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Cross-Sectional Regressions in Event Studies

Jim MusumeciDepartment of Finance, MOR 107

Bentley UniversityWaltham, MA 02452

[email protected]

Mark PetersonDepartment of Finance, Rehn 134A

Gordon and Sharon Teel Professor of FinanceSouthern Illinois UniversityCarbondale, IL 62901-4626

[email protected]

Current Draft: September, 2015

The authors are grateful to Claude Cicchetti, Marcia Millon Cornett, Dhaval Dave, Otgo Erhemjamts, Atul Gupta, Kartik Raman, Len Rosenthal, Richard Sansing, and Aimee Smith for their helpful comments. The usual disclaimer applies.

mailto:[email protected]

mailto:[email protected]

Cross-Sectional Regressions in Event StudiesAbstract

Christie [1987] demonstrated that when regressing abnormal returns on various firm characteristics, the “correct deflator [for those firm characteristics] is the market value of equity at the beginning of the period.” Despite this, many researchers deflate a variable of interest by total assets, and then add leverage, a ratio of book equity to market equity (or its reciprocal), or other independent variables. We show that such a method regularly produces relationships that appear to be statistically significant, but which in fact are spurious and attributable only to a mathematical artifact, not to any causal effects.

Cross-Sectional Regressions in Event Studies

“You keep using that method. I do not think it measures what you think it measures.”—with apologies to Inigo Montoya and William Goldman.

I. Introduction

Most modern event studies test not only whether average abnormal return is equal to zero, but also how abnormal returns are related to firm characteristics. For example, if the event under consideration pertains to the mortgage crisis, it is natural to conjecture that the firm’s response to the event is larger when it owns more mortgages. If V denotes the value of the presumably affected assets, TA the value of total assets, ME the market value of the firm’s equity, D the value of the firm’s debt, and ∆ denotes “change in,” and if everything else is held constant, then the fundamental accounting identity Total Assets = Total Liabilities + Stockholders’ Equity tells us to expect ∆V = ∆TA = ∆D + ∆ME. Later in the paper we consider the implications of risky debt, but for now we assume the firm’s debt is riskless, in which case ∆D = 0 and so ∆V = ∆ME. It is often difficult to measure ∆V on a regular basis, and so we typically assume it is proportional to V, or equivalently that ∆ME = kV for some presumably unknown constant k. A typical null hypothesis would be that k = 0; the alternative might be k ≠ 0, k > 0, or k < 0, depending on context.

A direct ordinary least squares (OLS) regression of ∆ME on V is inappropriate for at least a couple of reasons. First, a heteroscedasticity problem likely exists, with larger firms having larger error terms. While under this condition OLS will still produce unbiased estimates, these estimates will no longer have minimum variance within the set of linear estimates, and tests of statistical significance will be undependable. Second, both ∆ME and V are generally related to firm size, in which case we will find a significant relationship even if we mistakenly choose V to be some irrelevant variable that also tends to increase with firm size, i.e., almost any balance sheet or income statement item. To eliminate these problems, we typically scale ∆ME and V by some variable related to size before we estimate our regression. Christie [1987] showed that

the dependent variable, abnormal return, is essentially a measure of ∆MEME , and so consistency

implies we need to scale V by ME as well. Normalizing V instead by Total Assets spreads the

gain (or loss) from V equally across all the firm’s claimants, which is not what ∆MEME does. This

necessarily produces extraneous noise and weaker tests unless there exists a universal constant c such that ME = c TA for each firm in the sample.∙

Christie formally demonstrated that use of normalizing variables other than the market value of equity results in misspecification problems, but did not elaborate on the nature or

severity of the misspecification. We demonstrate that the commonly used variables VTA ,

DTA ,

and BEME (or

MEBE ) will usually produce misleading statistical significance. Moreover, we show

that some seemingly reasonable (and some deliberately unreasonable) choices of variables produce not only spuriously significant relations, but even contradictory ones.

II. Modern Practice in Cross-Sectional Event-Study Analysis

Despite Christie’s admonition, few modern event studies scale their independent variables by market equity. To identify the false inferences to which alternative methods can lead, we consider a hypothetical unexpected change in tax law that would lead to lower taxes associated with greater cash and short-term receivables (Compustat item DATA1; henceforth “cash”). A typical hypothesis might be that, ceteris paribus, ∆ME = ∆V = kV for some positive constant k, where V in this case denotes cash. We select a random sample of 150 firms1 from the 2012 Compustat database and initially assume this relation holds exactly for all the firms in the sample. In practice, of course, error terms exist, but if an empirical method works poorly under perfect conditions with no error terms, then it cannot be expected to perform well when noise is present. Nevertheless, after the main points are established, we consider the more realistic case in which ∆ME = ∆V does not hold exactly, and we find similar results.

To make the main point, we assume the firm’s gain in market value is 5% of the cash balance, and that this entire increase in value accrues to equityholders. Thus ∆ME = .05V = .05Cash, and so dividing both sides by ME gives us

∆MEME =

.05VME =

.05CashME . [1]

Without loss of generality, we assume that the expected return of each firm conditioned on the

market return is zero, in which case the abnormal return, AR, exactly equals .05CashME . For this

sample, parameter estimation for the cross-sectional regression Christie shows to be correct,

AR = + CashME , would produce α=α=0 and β=β=.05 by construction.

A modern researcher unaware that the true state of nature is given by [1] might well estimate a cross-sectional regression with commonly used variables, specifically,1 We excluded financials (SIC Code 6000-6900) and utilities (SIC code 4900-4999) and firms with negative book

equity, leaving us with 144 firms. The latter exclusion was made because BEME

is a meaningless statistic when

book equity is less than zero.

2

AR = α+β1CashTA

+β2DTA

+β3BEME

+ε . [2]

Christie shows [2] is misspecified, yet regressions like this make frequent appearances in the literature. One motivation to use them is that they appear to separate the event’s effects into component parts. The question we address here is largely empirical in nature: do such regressions really explain the components of abnormal return, and how should we interpret the results?

We estimate [2]’s parameters for our sample and find [with t-values in brackets]

AR = -.009 +.039CashTA

+.014 DTA

+.010 BEME . [3]

[-3.46] [8.14] [3.45] [5.59] Adj R2 = .351

The regression appears to identify several different firm characteristics that are strongly related to AR. Proponents of this technique would presumably infer from the significant t-values that

firms’ abnormal returns are increasing in each of CashTA ,

DTA , and

BEME .

If no firms in the sample had outstanding preferred stock, then DTA = 1 −¿

BETA . Because

DTA

and BETA would have identical variances and a correlation of -1 with each other, and have

correlations with other variables that are identical except for sign, substituting BETA for

DTA

would change only the value and t-statistic of the intercept α and the signs (but not the magnitudes2) of β2 and, most importantly, the t-statistic of β2. In general, however, some firms

will have outstanding preferred stock, and so BETA and

DTA will have a strong negative

correlation, but not a perfect one. This is sufficient for their tests of significance to be very similar:

AR = .004+.043CashTA

−.019 BETA

+.011 BEME . [4]

2 If r2 is substituted for r1 as an independent variable, a perfect correlation is necessary and sufficient to ensure that the magnitude of the slope’s t-statistic remains unchanged. In general, however, the coefficient itself may change (for example, changing the units of measure of any independent variable from dollars to thousands of dollars in any regression will result in identical t-values, but coefficients that are different by a factor of 1000). What guarantees in this case that the magnitude of the coefficient also remains unchanged is that the absolute value of

the coefficient of DebtTA

in BETA

=a+b DebtTA

is 1.

3

[2.18] [9.02] [-4.83] [6.23] Adj R2 = .396

Compared with the results of [3], [4]’s larger R2 and t-values for the coefficients seem to suggest it is a more powerful and therefore presumably a better test.

We next check what happens if we alter the form of the model by substituting for each independent variable its logarithm instead. Estimating the parameters of that regression gives us

AR = .019+.004 ln(CashTA )−.006 ln ( BETA )+.007 ln ( BEME ). [5]

[12.92] [9.88] [-6.62] [7.60] Adj R2 = .449

Compared with [4], the larger t-values and R2 of [5] would seem to suggest it is a more powerful test, and therefore a better one.

Finally, to see if we can improve on this some more, we take the log of the abnormal returns as well.3 This gives us the following result:

ln (AR)=−2.996+1.00 ln(CashTA )−1.00 ln(BETA )+1.00 ln ( BEME ). [6]

[∞] [∞] [–∞] [∞] Adj R2 = 1.0

To see how such extreme statistics are possible, we exponentiate e by each side of [6], giving us

AR = ∆MEME

=e−2.996(CashTA )( BETA )−1

( BEME ), or

∆MEME

=.05 (CashME ), [7]

which is precisely equation [1]. The regression results from [3] appeared to suggest Total Assets and Book Equity were factors that contribute to an explanation of abnormal returns, but they are absent from [7]. While it is not necessarily incorrect to essentially add and then subtract the effects of Total Assets and Book Equity (as regression [6] basically does), Occam’s Razor implies it is inappropriate (and somewhat misleading) to do so. The results of [3]—[6] are summarized in Table 1.

3 This requires that all the abnormal returns be positive, but by assumption ∆ME = kV > 0, so this condition is met. Similarly, if ∆ME = kV < 0, we can apply the same analysis to the variables -∆ME = -kV > 0. The more realistic case in which AR may have different signs for different firms is addressed after we establish the basic intuition for this deterministic special case.

