Difference-in-differences estimation
Agniva Ganguly
2025-07-18
Simple DiD settings
The difference-in-differences design is the go-to research design in contemporary econometrics to examine the effects of some regulatory intervention, a new accounting standard, or any other event. In the most basic form, this design can be implemented with a simple two-by-two matrix that displays the average values of the outcome variable for four groups split by two dimensions: (i) a treatment dimension that indicates whether the observation belongs to the treatment group affected by the shock or to the control group not affected by the shock, and (ii) a time dimension that indicates whether the observation is from the period before or after the shock.
In our illustration, for an international sample of firm-year
observations around the mandatory adoption of IFRS in 2005, we can
examine pre- and post-IFRS average stock illiquidity of firms in
countries that are subject to IFRS adoption (ifrs==1)
versus those that are not (ifrs==0).
In the following code, we use the tabulate command for
such a sample to display the average of an illiquidity variable named
zeroreturn:
use "data_ifrs_liq.dta",clear
sum year zeroreturn
gen post=0 if year<2005
replace post=1 if year>2005
// Matrix:
tabulate ifrs post, summarize(zeroreturn) means
tabulate year ifrs, summarize(zeroreturn) means Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
year | 120,361 2004.827 4.362174 1997 2012
zeroreturn | 120,361 .2529683 .2416692 .0040161 .9585062
(64,098 missing values generated)
(55,038 real changes made)
Means of zeroreturn
| post
ifrs | 0 1 | Total
-----------+----------------------+----------
0 | .23238348 .21638861 | .22415495
1 | .42692598 .32413876 | .38461387
-----------+----------------------+----------
Total | .27632782 .23379685 | .25529639
Means of zeroreturn
| ifrs
year | 0 1 | Total
-----------+----------------------+----------
1997 | .26316084 .40436974 | .29443972
1998 | .26435275 .41703674 | .30416961
1999 | .23038658 .43127403 | .28014852
2000 | .2377778 .4109584 | .27814002
2001 | .24320788 .42387623 | .2857064
2002 | .23410513 .45509432 | .28343205
2003 | .21828692 .44823993 | .26681914
2004 | .19536001 .41468009 | .23724099
2005 | .1905788 .37712961 | .22436797
2006 | .18036518 .3250164 | .20567873
2007 | .17165488 .30147589 | .1938751
2008 | .20508533 .32021188 | .22403615
2009 | .24238386 .34643473 | .2590272
2010 | .24348337 .33805393 | .25822946
2011 | .23819951 .31802251 | .25028696
2012 | .24446376 .32334254 | .25616878
-----------+----------------------+----------
Total | .22159004 .38408544 | .25296829From the values displayed in the four cells that are separated by
ifrs and post, we can calculate the
difference-in-difference as the change in average outcomes (i.e., the
change from post==0 to post==1) for the
treatment (ifrs==1) group minus the change in average
outcomes for the control group (ifrs==0).
Alternatively, we can also recover this difference-in-difference estimate by estimating a linear regression with interactions. Specifically, consider the following model for firm \(i\) in year \(t\), where each firm is located in country \(c\). We can use this model to estimate the effects of country-level IFRS adoption on some outcome variable \(y\):
\[ y_{i c t}=\beta_0+\beta_1 I F R S_c+\beta_2 P O S T_t+\beta_3 I F R S_c \times P O S T_t+\varepsilon_{i c t} \]
Using the same sample of firm-year observations, the simplest way we can estimate this difference-in-difference model is by regressing the outcome variable on the two indicator variables and their interaction.
(9,060 missing values generated)
Linear regression Number of obs = 111,301
F(3, 20) = 80.52
Prob > F = 0.0000
R-squared = 0.0779
Root MSE = .2327
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ifrs | .1945425 .0436291 4.46 0.000 .1035337 .2855513
post | -.0159949 .0164295 -0.97 0.342 -.0502663 .0182766
post_ifrs | -.0867924 .0179522 -4.83 0.000 -.12424 -.0493447
_cons | .2323835 .0237324 9.79 0.000 .1828786 .2818883
------------------------------------------------------------------------------The results suggest that firms that were subject to the IFRS-shock experienced a change in illiquidity that was significantly lower than the change in illiquidity of the control firms. This result is consistent with results in the literature that suggest IFRS adoption was associated with improvements in stock liquidity.
