Difference-in-differences with matching is a popular method to measure the impact of an intervention in health as well as social sciences. This method requires baseline data, i.e., data before interventions, which are not always available in reality. Instead, panel data with two time periods are often collected after interventions begin. In this paper, a simple matching method is proposed to measure impact of an intervention using two-period panel data after the intervention.Impact evaluation, difference-in-differences, matching, propensity score, panel data

Nguyen Viet, Cuong

English

Research Papers in Economics

  1
 
A Matching Method with Panel Data 
 
 
Nguyen Viet Cuong
1 
 
 
Abstract 
Difference-in-differences with matching is a popular method to measure the impact of an 
intervention in health as well as social sciences. This method requires baseline data, i.e., 
data before interventions, which are not always available in reality. Instead, panel data 
with two time periods are often collected after interventions begin. In this paper, a simple 
matching  method  is  proposed  to  measure  impact  of  an  intervention  using  two-period 
panel data after the intervention.  
 
 
Keywords:  Impact  evaluation,  difference-in-differences,  matching,  propensity  score, 
panel data 
JEL classification: H43; C21; J41 
 
 
                                                 
1 National Economics University; and Indochina Research & Consulting, Hanoi, Vietnam.  
Email: c_nguyenviet@yahoo.com    2
1. Introduction 
 
Difference-in-differences with matching is a widely-used method to measure impact of 
interventions  such  policies,  programs  and  treatments.  However,  this  method  requires 
baseline data, i.e., data before interventions, which are not always available for impact 
evaluation in reality. Instead, panel data with two time periods are often collected after 
interventions  begin.  When  there  are  panel  data  without  baseline  data,  one  can  use 
parametric  fixed-effect  regressions.  Compared  to  matching  methods,  parametric 
regressions have limitation that they must impose functional assumptions on outcome.  
The objective of this paper is to discuss identification and estimation of impact of 
an  intervention  using  a  matching  method  with  two-period  panel  data  after  the 
intervention. The impact parameter of interest is Average Treatment Effect on the Treated 
(ATT).  
The  paper  is  structured  in  four  sections.  The  second  section  discusses  the 
matching method using two-period panel data. The third section illustrates this matching 
method  by  an  empirical  study  on  impact  evaluation  of  health  insurance  in  Vietnam. 
Finally, the fourth section concludes.  
 
2. Matching using Panel Data 
 
2.1. Parameter of Interest  
   3
The main objective of impact evaluation of an  intervention is to assess the extent to 
which the intervention has changed outcome of subjects. To make definition explicit, 
suppose that there is an intervention of interest, and denote by D the binary variable of 
participation in the intervention, i.e.  1 = D  if one participates in the intervention, and 
0 = D  otherwise. Let Y denote observed outcome. This variable can receive two values 
depending on D:  1 Y Y =  if  1 = D , and  0 Y Y =  if  0 = D . The most popular parameter in 
impact evaluation is ATT,
 which is defined as (Heckman et al., 1999):
2   
) 1 ( ) 1 ( 0 1 = − = = D Y E D Y E ATT .              (1) 
One can be interested in ATT conditional on observed variables X: 
( ) ) 1 , | ( ) 1 , | ( 0 1 = − = = D X Y E D X Y E ATT X .              (2) 
In  (2),  ) 1 , | ( 0 = D X Y E   which  is  the  expected  conditional  outcome  of  the 
participants had they not received the intervention is not observed. Thus, estimation of 
) (X ATT  is not straightforward. The following sections discuss how to estimate  ) (X ATT  and 
ATT using matching methods with panel data.  
   
2.2. Difference-in-differences with Matching 
 
When panel data on participants and non-participants before and after an intervention are 
available,  ATT  can  be  estimated  using  a  method  of  difference-in-differences  with 
matching.  The  basic  idea  of  matching  is  to  find  a  control  group  that  has  similar 
                                                 