4

Assuming [7] (or, equivalently, [1]) holds, the coefficients in regression [2] will typically show up as significant. The basic principles in play here are that if two candidates for an independent variable have a perfect correlation, use of either will produce identical magnitudes for that variable’s t-statistic and for the overall R2, and when two variables have a large correlation with each other, either will produce very similar t-statistics and R2 values.4 In our specific context, all that is required is that the proportion of the firm that is financed with preferred stock is relatively constant across firms, and that the correlation between each of the independent variables and its logarithm is fairly large. Even this second condition is not necessary for the t-statistics to be misleading; because the correlation between a variable and its logarithm is always positive, the stated t-statistics will always be distorted away from zero. It is when this correlation is large, however, that the distortion will be particularly egregious.

Christie showed that a regression like [7] is the correct specification, and regression [6] is mathematically identical to [7]. Working backwards from [6] to [3] simply substitutes for each variable another variable that has a high magnitude of correlation with it. Thus the parameters of equation [3] all show up as significant not because all the independent variables provide explanatory power, but as a mathematical consequence of the true state of nature given by [7].

While the first and third independent variables, ( VTA ) and ( BEME ), are at least related to the

correct variable, ( VME ), the middle independent variable, (BETA ), is not.5 Any rejection of the

null hypothesis that its coefficient is equal to zero, then, is a form of Type I error. This component of Type I error can be quite large and is in addition to whatever level is chosen for the significance level . To see in greater detail why an irrelevant independent variable can have a statistically significant coefficient in a multiple regression, we first consider two effects of multicollinearity.

III. Multicollinearity

The most commonly appreciated aspect of multicollinearity is that it can obscure a true relationship between the dependent variable and one or more independent variables, leading

4 The inverse is not necessarily true, i.e., if two variables have a low correlation with each other, the t-statistics are not necessarily substantially different when one of these variables is substituted for the other. For example, if W1 and W2 are independent and identically distributed and Y = W1 + W2 + , then the regressions Y= α1+ β1W 1+ε

and Y= α2+ β2W 2+ε will on average produce similar estimates and similar t-statistics for their respective parameters, despite the fact that W1 and W2 are uncorrelated.

5 If (BETA ) happens to be correlated with ( VME ), it may appear to be significant in a simple regression. However,

by construction, its significance would disappear in a multivariable regression with both variables.

5

to less powerful tests. For example, consider a dependent variable Y, independent variables W1

and W2, and “building blocks” Zi, where the Zi are independent of each other. If

Y = Z1 + Z2

W1 = Z1 + Z3

and W2 = Z1 + Z4,

then simple regressions of Y on either W1 or W2 are likely to reveal significant coefficients provided the sample is sufficiently large or the variances of Z2, Z3, and Z4 are sufficiently small relative to that of Z1. A multiple regression of Y on W1 and W2 is problematic, though, because while an F-test will reveal that at least one of W1 or W2 is related to Y, it is difficult to ascertain which of W1 or W2 is the culprit.6

A less-appreciated consequence of multicollinearity is that it may give the appearance of a significant relationship, even when none exists. For example, consider

Y = Z1 + Z2

W1 = Z1 + Z3

and W2 = + Z3.

Now a simple regression of Y on W1 will typically produce a significant coefficient as before, but Y is independent of W2 and the results of that simple regression will reveal this. However, W1 and W2 have a positive correlation because of the common component Z3, and in a multivariate regression featuring W1 and W2 as independent variables, W2 will have the effect of “cleaning up” the noise induced by the presence of Z3 as a component of W1.7 Specifically, we will find that for the parameters of the sample regression line

Y i= α+ β1W 1 ,i+ β2W 2 , i+eiβ1→1 and β2→ -1 as sample size increases. However, W2 is not in any way related to Y, and its coefficient appears to be significant not because of any direct relationship with Y, but because it counteracts the Z3 noise term in W1.

The havoc that can be created by the introduction of superfluous independent variables is reminiscent of one version of Griliches’ Law: “Any cross-sectional regression with more than 6 In the belief that orthogonalization solves this multicollinearity problem, some researchers first orthogonalize one of the independent variables, say W1, by regressing it on another, W2, with which it is highly correlated, and then replace W1 in a multiple regression with the residuals e1 from the regression W 1= α+ β W 2+e1. Mitchell [1991], however, shows that this does not solve the multicollinearity problem at all. The coefficient and t-statistic for e1 in the ensuing multiple regression are identical to what they would have been for W1, while those for W2 are the same as they would have been in a simple regression without W1 or e1.7 We are hardly the first to make this observation that superfluous independent variables may spuriously appear to be significant if they have this “cleaning up” effect, or even to use that expression. Griliches and Wallace [1965] note the same possibility in their footnote 7.

6

five variables produces garbage.”8 It is for this reason that estimating a series of simple regressions as a complement to any multiple regression is desirable. When a multiple regression suggests an independent variable is significant, it is important to know whether the significance is due to a direct effect on the dependent variable, or to a reduction of noise in one or more other independent variables; simple regressions are not a panacea,9 but they can give us some indication of this. Additionally, it is a good idea to have a solid model a priori that is used as a basis for including an independent variable, rather than to add variables because they might produce better results.

IV. A Reconsideration of Cross-Sectional Regressions

There is nothing special about the choice of Total Assets and Book Equity as scaling variables in the Section II. We can obtain similar results for most strictly positive firm attributes. In this section we examine portfolios with a variety of possible Compustat values in addition to Book Equity and Total Assets. For now, we continue to assume that abnormal

returns are deterministic, specifically, AR = .05Cash

ME . For each such pair of Compustat values,

which we designate X1 and X2, we estimate parameters of the sequence of regressions

AR = α+βX1

X2+ε [8]

AR = α+β ln (X1

X2) +ε [8a]

AR = α+β1CashX1

+β2X1X2

+β3X2

ME+ε [9]

AR = α+β1CashX1

+β2 ln (X1X2

)+β3X2ME

+ε [10]

8 While this is the version attributed to Griliches on p. 28 of McCloskey’s The Writing of Economics, the only version we have been able to track down is “any time series regression containing more than four independent variables results in garbage” in Griliches’ comments on p. 335 of Intriligator and Kendrick [1974].9 For example, if Y = Z1 + Z2 + Z3, W1 = Z1, and W2 = Z2, and if the variance of Z1 is large relative to those of the other Zi, then W2 does have some explanatory power (through Z2) that might be revealed in a multiple regression, but might not be apparent in a simple regression. The reason is that, if the variance Z1 is sufficiently large, then Y = W2 + will have a great deal of noise (due to the presence of Z1 in the error term),while Y = 1W12W2 + provides a more powerful test (because Z1 will be removed from the error term).

7

AR = α+β1ln (CashX1

)+ β2 ln (X1X2

)+ β3 ln (X2ME

) +ε [11]

In an effort to find ratios X1

X2 that would be unrelated to our portfolios’ dependent variable,

.05CashME , we formed ratios with numerators and denominators taken from thirteen Compustat

database variables from 1995 to 2014.10 This gave us 156 ratios, and for our entire dataset we

found the correlation between .05Cash

ME and each ratio X1

X2. We selected the eight ratios that

had the lowest correlations within the entire database with the expectations that they would also have the lowest correlations within the portfolios of 150 firm-years, and so in a simple

regression might be expected to reject a null hypothesis of zero slope for X1

X2 (or its log) at a

frequency equal to the significance level.11 Thus we would expect any statistical significance of X1X2

in a multiple regression to be solely attributable to the “cleaning up” effect described in

Section III. Table 2 shows the results of 1000 portfolios of 150 firm-years each. We analyzed each

portfolio using actual ratios described above (the eight having the lowest correlation with Cash/ME, plus X1 = BE, X2 = TA and X1 = TA, X2 = BE12). Each panel shows the progression for one ratio from simple regression to a multivariate regression featuring logs of all three independent variables.

For example, Panel A of Table 2 features X1 = Operating Income before Depreciation = OIBDP and X2 = Capital Expenditures = CapEx, and is fairly typical. In the simple regression of

the first row, we reject the null hypothesis that the coefficient of X1X2

is zero 11.2% of the time.

This is substantially greater than the 5% we were expecting based on the fact that we chose this

ratio because of its low correlation with CashME for the full sample from which the portfolios

were drawn. It is not clear why this is so much greater than 5%, but may be due to the fact that

10 The variables (with Compustat Data Item numbers) are Total Receivables (DATA2), Total Inventories (DATA3), Total Current Assets (DATA4), Total Current Liabilities (DATA5), Total Assets (DATA6), Gross PP&E (DATA7), Net Sales (DATA12), Operating Income Before Depreciation (DATA13), Depreciation and Amortization (DATA14), Interest Expense (DATA15), PP&E Capital Expenditures (DATA30), Cost of Goods Sold (DATA41), Accounts Payable (DATA70). In all cases we trimmed from any regression any observation that had a negative component of a ratio.11 Alternatively, we can think of these as the eight ratios with the largest p-values. The smallest p-value of this set of eight was depreciation/total inventory, with a p-value of .8259 and a correlation of only -0.00111.12 Note that in Section II, we always used TA in the denominator to conform to common usage. Here, to be consistent with the other eight pairs of ratios, we let it appear in numerator or denominator, depending on whether it is X1 or X2.

