Firm and Industry Effects Together?

[Prateek Bedi]

Hi,

In the context of corporate finance, some studies claim to use firm and industry fixed effects together in panel data regressions. However, since inclusion of firm effects takes care of all time invariant variables together, how is it possible for a researcher to include industry effects (the industry of a firm remains the same over generally) also in the same regression? I understand that if at all the industry of even a single firm in the dataset changes from one year to another, it would be mechanically possible to obtain results for fixed effects regression. But since the industry of a firm usually remains same across time for almost the entire set of firms in the sample, how reliable are beta coefficients of independent variables in case of a regression with firm and industry effects?

Here are some papers which employ firm and industry effects together:

Thakur, B., & Kannadhasan, M. (2018). Corruption and cash holdings: Evidence from emerging market economies. Emerging Markets Review, 38, 1-17. doi: 10.1016/j.ememar.2018.11.008

Venkiteshwaran, V. (2011). Partial adjustment toward optimal cash holding levels. Review Of Financial Economics, 20(3), 113-121. doi: 10.1016/j.rfe.2011.06.002

Thanks!

[Justin Niakamal]

...how is it possible for a researcher to include industry effects (the industry of a firm remains the same over generally) also in the same regression?

Add a dummy variable to your specification.

xtreg y x i.industry, fe

As for why, I'd imagine you'd want to control for variation in firms across different industries. The fixed effect will take into account time invariant characteristics for a particular firm, but there might be differences to control for amongst firms in different industries. For example, manufacturing firms versus firms in the health care industry. As mentioned in the paper you've referenced

Having established that firm behavior is consistent with the existence of a target cash holding level, in this section we control for firm and industry specific determinants of optimal level of cash holdings and then examine the adjustment behavior

As for your last question,

how reliable are beta coefficients of independent variables in case of a regression with firm and industry effects?

Like in the aforementioned paper, don't concern your analysis with one model. Try different models/specifications (eg. With or without fixed effects/industry effects/ time fixed effects, etc.) and compare the differences amongst them.

[William Lisowski]

Let me expand slightly on Justin's answer in post #2.

However, since inclusion of firm effects takes care of all time invariant variables together, how is it possible for a researcher to include industry effects (the industry of a firm remains the same over generally) also in the same regression?

The firm fixed effect takes care of all time invariant variables that are specific to the firm - as Justin says - because you have effectively only one measurement of the time invariant variable for each firm, so there's no variability across time within the firm. The industry fixed effect is not specific to the firm - multiple firms have the same industry - so there's variability in the time invariant variables across the firms, if not across time, within each industry.

Think of the the firm effects as measuring the deviation of each firm from the industry effect for the industry that the firm is in.

[Prateek Bedi]

Thanks Justin Blasongame and William Lisowski for your replies. I think I have not been able to state my query clearly. Let me put it this way: Once we include firm effects in a panel regression, it becomes virtually impossible to introduce industry effects in the same regression (via i.industry or any other method). This is because by construction, firm effects control for all time-invariant variables and if any time-invariant variable is introduced in the model (for eg. industry of the firm), STATA omits it and does not report output for the same. This is based on the notion that the industry of a firm does not change over time. I believe that inclusion of firm effects automatically controls for the industry effects as well because industry of a firm is also time-invariant. However, I understand that if at all the industry of even a single firm in the dataset changes from one year to another, it would be mechanically possible to obtain results for regression with firm effects because now, STATA won't omit this variable.

Now, papers which report firm-effects and industry-effects together in a regression seem to obtain results (which is very strange) of two fixed effects together despite one being nested in the other. Finally, the question is: How do they report such results given the fact that it is not possible to obtain such results by construction?

[William Lisowski]

You are correct, I was not thinking correctly yesterday. My sincere apologies.

I have downloaded the PDF of the first paper you cite. Searching the PDF for "industry" I see that the models report robust standard errors that are clustered by firm and industry. Searching the PDF for "fixed effect" I see "Country fixed effects and firm fixed effects are included" in the models. So I think that for that paper you have let the treatment of standard errors mislead your interpretation of the fixed effects of the model. With that said, though, I would expect the same problem for country fixed effects. But the discussion below addresses this.

I was not able to make sense of the second paper, which was complicated, for me, by my inability to download the PDF version. I agree that for the model described in column 3 of Panel A of Table 2, they "estimate a firm fixed effects model with year and industry dummies". This compares to column 2 which includes "dummy variables to remove the year and industry fixed effects" lacking the firm fixed effect. My guess is that when the author transitioned to a fixed effects model in column 3 he did not report that the industry dummies were consequently omitted in the fitting of the model, because to him they were "nuisance parameters", but that is only a guess.