What is important to add here is that the coefficient of-0.0868 on the interaction term is exactly the same number as we would get from calculating the difference-in-difference from the two-by-two matrix displayed above. The benefit of the regression over the tabulation is that we now also obtain an estimate of the standard error associated with the difference-in-difference estimator.
A variant of this type of regression that is often estimated in
practice is one in which the main effects IFRS and POST are “subsumed”
by unit and time fixed effects. Specifically, in the same example, the
IFRS indicator can be replaced either by country- of firm-fixed effects,
while the POST variable can be replaced by year fixed effects. Using the
reghdfe program, we can absorb both firm- and year-fixed
effects as follows. Here, firms are identified by variable gvkey and the
interaction between IFRS and POST is again captured by the variable
post_ifrs that we created above.
First we illustrate the results we obtain on adding time and country fixed effects.
reg zeroreturn i.year i.countryid ifrs post post_ifrs, cluster(countryid)
reg zeroreturn i.year i.countryid post_ifrs, cluster(countryid) note: ifrs omitted because of collinearity.
note: post omitted because of collinearity.
Linear regression Number of obs = 111,301
F(14, 20) = .
Prob > F = .
R-squared = 0.2874
Root MSE = .20459
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
year |
1998 | -.0006483 .0046122 -0.14 0.890 -.0102691 .0089725
1999 | -.0352385 .0236309 -1.49 0.152 -.0845316 .0140546
2000 | -.032459 .021329 -1.52 0.144 -.0769504 .0120325
2001 | -.0241229 .0292886 -0.82 0.420 -.0852177 .036972
2002 | -.0239004 .0344396 -0.69 0.496 -.09574 .0479393
2003 | -.0387805 .0373613 -1.04 0.312 -.1167148 .0391538
2004 | -.0591383 .0456521 -1.30 0.210 -.1543669 .0360903
2006 | -.0718861 .0506407 -1.42 0.171 -.1775208 .0337485
2007 | -.0828033 .0375718 -2.20 0.039 -.1611766 -.00443
2008 | -.0515585 .0458215 -1.13 0.274 -.1471405 .0440234
2009 | -.0165657 .0358965 -0.46 0.649 -.0914445 .0583132
2010 | -.0170998 .0298396 -0.57 0.573 -.0793442 .0451445
2011 | -.024269 .0328281 -0.74 0.468 -.0927472 .0442092
2012 | -.0179488 .0372099 -0.48 0.635 -.0955674 .0596697
|
countryid |
2 | .1446156 .0008542 169.30 0.000 .1428338 .1463974
6 | .1056278 .0025726 41.06 0.000 .1002615 .1109941
12 | .1818235 .0025161 72.26 0.000 .176575 .187072
18 | .2483825 .0011592 214.27 0.000 .2459644 .2508005
24 | .18966 .0119885 15.82 0.000 .1646525 .2146675
25 | .1597764 .0110417 14.47 0.000 .1367439 .182809
26 | .1663577 .0100518 16.55 0.000 .1453901 .1873254
29 | .4185767 .0112388 37.24 0.000 .395133 .4420204
32 | -.1293801 .0186327 -6.94 0.000 -.1682472 -.0905131
34 | .351265 .0109491 32.08 0.000 .3284255 .3741045
35 | -.1604796 .0119335 -13.45 0.000 -.1853725 -.1355867
36 | -.0086126 .0085571 -1.01 0.326 -.0264624 .0092373
37 | -.1222561 .0102385 -11.94 0.000 -.1436133 -.100899
38 | .1378336 .0109803 12.55 0.000 .1149292 .160738
42 | .215822 .0102241 21.11 0.000 .1944949 .2371492
44 | .0753867 .0136506 5.52 0.000 .046912 .1038614
45 | -.0899285 .014831 -6.06 0.000 -.1208654 -.0589917
47 | -.0355286 .0128306 -2.77 0.012 -.0622927 -.0087644
48 | .1340392 .0142955 9.38 0.000 .1042193 .1638591
49 | -.0536608 .0085926 -6.24 0.000 -.0715847 -.0357369
|
ifrs | 0 (omitted)
post | 0 (omitted)
post_ifrs | -.089237 .0166074 -5.37 0.000 -.1238795 -.0545945
_cons | .2508545 .024029 10.44 0.000 .2007308 .3009782
------------------------------------------------------------------------------
Linear regression Number of obs = 111,301
F(14, 20) = .
