An interaction effect

[ALKEBSEE RADWAN]

Hello everyone,
I know you will say why I am asking too much these days. I actually got a revision that needs addressing.
I got the following comment from a reviewer
"The linear model is used in Section 5.3, but is there an interaction effect for the independent variables? Like second-order interaction effects? If there is an interaction effect, how does the way for OLS ensure that the statistical significance of its parameters is correct under the current model? "

To understand this comment I will explain a bit more.
I have used several linear regressions (OLS), two of them consists of an interaction effect ( IV*moderator variable). Both models have a significant coefficient for the interaction effect.

Here, a reviewer asks for how does the way for OLS ensure that the statistical significance of its parameters is correct under the current model?

Kindly can anyone help me addressing this comment?

[Carlo Lazzaro]

Alkebsee:
I'm not sure I got reviewer's comment right, but I'm under the impression that two issues need to be addressed:
1) statistical significance between no interaction vs. interaction:

. sysuse auto.dta
(1978 automobile data)

. regress price i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(7, 61)        =      0.39
       Model |    24684607         7  3526372.43   Prob > F        =    0.9049
    Residual |   552112352        61  9051022.16   R-squared       =    0.0428
-------------+----------------------------------   Adj R-squared   =   -0.0670
       Total |   576796959        68  8482308.22   Root MSE        =    3008.5

-------------------------------------------------------------------------------
        price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
      foreign |
     Foreign  |   2088.167   2351.846     0.89   0.378     -2614.64    6790.974
              |
        rep78 |
           2  |   1403.125   2378.422     0.59   0.557    -3352.823    6159.073
           3  |   2042.574   2204.707     0.93   0.358    -2366.011    6451.159
           4  |   1317.056   2351.846     0.56   0.578    -3385.751    6019.863
           5  |       -360   3008.492    -0.12   0.905    -6375.851    5655.851
              |
foreign#rep78 |
   Foreign#1  |          0  (empty)
   Foreign#2  |          0  (empty)
   Foreign#3  |  -3866.574   2980.505    -1.30   0.199    -9826.462    2093.314
   Foreign#4  |  -1708.278   2746.365    -0.62   0.536    -7199.973    3783.418
   Foreign#5  |          0  (omitted)
              |
        _cons |     4564.5   2127.325     2.15   0.036      310.651    8818.349
-------------------------------------------------------------------------------

. mat list e(b)

e(b)[1,18]
             0b.           1.          1b.           2.           3.           4.           5.  0b.foreign#  0b.foreign#  0b.foreign#
        foreign      foreign        rep78        rep78        rep78        rep78        rep78     1b.rep78     2o.rep78     3o.rep78
y1            0    2088.1667            0     1403.125    2042.5741    1317.0556         -360            0            0            0

     0b.foreign#  0b.foreign#  1o.foreign#  1o.foreign#   1.foreign#   1.foreign#  1o.foreign#            
       4o.rep78     5o.rep78     1b.rep78     2o.rep78      3.rep78      4.rep78     5o.rep78        _cons
y1            0            0            0            0   -3866.5741   -1708.2778            0       4564.5

. lincom 1.foreign - (1.foreign#1.rep78)

 ( 1)  1.foreign - 1o.foreign#1b.rep78 = 0

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   2088.167   2351.846     0.89   0.378     -2614.64    6790.974
------------------------------------------------------------------------------

*There's no evidence of and difference*

2) possible misspecification of the functional form of the regressand:

. sysuse auto.dta
(1978 automobile data)

. regress price i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(7, 61)        =      0.39
       Model |    24684607         7  3526372.43   Prob > F        =    0.9049
    Residual |   552112352        61  9051022.16   R-squared       =    0.0428
-------------+----------------------------------   Adj R-squared   =   -0.0670
       Total |   576796959        68  8482308.22   Root MSE        =    3008.5

-------------------------------------------------------------------------------
        price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
      foreign |
     Foreign  |   2088.167   2351.846     0.89   0.378     -2614.64    6790.974
              |
        rep78 |
           2  |   1403.125   2378.422     0.59   0.557    -3352.823    6159.073
           3  |   2042.574   2204.707     0.93   0.358    -2366.011    6451.159
           4  |   1317.056   2351.846     0.56   0.578    -3385.751    6019.863
           5  |       -360   3008.492    -0.12   0.905    -6375.851    5655.851
              |
foreign#rep78 |
   Foreign#1  |          0  (empty)
   Foreign#2  |          0  (empty)
   Foreign#3  |  -3866.574   2980.505    -1.30   0.199    -9826.462    2093.314
   Foreign#4  |  -1708.278   2746.365    -0.62   0.536    -7199.973    3783.418
   Foreign#5  |          0  (omitted)
              |
        _cons |     4564.5   2127.325     2.15   0.036      310.651    8818.349

 predict fitted, xb
(5 missing values generated)

