Interpretation of Data

[Guest]

Dear all,

I have a set of financial market data sets and I am testing if herding is present in different markets. As a matter of precaution, I have run the regression along with the -i.year- and -robust- commands to detrend my data and make sure they are robust to heteroskedasticity. I have had no issues executing the said commands, but I do have a trouble interpreting the results, I will show my first example below:

Here's my second example:

If I may draw your attention to the 'improved' R-squared, interpreting an increase in the R-squared as an 'improvement' is, of course, shallow, however, the increase is rather significant and persistent for not just the two examples I have shown above but most of my data sets, have I done the right thing by controlling for yearly trend and detrending? Another phenomenon worth pointing out is that the coefficient of 2007 is not only large, but significant in both examples, what might be the implication?

[Carlo Lazzaro]

Guest::
- usually adjusted R-sq should be used to make comparisons across different regression models;
- "the best" model (whatever that means) is the one that gives the truest and fairest view of the data generating process underlying your data (the literature in your research field can help you out in this respect);
you can test the significance of -i.yerar- via the folowing:

testparm(i.year)

[Guest]

Dear Carlo,

I am also dubious about the presence of heteroskedasticity in my data.

As you can see, I first ran my regression using robust standard errors, and the second time without. The standard errors and the robust standard errors did not differ significantly, does that not mean that there's no heteroskedasticity? My uncertainty arose because I also ran a Breusch-Pagan test and plotted graphs that indicate heteroskedasticity, which you will see below

Above is my rvfplot and I do see a trend. My BP test pointed out that there's heteroskedasticity too:





I then proceeded to scatter my residuals against my two x variables, one at a time, below are the results:
scatter uhat r_mt:


scatter uhat abs_r_mt:



Only in the second graph do I detect heteroskedasticity. What might have gone wrong?

[Carlo Lazzaro]

Guest:
please use CODE delimiters to post what you typed and what Stata gave you back; screenshots are hard to read and difficult to comment on.
If you suspect heteroskedasticity in your data (with several observations, visual inspection can easily outperforms BP test), you can well use -robust- standard errors.

[Guest]

Dear Carlo,

I apologise for the screenshots. Following are my results:

. . reg csad abs_r_mt r_mt_2

      Source |       SS           df       MS      Number of obs   =     2,609
-------------+----------------------------------   F(2, 2606)      =    369.70
       Model |  491.324057         2  245.662029   Prob > F        =    0.0000
    Residual |  1731.65225     2,606  .664486665   R-squared       =    0.2210
-------------+----------------------------------   Adj R-squared   =    0.2204
       Total |  2222.97631     2,608  .852368215   Root MSE        =    .81516

------------------------------------------------------------------------------
        csad |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    abs_r_mt |   .6120725    .028192    21.71   0.000     .5567915    .6673534
      r_mt_2 |  -.0587181   .0049878   -11.77   0.000    -.0684984   -.0489377
       _cons |   1.051306   .0260192    40.41   0.000     1.000286    1.102327
------------------------------------------------------------------------------


. reg csad abs_r_mt r_mt_2, robust

Linear regression                               Number of obs     =      2,609
                                                F(2, 2606)        =     309.63
                                                Prob > F          =     0.0000
                                                R-squared         =     0.2210
                                                Root MSE          =     .81516

------------------------------------------------------------------------------
             |               Robust
        csad |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    abs_r_mt |   .6120725   .0313965    19.49   0.000     .5505078    .6736372
      r_mt_2 |  -.0587181   .0058612   -10.02   0.000    -.0702111    -.047225
       _cons |   1.051306   .0267547    39.29   0.000     .9988437    1.103769
------------------------------------------------------------------------------

. . hettest abs_r_mt r_mt_2

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: abs_r_mt r_mt_2

         chi2(2)      =   108.45
         Prob > chi2  =   0.0000

. hettest abs_r_mt

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: abs_r_mt

         chi2(1)      =    19.32
         Prob > chi2  =   0.0000

. hettest r_mt_2

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
         Ho: Constant variance
         Variables: r_mt_2

         chi2(1)      =    67.88
         Prob > chi2  =   0.0000

Thanks for your advice Carlo, its a common practice in my workplace to use robust standard errors as default, however, the difficulty is that I have to justify why do I use it, that is also why heteroskedasticity tests were done. In short, I have to prove that there is heteroskedasticity, only then I could proceed to use -robust-.