2 Another popular parameter is Average Treatment Effect: ) ( ) ( 0 1 Y E Y E ATE − = . The identification and 
estimation of ATE and ATT using the matching methods are very similar. Thus this paper discusses only 
ATT.   4
distribution  of  X  as  the  treatment  group.
3  Matching  is  combined  with  difference-in-
difference estimation to allow intervention selection to be based on unobserved variables. 
However, this method requires the unobserved variables be time-invariant.  
Let  F Y0  denote pre-intervention outcome. After the intervention, let  S Y1  and  S Y0  
denote  potential  outcomes  in  states  of  intervention  and  no-intervention,  respectively.
4 
(X) ATT  after the intervention is defined as: 
) |X, D ) - E(Y |X, D   E(Y ATT S S (X) 1 1 0 1 = = =             (3) 
The difference-in-differences with matching method relies on an assumption that 
conditional on X, difference in outcome expectations between the participants and non-
participants is time-invariant:   
) 0 , | ( ) 1 , | ( ) 0 , | ( ) 1 , | ( 0 0 0 0 = − = = = − = D X Y E D X Y E D X Y E D X Y E S S F F .    (4) 
Then, ATT(X) can be identified, since: 
[ ]
[ ]
[ ][ ] ) |X,D E(Y ) |X,D E(Y - ) |X,D )-E(Y |X, D E(Y                     
) |X,D E(Y ) |X,D E(Y                          
) |X,D E(Y ) |X,D E(Y )-  |X, D ) - E(Y |X, D   E(Y ATT
F F S S
S S
F F S S (X)
0 1 0 1
0 1
0 1 1 1
0 0 0 1
0 0
0 0 0 1
= − = = = =
= − = +
= − = = = =
   (5) 
ATT is also identified, since: 
∫ =
= =
1
1
X|D
(X) ) dF(X|D ATT ATT  .              (6) 
 The  matching  estimator  is  based  on  equation  (5).  It  is  equal  to  difference  in 
differences in outcomes between the treatment and control groups before and after the 
intervention.  
                                                 
3 There is large literature on matching methods, e.g., Rubin (1979), Rosenbaum and Rubin (1983), and 
Smith and Todd (2005).  
4 Superscripts “F” and “S” mean “first” and “second” periods, respectively.   5
 
2.3. Matching using Panel Data without Baseline Data 
 
In  reality,  baseline  data  are  not  always  available  for  intervention  evaluation.  Instead, 
panel data with two time periods are often collected after the intervention begins. An 
intervention can take place continuously. There can be not only people leaving but also 
ones newly entering the intervention. Assume that there are two time periods, and let D1 
and  D2  denote  the  binary  variables  of  the  intervention  status  in  the  first  and  second 
periods, respectively. In the first period, let Y1F and Y0F denote potential outcomes with 
and without the intervention, respectively. Further, let Y1S and Y0S denote the potential 
outcomes with and without the intervention in the second period, respectively. Suppose 
that we are interested in ATT(X) in the second period, which is expressed as follows: 
) 1 , | ( ) 1 , | ( 2 0 2 1 ) ( = − = = D X Y E D X Y E ATT S S
S
X .          (7) 
Note that we cannot observe  ) 1 , | ( 2 0 = D X Y E S . The single matching method assumes 
that: 
) 0 , | ( ) 1 , | ( 2 0 2 0 = = = D X Y E D X Y E S S ,             (8) 
which  eliminates  any  correlation  between  the  intervention  and  unobserved  variables 
affecting the outcomes. Using panel data, we can identify the intervention impact without 
the assumption specified by (8). Rewrite (7) as follows: 
[ ]
[ ]. ) 1 , 0 , | ( ) 1 , 0 , | ( ) 1 , | 0 Pr(
) 1 , 1 , | ( ) 1 , 1 , | ( ) 1 , | 1 Pr(
2 1 0 2 1 1 2 1
2 1 0 2 1 1 2 1 ) (
= = − = = = = +
= = − = = = = =
D D X Y E D D X Y E D X D                 
D D X Y E D D X Y E D X D ATT
S S
S S
S
X       (9) 
Two identification assumptions are:   6
[ ]
[ ], ) 0 , 0 , | ( ) 1 , 0 , | (
) 0 , 0 , | ( ) 1 , 0 , | (
2 1 0 2 1 0
2 1 0 2 1 0
= = − = = =
= = − = =
D D X Y E D D X Y E
D D X Y E D D X Y E
F F
S S         (10) 
 [ ]
[ ]. ) 0 , 1 , | ( ) 0 , 1 , | (
) 1 , 1 , | ( ) 1 , 1 , | (
2 1 1 2 1 0
2 1 1 2 1 0
= = − = = =
= = − = =
D D X Y E D D X Y E
D D X Y E D D X Y E
F S
F S         (11) 
The first assumption means that difference in the no-intervention outcome (conditional on 
X) between people who do not participate in the intervention in both periods and those 
who participate in the intervention only in the second period is unchanged overtime. This 
assumption is similar to the assumption of the method of difference-in-differences with 
matching.  The  second  assumption  means  that  difference  between  the  no-intervention 
outcome in the second period and the intervention outcome in the first period is the same 
for people who participate in the intervention in both periods and those who participate in 
the intervention in the first period but not in the second one.   
  Substitute (10) and (11) into (9) and rewrite (9) as follows: 
[ ] {
[ ]}
[ ] {
[ ]} ) 0 , 0 , | ( ) 1 , 0 , | (
) 0 , 0 , | ( ) 1 , 0 , | ( ) 1 , | 0 Pr(
) 0 , 1 , | ( ) 1 , 1 , | (
) 0 , 1 , | ( ) 1 , 1 , | ( ) 1 , | 1 Pr(
2 1 0 2 1 0
2 1 0 2 1 1 2 1
2 1 1 2 1 1
2 1 0 2 1 1 2 1 ) (
= = − = =
= = − = = = = +
= = − = =
= = − = = = = =
D D X Y E D D X Y E -                 
D D X Y E D D X Y E D X D                 
D D X Y E D D X Y E -                 
D D X Y E D D X Y E D X D ATT
F F
S S  
F F
S S
S
X
   