8

while such tests are asymptotically well-specified, they may not be well-specified in samples of

only 150. Whether we use OIBDPCapEx or ln(

OIBDPCapEx ) does not make a large difference in overall

rejection rates of H0: = 0, except that OIBDPCapEx ’s overall rejection rate of 11.2% featured 11.1%

with a positive and significant t-statistic, and only .1% with a negative and significant t-statistic,

while the overall rejection rate of 11.9% for ln(OIBDPCapEx ) was more symmetrically distributed,

with a 5.8% frequency of positive rejections and a 6.1% frequency of negative rejections.

Because the distortion in the coefficient of X1

X2 (in this case,

OIBDPCapEx

¿or its log described in

Sections II and III is positive, we focus mainly on the rejections of H0 due to positive and significant t-statistics when we consider the multiple regressions. In the first multiple

regression [9] featuring CashOIBDP ,

OIBDPCapEx , and

CapExME as independent variables, we find the

positive and significant rejection rate for the coefficient of OIBDPCapEx has more than doubled (to

23.7% from 11.1%) when compared with that of the simple regression [8]. When we proceed

to an identical regression [10] except OIBDPCapEx is replaced by ln(

OIBDPCapEx ), this rejection rate

again more than doubles, from 23.7% to 61.0%. Finally, when we take the log of each of the three independent variables [11], this rejection rate increases from 61.0% to 99.3%, or nearly all the time. Thus, in a fashion analogous to the progression from regression [4] to [5] in section II, and consistent with the framework of Section III, we find that a variable that was selected because it had no apparent effect on the dependent variable appears to be significant in a multiple regression (with logs) almost all the time. As before, this is not because the variable is conveying information on its own, but rather because it is “cleaning up” the error created by weak choices for the other two independent variables.

Panel B features the same two variables, but in the opposite order—X1 = Capital Expenditures and X2 = Operating Income before Depreciation—and contains another surprising result. All five rows feature rejection rates that are fairly similar to those of Panel A. Why is

this surprising? Because we would expect any dependent variable that is increasing in OIBDPCapEx

(or its log) to be decreasing in its reciprocal, CapExOIBDP (or its log). However, that is not what the

penultimate multivariate regressions of Panels A and B show. When using ln(OIBDPCapEx ) as the

middle independent variable in Panel A, 61% of the time we find a positive and significant t-statistic, but for the same portfolios we find in Panel B that the dependent variable is also

9

increasing in ln(CapExOIBDP ) and has a positive and significant t-statistic 52.7% of the time. The last

regressions of Panels A and B—featuring logs of all three ratios used as independent variables—are even more damning. Here we have a 99.3% rejection rate suggesting the coefficient of ln(OIBDPCapEx ) is positive, but also a 99.2% rejection rate suggesting the coefficient of ln(

CapExOIBDP ) is

positive. Ln(OIBDPCapEx ) = - ln(

CapExOIBDP ), so clearly if abnormal returns are increasing in one of

these variables, they must be decreasing in the other. What purports to be information regarding those two variables is really indicative of the fact that we made poor choices of

independent variables in CashOIBDP and

CapExME , and also in

CashCapEx and

OIBDPME . It also

emphasizes that coefficients in multiple regressions have meaning only in context; multicollinearity creates a type of entanglement which implies the results do not have any generalizable interpretation, but instead have significance only in the context of the specific forms of the other variables.

Panels C—H in Table 2 are fairly similar, and are summarized in Table 3. Panels I (X1 = Book Equity, X2 = Total Assets) and J (X1 = Total Assets, X2 = Book Equity) are a bit different and merit extra discussion. First of all we note that these are similar to Panels A and B in that together

the multiple regressions [9]—[11] suggest the dependent variable is increasing in Book EquityTotal Assets

and also in Total AssetsBook Equity . As discussed in the last paragraph, this is implausible. Second, the

22.9% rejection rate for the coefficient of Book EquityTotal Assets in the simple regression [8], for

example, suggests that indeed there might be a weak relationship between Cash

Market Equity and

Book EquityTotal Assets . The fact that this rejection rate increases to 94.3% in the multivariate regression

[9] might seem to suggest [9] is a more powerful test. Unfortunately, when the significance levels are substantially misaligned, as Panels A—H show them to be, it is impossible to draw any meaningful inference when the null is rejected. For example, suppose a colleague presents a new test that he alleges is substantially more powerful than the generally accepted method. The only problem, he acknowledges, is that the test is misspecified—at a significance level of = 5%, it actually rejects true null hypotheses at a rate of about 19%. Unbeknownst to you, your colleague’s method is simply to calculate the t-statistic the standard way and multiply it by 1.5 before comparing it with the critical value. This test is obviously more powerful (compared with the standard test, it will reject more frequently when the null is false), but the problem is that the misspecification (rejecting too frequently when the null is true) makes the test results

10

unreliable. So it is with the multiple regressions involving Book Equity and Total Assets in Panels I and J. Because the results of Panels A—H demonstrate substantial misspecification, we cannot draw any meaningful inference from any results of Panels I and J except that the multivariate regressions reject much more frequently than the simple regressions as the discussion in Sections II and III demonstrates they would.

Panel A of Table 3 summarizes the frequency with which the coefficient of X1X2

[or ln(X1X2

)] is

found to have a significant positive t-statistic for the sequence of regressions [8]—[11]. Generally we find that the rejection rate is increasing as we move from a simple regression [8] to a multiple regression which uses the logs of each of the three independent variables [11]. These changes in rejection rates are reported in Panel B of Table 3. For example, for all ten ratios considered, the average rejection rate for simple regressions (Panel A’s 10.1%) increases by 18.0% to 28.1% when we consider a multivariate regression with all three (unlogged)

variables. This rejection rate increases by another 18.9% when we use the log of X1X2

(but not of

the other two variables), and increases by an additional 51.7% (to an average rejection rate of 98.7%) when we take logs of each of the three independent variables.

Not all pairs of X1 and X2, however, produce increases at the same rates. For example, Table 3’s Panel B shows that when we move from the simple, unlogged regression [8] to the multivariate regression [9] (with no logs of any of the three independent variables), the rejection rate for X1 = Book Equity, X2 = Total Assets shows the biggest increase in positive rejections at 71.4% (from 22.9% to 94.3%). In contrast, X1 = Accounts Payable, X2 = Total Receivables features an increase in rejection rates of only 4.2%. What accounts for this difference? Basically, when any independent variable is replaced with another independent variable with which it has a perfect correlation, the rejection rates for the slope coefficient will

be identical.13 Continuity implies that using X1X2

or ln(X1X2

) will produce very similar results when

their correlation is large. We have already seen that the rejection rates when we use logs of all three independent variables are consistently near 100%, and average almost 50% when we take

the log only of X1X2

. Because the correlation between BETA and its log is quite large (0.892),

almost all the bump in rejection rates of H0: 2 = 0 in moving from [8] to [10] occurs in the first

stage, from [8] to [9]. The same is not true of Accounts PayableTotal Receivables , which has a correlation with

its log of only 0.121. For this variable, the total increase in rejection rates of 29.6% from [8] to

13 Two variables W1 and W2 will have a perfect correlation if and only if W1,i = a + bW2,i for every observation i, and so substituting one for the other will change the slope coefficient itself by a factor of b; and will result in identical t-statistics for the slopes. The substitution will leave the intercept unchanged if and only if a = 0.

11

[10] does not come mostly in the move from [8] to [9], but rather in the move from [9] to [10], with its increase of 25.4%. Consistent with this, we find for the ten ratios the correlation

between changes in positive rejection rates one the one hand and the correlation between X1X2

and its log on the other is 0.768. Similarly, if the correlation between X1X2

and its log is a driving

factor here, we should also expect to find that variables with a low correlation get a bigger bump in rejection rates as they move from [9] to [10], and indeed we find the correlation

between changes in rejection rates and the correlation between X1X2

and its log is negative (-

0.392).The other results in Table 3 are also disturbing. We selected the (X1, X2) pairs based on their

low correlations with the dependent variable, and so any method suggesting they are significant—as do the multivariate regressions [9]-[11]—is seriously flawed. However, if the method is flawed, then it can (and does) produce erroneous results even when it is used with seemingly meaningful variables such as the firm’s total assets or book equity. Nevertheless, this method is commonly seen.

How does multicollinearity cause these results? For the relationship in question, for

example, Christie showed that the correct independent variable is VME , and when we instead

used VX1

in regression [9] we not only left out the relevant term ME, but also introduced some

noise in the form of X1. When we added the third variable, X2ME

, we restored the relevant term

ME, but we introduced more noise in the form of X2. Finally, when we added the term X1X2

we

mitigated both noise terms at once. Thus the term X1

X2 serves the same role in regressions [9]—

[11] that W2 serves in the second example of section III. It does not in any way directly explain the dependent variable, but it does reduce the noise that was introduced by the erroneous inclusion of X1 and X2 in the first and third independent variables of [9]—[11].

Panels C and D focus on rejections of H0: = 0 due to any significant t-statistic, whether positive or negative. The results are quite similar to those of Panels A and B.

