testing inequality between coefficients

[chiara piccardo]

Hello everyone,
I have a question that is related to hypoteses testing in stata

I run the following Pooled OLS regression (Stata 14.2):

regress Y   L.nopattm L.onlytm L.onlypat L.pattm L.X1 L.X2 i.X4 X5 X6 X7 i.year,noconstant  vce(robust)

where nopattm, onlypat, onlytm and pattm are dummy variables identifying exclusive combinations of patents and trademarks; Y is a measure for firm's performance and Xs are control variables.

I would need to test if performance is higher if the firm uses IP right protection. Thus, I would test the following hypotheses (where b1, b2, b3 and b4 are the estimated coefficients for the four dummy variables)
1) H0: b4>b1

2) H0: b2>b1

3) H0: b3>b1

I would test if performance is higher if the firm chooses to use both a patents and a trademarks than only one type of IP right
4)H0: b4>b2

5)Ho: b4>b3

If I have understood correctly from a previous discussion (see the following link : https://www.statalist.org/forums/for...then-the-other), in order to test these hypotheses, I should perform one side t tests. For example, I should use the following code in order to test the hypothesis number 4

 
test L.onlytm L.pattm
 
 ( 1)  L.onlytm = 0
 ( 2)  L.pattm_d = 0
 
       F(  2,686216) =45442.44
            Prob > F =    0.0000
 
. test L.onlytm -L.pattm=0
 
 ( 1)  L.onlytm - L.pattm_d = 0
 
       F(  1,686216) =    7.96
            Prob > F =    0.0048
 
. local sign_car = sign(_b[L.pattm]-_b[L.onlytm])
 
. display "H_0: PAT TM coef >=  TM coef. p-value = " normal(`sign_car'*sqrt(r(F)))
H_0: PAT TM coef >=  TM coef. p-value = .99760379

Tests suggest that: b4 and b2 are not jointly equal to zero; are not equal and b4>b2
Is this the properly interpretation?


Furthermore, I would test if adding an activity (i.e patents) while the other activity (i.e trademarks) is already being performed has a higher incremental effect on performance than adding the activity (patents) in isolation. Thus, I need to test the following hypothesis:

6) b4-b2>b3-b1

Can I follow the same approach as above and use the following code to test this hypothesis?

test L.nopattm L.onlytm L.onlypat L.pattm
 
 ( 1)  L.nopattm = 0
 ( 2)  L.onlytm = 0
 ( 3)  L.onlypat = 0
 ( 4)  L.pattm_d = 0
 
       F(  4,686216) =25374.94
            Prob > F =    0.0000
 
.
. test  L.nopattm- L.onlytm- L.onlypat- L.pattm=0
 
 ( 1)  L.nopattm - L.onlytm - L.onlypat - L.pattm_d = 0
 
       F(  1,686216) =82696.42
            Prob > F =    0.0000
 
.
. local sign_ip = sign(_b[L.pattm]-_b[L.onlytm]-_b[L.onlypat]+_b[L.nopattm])
 
.
. display "H_0: pattm-onytm >onlypat-nopattm p-value = " normal(`sign_ip'*sqrt(r(F)))
H_0: pattm-onytm >onlypat-nopattm p-value = 0

Is it still a one-side t test?
The four coefficients are not jointly equal to zero, are not equal and the test rejects the H0(H0 states that b4-b2>b3-b1). Is this interpretation right?
How can I obtain the value of the t statistic?

I thank you all in advance for your help.
Chiara