Difference in Differences, Ratio in Ratios, and Ratio in Odds Ratios for Limited Dependent Variables: A Review and More

2.1 Early Papers on DD with LDV

Despite the frequent appearance of LDV’s in reality, studies on DD with LDV are fairly scarce. The first highly cited paper is Ai and Norton (2003) who suggested to use the cross-derivative as the treatment effect: with our notation,

Ai and Norton (2003) has been cited about 7,500 times in Google Scholar.

Ai and Norton (2003) is correct when the interest is in the interaction of two genuine treatments. However, QS itself is not a treatment in DD. Rather, QS just represents the way how the single, not double, treatment D is administered: only to the Q = 1 group at t = 3. Hence, as was noted in Section 1, Puhani (2012, eq. (10)) showed that Φ(β₂ + β _τ  + β _q  + β _d ) − Φ(β₂ + β _τ  + β _q ) is an appropriate treatment effect. Puhani (2012) has been cited about 880 times in Google Scholar.

Wooldridge (2023, p. C39) concurred with Puhani: “Puhani’s definition is the correct one for identifying τ₂ when the PT assumption is stated in terms of the linear index”, where τ₂ is the average effect on the treated at the post-treatment period and ‘PT’ means parallel trends. Puhani (2012) addressed only the effect identification, whereas Ai and Norton (2003) discussed how to estimate (2.1) as well, and showed with an empirical example that (2.1) can differ from the slope of D = QS in sign.

Unaware of the early version of our paper in 2021, Taddeo et al. (2022) applied GLM to DD with LDV, as they proposed to use RR for count outcomes, and ROR for binary outcomes. Differently from our paper, however, Taddeo et al. (2022) considered neither zero-censored nor fractional outcome, although they mentioned categorical outcomes briefly. Also, the word ‘QMLE’ never appeared, and the types of covariates were not discussed although they matter much for categorical outcomes.

Taddeo et al. (2022, p. 404) also suggested a sensitivity analysis for parallel trend assumption. E.g., as to be seen later, the multiplicative form of parallel trends requires a RR form with only Y t 0 ’s (analogous to the RR in (1.6)) being equal to one. Replacing the constant one with another number can reveal how the treatment effect is affected when the number deviates from one. For instance, if an empirical finding gets reversed when the constant is 1.5, then it takes a 50 % violation of the parallel trends, and thus the initial finding may be deemed to be robust to violations of parallel trends.

2.2 DD and Staggered DD with LDV

The paper that overlaps most with our paper is Wooldridge (2023), which was written to generalize Wooldridge (2021) for linear DD models to nonlinear ones; for this reason, the working paper version of Wooldridge (2023) was available only in 2022. Wooldridge (2023) initially examined the canonical two-group two-period DD, and then moved onto staggered DD which has been gaining popularity these days; see, e.g. Callaway and Sant’Anna (2021), Goodman-Bacon (2021), Sun and Abraham (2021), Athey and Imbens (2022), Liu et al. (2024), and Borusyak et al. (2024).

Table 1 at the end of Section 2 of Wooldridge (2023) summarizes the recommendation for the appropriate estimator for each type of LDV: Poisson QMLE for count, zero-censored and non-negative outcomes; logit for binary and fractional outcomes; and multinomial logit for multinomial/categorical outcomes. These recommendations agree with our paper, which is why Wooldridge (2023) overlaps much with our paper. A word of caution for those trying to read our paper and Wooldridge’s: our (Q _i , D _it , W _it ) is (D _i , W _it , X _it ) in the notation of Wooldridge (2023), which can be confusing.

Given the similarity between our paper and Wooldridge (2023), the reader may wonder what the differences are between the two papers. Wooldridge (2023) is more general than our paper, as it addresses staggered DD with LDV in multi-period settings. Wooldridge (2023) also compares ‘pooled estimators using all observations together’ to ‘imputation estimators using only the untreated observations to impute the counterfactual untreated outcomes of the Q = 1 group at t = 3’. In imputation estimators, the imputed entity is subtracted from the outcome average of the Q = 1 group at t = 3, which our paper using all observations together does not examine. Wooldridge (2023) further provides conditions to make the two types of estimators the same. In the following, we show what our paper has that Wooldridge (2023) does not.