Tables 2 and 3 raise an interesting issue about the correct choice between W and ln(W) as an independent variable in regressions. Casual observation suggests many finance researchers believe that the reason for using ln(W) is that it will make a skewed distribution of W more symmetric. It is true that the distribution of ln(W) is likely to be more symmetric than that of W if W is skewed, but symmetry of the independent variable’s distribution is not one of the OLS

12

assumptions; the only distributional assumptions made of the independent variable are that it has a positive sample variance and a population variance that is finite (e.g., Kmenta (1997), p. 208). Apart from this, the distributional assumptions required for OLS all pertain to the disturbance term ε , not to the distribution of the independent variable itself (e.g., see Kmenta or Kennedy (2008), p. 41). Of course, it is possible that the distributions of the independent variable and the error term are related, but this is not necessarily so. For example, consider an institution that always has a substantial long position in options. Its return relative for the options division will be heavily skewed, as will its total return relative if options constitute most of its trading. However, there is no reason to suspect the disturbance term in a regression of total return relative on the option division’s return relative would be anything but normally distributed, and taking the log of the independent variable because it is skewed would be inappropriate.

Econometrics texts are not silent on the issue of log transformations; however, the main application has nothing to do with the distribution of an independent variable, but with the functional form. If it is additive, then OLS is used; logs are generally recommended only if the functional form is multiplicative, e.g., a Cobb-Douglas production function. Because our dependent variable is a multiplicative function of the independent variables, it should be no surprise that the t-statistics improve (in our case, misleadingly so) as we move from [9] to [10] to [11].

As a final example before proceeding, consider the DuPont formula, ROE = Return on Equity = Profit Margin*Total Asset Turnover*Equity Multiplier. As it is, the formula is a tautology, but suppose there is some measurement error so that it is only an approximation. If we wanted to see the relative contributions of the right-hand side ratios on Return on Equity for a sample with positive net incomes, it would be inappropriate to estimate parameters of the regression

ROE = + 1*Profit Margin + 2*Total Asset Turnover + 3*Equity Multiplier +

because the relationship is multiplicative, not additive. Instead, it would be more accurate to estimate the parameters of

ln(ROE) = +1*ln(Profit Margin)+2*ln(Total Asset Turnover)+3*ln(Equity Multiplier) + .

The choice of logs has nothing to do with skewness of Profit Margin, Total Asset Turnover, or the Equity Multiplier. It is driven by the correct choice of a model, not on the distribution of the sample observations. Similarly, even if the context were different, the effect of leverage (more

accurately, of 1 – leverage, or BETA ) on ROE is known to be multiplicative, so if ROE is the

13

dependent variable, ln(BETA ) is a better choice for an independent variable in a multivariate

regression than is simply BETA .

Tables 2 and 3 consider a deterministic abnormal return, specifically, AR = ∆MEME =

.05CashME .

In practice, of course, abnormal returns are noisy. Consequently, we extend the noiseless equation [1] to a more practical example in which an error term is present, specifically,

AR = ∆MEME =

.05CashME

+δ . [12]

Now the left side of [12] can be negative, and we can no longer take the log of both sides as we did to get from equation [2] to equation [7]. Nevertheless, the same intuition we developed in the deterministic case will apply here. For example, consider any regression (simple or multiple) and what happens if for every observation a constant is added to the dependent variable. This will affect only the intercept, and will leave the slope(s) unchanged.14 With that in mind, we can replace the dependent variable with its return relative, or one plus the abnormal return. Because the problems we have identified pertain only to slopes, and because the slopes remain unchanged, the difficulties when all abnormal returns are positive will persist when some are negative as well.

The extent of the misspecification problem in the presence of a stochastic error term is an empirical issue, and we address it through simulations. To avoid unnecessary complications, we simulate δas normally distributed with a mean of zero and a standard deviation of 2.5%.15 We find results that are not as strong as those in Table 2, but which nevertheless exemplify the main problem, namely, that variables not involving V or Market Equity may spuriously appear

to be significant. The correct regression AR = α+ β CashME

+e is very powerful. We found the

average t-statistic for β to be 7.39, and we rejected H0: = 0 in all but one of the 1000 simulated portfolios. Table 4 reports estimates the parameters of each of regressions [8]—[11], and Table 5 shows the differences in the rejection rates of H0: = 0 (simple) and H0: 2 = 0 (multivariate).

14 The same is true if we add any constant(s) to one or more independent variables, but we do not use this fact here.15 We chose these values because they are approximately the average parameters estimated by Brown and Warner [1985]. A main alternative would be to simply let ε be the stock’s market-model residual that day, but this would necessarily lead to heteroscedasticity and require the use of weighted least squares or generalized least squares, as pointed out by Karafiath et al [1991]. To avoid this additional complication, we opted for a homoscedastic error term, which allows us to use ordinary least squares.

14

The results are fairly similar to those of Tables 2 and 3, but not as dramatic. Still, when ln(X1/X2) is used, Table 5’s Panel A shows the average rejection rate of H0: 2 = 0 in favor of a positive 2 is 17.6%, substantially larger than the 3.2% rejection rate for the simple regression [8a] featuring ln(X1/X2) as the only independent variable. Even worse, when all three ratios are logged, the rejection rate of H0: 2 = 0 in favor of a positive 2 is 64.6%. The results of Table 5’s

Panels C and D are quite similar; the middle independent variable [X1X2

or ln(X1X2

)] appears to be

significant substantially more often than the 5% significance level or even regression [8a]’s 7.7%, again suggesting that its role in the multivariate regression is primarily “cleaning up” the noise created by using as normalizing variables something other than what Christie shows to be the correct normalizer, market equity.

V. Other Issues

Because it is well-established that leverage is usually associated with an increase in the

dispersion of returns, the reason it provides no marginal explanatory power for ∆MEME above

and beyond what is provided by VME merits a bit of extra explanation. Basically, the familiar

result that additional debt increases equityholders’ risk is based on the assumption that such debt is used to increase the firm’s assets to scale. However, if the increase in debt is not

associated with a proportional increase in risky assets, it is important to use VME rather than

VTA .

As a thought experiment, we consider two cases for a firm that starts as 100% equity and doubles its size by borrowing; in the first case, the firm increases its assets to scale, while in the second it invests the proceeds from the debt in a riskless asset. In the first case, the firm’s equity is indeed more risky. In the second case, however, the issuance of more debt and the purchase of riskless assets form a perfect hedge; the firm’s equityholders are in no more risky a

position than before. Use of VME as the independent variable captures this; in the first case,

equity is more risky because V has doubled, while in the second case, the risk to equityholders

has not changed because V has not changed. Use of DTA (or, equivalently,

BETA ), however, is

misleading because it suggests equityholders’ risk has increased in both cases due to the

increase in debt. Thus VME works correctly in both cases, while

DTA erroneously implies

15

equityholders always bear more risk, even if V remains unchanged. In either case, leverage adds no explanatory power for equityholders’ returns above and beyond that which was

provided by VME

.

The analysis above (and indeed throughout the paper) has been based on the assumption of riskless debt, but we now briefly consider the impact of risky debt. If debt is risky, then ∆V = kV

= ∆ME + ∆D, or equivalently,∆MEME

= kVME

− ∆DME . Now if we estimate the parameters of the

regression

AR = α+β1VME

+β2DME

+ε, [13]

we would expect β1 = k to have a sign opposite that of β2. If k > 0, for example, and the firm’s debt is risky, then debtholders share some part of the gain ∆V. The more debt there is to share that gain, the smaller the gain that accrues to the equityholders, and thus E(β2) < 0.16 This is

superficially the opposite of the result found in [3], where DTA had a positive coefficient, but [3]

assumed riskless debt and used the erroneous variables VTA and

DTA , while here we are

explicitly allowing risky debt and using the correct variables VME and

DME . The result here is

that, if debt is risky, then it mitigates gains and losses to shareholders. This is similar to a consideration of risky debt in the Miller and Miller Capital Structure Proposition; risky debt itself does not create extra risk, but rather shares (and thus mitigates) risk that would otherwise accrue to equityholders. More importantly here, the terms of [13] are normalized by market equity for the same reasons as discussed in Christie [1987] and in the introduction.

While not explicitly stated, we have assumed so far that V is an entry from a market value balance sheet. As a consequence, it could also be some capitalized value from an income statement, and such variables as sales, cost of goods sold, or net income are also candidates for V. In practice, however, market value balance sheets are generally academic fictions. Only book values are typically available, and we simply make the assumption that they are closely related to analogous market-value balance sheet entries. Provided any discrepancies between relevant book-value entries and market-value entries are proportional across firms, this changes the slope coefficients themselves but does not alter the t-statistics in any way. For example, suppose the true dependence of abnormal returns on the variable of interest is c(VMarket Value

ME) for some constant c, but all we can measure is

V BookValue

ME. If for all firms book values

16 A result of β2 = 0 would indicate the debt is riskless, at least as far as changes in V are concerned.

16

are overstated (or understated) by the same proportion d, then the coefficient we measure will

be equal to cd instead of c, but the t-values will be identical.