Prob > F = .
R-squared = 0.2874
Root MSE = .20459
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
year |
1998 | -.0006483 .0046122 -0.14 0.890 -.0102691 .0089725
1999 | -.0352385 .0236309 -1.49 0.152 -.0845316 .0140546
2000 | -.032459 .021329 -1.52 0.144 -.0769504 .0120325
2001 | -.0241229 .0292886 -0.82 0.420 -.0852177 .036972
2002 | -.0239004 .0344396 -0.69 0.496 -.09574 .0479393
2003 | -.0387805 .0373613 -1.04 0.312 -.1167148 .0391538
2004 | -.0591383 .0456521 -1.30 0.210 -.1543669 .0360903
2006 | -.0718861 .0506407 -1.42 0.171 -.1775208 .0337485
2007 | -.0828033 .0375718 -2.20 0.039 -.1611766 -.00443
2008 | -.0515585 .0458215 -1.13 0.274 -.1471405 .0440234
2009 | -.0165657 .0358965 -0.46 0.649 -.0914445 .0583132
2010 | -.0170998 .0298396 -0.57 0.573 -.0793442 .0451445
2011 | -.024269 .0328281 -0.74 0.468 -.0927472 .0442092
2012 | -.0179488 .0372099 -0.48 0.635 -.0955674 .0596697
|
countryid |
2 | .1446156 .0008542 169.30 0.000 .1428338 .1463974
6 | .1056278 .0025726 41.06 0.000 .1002615 .1109941
12 | .1818235 .0025161 72.26 0.000 .176575 .187072
18 | .2483825 .0011592 214.27 0.000 .2459644 .2508005
24 | .18966 .0119885 15.82 0.000 .1646525 .2146675
25 | .1597764 .0110417 14.47 0.000 .1367439 .182809
26 | .1663577 .0100518 16.55 0.000 .1453901 .1873254
29 | .4185767 .0112388 37.24 0.000 .395133 .4420204
32 | -.1293801 .0186327 -6.94 0.000 -.1682472 -.0905131
34 | .351265 .0109491 32.08 0.000 .3284255 .3741045
35 | -.1604796 .0119335 -13.45 0.000 -.1853725 -.1355867
36 | -.0086126 .0085571 -1.01 0.326 -.0264624 .0092373
37 | -.1222561 .0102385 -11.94 0.000 -.1436133 -.100899
38 | .1378336 .0109803 12.55 0.000 .1149292 .160738
42 | .215822 .0102241 21.11 0.000 .1944949 .2371492
44 | .0753867 .0136506 5.52 0.000 .046912 .1038614
45 | -.0899285 .014831 -6.06 0.000 -.1208654 -.0589917
47 | -.0355286 .0128306 -2.77 0.012 -.0622927 -.0087644
48 | .1340392 .0142955 9.38 0.000 .1042193 .1638591
49 | -.0536608 .0085926 -6.24 0.000 -.0715847 -.0357369
|
post_ifrs | -.089237 .0166074 -5.37 0.000 -.1238795 -.0545945
_cons | .2508545 .024029 10.44 0.000 .2007308 .3009782
------------------------------------------------------------------------------Now, let’s absorb the fixed effects.
