Linearity panel data

[Femke Versluis]

Hi everyone.
I am working on a regression on the overestimation of companies, and I'm working with panel data. I want to test for linearity, but all the scatter plots I do aren't of much help. The graph matrix gives me this outcome:

and this is the regression I did:

Does anyone know why it doesn't show linearity? And what I am doing wrong or another way to test it?

[Carlo Lazzaro]

Femke:
-graph Matrix- is good at showing correlation among variables (among other interesting features).
That said, I would interpret your striving for linearity as a way to test whether non-linera relationships among regressors and regressand do exist.
The first thing that springs to my mind is to test whether squared age makes sense in your regrrssion model:

xtreg overestimation c.age##c.age entrybarrier index i.region i.cagr

The second step would be to test if squared fitted values highlight the lack of further predictors and/or interactions in the right-hand side of your regerssion equation.
You can take a look at the following toy-example:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. xtreg ln_wage age i.year i.south

Random-effects GLS regression                   Number of obs     =     28,502
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1071                                         min =          1
     between = 0.1371                                         avg =        6.1
     overall = 0.1194                                         max =         15

                                                Wald chi2(16)     =    3533.31
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .0139268   .0018431     7.56   0.000     .0103144    .0175392
             |
        year |
         69  |   .0748568   .0124895     5.99   0.000     .0503779    .0993358
         70  |   .0449056   .0120106     3.74   0.000     .0213653    .0684459
         71  |   .0823852   .0124649     6.61   0.000     .0579545     .106816
         72  |   .0836341   .0134961     6.20   0.000     .0571823     .110086
         73  |    .085007   .0141998     5.99   0.000      .057176     .112838
         75  |   .0716501   .0164903     4.34   0.000     .0393297    .1039705
         77  |   .1038729   .0193532     5.37   0.000     .0659413    .1418046
         78  |   .1296735   .0210804     6.15   0.000     .0883567    .1709904
         80  |   .1093764   .0242817     4.50   0.000     .0617851    .1569676
         82  |   .0993104   .0274684     3.62   0.000     .0454734    .1531475
         83  |   .1117414   .0291919     3.83   0.000     .0545264    .1689563
         85  |   .1368575   .0325795     4.20   0.000     .0730028    .2007121
         87  |    .126616   .0360617     3.51   0.000     .0559364    .1972956
         88  |   .1638151   .0384291     4.26   0.000     .0884955    .2391347
             |
     1.south |   -.132044   .0082467   -16.01   0.000    -.1482073   -.1158808
       _cons |   1.209158   .0370906    32.60   0.000     1.136462    1.281854
-------------+----------------------------------------------------------------
     sigma_u |  .35537854
     sigma_e |  .30272588
         rho |  .57949767   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. predict fitted, xb
(32 missing values generated)

. g sq_fitted=fitted^2
(32 missing values generated)

. xtreg ln_wage fitted sq_fitted

Random-effects GLS regression                   Number of obs     =     28,502
Group variable: idcode                          Number of groups  =      4,710

R-sq:                                           Obs per group:
     within  = 0.1085                                         min =          1
     between = 0.1413                                         avg =        6.1
     overall = 0.1222                                         max =         15

                                                Wald chi2(2)      =    3599.09
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      fitted |    3.33779   .3087343    10.81   0.000     2.732682    3.942898
   sq_fitted |  -.7015439   .0925002    -7.58   0.000    -.8828409   -.5202469
       _cons |  -1.933331   .2565439    -7.54   0.000    -2.436148   -1.430514
-------------+----------------------------------------------------------------
     sigma_u |  .35642505
     sigma_e |  .30252041
         rho |  .58126061   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test sq_fitted

 ( 1)  sq_fitted = 0

           chi2(  1) =   57.52
         Prob > chi2 =    0.0000

.
*as -test- outcome reaches statistical significance, the regression model is ill-specified
Maybe non-lineraity is an issue deserving further investigation*

[Femke Versluis]

Dear Carlo, thank you very much for your explanation. The only part that I was wondering about is the second to last sentence. Does this mean that Prob > chi2 = 0.000 means it's insignificant and so the model is not ill-specified?

[Carlo Lazzaro]

Femke:
not quite.
As the H0 is rejected, .-test- oucome warns us about possible non-linearity and/or omitted predictors in the original regression model. Put differently, -test- outcome supports the evidence that the original regression model is ill-specified..