VI. Extensions

Apart from use of Book Equity and Total Assets as X1 and X2, Tables 2—5 may appear to be contrived, and indeed to some extent they are. However, the problems they identify can occur even in more common settings. Specifically, is not strictly necessary for the regression to include X1 and X2 in multiple places. Suppose, for example, there existed another balance sheet item, A, whose value was a constant proportion of, say, total assets, for every firm, or A i = cTAi. In this case, substitution of Ai for TAi (or vice versa) as a component of any variable would result in beta coefficients that were different by a factor of c, but which would have identical t-statistics. If there is a slight perturbation so that Ai = cTAi + i (where i has a small variance relative to the dispersion of TAi), then substitution of Ai for TAi (or vice versa) in any ratio will produce very similar t-statistics. Since Ai = cTAi + i is essentially a regression of Ai on TAi that is constrained to go through the origin, one measure of how slight the perturbations i really are is the R2 of this constrained regression. While the R2 of a regression with an intercept is uniquely defined, there are no fewer than eight17 common ways of measuring the R2 of a regression constrained to go through the origin. We choose the one SAS uses, 1−∑ ¿¿¿ and dub it the constrained R2 for the remainder of this paper.18 We note in passing that when it comes to the entire ratio that serves as an independent variable, that a substitute have a very large linear correlation is sufficient to produce similar t-statistics for the slope coefficients. However, when we are looking at either the numerator or denominator of an independent variable, a high linear correlation is insufficient to produce comparable results; instead, the constrained correlation must be very large to ensure similar t-statistics. The reason is that a high linear correlation between X1 and X2 implies that there exist a and b such that X2 ≅ a + bX1, but if a ≠ 0 and if X2 appears in a ratio in lieu of X1, there will not be a cancelling out effect because the constant term a will not cancel out. However, if the constrained correlation is large, there will be a cancelling out effect because X2 ≅ bX1 is ensured.19

For the full set of observations, one of the largest constrained correlations was that between Operating Income Before Depreciation and Total Assets, at 0.9741. This suggests that

17 E.g., see Kvalseth (1985).18 In mathematics, Y = X is termed a linear transformation and Y = X an affine transformation, and thus it seems natural to call their measures of R2 linear and affine, respectively. However, the use of the expression “linear correlation” to describe the affine transformation is already widespread, so we resort to this alternative.

17

we should obtain similar results when the value for Total Assets is substituted for one of the values of Operating Income Before Depreciation in Panel A of Tables 2 and 4. We made this substitution and report the results in Table 6.

As Table 6 suggests, the problem of misleading significance is somewhat reduced but nevertheless persists when Total Assets replace Operating Income Before Depreciation in either

the first or third variable (because the middle variables, OIBDPCapEx or

CapExOIBDP are the ones

producing misleading significant slope coefficients, we left them intact). Panels A and B show

the results for the deterministic AR = ∆MEME =

.05CashME , while C and D show them for the

stochastic AR = ∆MEME =

.05CashME

+δ . In the first three rows Panel A (with no substitutions), for

example, frequency of positive and significant rejections that the coefficient of the middle term

(OIBDPCapEx or its logarithm) equals zero range from 23.7% to 61.0% to 99.3% (as we proceed from

taking no logs to taking the log of the middle independent variable only to taking logs of all three independent variables, i.e., regressions [9]—[11]). When we substitute Total Assets for the first independent variable’s denominator of OIBDP in the last three rows, these rejection rates are significantly smaller, proceeding from 11.3% to 22.2% to 73.8%. Still, all are larger than the 5.8% rejection rate we actually found in Table 2, Panel A’s simple regression [8a], AR =

α+β ln (OIBDPCapEx

)+ε . A similar result is found in Panel B of Table 6, which uses CapExOIBDP (or its

logarithm) as the middle variable. The original frequencies of positive and significant t-values for [9]—[11] of 24.1%, 52.7%, and 99.2% drop to 14.2%, 33.7%, and 84.9% when Total Assets are substituted for Operating Income Before Depreciation in the numerator of the third independent variable. Again, while the rejection rates are somewhat lower after the substitution is made, all are substantially larger than the 6.1% we found for the actual simple regression [8a] in Panel B of Table 2. When the abnormal return AR is not deterministic and

instead includes an error term, Panel C of Table 6 shows similar results: for OIBDPCapEx (or its

logarithm) the original positive and significant rejection rates of 9.5%, 20.4%, and 50.3% become 5.2%, 9.9%, and 24.0% after the substitution is made. All exceed the 3.4% rejection

19 For example, suppose X ~U[5, 10]. The correlation between X and the simple translation 6.02*1023 + X is 1, and

yet, because 6.02*1023 dwarfs X, 1

6.02∗1023+X will have approximately the same value for any values of X that

are orders of magnitude smaller, and so will not produce the desired “cancelling out” effect with any X that may appear in a numerator. A high constrained R2, however, will necessarily produce a cancelling out effect. A similar result would occur if X in a numerator is replaced with 6.02*1023 + X.

18

rate for Table 4, Panel A’s simple regression [8a], the last two by substantial amounts. Panel D of Table 6 shows similar results.

In all cases, Table 6 suggests the problem of distorted t-statistics stemming from the fact that an irrelevant independent variable may “clean up” poor measures of other (relevant) independent variables can remain even when the irrelevant component appears in only one ratio. Provided they have a high constrained correlation, one variable may be substituted for another in either a numerator or a denominator and produce deceptive significance levels. This can be a substantial problem for regressions with a large number of independent variables, as any set of three or more independent variables can combine in such a way that one (or more) of them appear to be significant even though their only role is cleaning up noise created by poor choices of other independent variables.20

VII. Conclusions

Christie [1987] showed that market equity [ME] is the correct scaling variable for any cross-sectional regression of abnormal returns on firm characteristics, but many researchers scale by such variables as Total Assets or Book Equity instead. We demonstrate that the apparent significance of the resulting variables can be a mathematical artifact that is unrelated to their true significance. Not only can true (and sometimes clearly so) null hypotheses frequently be rejected, but depending on the initial setup, they can be rejected in favor of contradictory alternatives. Moreover, because the effect can occur whenever two variables have a high constrained R2, detection of false inferences is quite difficult. One (imperfect) step towards confirming coefficients in a multivariate regression are truly what we purport them to be is to test them in simple regressions as well. These results also highlight the importance of having a specific model and using it to determine the exact variables and their appropriate form rather than let these choices be made by the data. Finally, we can conclude it is inappropriate to add independent variables based on the assumptions that they might matter, and if they don’t they will not cause any damage. They may well cause damage by creating (or perhaps resolving) needless noise that makes them and other irrelevant variables appear to be significant. Estimates of multivariate regressions have meaning only in the context of all the independent variables selected, and adding, deleting, or changing variables can completely alter the meaning of other variables’ coefficients.

Moreover, while our framework is based on event-study cross-sectional regressions, this condition was only invoked because in this setting Christie has identified the correct functional

20 For example, a regression with 10 independent variables will have 210−11−(102 )=968 subsets of three or

more independent variables, any of which can be plagued by this problem.

19

form. The general principle could be shown to apply to other cross-sectional regressions as well, but a proof would require that we knew the correct functional form. While such a correct functional form may exist, we will rarely know what it is; nevertheless, its existence implies that estimates of other multiple regressions will be subject to the same problem we have identified for event-study cross-sectional regressions. Thus this paper provides evidence in support of Griliches’ Law: “Any cross-sectional regression with more than five variables produces garbage.” The more independent variables that appear in a regression, the greater the chance that some subset of them will combine to make an irrelevant variable appear to be significant because it is “cleaning up” the noise created by an incorrect choice of other variables.

References

Brown, S., and J. Warner, 1985, Using daily stock returns: The case of event studies,” Journal of Financial Economics, 14(1): 3—31.

Christie, A. A., 1987, On Cross-Sectional Analysis in Accounting Research, Journal of Accounting and Economics, 9(3):231—258.

Goldman, W., 1973, The Princess Bride, Harcourt Brace Jovanovich (San Diego).

Griliches, Z, and N. Wallace, 1965, The Determinants of Investment Reinvestigated, International Economic Review, 6(3): 311—329.

Kendrick, D., and M. Intriligator, 1974, Frontiers of quantitative economics: Papers invited for presentation at the Econometric Society Winter Meetings, New York, 1969 [and] Toronto, 1972. Vol. 2. North-Holland Pub. Co.

Karafiath, I., R. Mynatt, and K. Smith, 1991, The Brazilian default announcement and the contagion effect hypothesis, Journal of Banking and Finance, 15(3): 699—716.

Kennedy, P., 2008, A Guide to Econometrics, 6th Edition, Blackwell Publishing.

Kmenta, J., 1997, Elements of Econometrics, Second Edition, The University of Michigan Press.

Kvalseth, T., 1985, Cautionary Note about R2, The American Statistician, 39(4): 279-285.

McCloskey, D., 1987, The Writing of Economics, MacMillan Publishing Co., New York.

20

Mitchell, D., 1991, Invariance of Results Under a Common Orthogonalization, Journal of Economics and Business, 43: 193-196.

21

Table 1—Summary of successive regression forms from equations (3)—(6) and related multivariate regressions.