                      (12) 
Now, 
S
X ATT ) ( is  identified  since  all  terms  in  (12)  can  be  observed.  The  unconditional 
parameter is also identified by (6). Matching can be performed according to (12): (i) 
people who participate in the intervention in both periods are matched with those who 
participate in the intervention only in the first period, (ii) people who participate in the 
intervention only in the second period are matched with those who do not participate in 
the intervention in both periods.    7
To find the control groups who have similar variables X, we requires common 
support assumptions as follows: 
1 ) 1 , | 1 ( 0 1 2 < = = < D X D P                  (13) 
1 ) 0 , | 1 ( 0 1 2 < = = < D X D P                 (14) 
These assumptions mean that given the intervention status in the first period there are 
non-participants  who  have  the  X  variables  similar  to  those  of  the  participants  in  the 
second period.    
A problem is how to match non-participants with participants. Since a paper by 
Rosenbaum and Rubin (1983), the matching is often conducted based on the probability 
of being assigned into the intervention, which is called the propensity score.
5 In our case, 
the propensity score is the probability of participating in the intervention in the second 
period  given  variables  X  and  D1.  We  can  use  logit  or  probit  regressions  to  predict 
) | 1 ( ˆ
2 X D P =  in the separate samples of people with  1 1 = D  and  0 1 = D . 
After the treatment and control groups are constructed, ATT can be estimated by 
differences in outcomes between the treatment and control groups as specified by (12). 
The standard errors are calculated using bootstrap techniques.  
  Finally, one can be interested in  ) (X ATT for the first period: 
) 1 , | ( ) 1 , | ( 1 0 1 1 ) ( = − = = D X Y E D X Y E ATT F F
F
X .            (15) 
which  is  estimated  very  similarly  by  reversing  the  first  and  second  periods  in  the 
estimation of 
S
X ATT ) ( .   
                                                 
5 Other matching methods can be subclassification and covariate matching (Rubin, 1979).   8
3. Empirical Example 
 
This section illustrates estimation of impact of health insurance on the number of annual 
healthcare contacts in Vietnam using the matching method. In Vietnam, health insurance 
has been implemented since 1992, and there are no baseline data for health insurance.  To 
measure impacts of health insurance, the paper uses data from Vietnam Household Living 
Standard Surveys (VHLSS) in 2004 and 2006. These surveys were conducted by General 
Statistical Office of Vietnam. These surveys set up panel data, which are representative 
for national, rural and urban levels. The number of individuals in the panel data used is 
16685.  
  Table 1 presents the distribution of sample individuals in the panel data of the 
surveys  by  health  insurance.  Not  all  Vietnamese  people  were  covered  by  health 
insurance. There were 4802 and 6337 people having health insurance in 2004 and 2006, 
respectively. There were 3401 people having health insurance in both 2004 and 2006.  
Table 1: Distribution of sampled individuals by health insurance 
 