areg zeroreturn ifrs post post_ifrs, cluster(countryid) absorb(gvkey)
areg zeroreturn i.year i.countryid ifrs post post_ifrs, cluster(countryid) absorb(gvkey)
reghdfe zeroreturn post_ifrs, cluster(countryid) absorb(countryid year)
reghdfe zeroreturn post_ifrs, cluster(countryid) absorb(gvkey year)
reghdfe zeroreturn ifrs post post_ifrs, cluster(countryid) absorb(gvkey year)note: ifrs omitted because of collinearity
Linear regression, absorbing indicators Number of obs = 111,301
Absorbed variable: gvkey No. of categories = 11,134
F(2, 20) = 27.18
Prob > F = 0.0000
R-squared = 0.8075
Adj R-squared = 0.7861
Root MSE = 0.1121
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ifrs | 0 (omitted)
post | -.0109342 .0122792 -0.89 0.384 -.0365483 .0146798
post_ifrs | -.100076 .0195148 -5.13 0.000 -.1407832 -.0593687
_cons | .2686985 .0052333 51.34 0.000 .2577822 .2796149
------------------------------------------------------------------------------
note: ifrs omitted because of collinearity.
note: post omitted because of collinearity.
note: 2.countryid omitted because of collinearity
note: 6.countryid omitted because of collinearity
note: 12.countryid omitted because of collinearity
note: 18.countryid omitted because of collinearity
note: 24.countryid omitted because of collinearity
note: 25.countryid omitted because of collinearity
note: 26.countryid omitted because of collinearity
note: 29.countryid omitted because of collinearity
note: 32.countryid omitted because of collinearity
note: 34.countryid omitted because of collinearity
note: 35.countryid omitted because of collinearity
note: 36.countryid omitted because of collinearity
note: 37.countryid omitted because of collinearity
note: 38.countryid omitted because of collinearity
note: 42.countryid omitted because of collinearity
note: 44.countryid omitted because of collinearity
note: 45.countryid omitted because of collinearity
note: 47.countryid omitted because of collinearity
note: 48.countryid omitted because of collinearity
note: 49.countryid omitted because of collinearity
Linear regression, absorbing indicators Number of obs = 111,301
Absorbed variable: gvkey No. of categories = 11,134
F(15, 20) = 124.14
Prob > F = 0.0000
R-squared = 0.8184
Adj R-squared = 0.7982
Root MSE = 0.1089
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
year |
1998 | -.0116766 .0068022 -1.72 0.102 -.0258657 .0025125
1999 | -.0507722 .019359 -2.62 0.016 -.0911543 -.01039
2000 | -.049606 .0192799 -2.57 0.018 -.0898231 -.0093889
2001 | -.0444193 .0278522 -1.59 0.126 -.1025179 .0136793
2002 | -.0471566 .0306915 -1.54 0.140 -.111178 .0168648
2003 | -.0628965 .0330108 -1.91 0.071 -.1317559 .0059628
2004 | -.0838732 .0409612 -2.05 0.054 -.1693168 .0015704
2006 | -.0928505 .0446475 -2.08 0.051 -.1859835 .0002826
2007 | -.1036812 .0309445 -3.35 0.003 -.1682304 -.0391321
2008 | -.0719034 .0399061 -1.80 0.087 -.1551461 .0113392
2009 | -.0353541 .0299029 -1.18 0.251 -.0977303 .0270222
2010 | -.0349401 .023678 -1.48 0.156 -.0843316 .0144513
2011 | -.0414005 .0269526 -1.54 0.140 -.0976227 .0148217
2012 | -.0338997 .031699 -1.07 0.298 -.1000227 .0322234
|
countryid |
2 | 0 (omitted)
6 | 0 (omitted)
12 | 0 (omitted)
18 | 0 (omitted)
24 | 0 (omitted)
25 | 0 (omitted)
26 | 0 (omitted)
29 | 0 (omitted)
32 | 0 (omitted)
34 | 0 (omitted)
35 | 0 (omitted)
36 | 0 (omitted)
37 | 0 (omitted)
38 | 0 (omitted)
42 | 0 (omitted)
44 | 0 (omitted)
45 | 0 (omitted)
47 | 0 (omitted)
48 | 0 (omitted)
49 | 0 (omitted)
|
ifrs | 0 (omitted)
post | 0 (omitted)
post_ifrs | -.0991684 .0188044 -5.27 0.000 -.1383936 -.0599431