MultivariateRegression Constant

(t-statistic)Coefficient (t-statistic) of Cash/TA

or ln(Cash/TA)

Coefficient (t-statistic)

of D/TA, BE/TA, or ln(BE/TA)

Coefficient (t-statistic) of BE/ME

or ln(BE/ME)

[3].05Cash/ME = + 1(Cash/TA) +

2(D/TA) + 3(BE/ME)-.0090(-3.46)

.0386(8.14)

.0143(3.45)

.0098(5.59)

[4].05Cash/ME = + 1(Cash/TA) +

2(BE/TA) + 3(BE/ME).0040(2.18)

.0431(9.02)

-.0186(-4.83)

.0107(6.23)

[5].05Cash/ME = + 1ln(Cash/TA) +

2ln(BE/TA) + 3ln(BE/ME).0185

(12.92).0040(9.88)

-.0061(-6.62)

.0068(7.60)

[6]Ln(.05Cash/ME) = + 1ln(Cash/TA) +

2ln(BE/TA) + 3ln(BE/ME)-2.9957(−∞)

1.000(∞)

-1.000(−∞)

1.000(∞)

In the multivariate regressions, t-values for the three variable’s coefficients improve as we move from

using DebtTA [equation (3)] to

BETA [equation (4)], and then improve more as when we take natural

logarithms of the three independent variables [equation (5)], and finally become infinite when we take the logarithm of the dependent variable as well [equation (6)]. The multivariate t-statistics also generally improve [except when we take logarithms to move from (4) to (5)], but not as dramatically.

22

Table 2: Simulated Portfolios for various choices of X1 and X2 in sequence of regressions from

.05(Cash)ME

= α+βX1X2

to .05(Cash)ME

= α+β1ln ( VX1

)+ β2 ln(X1X2

)+β3 ln (X2ME

).

Panel A: X1 = Operating Income Before Depreciation, X2 = Capital Expenditures

Independent Variables

CashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1X2

only

[8]

(0.50)[0.112] <0.111>{0.516}

ln (X1X2

) only

[8a]

(-0.10)[0.119]<0.058>{0.437}

All three (unlogged)

[9]

(4.18)[0.829]<0.828>{0.053}

(1.36)[0.237]<0.237>{0.367}

(4.15)[0.604]<0.603>{0.176}

All three

(log of X1X2

only)[10]

(4.76)[0.882]<0.881>{0.028}

(2.48)[0.610]<0.610>{0.138}

(4.88)[0.704]<0.704>{0.100}

All three logged

[11]

(10.87)[1.000]<1.000>{0.000}

(6.16)[0.993]<0.993>{0.001}

(7.78)[1.000]<1.000>{0.000}

(average t) [total rejection rate] <positive and significant rejection rate>{average p-value}

23

Table 2, Panel B: X1 = Capital Expenditures, X2 = Operating Income Before DepreciationIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.51)[0.104]<0.103>{0.500}

ln (X1X2

) only

[8a]

(0.10)[0.119]<0.061>{0.437}


[9]

(4.48)[0.885]<0.885>{0.039}

(1.31)[0.242]<0.241>{0.362}

(5.83)[0.754]<0.754>{0.086}

All three

(log of X1

X2

only)[10]

(4.86)[0.898]<0.898>{0.031}

(2.03)[0.531]<0.527>{0.186}

(5.96)[0.775]<0.775>{0.082}

All three logged

[11]

(10.87)[1.000]<1.000>{0.000}

(7.09)[0.992]<0.992>{0.002}

(7.78)[1.000]<1.000>{0.000}


24

Table 2, Panel C: X1 = Accounts Payable, X2 = Total ReceivablesIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.32)[0.082]<0.081>{0.575}

ln (X1X2

) only

[8a]

(0.28)[0.099]<0.079>{0.453}


[9]

(3.79)[0.799]<0.799>{0.059}

(0.77)[0.127]<0.123>{0.519}

(4.62)[0.722]<0.722>{0.105}

All three

(log of X1

X2

only)[10]

(4.08)[0.839]<0.839>{0.044}

(1.63)[0.377]<0.377>{0.243}

(4.77)[0.748]<0.748>{0.091}

All three logged

[11]

(11.14)[1.000]<1.000>{0.000}

(7.48)[1.000]<1.000>{0.000}

(9.30)[1.000]<1.000>{0.000}


25

Table 2, Panel D: X1 = Depreciation and Amortization, X2 = Total InventoryIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.07)[0.061]<0.061>{0.542}

ln (X1X2

) only

[8a]

(0.33)[0.100]<0.075>{0.460}


[9]

(3.76)[0.807]<0.807>{0.053}

(0.63)[0.115]<0.115>{0.531}

(3.49)[0.488]<0.488>{0.217}

All three

(log of X1

X2

only)[10]

(4.09)[0.873]<0.873>{0.033}

(2.00)[0.482]<0.482>{0.172}

(4.09)[0.626]<0.626>{0.129}

All three logged

[11]

(10.40)[1.000]<1.000>{0.000}

(8.09)[1.000]<1.000>{0.000}

(8.73)[1.000]<1.000>{0.000}


26

Table 2, Panel E: X1 = Inventory, X2 = Cost of Goods SoldIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(-0.30)[0.077]<0.045>{0.475}

ln (X1X2

) only

[8a]

(-0.30)[0.099]<0.018>{0.452}


[9]

(2.47)[0.550]<0.550>{0.154}

(0.75)[0.142]<0.136>{0.490}

(4.43)[0.619]<0.618>{0.158}

All three

(log of X1

X2

only)[10]

(2.58)[0.596]<0.596>{0.138}

(0.97)[0.200]<0.194>{0.360}

(4.48)[0.630]<0.630>{0.151}

All three logged

[11]

(10.44)[1.000]<1.000>{0.000}

(6.51)[0.994]<0.994>{0.001}

(8.96)[1.000]<1.000>{0.000}


27

Table 2, Panel F: X1 = Interest Expense, X2 = Cost of Goods SoldIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.40)[0.119]<0.116>{0.484}

ln (X1X2

) only

[8a]

(-0.14)[0.096]<0.034>{0.449}


[9]

(1.48)[0.271]<0.271>{0.313}

(1.01)[0.205]<0.205>{0.443}

(4.62)[0.639]<0.639>{0.141}

All three

(log of X1

X2

only)[10]

(1.53)[0.285]<0.285>{0.283}

(0.620)[0.166]<0.142>{0.398}

(4.57)[0.626]<0.626>{0.148}

All three logged

[11]

(10.14)[1.000]<1.000>{0.000}

(8.16)[1.000]<1.000>{0.000}

(8.97)[1.000]<1.000>{0.000}

(average t) [total rejection rate] <positive and significant rejection rate>|negative and significant rejection rate|{average p-value}

28

Table 2, Panel G: X1 = Capital Expenditures, X2 = Accounts PayableIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(-0.27)[0.067]<0.044>{0.470}

ln (X1X2

) only

[8a]

(-0.90)[0.178]<0.009>{0.366}


[9]

(3.30)[0.707]<0.707>{0.102}

(0.46)[0.093]<0.092>{0.525}

(4.27)[0.621]<0.621>{0.159}

All three

(log of X1

X2

only)[10]

(3.44)[0.737]<0.737>{0.082}

(1.06)[0.259]<0.249>{0.345}

(4.43)[0.644]<0.644>{0.142}

All three logged

[11]

(11.36)[1.000]<1.000>{0.000}

(7.55)[0.999]<0.999>{0.000}

(9.19)[1.000]<1.000>{0.000}


29

Table 2, Panel H: X1 = Capital Expenditures, X2 = Interest ExpenseIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.19)[0.042]<0.042>{0.657}

ln (X1X2

) only

[8a]

(-0.50)[0.107]<0.013>{0.432}


[9]

(3.45)[0.728]<0.728>{0.090}

(0.60)[0.079]<0.079>{0.576}

(5.58)[0.724]<0.724>{0.107}

All three

(log of X1

X2

only)[10]

(3.69)[0.789]<0.789>{0.066}

(1.43)[0.302]<0.301>|{0.282}

(5.85)[0.769]<0.769>{0.081}

All three logged

[11]

(10.41)[1.000]<1.000>{0.000}

(7.19)[0.999]<0.999>{0.000}

(8.53)[1.000]<1.000>{0.000}


30

Table 2, Panel I: X1 = Book Equity, X2 = Total AssetsIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.94)[0.253]<0.229>{0.326}

ln (X1X2

) only

[8a]

(-0.03)[0.228]<0.097>{0.349}


[9]

(6.38)[0.920]<0.918>{0.024}

(4.26)[0.943]<0.943>{0.015}

(7.74)[0.988]<0.988>{0.003}

All three

(log of X1

X2

only)[10]

(7.17)[0.973]<0.972>{0.007}

(4.54)[0.946]<0.945>{0.014}

(7.99)[0.990]<0.990>{0.002}

All three logged

[11]

(10.88)[1.000]<1.000>{0.000}

(4.53)[0.909]<0.906>{0.029}

(9.41)[1.000]<1.000>{0.000}


31

Table 2, Panel J: X1 = Total Assets, X2 = Book Equity Independent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.85)[0.188]<0.179>{0.442}

ln (X1X2

) only

[8a]

(0.03)[0.228]<0.131>{0.349}


[9]

(9.54)[1.000]<1.000>{0.000}

(3.30)[0.643]<0.643>{0.112}

(9.27)[0.999]<0.999>{0.000}

All three

(log of X1

X2

only)[10]

(9.95)[1.000]<1.000>{0.000}

(3.87)[0.872]<0.872>|{0.034}

(9.44)[0.999]<0.999>{0.000}

All three logged

[11]

(10.88)[1.000]<1.000>{0.000}

(5.72)[0.988]<0.988>{0.002}

(9.41)[1.000]<1.000>{0.000}


32

Table 3: Summary of Rejection Rates of the Coefficient of X1/X2, Deterministic ARs

Panel A: Only positive rejections (t-statistic > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.

