Large coefficient compared to mean of dependent variable; IV; Interpretation of coefficient

[Shisho Jakas]

Dear Statalist,

I am writing to ask for help interpreting the coefficient with the mean dependent variable.
In my panel (id, year) dataset, I have yvar (number of accident), xvar (treatment), and instrument.
Given that all the assumption holds for IV, how can I interpret the coefficient from the second stage compared to mean dependent value?

As far as I know, in OLS or DID, mean dependent variable is reported to interpret the coefficient compared to the mean value.
Does this apply to IV as well?
What if the coefficient is way l larger than mean of dependent value then?


For example, from my result shows coefficient of xvar = -0.31 and mean of the dependent value = 0.09

So, if the value of xvar increases by one unit (treated), yvar (accident) goes down by 0.31?
And, compared to control group, how can I use mean value to interpret this coefficient?


All the data, code, and results are following:

Thank you so much for your help in advance.

copy starting from the next line ------- ---------------

* Example    generated by -dataex-. For more info, type help dataex
clear
input int    id float year byte yvar float(instrument xvar)
7 1986 0    189.87224 0
7 1987 0    192.2707 0
7 1988 0    192.7204 0
7 1989 0    191.68484 0
7 1990 0    193.5854 0
7 1991 0    190.0769 0
7 1992 0    194.39555 0
7 1993 0    188.67802 1
7 1994 0    193.45345 1
7 1995 0    192.5287 1
7 1996 0    191.7457 1
8 1986 0    196.9478 0
8 1987 0    199.43567 0
8 1988 0    199.9021 0
8 1989 0    198.82797 0
8 1990 0    200.79936 0
8 1991 0    197.16013 0
8 1992 0    201.6397 0
8 1993 0    195.7091 0
8 1994 0    200.66248 0
8 1995 0    199.70326 1
8 1996 0    198.8911 1
19 1986 0    238.9022 1
19 1987 0    241.92 1
19 1988 0    242.4858 1
19 1989 0    241.18288 1
19 1990 0    243.5742 1
19 1991 0    239.15973 1
19 1992 0    244.59355 1
19 1993 0    237.3996 1
19 1994 0    243.40817 1
19 1995 0    242.24463 1
19 1996 0    241.2594 1
20 1986 0    225.95587 0
20 1987 1    228.81015 0
20 1988 0    229.3453 0
20 1989 0    228.11296 1
20 1990 0    230.3747 1
20 1991 0    226.19945 1
20 1992 0    231.3388 1
20 1993 0    224.5347 1
20 1994 0    230.21767 1
20 1995 0    229.11717 1
20 1996 0    228.18536 1
21 1986 0    223.52946 0
21 1987 0    226.3531 0
21 1988 0    226.8825 0
21 1989 0    225.66338 0
21 1990 0    227.90085 0
21 1991 0    223.7704 0
21 1992 0    228.8546 0
21 1993 0    222.12357 0
21 1994 0    227.7455 0
21 1995 0    226.6568 0
21 1996 0    225.735 0
22 1986 0    214.83475 0
22 1987 0    217.54854 0
22 1988 0    218.05734 0
22 1989 0    216.88565 0
22 1990 0    219.0361 0
22 1991 0    215.06633 0
22 1992 0    219.95274 0
22 1993 0    213.4835 0
22 1994 1    218.8868 0
22 1995 0    217.84044 0
22 1996 0    216.9545 1
23 1986 0    259.33914 1
23 1987 1    262.6151 1
23 1988 0    263.22934 1
23 1989 0    261.8149 1
23 1990 0    264.41083 1
23 1991 0    259.6187 1
23 1992 0    265.51736 1
23 1993 1    257.70804 1
23 1994 0    264.2306 1
23 1995 0    262.9675 1
23 1996 0    261.898 1
24 1986 1    261.66428 0
24 1987 0    264.96964 1
24 1988 1    265.58936 1
24 1989 0    264.16226 1
24 1990 0    266.78143 1
24 1991 0    261.94635 1
24 1992 0    267.8979 1
24 1993 0    260.01852 1
24 1994 0    266.59958 1
24 1995 0    265.32516 1
24 1996 0    264.2461 1
25 1986 3    249.57635 0
25 1987 0    252.729 0
25 1988 1    253.3201 1
25 1989 1    251.95892 1
25 1990 1    254.4571 1
25 1991 1    249.8454 1
25 1992 1    255.522 1
25 1993 0    248.0066 1
25 1994 0    254.28365 1
25 1995 0    253.0681 1
25 1996 0    252.0389 1
26 1986 0    249.85435 0
end