_cons | .3173257 .0268852 11.80 0.000 .261244 .3734073
------------------------------------------------------------------------------
(MWFE estimator converged in 4 iterations)
HDFE Linear regression Number of obs = 111,301
Absorbing 2 HDFE groups F( 1, 20) = 28.88
Statistics robust to heteroskedasticity Prob > F = 0.0000
R-squared = 0.2874
Adj R-squared = 0.2872
Within R-sq. = 0.0071
Number of clusters (countryid) = 21 Root MSE = 0.2046
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
post_ifrs | -.089237 .0166059 -5.37 0.000 -.1238764 -.0545976
_cons | .2624257 .0013267 197.81 0.000 .2596583 .2651931
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
countryid | 21 21 0 *|
year | 15 1 14 |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
(dropped 364 singleton observations)
(MWFE estimator converged in 6 iterations)
HDFE Linear regression Number of obs = 110,937
Absorbing 2 HDFE groups F( 1, 20) = 30.90
Statistics robust to heteroskedasticity Prob > F = 0.0000
R-squared = 0.8175
Adj R-squared = 0.7978
Within R-sq. = 0.0260
Number of clusters (countryid) = 21 Root MSE = 0.1089
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
post_ifrs | -.0991684 .017839 -5.56 0.000 -.1363798 -.0619569
_cons | .2628359 .0014286 183.98 0.000 .2598559 .2658158
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
gvkey | 10770 10770 0 *|
year | 15 1 14 |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
(dropped 364 singleton observations)
(MWFE estimator converged in 7 iterations)
note: ifrs is probably collinear with the fixed effects (all partialled-out val
> ues are close to zero; tol = 1.0e-09)
note: post is probably collinear with the fixed effects (all partialled-out val
> ues are close to zero; tol = 1.0e-09)
HDFE Linear regression Number of obs = 110,937
Absorbing 2 HDFE groups F( 1, 20) = 30.90
Statistics robust to heteroskedasticity Prob > F = 0.0000
R-squared = 0.8175
Adj R-squared = 0.7978
Within R-sq. = 0.0260
Number of clusters (countryid) = 21 Root MSE = 0.1089
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ifrs | 0 (omitted)
post | 0 (omitted)
post_ifrs | -.0991684 .017839 -5.56 0.000 -.1363798 -.0619569
_cons | .2628359 .0014286 183.98 0.000 .2598559 .2658158
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
gvkey | 10770 10770 0 *|
year | 15 1 14 |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computationThe final difference-in-difference coefficient of-0.0992 supports the
same conclusion as before that IFRS adoption is associated with improved
stock liquidity. Compared with the simple estimation performed earlier,
we can see that inclusion of the firm- and year-fixed effects indeed
causes the main effects of post and ifrs to
drop out of the estimation, because these are perfectly collinear with
the fixed effects. Also, we can see the substantial increase in
explanatory power(\(R^2\))as a result
of the inclusion of the fixed effects instead of the simple main effect
variables.
Testing for parallel trends
The key identifying assumption underlying the use of a difference-in-difference estimator is that the average changes in the outcome variable would have been the same for the treatment and control groups in the absence of treatment. This assumption is known as the parallel trends assumption. Unfortunately, because the parallel trends assumption focuses on unobservable counterfactual outcomes (e.g., the change in liquidity post-2005 for IFRS-adoption firms in the hypothetical case that they had not adopted IFRS), this assumption is not testable.