Let's look at the following, where I fit the same model three different ways.

clear
set obs 9
generate byte firm = _n
expand 5
bysort firm: generate int year = _n
recode firm (1/3 = 1) (4/6 = 2) (7/9 = 3), generate(industry)
sort firm year

set seed 42
generate float y = 100*(firm-1) + 100*(industry-1) +100*(year-1) + rnormal()

xtset firm year
regress y i.firm i.industry i.year
regress y i.industry i.year i.firm
xtreg y i.industry i.year, fe

The first two models fit as pooled OLS with firm, industry, and year fixed effects, show that the order in which the variables are specified determines the choice of variables to omit. The first model omits the industry effects, because industry is totally determined by the firm variables appearing before it in the model.

. regress y i.firm i.industry i.year
note: 2.industry omitted because of collinearity
note: 3.industry omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =        45
-------------+----------------------------------   F(12, 32)       >  99999.00
       Model |  6000091.67        12   500007.64   Prob > F        =    0.0000
    Residual |   26.264716        32  .820772374   R-squared       =    1.0000
-------------+----------------------------------   Adj R-squared   =    1.0000
       Total |  6000117.94        44  136366.317   Root MSE        =    .90596

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        firm |
          2  |   100.8941   .5729825   176.09   0.000     99.72699    102.0612
          3  |   199.7856   .5729825   348.68   0.000     198.6185    200.9527
          4  |   400.1137   .5729825   698.30   0.000     398.9465    401.2808
          5  |   500.1389   .5729825   872.87   0.000     498.9718    501.3061
          6  |   600.7585   .5729825  1048.48   0.000     599.5914    601.9257
          7  |   800.0716   .5729825  1396.33   0.000     798.9045    801.2388
          8  |   900.5123   .5729825  1571.62   0.000     899.3452    901.6795
          9  |   999.9758   .5729825  1745.21   0.000     998.8087    1001.143
             |
    industry |
          2  |          0  (omitted)
          3  |          0  (omitted)
             |
        year |
          2  |   100.2236   .4270759   234.67   0.000      99.3537    101.0936
          3  |   199.8954   .4270759   468.06   0.000     199.0255    200.7654
          4  |   300.4905   .4270759   703.60   0.000     299.6206    301.3604
          5  |   399.9301   .4270759   936.44   0.000     399.0602    400.8001
             |
       _cons |  -.4335776   .4869415    -0.89   0.380    -1.425445    .5582898
------------------------------------------------------------------------------

The second model omits two of the firm categories, because they are fully determined by the industry and the firm categories appearing before them. Otherwise, the results of the model are identical to the first, and it generates identical fitted values. This gives the semblance of showing industry fixed effects, at the cost of omitting multiple firm fixed effects and complicating their interpretation, because the regress command cannot "know" that firms are nested within industries and thus firm determines industry. (One could argue that it should be able to figure that out, but this was just a trivial example of joint determination.)

note: 6.firm omitted because of collinearity
note: 9.firm omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =        45
-------------+----------------------------------   F(12, 32)       >  99999.00
       Model |  6000091.67        12   500007.64   Prob > F        =    0.0000
    Residual |   26.264716        32  .820772374   R-squared       =    1.0000
-------------+----------------------------------   Adj R-squared   =    1.0000
       Total |  6000117.94        44  136366.317   Root MSE        =    .90596

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    industry |
          2  |   600.7585   .5729825  1048.48   0.000     599.5914    601.9257
          3  |   999.9758   .5729825  1745.21   0.000     998.8087    1001.143
             |
        year |
          2  |   100.2236   .4270759   234.67   0.000      99.3537    101.0936
          3  |   199.8954   .4270759   468.06   0.000     199.0255    200.7654
          4  |   300.4905   .4270759   703.60   0.000     299.6206    301.3604
          5  |   399.9301   .4270759   936.44   0.000     399.0602    400.8001
             |
        firm |
          2  |   100.8941   .5729825   176.09   0.000     99.72699    102.0612
          3  |   199.7856   .5729825   348.68   0.000     198.6185    200.9527
          4  |  -200.6449   .5729825  -350.18   0.000     -201.812   -199.4778
          5  |  -100.6196   .5729825  -175.61   0.000    -101.7867   -99.45247
          6  |          0  (omitted)
          7  |  -199.9042   .5729825  -348.88   0.000    -201.0713    -198.737
          8  |  -99.46346   .5729825  -173.59   0.000    -100.6306   -98.29634
          9  |          0  (omitted)
             |
       _cons |  -.4335776   .4869415    -0.89   0.380    -1.425445    .5582898
------------------------------------------------------------------------------

Finally, a fixed effects regression is a different thing than including fixed effects in a pooled OLS regression; there are assumptions about the variance structure.