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

. regress price fitted sq_fitted

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(2, 66)        =      1.48
       Model |    24684607         2  12342303.5   Prob > F        =    0.2361
    Residual |   552112352        66  8365338.66   R-squared       =    0.0428
-------------+----------------------------------   Adj R-squared   =    0.0138
       Total |   576796959        68  8482308.22   Root MSE        =    2892.3

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
      fitted |   1.000001   8.993582     0.11   0.912    -16.95627    18.95627
   sq_fitted |  -1.02e-10   .0008012    -0.00   1.000    -.0015997    .0015997
       _cons |  -.0027133   24867.64    -0.00   1.000    -49649.85    49649.85
------------------------------------------------------------------------------

.
*There's no evidence of misspecification of the functional form of the regressand*

[ALKEBSEE RADWAN]

Alkebsee:
I'm not sure I got reviewer's comment right, but I'm under the impression that two issues need to be addressed:
1) statistical significance between no interaction vs. interaction:

. sysuse auto.dta
(1978 automobile data)

. regress price i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

Source | SS df MS Number of obs = 69
-------------+---------------------------------- F(7, 61) = 0.39
Model | 24684607 7 3526372.43 Prob > F = 0.9049
Residual | 552112352 61 9051022.16 R-squared = 0.0428
-------------+---------------------------------- Adj R-squared = -0.0670
Total | 576796959 68 8482308.22 Root MSE = 3008.5

-------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
--------------+----------------------------------------------------------------
foreign |
Foreign | 2088.167 2351.846 0.89 0.378 -2614.64 6790.974
|
rep78 |
2 | 1403.125 2378.422 0.59 0.557 -3352.823 6159.073
3 | 2042.574 2204.707 0.93 0.358 -2366.011 6451.159
4 | 1317.056 2351.846 0.56 0.578 -3385.751 6019.863
5 | -360 3008.492 -0.12 0.905 -6375.851 5655.851
|
foreign#rep78 |
Foreign#1 | 0 (empty)
Foreign#2 | 0 (empty)
Foreign#3 | -3866.574 2980.505 -1.30 0.199 -9826.462 2093.314
Foreign#4 | -1708.278 2746.365 -0.62 0.536 -7199.973 3783.418
Foreign#5 | 0 (omitted)
|
_cons | 4564.5 2127.325 2.15 0.036 310.651 8818.349
-------------------------------------------------------------------------------

. mat list e(b)

e(b)[1,18]
0b. 1. 1b. 2. 3. 4. 5. 0b.foreign# 0b.foreign# 0b.foreign#
foreign foreign rep78 rep78 rep78 rep78 rep78 1b.rep78 2o.rep78 3o.rep78
y1 0 2088.1667 0 1403.125 2042.5741 1317.0556 -360 0 0 0

0b.foreign# 0b.foreign# 1o.foreign# 1o.foreign# 1.foreign# 1.foreign# 1o.foreign#
4o.rep78 5o.rep78 1b.rep78 2o.rep78 3.rep78 4.rep78 5o.rep78 _cons
y1 0 0 0 0 -3866.5741 -1708.2778 0 4564.5

. lincom 1.foreign - (1.foreign#1.rep78)

( 1) 1.foreign - 1o.foreign#1b.rep78 = 0

------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
(1) | 2088.167 2351.846 0.89 0.378 -2614.64 6790.974
------------------------------------------------------------------------------

*There's no evidence of and difference*

2) possible misspecification of the functional form of the regressand:

. sysuse auto.dta
(1978 automobile data)

. regress price i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

Source | SS df MS Number of obs = 69
-------------+---------------------------------- F(7, 61) = 0.39
Model | 24684607 7 3526372.43 Prob > F = 0.9049
Residual | 552112352 61 9051022.16 R-squared = 0.0428
-------------+---------------------------------- Adj R-squared = -0.0670
Total | 576796959 68 8482308.22 Root MSE = 3008.5

