning conditioning diagnostics through centering, The American Statistician 38, 73-82. Belsley, D.A., 1986, Centering, the constant, first-differencing, and assessing conditioning, in: D.A. Belsley and K. Kuh, eds., Model reliability (MIT Press, Cambridge, MA). Belsley, D.A. and R.W. Oldford, 1986, The general problem of ill-conditioning and its role in statistical analysis, Computational Statistics and Data Analysis 4, 103-120. Belsley, D.A., E. Kuh and R.E. Welsch, 1980, Regression diagnostics: Identifying influential observations and sources of collinearity (Wiley, New York). Kuh, E. and R. Schmalensee, 1972, An introduction to applied macroeconomics (North-Holland, Amsterdam).