. xtreg y i.industry i.year, fe
note: 2.industry omitted because of collinearity
note: 3.industry omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =         45
Group variable: firm                            Number of groups  =          9

R-sq:                                           Obs per group:
     within  = 1.0000                                         min =          5
     between =      .                                         avg =        5.0
     overall = 0.1500                                         max =          5

                                                F(4,32)           =  274202.42
corr(u_i, Xb)  = 0.0000                         Prob > F          =     0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    industry |
          2  |          0  (omitted)
          3  |          0  (omitted)
             |
        year |
          2  |   100.2236   .4270759   234.67   0.000      99.3537    101.0936
          3  |   199.8954   .4270759   468.06   0.000     199.0255    200.7654
          4  |   300.4905   .4270759   703.60   0.000     299.6206    301.3604
          5  |   399.9301   .4270759   936.44   0.000     399.0602    400.8001
             |
       _cons |   499.8165   .3019883  1655.09   0.000     499.2014    500.4316
-------------+----------------------------------------------------------------
     sigma_u |  357.06654
     sigma_e |  .90596489
         rho |  .99999356   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(8, 32) = 7.8e+05                    Prob > F = 0.0000

[Joro Kolev]

Prateek Bedi , I think you pretty much gave yourself the answer while asking your question

You can not include industry fixed effects on the top of firms fixed effects, if firms do not change industries over time. If you try to do so, Stata will drop those industry fixed effects due to collinearity. Here

. webuse grunfeld, clear

. gen industry = company<4

. replace industry = 3 if company>7
(60 real changes made)

. xtset company year
       panel variable:  company (strongly balanced)
        time variable:  year, 1935 to 1954
                delta:  1 year

. xtreg invest mvalue kstock i.industry, fe
note: 1.industry omitted because of collinearity
note: 3.industry omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =        200
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.7668                                         min =         20
     between = 0.8194                                         avg =       20.0
     overall = 0.8060                                         max =         20

                                                F(2,188)          =     309.01
corr(u_i, Xb)  = -0.1517                        Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .1101238   .0118567     9.29   0.000     .0867345    .1335131
      kstock |   .3100653   .0173545    17.87   0.000     .2758308    .3442999
             |
    industry |
          1  |          0  (omitted)
          3  |          0  (omitted)
             |
       _cons |  -58.74393   12.45369    -4.72   0.000    -83.31086     -34.177
-------------+----------------------------------------------------------------
     sigma_u |  85.732501
     sigma_e |  52.767964
         rho |  .72525012   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 188) = 49.18                     Prob > F = 0.0000

.

[Joro Kolev]

And the result above by the way is numerically equivalent to the result you would obtain by not including the collinear industry fixed effects:

. xtreg invest mvalue kstock, fe

Fixed-effects (within) regression               Number of obs     =        200
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.7668                                         min =         20
     between = 0.8194                                         avg =       20.0
     overall = 0.8060                                         max =         20

                                                F(2,188)          =     309.01
corr(u_i, Xb)  = -0.1517                        Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .1101238   .0118567     9.29   0.000     .0867345    .1335131
      kstock |   .3100653   .0173545    17.87   0.000     .2758308    .3442999
       _cons |  -58.74393   12.45369    -4.72   0.000    -83.31086     -34.177
-------------+----------------------------------------------------------------
     sigma_u |  85.732501
     sigma_e |  52.767964
         rho |  .72525012   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 188) = 49.18                     Prob > F = 0.0000

[Prateek Bedi]

You are correct, I was not thinking correctly yesterday. My sincere apologies.

I have downloaded the PDF of the first paper you cite. Searching the PDF for "industry" I see that the models report robust standard errors that are clustered by firm and industry. Searching the PDF for "fixed effect" I see "Country fixed effects and firm fixed effects are included" in the models. So I think that for that paper you have let the treatment of standard errors mislead your interpretation of the fixed effects of the model. With that said, though, I would expect the same problem for country fixed effects. But the discussion below addresses this.

I was not able to make sense of the second paper, which was complicated, for me, by my inability to download the PDF version. I agree that for the model described in column 3 of Panel A of Table 2, they "estimate a firm fixed effects model with year and industry dummies". This compares to column 2 which includes "dummy variables to remove the year and industry fixed effects" lacking the firm fixed effect. My guess is that when the author transitioned to a fixed effects model in column 3 he did not report that the industry dummies were consequently omitted in the fitting of the model, because to him they were "nuisance parameters", but that is only a guess.

