# How to interpret Whited Wu index (WW-index)

I want to study financial constraints on companies of different groups and for that I try to use Whited-Wu index: WW = - 0.091CF - 0.062DIVPOS + 0.021*TLTD - 0.044*LNTA + 0.102*ISG - 0.035*SG Where TLTD is the ratio of the long-term debt to total assets; DIVPOS is an indicator that takes the value of one if the firm pays cash dividends; SG is firm sales growth; LNTA is the natural log of total assets; ISG is the firm’s three-digit industry sales growth; CASH is the ratio of liquid assets to total assets; CF is the ratio of cash flow to total assets (see Whited and Wu. Financial Constraints Risk. 2006).

Although I managed to compute it, I have difficulties with results interpretation. My idea was to compare different companies/groups of firms using this index. As written in one of the papers, "higher index values can be associated to higher need of external capital". So if in 2010, for "Toyota" I got value -0.9 and for "Fuji Technica Inc." -0.53, may I conclude that Toyota is more financially constrained (requires more external capital)? Similarly, if I get two industries - food and car producing - then the first have higher values of WW.

This is equation $(13)$ of Whited, T. M., & Wu, G. (2006). Financial constraints risk. Review of Financial Studies, 19(2), 531-559.

It is empirically estimated as regards the specific coefficient values. The important question is,

What is the left-hand-side?

Looking at eq. $(12)$ of the paper the left-hand-side of $(13)$ is $\lambda_{i,t+1}$ which in turn is part of

$$\Lambda_{i,t+1} = \frac {1+\lambda_{i,t+1}}{1+\lambda_{i,t}}$$

$\Lambda_{i,t+1}$ is discussed in the middle of p. 536 (between eq. 5 and eq. 6). As the authors write,

If the outside equity constraint is binding, the effects of external finance constraints show up in the term $\Lambda_{i,t+1} = (1+\lambda_{i,t+1})/(1+\lambda_{i,t})$, which is the relative shadow cost of external finance. In the absence of finance constraints, $\Lambda_{i,t+1}=1$. On the other hand, if the equity constraint binds, then generally $\Lambda_{i,t+1}\neq1$, unless $\lambda_{i,t+1} = \lambda_{i,t}$. As also noted in Gomes, Yaron, and Zhang (2004), this last observation implies that finance constraints can only affect investment if they are time varying. It is the shadow value of the constraint today, relative to tomorrow, that is important.

So $\lambda_{i,t+1}$ is the shadow cost of external finance at period $t+1$. Near the end of p. 538 the authors write

The higher $\lambda_{i,t+1}$, the greater is the effect of finance constraints.

This is translated as "the higher $\lambda_{i,t+1}$, the more difficult (or costly) is for a firm to obtain external financing". It does not appear to relate to whether it needs more external financing or not.

Moreover, as the authors write in p. 540, it being a shadow value, it must be non-negative. So what do you make of the fact that you obtained negative values?

• Regarding negative values - didn't payed attention to this since some authors have negative values (which looks weird). The only thing where it might be a mistake in my calculations is units. I suppose that all ratios are in raw numbers (not percentage) and the unit of money is $thousands (used in ln(TotalAssets)), am I right? – Kirill Lykov Jan 3 '15 at 22:34 • All variables in the definition of the index are pretty straightforward: obtain magnitudes from balance sheets, use percentages for growth rates. The authors refer to an older paper of one of them, where one can find that "Cash Flow" is defined as "Income + Depreciation". – Alecos Papadopoulos Jan 3 '15 at 23:18 • Not sure - they don't write explicitly that it is in percentages or in other units. More than that, Tables 2 and 3 suggest that everything is not in percent. Plus they write that assets are in$ millions in these tables. Do you think that they use the same numbers (units) in the calculations as for these tables? – Kirill Lykov Jan 3 '15 at 23:41
• All variables are ratios, percentages, dummies (so original units have been neutralized) except one which is in logarithms (and here units do matter). And in Table 2 they say that total assets are measured in millions of USD. – Alecos Papadopoulos Jan 4 '15 at 0:08

In the question, answers, and comments, I saw a few references to the potential problem/issue with negative WW index values. Since this caused me some trouble years ago, let me share with you what I found out when I asked Toni Whited about this at a conference one time. I told her that all my WW index numbers were coming out negative--why would this be? She told me that in the editing process for the manuscript at RFS, a constant that belongs in the index was inadvertently omitted. She said to add 0.65 to the formula (13) as listed in the paper and you will have the "true" WW index. (This number is visible in their Table 1, column 4, first row, for $\alpha_1$. This constant also shows up as $b_0$ on the right hand side in equation 12.)

I know this is a very late answer, but maybe it will help out some other poor researcher down the line who is also beating their head against a wall.

Hey I think I can guess your mistake... you have to insert the numbers as full percentages inside the WW-Index Formula (so 4.33 and not 0.0433). This is the difference to the KZ index. you have to do it for the "CF","TLTD","ISG" and "SG". then you should not receive negative values.

• It is not relevant for me anymore, but for other people might be useful. Since I saw at least 2 papers with strictly negative WW-index numbers. Is this percentage story written in the WW paper somewhere? – Kirill Lykov Apr 10 '15 at 9:38
• Unfortunately it is not written in the WW paper. It is just how I did it during my research, without receiving negative values. However, there is still a possibility to receive negative values even with this approach, it just much minor. Excuse me but do you maybe remember the authors of the paper with the negative WW-indexes? The WW-Indexes in the WW paper are all positive and close to one. – AlexWeiß Apr 10 '15 at 9:50
• – Kirill Lykov Apr 11 '15 at 8:52