A possible solution to this problem is to take a visual approach in which we compare the relative trends in the outcome variable for the treatment and control groups in the pre-treatment period. If the trends already diverge in the pre-treatment period in a way similar to the difference-in-difference effect we find, we have a problem because the parallel trends assumption is unlikely to hold. For our previous example, we can do this by separately plotting the average values of the illiquidity variable for the treatment and control groups over time.
tempfile original
save `original'
collapse (mean) zeroreturn, by(ifrs year)
global options "xline(2005) legend(on order(1 2) label(1 'IFRS==1') label(2 'IFRS==0') cols(2) position(12))"
sort year
graph twoway connect zeroreturn year if ifrs==1, $options || ///
connect zeroreturn year if ifrs==0, lpattern(dash)|| ///
lfit zeroreturn year if ifrs==1 & year<2005 || ///
lfit zeroreturn year if ifrs==0 & year<2005 || ///
lfit zeroreturn year if ifrs==1 & year>2005 || ///
lfit zeroreturn year if ifrs==0 & year>2005
graph export "fig_did_parallel.png", replace
use `original', clearfile C:\Users\User\AppData\Local\Temp\ST_8e70_000001.tmp saved as .dta format
file fig_did_parallel.png saved as PNG format
A comparison of the trends suggests that the negative difference-in-difference effect found earlier is unlikely to be caused by a violation of parallel trends. Before treatment year 2005, we can actually see that the trends move in opposite direction of the negative effect found in the difference-in-difference estimation. Although this result suggests that the pre-trends are not exactly parallel, the visual inspection does provide us with some confidence that the divergence in the trends after treatment year 2005 is unlikely to be the result from a continuation of the trends in pre-treatment outcomes for the treatment and control groups.
We can also test the difference in pre-trends using significance
tests. For example, we can test whether the time trend in average
outcomes differs significantly between the two groups in the years
before 2005.The following code shows how we can assess the divergence in
pre-trends statistically using interactions between the
ifrs indicator and a continuous (categorical)
time variable. The results indicate that we cannot rejected
the null hypothesis of parallel pre-trends, as the coefficient on the
interaction between ifrs and time is not
statistically significant at conventional levels (\(p\) = 0.102):
sum year
gen time=1+year-r(min)
reg zeroreturn i.ifrs##c.time if year<2005,cluster(countryid)
reg zeroreturn i.ifrs##c.time if year>2005, cluster(countryid) Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
year | 120,361 2004.827 4.362174 1997 2012
Linear regression Number of obs = 56,263
F(3, 20) = 18.01
Prob > F = 0.0000
R-squared = 0.1145
Root MSE = .23115
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
1.ifrs | .1370058 .052788 2.60 0.017 .026892 .2471197
time | -.0083804 .0067808 -1.24 0.231 -.0225249 .0057642
|
ifrs#c.time |
1 | .0118943 .0069317 1.72 0.102 -.0025649 .0263535
|
_cons | .273432 .0327904 8.34 0.000 .2050325 .3418315
------------------------------------------------------------------------------
Linear regression Number of obs = 55,038
F(3, 20) = 16.01
Prob > F = 0.0000
R-squared = 0.0384
Root MSE = .23239
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
1.ifrs | .2534664 .0671206 3.78 0.001 .1134553 .3934774
time | .0131765 .0027111 4.86 0.000 .0075211 .0188318
|
ifrs#c.time |
1 | -.0112905 .0028623 -3.94 0.001 -.0172612 -.0053198
|
_cons | .0466302 .051473 0.91 0.376 -.0607407 .1540011
------------------------------------------------------------------------------Because the parallel trends assumption is not testable, an alternative approach that re searchers often take is to perform placebo tests in which the treatment effect is tested around a year in which treatment should not have an effect. In the sample used for the IFRS example above, the pre-IFRS period runs from 1997–2004. Hence, we could for example split the sample into two periods 1997–2000 and 2001-2004 and pretend as if treatment would take place at the end of the year 2000. If a violation of parallel trends (or related, any other endogeneity concern) causes the difference-in-difference estimate found earlier to be biased, we might observe a similar difference-in-difference estimate for this placebo treatment. The results below suggest this is not the case and provide some more assurance regarding the liquidity effects of IFRS documented earlier. In fact, the difference-in-difference appears to be positive instead of negative (albeit only borderline significant at \(p\) < 0.10):
gen placebo=0 if year<2005
replace placebo=1 if ifrs==1 & year>2000 & year<2005
reghdfe zeroreturn placebo, cluster(countryid) absorb(gvkey year)(64,098 missing values generated)
(6,793 real changes made)
(dropped 1296 singleton observations)
(MWFE estimator converged in 6 iterations)
HDFE Linear regression Number of obs = 54,967
Absorbing 2 HDFE groups F( 1, 20) = 3.13
Statistics robust to heteroskedasticity Prob > F = 0.0922
R-squared = 0.8714
Adj R-squared = 0.8453
Within R-sq. = 0.0091
Number of clusters (countryid) = 21 Root MSE = 0.0966
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
placebo | .0460763 .0260493 1.77 0.092 -.0082616 .1004143
_cons | .2709885 .0031581 85.81 0.000 .2644007 .2775762
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
gvkey | 9248 9248 0 *|
year | 8 1 7 |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computationFinally, another common approach that is often used to address the
parallel trends assump tion is the following. Instead of testing the
difference-in-difference using the interaction between the treatment
variable and the variable indicating the post-treatment period (in the
example here: ifrs and post), it is common for
researchers to replace the \(post\)
variable by separate indicators for years in the pre- and post-treatment
windows. For example, the following code shows how we can obtain annual
estimates of the difference in average outcomes between the treatment
and control group.