Let's look at the following, where I fit the same model three different ways.

clear
set obs 9
generate byte firm = _n
expand 5
bysort firm: generate int year = _n
recode firm (1/3 = 1) (4/6 = 2) (7/9 = 3), generate(industry)
sort firm year

set seed 42
generate float y = 100*(firm-1) + 100*(industry-1) +100*(year-1) + rnormal()

xtset firm year
regress y i.firm i.industry i.year
regress y i.industry i.year i.firm
xtreg y i.industry i.year, fe

The first two models fit as pooled OLS with firm, industry, and year fixed effects, show that the order in which the variables are specified determines the choice of variables to omit. The first model omits the industry effects, because industry is totally determined by the firm variables appearing before it in the model.

. regress y i.firm i.industry i.year
note: 2.industry omitted because of collinearity
note: 3.industry omitted because of collinearity

Source | SS df MS Number of obs = 45
-------------+---------------------------------- F(12, 32) > 99999.00
Model | 6000091.67 12 500007.64 Prob > F = 0.0000
Residual | 26.264716 32 .820772374 R-squared = 1.0000
-------------+---------------------------------- Adj R-squared = 1.0000
Total | 6000117.94 44 136366.317 Root MSE = .90596

------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
firm |
2 | 100.8941 .5729825 176.09 0.000 99.72699 102.0612
3 | 199.7856 .5729825 348.68 0.000 198.6185 200.9527
4 | 400.1137 .5729825 698.30 0.000 398.9465 401.2808
5 | 500.1389 .5729825 872.87 0.000 498.9718 501.3061
6 | 600.7585 .5729825 1048.48 0.000 599.5914 601.9257
7 | 800.0716 .5729825 1396.33 0.000 798.9045 801.2388
8 | 900.5123 .5729825 1571.62 0.000 899.3452 901.6795
9 | 999.9758 .5729825 1745.21 0.000 998.8087 1001.143
|
industry |
2 | 0 (omitted)
3 | 0 (omitted)
|
year |
2 | 100.2236 .4270759 234.67 0.000 99.3537 101.0936
3 | 199.8954 .4270759 468.06 0.000 199.0255 200.7654
4 | 300.4905 .4270759 703.60 0.000 299.6206 301.3604
5 | 399.9301 .4270759 936.44 0.000 399.0602 400.8001
|
_cons | -.4335776 .4869415 -0.89 0.380 -1.425445 .5582898
------------------------------------------------------------------------------

The second model omits two of the firm categories, because they are fully determined by the industry and the firm categories appearing before them. Otherwise, the results of the model are identical to the first, and it generates identical fitted values. This gives the semblance of showing industry fixed effects, at the cost of omitting multiple firm fixed effects and complicating their interpretation, because the regress command cannot "know" that firms are nested within industries and thus firm determines industry. (One could argue that it should be able to figure that out, but this was just a trivial example of joint determination.)

note: 6.firm omitted because of collinearity
note: 9.firm omitted because of collinearity

Source | SS df MS Number of obs = 45
-------------+---------------------------------- F(12, 32) > 99999.00
Model | 6000091.67 12 500007.64 Prob > F = 0.0000
Residual | 26.264716 32 .820772374 R-squared = 1.0000
-------------+---------------------------------- Adj R-squared = 1.0000
Total | 6000117.94 44 136366.317 Root MSE = .90596

------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
industry |
2 | 600.7585 .5729825 1048.48 0.000 599.5914 601.9257
3 | 999.9758 .5729825 1745.21 0.000 998.8087 1001.143
|
year |
2 | 100.2236 .4270759 234.67 0.000 99.3537 101.0936
3 | 199.8954 .4270759 468.06 0.000 199.0255 200.7654
4 | 300.4905 .4270759 703.60 0.000 299.6206 301.3604
5 | 399.9301 .4270759 936.44 0.000 399.0602 400.8001
|
firm |
2 | 100.8941 .5729825 176.09 0.000 99.72699 102.0612
3 | 199.7856 .5729825 348.68 0.000 198.6185 200.9527
4 | -200.6449 .5729825 -350.18 0.000 -201.812 -199.4778
5 | -100.6196 .5729825 -175.61 0.000 -101.7867 -99.45247
6 | 0 (omitted)
7 | -199.9042 .5729825 -348.88 0.000 -201.0713 -198.737
8 | -99.46346 .5729825 -173.59 0.000 -100.6306 -98.29634
9 | 0 (omitted)
|
_cons | -.4335776 .4869415 -0.89 0.380 -1.425445 .5582898
------------------------------------------------------------------------------

Finally, a fixed effects regression is a different thing than including fixed effects in a pooled OLS regression; there are assumptions about the variance structure.