A/P Deprec. Total Inventory

Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold

A/P Interest Expense

Total Assets

Book Equity

Simple, not logged [8] 0.111 0.103 0.081 0.061 0.045 0.116 0.044 0.042 0.229 0.179 0.101

Simple, logged [8a] 0.058 0.061 0.079 0.075 0.018 0.034 0.009 0.013 0.097 0.131 0.058

Multivariate, none logged

[9] 0.237 0.241 0.123 0.115 0.136 0.205 0.092 0.079 0.943 0.643 0.281Multivariate,

only X1/X2

logged [10] 0.61 0.527 0.377 0.482 0.194 0.142 0.249 0.301 0.945 0.872 0.470Multivariate,

all logged [11] 0.993 0.992 1 1 0.994 1 0.999 0.999 0.906 0.988 0.987

Table 3, Panel B: Increases in rejection rates: Only positive rejections (t-statistic > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


minus Simple, not

logged 0.126 0.138 0.042 0.054 0.091 0.089 0.048 0.037 0.714 0.464 0.180Multivariate,

only X1/X2

logged minus

Multivariate, none logged 0.373 0.286 0.254 0.367 0.058 -0.063 0.157 0.222 0.002 0.229 0.189Multivariate,

all logged minus

Multivariate, only X1/X2

logged 0.383 0.465 0.623 0.518 0.8 0.858 0.75 0.698 -0.039 0.116 0.517Correlation

between X1/X2 and ln(X1/X2) 0.065 0.071 0.121 0.253 0.173 0.069 0.169 0.162 0.892 0.11

34

Table 3, Panel C: All rejections (|t-statistic| > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


Simple, logged [8a] 0.119 0.119 0.099 0.1 0.099 0.096 0.178 0.107 0.228 0.228 0.137


[9] 0.237 0.242 0.127 0.115 0.142 0.205 0.093 0.079 0.943 0.643 0.283Multivariate,

only X1/X2


all logged [11] 0.993 0.992 1 1 0.994 1 0.999 0.999 0.909 0.988 0.987

35

Table 3, Panel D: Increases in rejection rates (all |t-statistic| > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


minus Simple, not


only X1/X2

logged minus


all logged minus




36

Table 4: Simulated Portfolios for various choices of X1 and X2 in sequence of regressions from

.05(Cash)ME

+δ = α+βX1X2

+ε to

.05(Cash)

ME+δ = α+β1ln ( VX1 )+β2ln( X1X2 )+ β3 ln( X2

ME )+ε .

Panel A: X1 = Operating Income Before Depreciation, X2 = Capital Expenditures

Independent Variables

CashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.21)[0.076]<0.059>{0.501}

ln (X1X2

) only

[8a]

(-0.02)[0.070]<0.034>{0.488}


[9]

(1.47)[0.326]<0.324>{0.269}

(0.54)[0.102]<0.095>{0.453}

(1.72)[0.314]<0.310>{0.323}

All three

(log of X1

X2

only)[10]

(1.67)[0.398]<0.398>{0.234}

(0.99)[0.209]<0.204>{0.360}

(1.96)[0.352]<0.348>{0.286}

All three logged

[11]

(3.30)[0.870]<0.870>{0.028}

(2.02)[0.503]<0.503>{0.186}

(2.56)[0.619]<0.619>{0.121}


Table 4, Panel B: X1 = Capital Expenditures, X2 = Operating Income Before DepreciationIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.19)[0.074]<0.053>{0.494}

ln (X1X2

) only

[8a]

(0.02)[0.070]<0.036>{0.488}


[9]

(1.58)[0.322]<0.322>{0.242}

(0.45)[0.088]<0.074>{0.452}

(2.33)[0.419]<0.418>{0.254}

All three

(log of X1

X2

only)[10]

(1.68)[0.382]<0.382>{0.224}

(0.68)[0.132]<0.126>{0.413}

(2.35)[0.423]<0.423>{0.249}

All three logged

[11]

(3.30)[0.870]<0.870>{0.029}

(2.15)[0.548]<0.548>{0.139}

(2.56)[0.619]<0.619>{0.121}


38

Table 4, Panel C: X1 = Accounts Payable, X2 = Total ReceivablesIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.13)[0.063]<0.049>{0.484}

ln (X1X2

) only

[8a]

(0.15)[0.081]<0.051>{0.480}


[9]

(1.73)[0.392]<0.392>{0.239}

(0.34)[0.090]<0.078>{0.468}

(2.32)[0.403]<0.400>{0.271}

All three

(log of X1

X2

only)[10]

(1.86)[0.436]<0.436>{0.212}

(0.78)[0.152]<0.144>{0.411}

(2.39)[0.408]<0.406>{0.259}

All three logged

[11]

(4.29)[0.975]<0.975>{0.006}

(2.92)[0.769]<0.769>{0.057}

(3.67)[0.896]<0.896>{0.027}


39

Table 4, Panel D: X1 = Depreciation and Amortization, X2 = Total InventoryIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.05)[0.046]<0.033>{0.534}

ln (X1X2

) only

[8a]

(0.15)[0.063]<0.040>{0.486}


[9]

(1.54)[0.338]<0.338>{0.254}

(0.31)[0.054]<0.046>{0.501}

(1.70)[0.272]<0.270>{0.338}

All three

(log of X1

X2

only)[10]

(1.67)[0.379]<0.379>{0.229}

(0.91)[0.175]<0.170>{0.386}

(1.93)[0.330]<0.328>{0.304}

All three logged

[11]

(3.66)[0.924]<0.924>{0.017}

(2.97)[0.787]<0.787>{0.064}

(3.26)[0.818]<0.818>{0.049}


40

Table 4, Panel E: X1 = Inventory, X2 = Cost of Goods SoldIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(-0.19)[0.070]<0.028>{0.491}

ln (X1X2

) only

[8a]

(-0.20)[0.062]<0.014>{0.481}


[9]

(1.08)[0.211]<0.209>{0.364}

(0.29)[0.080]<0.062>{0.482}

(2.16)[0.351]<0.347>{0.293}

All three

(log of X1

X2

only)[10]

(1.11)[0.224]<0.223>{0.358}

(0.39)[0.074]<0.066>{0.471}

(2.18)[0.353]<0.349>{0.286}

All three logged

[11]

(3.70)[0.931]<0.931>{0.016}

(2.27)[0.605]<0.605>{0.121}

(3.28)[0.838]<0.838>{0.047}


41

Table 4, Panel F: X1 = Interest Expense, X2 = Cost of Goods SoldIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.23)[0.077]<0.059>{0.477}

ln (X1X2

) only

[8a]

(-0.01)[0.064]<0.029>{0.477}


[9]

(0.64)[0.099]<0.094>{0.438}

(0.49)[0.111]<0.102>{0.455}

(2.32)[0.386]<0.383>{0.278}

All three

(log of X1

X2

only)[10]

(0.68)[0.115]<0.106>{0.425}

(0.32)[0.099]<0.076>{0.452}

(2.30)[0.379]<0.376>{0.281}

All three logged

[11]

(3.83)[0.944]<0.944>{0.016}

(3.14)[0.824]<0.824>{0.054}

(3.48)[0.859]<0.859>{0.038}

(average t) [total rejection rate] <positive and significant rejection rate>|negative and significant rejection rate|{average p-value}

42

Table 4, Panel G: X1 = Capital Expenditures, X2 = Accounts PayableIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(-0.17)[0.056]<0.030>{0.492}

ln (X1X2

) only

[8a]

(-0.46)[0.093]<0.009>{0.449}


[9]

(1.54)[0.365]<0.362>{0.269}

(0.20)[0.069]<0.056>{0.500}

(2.108)[0.379]<0.372>{0.276}

All three

(log of X1

X2

only)[10]

(1.61)[0.389]<0.386>{0.251}

(0.53)[0.101]<0.091>{0.438}

(2.25)[0.379]<0.375>{0.269}

All three logged

[11]

(4.39)[0.979]<0.979>{0.005}

(2.97)[0.783]<0.783>{0.054}

(3.68)[0.897]<0.897>{0.027}


43

Table 4, Panel H: X1 = Capital Expenditures, X2 = Interest ExpenseIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.04)[0.035]<0.027>{0.522}

ln (X1X2

) only

[8a]

(-0.30)[0.071]<0.013>{0.471}


[9]

(1.38)[0.311]<0.309>{0.291}

(0.21)[0.048]<0.041>{0.500}

(2.54)[0.432]<0.425>{0.249}

All three

(log of X1

X2

only)[10]

(1.47)[0.326]<0.325>{0.269}

(0.59)[0.098]<0.095>{0.444}

(2.62)[0.451]<0.447>{0.229}

All three logged

[11]

(3.86)[0.947]<0.947>{0.013}

(2.74)[0.737]<0.737>{0.071}

(3.30)[0.830]<0.830>{0.045}


44

Table 4, Panel I: X1 = Book Equity, X2 = Total AssetsIndependent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.37)[0.100]<0.081>{0.446}

ln (X1X2

) only

[8a]

(-0.11)[0.098]<0.039>{0.425}


[9]

(2.46)[0.646]<0.643>{0.126}

(1.66)[0.394]<0.394>{0.222}

(3.14)[0.672]<0.672>{0.114}

All three

(log of X1

X2

only)[10]