forvalues i =2000(1)2010{
gen p_`i'=0
replace p_`i'=ifrs if year==`i'
}
reghdfe zeroreturn p_* if year>=2000 & year<=2010, cluster(countryid) absorb(gvkey year) noconstant
coefplot, vertical yline(0)
graph export "fig_did_parallel_dyn.png", replace(1,617 real changes made)
(1,712 real changes made)
(1,733 real changes made)
(1,688 real changes made)
(1,660 real changes made)
(1,641 real changes made)
(1,530 real changes made)
(1,441 real changes made)
(1,333 real changes made)
(1,247 real changes made)
(1,176 real changes made)
(dropped 350 singleton observations)
(MWFE estimator converged in 6 iterations)
note: p_2010 omitted because of collinearity
HDFE Linear regression Number of obs = 87,979
Absorbing 2 HDFE groups F( 10, 20) = 50.45
Statistics robust to heteroskedasticity Prob > F = 0.0000
R-squared = 0.8458
Adj R-squared = 0.8255
Within R-sq. = 0.0347
Number of clusters (countryid) = 21 Root MSE = 0.1013
(Std. err. adjusted for 21 clusters in countryid)
------------------------------------------------------------------------------
| Robust
zeroreturn | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
p_2000 | .0835553 .0154256 5.42 0.000 .0513779 .1157326
p_2001 | .0942573 .0155947 6.04 0.000 .0617274 .1267873
p_2002 | .1288324 .0148114 8.70 0.000 .0979364 .1597284
p_2003 | .138205 .0151276 9.14 0.000 .1066493 .1697607
p_2004 | .1156583 .0261436 4.42 0.000 .0611237 .1701929
p_2005 | .0736742 .029243 2.52 0.020 .0126744 .1346739
p_2006 | .0351989 .0246496 1.43 0.169 -.0162192 .086617
p_2007 | .0200444 .0141397 1.42 0.172 -.0094505 .0495394
p_2008 | .0069627 .0218437 0.32 0.753 -.0386025 .0525279
p_2009 | .003151 .013565 0.23 0.819 -.0251452 .0314471
p_2010 | 0 (omitted)
------------------------------------------------------------------------------
Absorbed degrees of freedom:
-----------------------------------------------------+
Absorbed FE | Categories - Redundant = Num. Coefs |
-------------+---------------------------------------|
gvkey | 10230 10230 0 *|
year | 11 1 10 |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation
file fig_did_parallel_dyn.png saved as PNG format
The results displayed above suggest that a significant liquidity effect of IFRS adoption appears to materialize as of year 2007, where the 95%-confidence interval associated with the point estimate (the vertical line) no longer crosses the zero line.
It is important to note for any similar analysis that is used to identify “dynamic” treatment effects like these—is typically highly sensitive to the choice of the time period that is omitted to provide a baseline. We can see above that all coefficient estimates (the dots) and their 95% confidence intervals (the vertical lines) are obtained relative to the baseline year 2000. If we would have chosen a different baseline year, the plot would look very different.