. xtreg y i.industry i.year, fe
note: 2.industry omitted because of collinearity
note: 3.industry omitted because of collinearity

Fixed-effects (within) regression Number of obs = 45
Group variable: firm Number of groups = 9

R-sq: Obs per group:
within = 1.0000 min = 5
between = . avg = 5.0
overall = 0.1500 max = 5

F(4,32) = 274202.42
corr(u_i, Xb) = 0.0000 Prob > F = 0.0000

------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
industry |
2 | 0 (omitted)
3 | 0 (omitted)
|
year |
2 | 100.2236 .4270759 234.67 0.000 99.3537 101.0936
3 | 199.8954 .4270759 468.06 0.000 199.0255 200.7654
4 | 300.4905 .4270759 703.60 0.000 299.6206 301.3604
5 | 399.9301 .4270759 936.44 0.000 399.0602 400.8001
|
_cons | 499.8165 .3019883 1655.09 0.000 499.2014 500.4316
-------------+----------------------------------------------------------------
sigma_u | 357.06654
sigma_e | .90596489
rho | .99999356 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(8, 32) = 7.8e+05 Prob > F = 0.0000

Sir, I completely agree with you now. Thanks a lot for putting in so much effort to verify and confirm me this. I sincerely appreciate your effort. Further, consider a case in which the industry of a single firm in the dataset changes from one year to another. Now, it would be mechanically possible to obtain results for regression with firm effects because STATA won't omit this variable (sensing a variation of industry across time). In this case, how reliable are estimates of independent variables in the regression output?

[Prateek Bedi]

Prateek Bedi , I think you pretty much gave yourself the answer while asking your question

You can not include industry fixed effects on the top of firms fixed effects, if firms do not change industries over time. If you try to do so, Stata will drop those industry fixed effects due to collinearity. Here

. webuse grunfeld, clear

. gen industry = company<4

. replace industry = 3 if company>7
(60 real changes made)

. xtset company year
panel variable: company (strongly balanced)
time variable: year, 1935 to 1954
delta: 1 year

. xtreg invest mvalue kstock i.industry, fe
note: 1.industry omitted because of collinearity
note: 3.industry omitted because of collinearity

Fixed-effects (within) regression Number of obs = 200
Group variable: company Number of groups = 10

R-sq: Obs per group:
within = 0.7668 min = 20
between = 0.8194 avg = 20.0
overall = 0.8060 max = 20

F(2,188) = 309.01
corr(u_i, Xb) = -0.1517 Prob > F = 0.0000

------------------------------------------------------------------------------
invest | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
mvalue | .1101238 .0118567 9.29 0.000 .0867345 .1335131
kstock | .3100653 .0173545 17.87 0.000 .2758308 .3442999
|
industry |
1 | 0 (omitted)
3 | 0 (omitted)
|
_cons | -58.74393 12.45369 -4.72 0.000 -83.31086 -34.177
-------------+----------------------------------------------------------------
sigma_u | 85.732501
sigma_e | 52.767964
rho | .72525012 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 188) = 49.18 Prob > F = 0.0000

.

Agreed. Thanks a lot!

[William Lisowski]

Further, consider a case in which the industry of a single firm in the dataset changes from one year to another. Now, it would be mechanically possible to obtain results for regression with firm effects because STATA won't omit this variable (sensing a variation of industry across time). In this case, how reliable are estimates of independent variables in the regression output?

This is a bit beyond my expertise. In general I expect that the effect will largely be limited to the firm and industry effects, with little effect on the other independent variables. But I will agree with what I see as your underlying sentiment, that including industry fixed effects in a model with firm fixed effects is not likely to be helpful and should be avoided. As indeed should be any variable that estimates an effect from a very small fraction of the observations.

[Prateek Bedi]

This is a bit beyond my expertise. In general I expect that the effect will largely be limited to the firm and industry effects, with little effect on the other independent variables. But I will agree with what I see as your underlying sentiment, that including industry fixed effects in a model with firm fixed effects is not likely to be helpful and should be avoided. As indeed should be any variable that estimates an effect from a very small fraction of the observations.

Thanks a lot for your guidance, Sir. Really appreciate!!