Coefficient Interpreation using proportions

[Nicole Kristine]

Hi everyone,

I am struggling with the interpretation of my regression results. I use stock returns as the dependent variable and the values range between -0.81 to 0.35. Most of my independent variables are also given in proportions e.g. LTDebt is Long-Term Debt/Total Assets and ranges between 0 and 1.

Since I do not use them as percentages but as proportions I am unsure how to interpet the coefficients for e.g. LTDebt, since the use of percentage points would probably be incorrect?

Here is an example of my regression output:

. reg CrisisPeriodReturn ES lnMarketCap LTDebt STDebt CashHoldings Profitability BM BMDummy Momentum
>  IdioVola SMB HML MOM MktRf, robust

Linear regression                               Number of obs     =        852
                                                F(14, 837)        =      14.49
                                                Prob > F          =     0.0000
                                                R-squared         =     0.1832
                                                Root MSE          =     .14356

-------------------------------------------------------------------------------
              |               Robust
CrisisPerio~n |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
           ES |  -.0288199   .0303761    -0.95   0.343    -.0884422    .0308024
  lnMarketCap |   .0088098    .005104     1.73   0.085    -.0012084     .018828
       LTDebt |  -.0999875   .0377497    -2.65   0.008    -.1740827   -.0258923
       STDebt |   .0136328   .0855839     0.16   0.873    -.1543514     .181617
 CashHoldings |   .0249738   .0579692     0.43   0.667    -.0888082    .1387558
Profitability |   -.010731   .0264634    -0.41   0.685    -.0626734    .0412113
           BM |  -.0076934   .0165694    -0.46   0.643     -.040216    .0248291
      BMDummy |  -.0529637   .0545169    -0.97   0.332    -.1599696    .0540422
     Momentum |    .013844   .0182355     0.76   0.448    -.0219487    .0496368
     IdioVola |  -.0493467   .2616701    -0.19   0.850    -.5629533    .4642599
          SMB |  -.0171075   .0069139    -2.47   0.014    -.0306781    -.003537
          HML |  -.0262343   .0079919    -3.28   0.001    -.0419208   -.0105478
          MOM |   .0708367   .0127377     5.56   0.000     .0458351    .0958384
        MktRf |  -.0811975   .0127559    -6.37   0.000    -.1062347   -.0561602
        _cons |  -.3801371   .0838043    -4.54   0.000    -.5446285   -.2156458
-------------------------------------------------------------------------------

Thank you in advance! I really appreciate your help.

Nicole

[Weiwen Ng]

Thanks for using code delimiters to present your results. They're clearly readable. The coefficient on long-term debt as a proportion of total assets means that when LTD/assets goes to 1 from 0, the average crisis period return decreases by 0.0999875.

You seem to be saying that crisis period return is formatted like a proportion, so 0 means a 0% return, 1 means a 100% return. If that's right, then that coefficient above is basically a 10 percentage point loss. (Technically, I'm not sure it qualifies as a proportion if it can take on negative values, but I think the concept is similar.)

Of course, we are talking about an expected loss of 10 percentage points when LTD/assets goes from 0 to 100%. I'm not in your field. Maybe a 100% ratio there isn't that bad. Maybe it is really bad. I don't know. If you turned LTD/assets into percentage points, i.e. you take that variable and multiply by 100, then you re-ran the regression, you'd now have the beta represent the change in expected return associated with a one percentage point increase in LTD/assets.

So, you could multiply your dependent variable by 100 to ease interpretation if you wanted. You could do the exact same for any of your independent variables that are denominated in proportions. A similar but not identical approach I've used with proportions is to turn them into 10 percentage point chunks, i.e. you take the variable and multiply by 10. In my field, I might be talking about the proportion of hospital patient-days that are covered by Medicare. Going from 0% to 100% is a) not something you'd see in reality and b) it could produce really big effects on the dependent variable, and casual readers might initially be stunned and confused.

[Nicole Kristine]

Thank you for your help Weiwen!

You are right, I could simply multiply my returns and LTDebt by 100. Since I would do it on both ends, the coefficient remains the same (-0.999875). However, I am still confused whith the correct interpretation.

Is it:
(1) A 1 percentage point increase in LTDebt is associated with a 10 percentage point decrease in returns

or is it:

(2) A 1 percetage point increase in LTDEbt is associated with a 0.1 percentage point decrease in returns.

[Joro Kolev]

The interpretation of the estimated coefficient in linear regression is always the same: One unit increase in the independent variable leads to (estimated coefficient) units increase in the dependent variable.

As Weiwen said, one unit is one unit, in your case means the LTD/Assets ratio increasing by one unit, that is from 0 to 1. If you do not like this because it is too big of a change, you can multiply 100*(LTD/Assets) and use this as a regressor, then one unit will be 1 percent increase in 100*(LTD/Assets).

You have not explained what is your dependent variable: is it gross (e.g., 1.05) or net return (e.g., 0.05)? Is it in decimal (e.g., 0.05) or in percentage form (e.g., 5%)? And per what period of time it is? Per day, per week, per year, per decade etc.? You do say "values range between -0.81 to 0.35" which kind of suggests net decimal returns, but then why are the values so big...

Thank you for your help Weiwen!

You are right, I could simply multiply my returns and LTDebt by 100. Since I would do it on both ends, the coefficient remains the same (-0.999875). However, I am still confused whith the correct interpretation.

Is it:
(1) A 1 percentage point increase in LTDebt is associated with a 10 percentage point decrease in returns

or is it:

(2) A 1 percetage point increase in LTDEbt is associated with a 0.1 percentage point decrease in returns.

[Nicole Kristine]

Thank you for the clarification.

My returns are net decimals returns calculated over 7 months of the stock market crash from the financial crisis, which is why the values are so big.

[Joro Kolev]

Very good then. We can imagine that you multiply both your dependent and independent variable by 100, you do not need to refit the regression the estimate on LTD/Assets will not change, and it seems that the interpretation is that

1% increase in LTD/Assets leads to .0999875% decrease in 7 months net returns.

Thank you for the clarification.

My returns are net decimals returns calculated over 7 months of the stock market crash from the financial crisis, which is why the values are so big.

[Nicole Kristine]

oh okay thank you!!

But now I am a little confused why it's percentages and not percentage points? Because percentages work like a multiplier and not like an addition?

[Nicole Kristine]

Is there someone who could help me out with my confusion? Would be super helpful, Thanks!

[Weiwen Ng]

oh okay thank you!!

But now I am a little confused why it's percentages and not percentage points? Because percentages work like a multiplier and not like an addition?

I am not completely sure what you mean, but I think you were referring to Joro's explanation:

1% increase in LTD/Assets leads to .0999875% decrease in 7 months net returns.

You are correct that the meaning of percent and percentage point can sometimes diverge. That's why I used the term percentage point in my post. Taking Joro's post in isolation, though, I think the meanings of percent and percentage point are identical in this case.

Say you were talking a (made up example) mutual fund expense ratio that was 1.0% in one year. If you say that the fund's expense ratio increased by 50%, most people will assume it increased to 1.5%. However, strictly speaking, that could mean that the expense ratio is now 51.0%, and I'd probably have said it increased by 0.5 percentage points (or I think the finance term of art is 50 basis points) instead. That's where the meaning of % and percentage point potentially diverge. In my made up example, the magnitude of the change does make the meaning clear in context (a mutual fund with even a high digit expense ratio is spending way too much money and will probably fold), but in other contexts it may be more ambiguous.