As you could see from above, the robust and general standard errors do not differ much, implying that there might not be heteroskedasticity. However, the BP test clearly tells us that the data is heteroskedastic, that is also when the confusion came upon.

[Marcos Almeida]

I think the large sample size may prompt the BP to give a significant p-value. Maybe the "proof of the pudding" lies in the (lack of significant change of) robust SEs, as you noticed.

Additionnaly, this thread might interest you.

[Carlo Lazzaro]

Guest:
as an aside to Marcos' helpful insight, what if you type:

estat hettest

after regression with default standard errors?

[Guest]

Thanks Carlo and Marcos, below is my -estat hettest- results.

. reg csad abs_r_mt rmt2

      Source |       SS           df       MS      Number of obs   =     2,609
-------------+----------------------------------   F(2, 2606)      =    369.70
       Model |  491.324057         2  245.662029   Prob > F        =    0.0000
    Residual |  1731.65225     2,606  .664486665   R-squared       =    0.2210
-------------+----------------------------------   Adj R-squared   =    0.2204
       Total |  2222.97631     2,608  .852368215   Root MSE        =    .81516

------------------------------------------------------------------------------
        csad |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    abs_r_mt |   .6120725    .028192    21.71   0.000     .5567915    .6673534
        rmt2 |  -.0587181   .0049878   -11.77   0.000    -.0684984   -.0489377
       _cons |   1.051306   .0260192    40.41   0.000     1.000286    1.102327
------------------------------------------------------------------------------

. hettest

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: fitted values of csad

         chi2(1)      =    10.41
         Prob > chi2  =   0.0013

. hettest r_mt abs_r_mt

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: r_mt abs_r_mt

         chi2(2)      =    16.56
         Prob > chi2  =   0.0003

. hettest abs_r_mt

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: abs_r_mt

         chi2(1)      =    16.06
         Prob > chi2  =   0.0001

. hettest rmt2

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: rmt2

         chi2(1)      =    15.75
         Prob > chi2  =   0.0001

.

[Carlo Lazzaro]

Guest:
-what if you log your dependent variable?
- I would also perform -estat ovtest- as a regerssion postestimation test;
- by the way: where did you interactions go?

[Guest]

Dear Carlo,

Did you mean that I should try to log my dependent variables when I am doing the regression? Below is my Ramsey RESET Test result:

. estat ovtest

Ramsey RESET test using powers of the fitted values of csad
       Ho:  model has no omitted variables
                F(3, 2594) =      1.52
                  Prob > F =      0.2084

I am sorry but I am not sure what interactions were you referring to.

Thanks for your inputs.

[Carlo Lazzaro]

Guest:
1) yes, I meant to log the dv;
2) the -ovtest- outcome does not show any evidence of non-linear relationships between your dv and predictors;
3) I refer to the interactions that you reported in your first post (screenshots).

[Guest]

Dear Carlo,

I think I understand what you meant now. Below is my regression result after logging the DVs:

. gen log1= log( abs_r_mt)
(192 missing values generated)

. gen log2=log( r_mt# r_mt)
r_mt#r_mt invalid name
r(198);

. gen rmt2= r_mt^2
(1 missing value generated)

. gen log2=log( rmt2)
(192 missing values generated)

. reg csad log1 log2
note: log1 omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =     2,418
-------------+----------------------------------   F(1, 2416)      =    223.02
       Model |  144.370867         1  144.370867   Prob > F        =    0.0000
    Residual |  1563.95462     2,416   .64733221   R-squared       =    0.0845
-------------+----------------------------------   Adj R-squared   =    0.0841
       Total |  1708.32549     2,417  .706795816   Root MSE        =    .80457

------------------------------------------------------------------------------
        csad |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        log1 |          0  (omitted)
        log2 |   .0970239   .0064968    14.93   0.000      .084284    .1097639
       _cons |   1.774776   .0170143   104.31   0.000     1.741412     1.80814
------------------------------------------------------------------------------

.

