Regression interpretation of results (linear/ushaped)

[Christoph Domnick]

Hello
I am new to Stata and I am trying to run some regressions for my bachelor thesis. First, I am using an unbalanced panel data with 2401 observations. The data is collected over a time span of 7 years whereas the only 6 years are used due to lagging the dependent variable one year behind the dependent variable. However, I have lagged the independent variable already in excel behind the dependent variable, which means that the 2401 observations only include the relevant 6 years. The data is collected from 463 different companies. My aim is it to make a regression analysis including the dependent variable ROANI, independent variables BVSTD BVLTD and BVTD and the control variables RK TAN GA GS and SZ. I used the fixed effect model, since the Hausman test rejected the null hypothesis. Furthermore, I tested the regression for autocorrelation and heteroskedasticity and it came out that both tests rejected the null hypothesis and therefore I included the term -robust- in the equation,

However, the thesis I am working on tests two different possibilities. First, if the relationship is inverted U -shaped, by adding the variables BVSTD^2, BVLTD^2 and LTD^2 to the equation and furthermore, if the relationship is linear.
Therefore I created the ouptups which can be seen in the attachements.





As it seems the relationship is not linear since the p values are really high. However, could someone help me to interprete the results of both outputs and tell me if the models are correctly entered? Or is is possible that I did some mistakes?

I am thankful for any advise

[Marcos Almeida]

Welcome to the Stata Forum/Statalist,

Please take a look at the FAQ, particularly the advice about sharing data/command/output. Also, the reason for avoiding snapshots.

That said, I just wish to add a couple of notes.

You don't need to create variables with the squared terms. Please see below an example, with the same results.

. sysuse auto
(1978 Automobile Data)

. gen price2 = price^2

. regress mpg price price2 foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(3, 70)        =     20.10
       Model |  1130.78787         3  376.929288   Prob > F        =    0.0000
    Residual |  1312.67159        70  18.7524513   R-squared       =    0.4628
-------------+----------------------------------   Adj R-squared   =    0.4398
       Total |  2443.45946        73  33.4720474   Root MSE        =    4.3304

------------------------------------------------------------------------------
         mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       price |  -.0039494   .0010082    -3.92   0.000    -.0059602   -.0019386
      price2 |   1.74e-07   5.77e-08     3.01   0.004     5.86e-08    2.89e-07
     foreign |   5.997569   1.130642     5.30   0.000     3.742573    8.252564
       _cons |   35.76619   3.569962    10.02   0.000     28.64612    42.88625
------------------------------------------------------------------------------

. regress mpg c.price##c.price foreign

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(3, 70)        =     20.10
       Model |  1130.78788         3  376.929292   Prob > F        =    0.0000
    Residual |  1312.67158        70  18.7524512   R-squared       =    0.4628
-------------+----------------------------------   Adj R-squared   =    0.4398
       Total |  2443.45946        73  33.4720474   Root MSE        =    4.3304

---------------------------------------------------------------------------------
            mpg |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
          price |  -.0039494   .0010082    -3.92   0.000    -.0059602   -.0019386
                |
c.price#c.price |   1.74e-07   5.77e-08     3.01   0.004     5.86e-08    2.89e-07
                |
        foreign |   5.997569   1.130642     5.30   0.000     3.742574    8.252564
          _cons |   35.76619   3.569962    10.02   0.000     28.64612    42.88625

I gather - margins - and - marginsplot would be great in terms of interpretation.

You may wish to check whether the model "improved" significantly with the addtiion of the squared terms, particularly BVSTD2.

Hopefully that helps.

[Carlo Lazzaro]

Christoph:
some remarks about your query:
- if you lag the dependent variable and you use it as a predictor, -xtreg- will give you biased results (see -xtabond- instead). Admittedly, I can't get from your post whether this is the case with your research or not;
- you can lag/leadf/differenciate variable with -tsvarlist- commands;
- you should prefere creating categorical variables and interactions (wich include squared terms) via -fvvarlist- rather than by hand (this approach is particularly useful if you plan to use -margins- and -marginsplot- thereafter);
- the results of your model do not show evidence of a quadratic relationship for -BVT- only, whose squared term is barely significant (however, the increase in withinh R-sq is pretty negligible vs previous model);
- I would test whether your models suffere from omitted variable bias.

[Christoph Domnick]

Dear Marcos and Carlo,

Thanks a lot for your responses. Marocs, I don’t understand the logic behind your advice :D As you have shown in your example, creating the variable or using the
regress mpg c.price##c.price foreign code produces the exact same results. Could you maybe further elaborate on that? And sorry for using a snapshot :D

Carlo, sorry I mentioned it wrong in my statement. The independent variables and control variables are lagged one year behind the dependent variable. Furthermore, due to multicollinearity, I am running the regressions separately for each independent variable. Do you mean that only for BVT there is evidence for a quadratic relationship? Why is it better to use fvvarlist and how does it help me for my regression? Sorry I am just confused, since I have only seen it before how I did it and not with the fvvarit option.

Finally, would someone be able to explain the use of margins and margins plot for me, since I have never come across those techniques during my courses or in the literature I am using.

Thanks a lot

[Marcos Almeida]

To have a grasp on margins and marginsplot, just type in Stata’s command window: help margins and help marginsplot. You may wish to read the Stata Manual as well. Last but not least, Richard William’s excellent texts, freely available on the Internet.

One of the advantages of the code I suggested is exactly related to them (I mean, margins and marginsplot). Also, both are great to provide a frienlier interpretation of the results.

[Carlo Lazzaro]

Christoph:
- I was too hectic in my previous reply: the results of your model show evidence of a quadratic relationship for -BVT- only, whose squared term is barely significant (however, the increase in withinh R-sq is pretty negligible vs previous model);
- -fvvarlist- notation will help you enormously in creating categorical variables (especially when it has >2 levels) and interactions. Echoing Marcos' wise comments, the relationship between -fvvarlist-, -margins- and -marginsplot- is also noteworthy.