Uninsured in 
2004 
Insured in 2004  Total 
Uninsured in 2006  8947  1401  10348 
Insured in 2006  2936  3401  6337 
Total  11883  4802  16685 
Source: Estimation from panel data of VHLSS 2004-2006. 
To estimate the intervention impact, we construct two treatment groups and two 
control groups. The first treatment group includes people having health insurance in both 
2004 and 2006. This group is matched with a control group who include people having 
health insurance in 2004 but not 2006. The second treatment group are those who are 
insured  in  2006  but  not  2004.  This  group  is  matched  with  a  control  group  who  are   9
uninsured in both 2004 and 2006. The treatment and control groups are matched based on 
the closeness of the propensity score. The propensity score are the probability of being 
insured  in  2006,  which  are  estimated  from  two  logit  regressions:  the  first  using  the 
sample of people insured in 2004, and the second using the sample of people uninsured in 
2004. Control variables in the logit regressions include per capita income in 2004 and 
2006,  age  in  2004,  sickness  in  2004  and  2006,  educational  degree  in  2004,  regional 
dummy variables, and urbanity. Once the treatment and control groups are setup, the 
intervention impacts can be estimated by differences in outcome between the treatment 
and control groups overtime (see equation (12)).  
  Table 2 presents impact estimates of health insurance on the number of annual 
healthcare contacts of the insured people in 2006. It presents all the estimates which are 
used to compute ATT. All the three matching estimators give similar results. The impact 
estimates of ATT are statistically significant at 10%. Health insurance helped the insured 
people increase the number of annual healthcare contacts by around 0.17 in 2006.       
Table 2: Impacts of Health Insurance 
Terms  1 nearest neighbor 
matching 
5 nearest neighbors 
matching 
Kernel matching 
with bandwidth of 
0.05 
1 , 1 | 2 1 1 = = D D Y S   1.319***  1.319***  1.319*** 
  [0.074]  [0.074]  [0.074] 
0 , 1 | 2 1 0 = = D D Y S   1.044***  1.096***  1.022*** 
  [0.161]  [0.154]  [0.154] 
1 , 1 | 2 1 1 = = D D Y F   1.019***  1.019***  1.019*** 
  [0.050]  [0.050]  [0.050] 
0 , 1 | 2 1 1 = = D D Y F   0.905***  0.957***  0.914*** 
  [0.119]  [0.123]  [0.109] 
1 , 0 | 2 1 1 = = D D Y S   1.602***  1.602***  1.602*** 
  [0.080]  [0.080]  [0.080]   10
Terms  1 nearest neighbor 
matching 
5 nearest neighbors 
matching 
Kernel matching 
with bandwidth of 
0.05 
0 , 0 | 2 1 0 = = D D Y S   1.231***  1.227***  1.229*** 
  [0.095]  [0.103]  [0.112] 
1 , 0 | 2 1 0 = = D D Y F   1.251***  1.251***  1.251*** 
  [0.083]  [0.083]  [0.083] 
0 , 0 | 2 1 0 = = D D Y F   1.054***  1.087***  1.068*** 
  [0.098]  [0.096]  [0.094] 
1 ˆT T A   0.161*  0.160*  0.191* 
  [0.095]  [0.091]  [0.110] 
0 ˆT T A   0.173*  0.211*  0.190* 
  [0.108]  [0.123]  [0.119] 
) 1 | 1 Pr( 2 1 = = D D   0.537***  0.537***  0.537*** 
  [0.010]  [0.010]  [0.010] 
T T A ˆ   0.166*  0.184*  0.190* 
  0.097  [0.108]  [0.113] 
 
Note: 
[ ] [ ] ) 0 , 1 | ( ) 1 , 1 | ( ) 0 , 1 | ( ) 1 , 1 | ( ˆ
2 1 1 2 1 1 2 1 0 2 1 1 1 = = − = = = = − = = = D D Y D D Y - D D Y D D Y T T A F F S S  
[ ] [ ] ) 0 , 0 | ( ) 1 , 0 | ( ) 0 , 0 | ( ) 1 , 0 | ( ˆ
2 1 0 2 1 0 2 1 0 2 1 1 0 = = − = = = = − = = = D D Y D D Y - D D Y D D Y T T A F F S S  
0 2 1 1 2 1 ˆ ) 1 | 0 Pr( ˆ ) 1 | 1 Pr( ˆ T T A D D T T A D D T T A   = = + = = =  
Figures in brackets are standard errors, which are corrected for sampling weights and estimated using non-
parametric bootstrap with 500 replications. 
* significant at 10%; ** significant at 5%; *** significant at 1% 
Source: Estimation from panel data of VHLSS 2004-2006. 
 
 
4. Conclusion 
 
In impact evaluation of an intervention, baseline data are not always available. Thus, the 
method  of  difference-in-differences  with  matching  cannot  be  applied  straightforward. 
Two-period panel data can be collected after interventions start. This paper discusses the 
identification and estimation of ATT using the matching with two-period panel data. It is 
shown that under some identification assumptions, ATT of an intervention of interest can   11
be  measured  as  a  weighted  average  of  intervention  impacts  on  groups  with  different 
intervention statuses in the two periods.    
 
 
 
References 
 
Heckman, J., R. Lalonde and J. Smith, 1999. The Economics and Econometrics of Active 
Labor Market Programs. Handbook of Labor Economics, Volume 3, Ashenfelter, A. and 
D. Card, eds.,  Elsevier Science. 
Rosenbaum,  P.  and  R.  Rubin,  1983.  The  Central  Role  of  the  Propensity  Score  in 
Observational Studies for Causal Effects. Biometrika 70 (1), 41-55. 
Rubin, D., 1979. Using Multivariate Sampling and Regression Adjustment to Control 
Bias in Observational Studies. Journal of the American Statistical Association. 74, 318–
328. 
Smith,  J.  and  P.  Todd,  2005.  Does  Matching  Overcome  LaLonde’s  Critique  of 
Nonexperimental Estimators? Journal of Econometrics 125 (1–2), 305–353. 
 

A Matching Method with Panel Data

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