-------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
--------------+----------------------------------------------------------------
foreign |
Foreign | 2088.167 2351.846 0.89 0.378 -2614.64 6790.974
|
rep78 |
2 | 1403.125 2378.422 0.59 0.557 -3352.823 6159.073
3 | 2042.574 2204.707 0.93 0.358 -2366.011 6451.159
4 | 1317.056 2351.846 0.56 0.578 -3385.751 6019.863
5 | -360 3008.492 -0.12 0.905 -6375.851 5655.851
|
foreign#rep78 |
Foreign#1 | 0 (empty)
Foreign#2 | 0 (empty)
Foreign#3 | -3866.574 2980.505 -1.30 0.199 -9826.462 2093.314
Foreign#4 | -1708.278 2746.365 -0.62 0.536 -7199.973 3783.418
Foreign#5 | 0 (omitted)
|
_cons | 4564.5 2127.325 2.15 0.036 310.651 8818.349

predict fitted, xb
(5 missing values generated)

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

. regress price fitted sq_fitted

Source | SS df MS Number of obs = 69
-------------+---------------------------------- F(2, 66) = 1.48
Model | 24684607 2 12342303.5 Prob > F = 0.2361
Residual | 552112352 66 8365338.66 R-squared = 0.0428
-------------+---------------------------------- Adj R-squared = 0.0138
Total | 576796959 68 8482308.22 Root MSE = 2892.3

------------------------------------------------------------------------------
price | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
fitted | 1.000001 8.993582 0.11 0.912 -16.95627 18.95627
sq_fitted | -1.02e-10 .0008012 -0.00 1.000 -.0015997 .0015997
_cons | -.0027133 24867.64 -0.00 1.000 -49649.85 49649.85
------------------------------------------------------------------------------

.
*There's no evidence of misspecification of the functional form of the regressand*

dear @Carlo Lazzaro Thank you so much
I have a question regarding this two tests as follows:
the first one (1) statistical significance between no interaction vs. interaction), If the Pvalue of lincom command is significant can I confirm that my OLS for the interaction ensures the significance level I got in the current level?
Because the outcome of my model was as follows

------------------------------------------------------------------------------
         real |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   .0053886   .0032907     1.68  0.092    -.0010619    .0118391
------------------------------------------------------------------------------

it means there is a difference
*****

Also, for the second test I got the following

------------------------------------------------------------------------------
         real |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      fitted |    .937996   .0249639    37.57   0.000     .8890614    .9869306
   sq_fitted |   1.480995   .1080274    13.71   0.000     1.269238    1.692751
       _cons |  -.0042517   .0016906    -2.51   0.012    -.0075656   -.0009378
------------------------------------------------------------------------------

Please explain it

best,

[Carlo Lazzaro]

Alkebsee:
1) the p-value=0.092 does not support the evidence of any difference;
2) as -sq_fitted- voefficient reaches statistical significance, your model is misspecified.

[ALKEBSEE RADWAN]

Alkebsee:
1) the p-value=0.092 does not support the evidence of any difference;
2) as -sq_fitted- voefficient reaches statistical significance, your model is misspecified.

Carlo Lazzaro
I actually want to ask you whether the two outcomes suggests that my OLS ensures the significance level of the interaction effect?
for example: As my interaction model got a significant coefficient of sq_fitted, can I say the model doesn't ensure the significant level of my current model?

once again thank you so much

[Carlo Lazzaro]

Alkebsee:
not quite.
1) the first outcome (p-value=0.092) does not support the evidence of any difference between no interaction and interaction;
2) the statistical significance reached by -sq-fitted- tells you that your model is misspecified (that is, you probably missed to add other poredictors and/or interactions).

[ALKEBSEE RADWAN]

Alkebsee:
not quite.
1) the first outcome (p-value=0.092) does not support the evidence of any difference between no interaction and interaction;
2) the statistical significance reached by -sq-fitted- tells you that your model is misspecified (that is, you probably missed to add other poredictors and/or interactions).

Thank you very much

[ALKEBSEE RADWAN]

Alkebsee:
not quite.
1) the first outcome (p-value=0.092) does not support the evidence of any difference between no interaction and interaction;
2) the statistical significance reached by -sq-fitted- tells you that your model is misspecified (that is, you probably missed to add other poredictors and/or interactions).