(2.76)[0.712]<0.711>{0.094}

(1.77)[0.445]<0.445>{0.209}

(3.22)[0.697]<0.697>{0.106}

All three logged

[11]

(3.88)[0.963]<0.963>{0.007}

(1.56)[0.367]<0.363>{0.242}

(3.46)[0.886]<0.886>{0.029}


45

Table 4, Panel J: X1 = Total Assets, X2 = Book Equity Independent

VariablesCashX1

(or its log)

X1X2

(or its log)

X2ME

(or its log)

X1

X2 only

[8]

(0.49)[0.133]<0.115>{0.438}

ln (X1X2

) only

[8a]

(0.11)[0.098]<0.059>{0.425}


[9]

(3.38)[0.898]<0.898>{0.025}

(1.27)[0.263]<0.257>{0.332}

(3.45)[0.775]<0.775>{0.066}

All three

(log of X1

X2

only)[10]

(3.53)[0.923]<0.923>{0.021}

(1.48)[0.343]<0.340>{0.276}

(3.49)[0.782]<0.782>{0.062}

All three logged

[11]

(3.88)[0.963]<0.963>{0.007}

(2.16)[0.542]<0.542>{0.159}

(3.46)[0.886]<0.886>{0.029}


46

Table 5: Summary of Rejection Rates of the Coefficient of X1/X2, Noisy ARs

Panel A: Only positive rejections (t-statistic > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend

.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


Simple, logged [8a] 0.034 0.036 0.051 0.04 0.014 0.029 0.009 0.013 0.039 0.059 0.032


[9] 0.095 0.074 0.078 0.046 0.062 0.102 0.056 0.041 0.394 0.257 0.121Multivariate,

only X1/X2


all logged [11] 0.503 0.548 0.769 0.787 0.605 0.824 0.783 0.737 0.363 0.542 0.646

Table 5, Panel B: Increases in rejection rates: Only positive rejections (t-statistic > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


minus Simple, not


only X1/X2

logged minus


all logged minus




48

Table 5, Panel C: All rejections (|t-statistic| > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


Simple, logged [8a] 0.07 0.07 0.081 0.063 0.062 0.064 0.093 0.071 0.098 0.098 0.077


[9] 0.102 0.088 0.09 0.054 0.08 0.111 0.069 0.048 0.394 0.263 0.130Multivariate,

only X1/X2


all logged [11] 0.503 0.548 0.769 0.787 0.605 0.824 0.783 0.737 0.363 0.542 0.646

49

Table 5, Panel D: Increases in rejection rates (all |t-statistic| > 1.96)A B C D E F G H I J average

X1 Oper. Inc.

before Deprec.

Capital Expend.


Interest Expense

Capital Expend.

Capital Expend.

Book Equity

Total Assets

X2 Capital Expend.

Oper. Inc.

before Deprec.

Total Receivables

Total Inventory

Cost of Goods Sold

Cost of Goods Sold


Total Assets

Book Equity


minus Simple, not


only X1/X2

logged minus

Multivariate, none logged 0.107 0.044 0.062 0.121 -0.006 -0.035 0.032 0.05 0.051 0.08 0.051Multivariate,

all logged minus




50

Table 6: A Comparison of Results when Substitutions Occur

Panel A: Dependent Variable = .05(Cash)ME

Regression R1 R2 R3 t-statistics for β1 t-statistics for β2 t-statistics for β3

[9]CashOIBDP

OIBDPCapEx

CapExME

(4.18)[0.829]<0.828>{0.053}

(1.36)[0.237]<0.237>{0.367}

(4.15)[0.604]<0.603>{0.176}

[10]CashOIBDP

ln (OIBDPCapEx

) CapExME

(4.76)[0.882]<0.881>{0.028}

(2.48)[0.610]<0.610>{0.138}

(4.88)[0.704]<0.704>{0.100}

[11]ln ( Cash

OIBDP) ln (OIBDP

CapEx) ln (CapEx

ME) (10.87)

[1.000]<1.000>{0.000}

(6.16)[0.993]<0.993>{0.001}

(7.78)[1.000]<1.000>{0.000}

[9]CashTA

OIBDPCapEx

CapExME

(6.60)[0.990]<0.990>{0.002}

(0.47)[0.131]<0.113>{0.461}

(5.34)[0.801]<0.801>{0.048}

[10]CashTA

ln (OIBDPCapEx

) CapExME

(6.62)[0.990]<0.990>{0.003}

(0.86)[0.244]<0.222>{0.353}

(5.32)[0.791]<0.791>{0.056}

[11]CashTA

ln (OIBDPCapEx

) CapExME

(7.62)[1.000]<1.000>{0.000}

(2.86)[0.739]<0.738>{0.093}

(5.06)[0.971]<0.971>{0.006}


51

Table 6, Panel B: Dependent Variable = .05(Cash)ME


[9]CashCapEx

CapExOIBDP

OIBDPME

(4.48)[0.885]<0.885>{0.039}

(1.31)[0.242]<0.241>{0.362}

(5.83)[0.754]<0.754>{0.086}

[10]CashCapEx

ln ( CapExOIBDP

) OIBDPME

(4.86)[0.898]<0.898>{0.031}

(2.03)[0.531]<0.527>{0.186}

(5.96)[0.775]<0.775>{0.082}

[11]ln ( Cash

CapEx) ln ( CapEx

OIBDP) ln (OIBDP

ME) (10.87)

[1.000]<1.000>{0.000}

(7.09)[0.992]<0.992>{0.002}

(7.78)[1.000]<1.000>{0.000}

[9]CashCapEx

CapExOIBDP

TAME

(4.60)[0.893]<0.893>{0.035}

(0.62)[0.164]<0.142>{0.437}

(7.53)[0.925]<0.925>{0.021}

[10]CashCapEx

ln ( CapExOIBDP

) TAME

(4.73)[0.890]<0.890>{0.040}

(1.29)[0.351]<0.337>{0.291}

(7.51)[0.923]<0.923>{0.022}

[11]ln ( Cash

CapEx) ln ( CapEx

OIBDP) ln ( TA

ME) (9.39)

[0.999]<0.999>{0.000}

(3.71)[0.852]<0.849>{0.047}

(7.40)[1.000]<1.000>{0.000}


52

Table 6, Panel C: Dependent Variable = .05(Cash)ME

+δ ,δ normally distributed with mean 0 and standard deviation 0.025


[9]CashOIBDP

OIBDPCapEx

CapExME

(1.47)[0.326]<0.324>{0.269}

(0.54)[0.102]<0.095>{0.453}

(1.72)[0.314]<0.310>{0.323}

[10]CashOIBDP

ln (OIBDPCapEx

) CapExME

(1.67)[0.398]<0.398>{0.234}

(0.99)[0.209]<0.204>{0.360}

(1.96)[0.352]<0.348>{0.286}

[11]ln ( Cash

OIBDP) ln (OIBDP

CapEx) ln (CapEx

ME) (3.30)

[0.870]<0.870>{0.028}

(2.02)[0.503]<0.503>{0.186}

(2.56)[0.619]<0.619>{0.121}

[9]CashTA

OIBDPCapEx

CapExME

(2.24)[0.605]<0.604>{0.118}

(0.20)[0.070]<0.052>{0.478}

(2.04)[0.379]<0.378>{0.271}

[10]CashTA

ln (OIBDPCapEx

) CapExME

(2.23)[0.594]<0.593>{0.122}

(0.398)[0.110]<0.099>{0.447}

(2.04)[0.379]<0.376>{0.272}

[11]CashTA

ln (OIBDPCapEx

) CapExME

(2.64)[0.738]<0.738>{0.068}

(1.14)[0.241]<0.240>{0.337}

(1.95)[0.443]<0.443>{0.211}


53

Table 6, Panel D: Dependent Variable = .05(Cash)ME

+δ , δ normally distributed with mean 0 and standard deviation 0.025


[9]CashCapEx

CapExOIBDP

OIBDPME

(1.58)[0.322]<0.322>{0.242}

(0.45)[0.088]<0.074>{0.452}

(2.33)[0.419]<0.418>{0.254}

[10]CashCapEx

ln ( CapExOIBDP

) OIBDPME

(1.68)[0.382]<0.382>{0.224}

(0.68)[0.132]<0.126>{0.413}

(2.35)[0.423]<0.423>{0.249}

[11]ln ( Cash

CapEx) ln ( CapEx

OIBDP) ln (OIBDP

ME) (3.30)

[0.870]<0.870>{0.029}

(2.15)[0.548]<0.548>{0.139}

(2.56)[0.619]<0.619>{0.121}

[9]CashCapEx

CapExOIBDP

TAME

(1.55)[0.318]<0.318>{0.246}

(0.17)[0.077]<0.045>{0.473}

(2.78)[0.500]<0.498>{0.211}

[10]CashCapEx

ln ( CapExOIBDP

) TAME

(1.56)[0.349]<0.348>{0.246}

(0.38)[0.093]<0.077>{0.452}

(2.77)[0.499]<0.498>{0.212}

[11]ln ( Cash

CapEx) ln ( CapEx

OIBDP) ln ( TA

ME) (2.82)

[0.768]<0.768>{0.052}

(1.05)[0.183]<0.180>{0.353}

(2.47)[0.629]<0.629>{0.129}


54

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