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Variable Obs Mean Std. dev. Min Max

id 11,858 897.3237 525.8204 7 1845
year 11,858 1991 3.162411 1986 1996
yvar 11,858 .0766571 .3330643 0 5
instrument 9,405 222.899 30.43519 100.4544 305.2475
xvar 11,858 .1174149 .2468511 0 1



The code I use for 2sls:

xtset id year
eststo: xtivreg yvar (xvar = instrument), fe vce(cluster id) 
estadd ysumm
esttab using t1.csv, se star(* 0.10 ** 0.05 *** 0.01)  stats(N N_g ymean)  b(%9.3f) replace

Result:

yvar Coefficient std. err. z P>z [95% conf. interval]

xvar -.3174492 .1749153 -1.81 0.070 -.6602769 .0253784
_cons .2543608 .0902381 2.82 0.005 .0774975 .4312242


e(ymean) = .09059011

[Joro Kolev]

I do not know where you have seen these comparisons/interpretations, but in linear models the meaning of the estimated parameter on a regressor is always the same:

it is by how much the dependent variable changes as a result of a unit change in the regressor.

[Shisho Jakas]

Dear Joro Kolev,

Thank you so much for your response. So, you mean that there is useless to report mean value of dependent variable? For example, Heath and Mobarak (2015) https://www.sciencedirect.com/scienc...04387815000085 reported them at the end of regression table, and give us some interpretation with estimated parameters.

[Joro Kolev]

Dear Joro Kolev,

Thank you so much for your response. So, you mean that there is useless to report mean value of dependent variable? For example, Heath and Mobarak (2015) https://www.sciencedirect.com/scienc...04387815000085 reported them at the end of regression table, and give us some interpretation with estimated parameters.

No, on the opposite, I think that reporting summary statistics of the dependent variable, including its mean, is a must. The summary statistics of the dependent variable are telling us what we are dealing with, what are the orders of magnitude we are trying to explain.

And yet when we are fitting a regression we are estimating a conditional mean

E(Y|X) = a + b*X, say, if we have only one regressor.

The interpretation of b is always the same, it is the derivative d[E(Y|X)]/dX = b. All the interpretations of our regression parameters come from here.

[Shisho Jakas]

Thank you so much for your kind explanation. If I understood correctly, the mean of the dependent value has nothing to do with the interpretation of the estimated parameter although it gives us information about the variables.

So, is there nothing wrong with having a bigger point estimate (in magnitude) than the mean of the dependent value?

From my example, the dependent variable changes by -0.31 when regressor changes by 1 (here, from 0 to 1, so if treated) even though the mean of the dependent variable is 0.09.

I get confused, because in my feeling, this is a BIG change compared to its mean value.

And the paper I mentioned includes the calculation with the mean dependent variable " The first column indicates that girls living in garment-proximate vil- lages where factories have been operating for 6.4 years (which is the average exposure in the garment-proximate villages across all units of observation in the regression) have a (6.4 × 0.00048) = 0.3 percentage point lower probability of getting marriedThe probability that a girl in our sample is married during the average sample year is 1.1%, and the 0.3 percentage point decrease therefore represents a 28% decrease in the hazard of getting married."

when point estimate: -0.00048, mean dependent variable: 0.0111 in Table 4, for example.