Dear Carlo Lazzaro I am actually still confused. Some aruge that Lincom command is same as the test command. So, when I use the Chow test (test command) I interpret the P-value as follows:
when P is significant (at 1, 5 10%) I decide that there is a significant diference between the two coefficient in favor of the higher one.
However, In the lincom command I got confused thereby, I will show you another outcomes to understand clearly
please have a look below

 lincom log_ID_pay - ( consh_id)

 ( 1)  [restate_dum]log_ID_pay - [restate_dum]consh_id = 0

------------------------------------------------------------------------------
 restate_dum |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         (1) |   -.323614   .1418672    -2.28   0.023    -.6016687   -.0455593
------------------------------------------------------------------------------

Sorry for disturbing you again

[Carlo Lazzaro]

Alkebsee:
1) if your arbitary value is 0.10 (instead of the usual but still arbitary 0.05), a 0,092 p-value supports the evidence of a (I would say bare) statistically signicìficant difference;
2) your last example is also statistically significant at 0.10 and 0.05 arbitrary values.
That said, it is good habit to define beforehand the arbitrary value for statistical significance instead of defining it after the results have come alive.

[ALKEBSEE RADWAN]

Alkebsee:
1) if your arbitary value is 0.10 (instead of the usual but still arbitary 0.05), a 0,092 p-value supports the evidence of a (I would say bare) statistically signicìficant difference;
2) your last example is also statistically significant at 0.10 and 0.05 arbitrary values.
That said, it ia good ahbit to define beforehand the arbitrary value for statistical significance instead of defining it after the results have come alive.

Ok Ok I got it clearly thank you so much

[Carlo Lazzaro]

Alkebsee:
my previous example about -lincom- shoud have been:

. regress price i.foreign##i.rep78
note: 1.foreign#1b.rep78 identifies no observations in the sample.
note: 1.foreign#2.rep78 identifies no observations in the sample.
note: 1.foreign#5.rep78 omitted because of collinearity.

      Source |       SS           df       MS      Number of obs   =        69
-------------+----------------------------------   F(7, 61)        =      0.39
       Model |    24684607         7  3526372.43   Prob > F        =    0.9049
    Residual |   552112352        61  9051022.16   R-squared       =    0.0428
-------------+----------------------------------   Adj R-squared   =   -0.0670
       Total |   576796959        68  8482308.22   Root MSE        =    3008.5

-------------------------------------------------------------------------------
        price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
      foreign |
     Foreign  |   2088.167   2351.846     0.89   0.378     -2614.64    6790.974
              |
        rep78 |
           2  |   1403.125   2378.422     0.59   0.557    -3352.823    6159.073
           3  |   2042.574   2204.707     0.93   0.358    -2366.011    6451.159
           4  |   1317.056   2351.846     0.56   0.578    -3385.751    6019.863
           5  |       -360   3008.492    -0.12   0.905    -6375.851    5655.851
              |
foreign#rep78 |
   Foreign#1  |          0  (empty)
   Foreign#2  |          0  (empty)
   Foreign#3  |  -3866.574   2980.505    -1.30   0.199    -9826.462    2093.314
   Foreign#4  |  -1708.278   2746.365    -0.62   0.536    -7199.973    3783.418
   Foreign#5  |          0  (omitted)
              |
        _cons |     4564.5   2127.325     2.15   0.036      310.651    8818.349
-------------------------------------------------------------------------------

. mat list e(b)

e(b)[1,18]
             0b.           1.          1b.           2.           3.           4.           5.  0b.foreign#  0b.foreign#  0b.foreign#
        foreign      foreign        rep78        rep78        rep78        rep78        rep78     1b.rep78     2o.rep78     3o.rep78
y1            0    2088.1667            0     1403.125    2042.5741    1317.0556         -360            0            0            0

     0b.foreign#  0b.foreign#  1o.foreign#  1o.foreign#   1.foreign#   1.foreign#  1o.foreign#            
       4o.rep78     5o.rep78     1b.rep78     2o.rep78      3.rep78      4.rep78     5o.rep78        _cons
y1            0            0            0            0   -3866.5741   -1708.2778            0       4564.5

. lincom 1.foreign - (1.foreign+1.foreign#1.rep78)

 ( 1)  - 1o.foreign#1b.rep78 = 0

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |          0  (omitted)
------------------------------------------------------------------------------

*Explanation:*
. di 2088.1667 - (2088.1667 + 0)
0

.

The revised version of the toy-example captures the interaction, whereas the previous one did not.