I think you are confused because I have changed the names of my variables, I have to assure you that these are innocuous changes, you will see why:

. gen abs_r_mt = abs(r_mt)
(1 missing value generated)

. regress csad c.abs_r_mt##c.abs_r_mt i.year, robust

Linear regression                               Number of obs     =      2,609
                                                F(11, 2597)       =     141.73
                                                Prob > F          =     0.0000
                                                R-squared         =     0.3903
                                                Root MSE          =     .72244

------------------------------------------------------------------------------
             |               Robust
        csad |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    abs_r_mt |   .4994904   .0292174    17.10   0.000     .4421987    .5567822
             |
  c.abs_r_mt#|
  c.abs_r_mt |   -.052799   .0051352   -10.28   0.000    -.0628686   -.0427295
             |
        year |
       2002  |  -.2258721   .0459779    -4.91   0.000    -.3160291   -.1357151
       2003  |    -.04762   .0484289    -0.98   0.326    -.1425832    .0473433
       2004  |   .0641004   .0472513     1.36   0.175    -.0285536    .1567545
       2005  |   .4017124   .0588411     6.83   0.000     .2863322    .5170927
       2006  |   .5451839   .0563295     9.68   0.000     .4347286    .6556392
       2007  |   1.051536   .0768241    13.69   0.000     .9008934    1.202179
       2008  |   .8957598   .0699791    12.80   0.000     .7585394     1.03298
       2009  |   .4675094   .0568641     8.22   0.000     .3560059    .5790129
       2010  |   .2445753   .0514202     4.76   0.000     .1437465    .3454041
             |
       _cons |   .8224699   .0380373    21.62   0.000     .7478834    .8970565
------------------------------------------------------------------------------

. 
end of do-file

. gen rmt2 = r_mt^2
(1 missing value generated)

. reg csad abs_r_mt rmt2 i.year, robust

Linear regression                               Number of obs     =      2,609
                                                F(11, 2597)       =     141.73
                                                Prob > F          =     0.0000
                                                R-squared         =     0.3903
                                                Root MSE          =     .72244

------------------------------------------------------------------------------
             |               Robust
        csad |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    abs_r_mt |   .4994904   .0292174    17.10   0.000     .4421987    .5567822
        rmt2 |   -.052799   .0051352   -10.28   0.000    -.0628686   -.0427295
             |
        year |
       2002  |  -.2258721   .0459779    -4.91   0.000    -.3160291   -.1357151
       2003  |    -.04762   .0484289    -0.98   0.326    -.1425832    .0473433
       2004  |   .0641004   .0472513     1.36   0.175    -.0285536    .1567545
       2005  |   .4017124   .0588411     6.83   0.000     .2863322    .5170927
       2006  |   .5451839   .0563295     9.68   0.000     .4347286    .6556392
       2007  |   1.051536   .0768241    13.69   0.000     .9008934    1.202179
       2008  |   .8957598   .0699791    12.80   0.000     .7585394     1.03298
       2009  |   .4675094   .0568641     8.22   0.000     .3560059    .5790129
       2010  |   .2445753   .0514202     4.76   0.000     .1437465    .3454041
             |
       _cons |   .8224699   .0380373    21.62   0.000     .7478834    .8970565
------------------------------------------------------------------------------

.

As you can tell, the results were identical, the only difference are the names of the variables.

Thank you.

[Carlo Lazzaro]

Guest:
1) you seemingly logged the independent variables (instead of the dv, that is -csad-);
2) as far as categorical variables and interactions creation is concerned, it's a safe (and rewarding) habit to rely always on -fvvarlist- notation.

[Guest]

Carlo:

. gen log1=log(csad)
(192 missing values generated)

. reg log1 abs_r_mt rmt2

      Source |       SS           df       MS      Number of obs   =     2,418
-------------+----------------------------------   F(2, 2415)      =    200.50
       Model |  73.7438233         2  36.8719117   Prob > F        =    0.0000
    Residual |  444.111582     2,415  .183897135   R-squared       =    0.1424
-------------+----------------------------------   Adj R-squared   =    0.1417
       Total |  517.855405     2,417  .214255443   Root MSE        =    .42883

------------------------------------------------------------------------------
        log1 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    abs_r_mt |   .2437952   .0158073    15.42   0.000     .2127979    .2747925
        rmt2 |  -.0220551   .0027192    -8.11   0.000    -.0273873   -.0167229
       _cons |   .1940486   .0152521    12.72   0.000       .16414    .2239573
------------------------------------------------------------------------------

.

[Carlo Lazzaro]

Guest:
now perform -estat ovtest- and -estat hettest- after log-linear regression.