First, when the adopted model is estimated for the LDV Y generated by its latent continuous version Y* with a linear index β 2 + β τ S + β q Q + β d D + β w ′ W , our paper shows that β _d can be viewed either as the effect of D on Y*, or as the proportional effect on Y for the log link as in (1.6) or the proportional-odds effect for the logit link. Even if Y* is inconceivable, still the proportional effect interpretation holds. In contrast to this, Wooldridge (2023) takes one extra step to estimate E Y 3 1 − Y 3 0 | Q = 1 , W (or E Y 3 1 − Y 3 0 | W ), which is not a necessity; e.g.,

is an estimator with the log link and QMLE estimates β ̂ 2 , β ̂ τ , β ̂ q , β ̂ d , β ̂ w ′ ′ .

Second, we also review the more recent literature on DD with LDV in 2024 and 2025: Tchetgen et al. (2024) and Kim and Lee (2025), who tried to reduce the regression function specification errors. Kim and Lee (2025) is reviewed shortly below, whereas Tchetgen et al. (2024) is reviewed in Section 4 after ROR is introduced because their approach is applicable only with ROR.

Third, covariates are often “brushed aside” in the DD literature. That is, a theory is developed without W first, and when W appears, a remark such as “The theory developed without W holds with W, either by conditioning on W or by controlling W linearly” is made. Wooldridge (2023) also assumes time-constant W, and either condition on W or control W linearly. Handling W “lightly” in this way can be problematic, because the nature of W matters much in panel data (i.e. unit-constant/varying and time-constant/varying), and it is more so for categorical/multinomial outcomes, as the nature of W is three-dimensional (unit-constant/varying, time-constant/varying, and category-constant/varying), where normalizations matter much even for RCS as can be seen in Lee (2015) and our Appendix for multinomial outcomes.

Fourth, in addition to the above three main contributions, there are also some minor ones: (i) DD identification conditions are usually stated and verified for panel data models, but since we use QMLE/MLE with RCS, we verify them with RCS; (ii) “triple ratios” are used to deal with non-parallel multiplicative trends; (iii) we provide the details on justifying RR for zero-censored outcomes; (iv) for proportional odds effect which is difficult to grasp, we show a ‘rare event condition’ which turns the proportional odds effect into a proportional effect; and (v) we present the exact maximand of the QMLE/MLE, so that it can be easily understood and implemented by practitioners, going well beyond presenting a summary table as in Table 1 of Wooldridge (2023).

2.3 Causal Reduced Form for OLS to DD with LDV

Recently, Kim and Lee (2025) showed that, for any form of Y (binary, count, continuous, …) with meaningful Y¹ − Y⁰, a linear model holds always for ΔY ≡ Y₃ − Y₂, to which ordinary least squares (OLS) can be applied. To see how, recall D _it  = Q _i 1[t = 3] and observe:

Under PT ΔY⁰⨿Q|W, take E(⋅|Q, W) on ΔY where ‘ ⨿’ stands for independence:

The ΔY equation linear in Q holds for any Y, and it is a “causal reduced form (CRF)” in the sense that it is a reduced/derived form with a causal parameter τ₁(W):

CRF may sound strange, but it has been used fruitfully in Lee (2018) and Lee et al. (2025) for binary exogenous D; Lee (2021), Lee et al. (2023), Choi et al. (2023), and Kim and Lee (2024) for binary endogenous D; Lee (2024) for binary exogenous D with a mediator; and Kim (2025) for network/spillover effect.

To remove E(ΔY⁰|W) from the CRF, take E(⋅|W) on the CRF:

Subtract this from the CRF to get ΔY − E(ΔY|W) = τ₁(W) ⋅ (Q − p _W ) + U. Now, OLS of ΔY − E(ΔY|W) on Q − p _W can be done, which is equivalent to OLS of ΔY on Q − p _W because Q − p _W is orthogonal to E(ΔY|W). The propensity score p _W can be estimated with probit or logit.

Kim and Lee (2025) showed that OLS of ΔY on Q − p _W is consistent to:

OW stands for “overlap weight”. CRF is nonparametric as no parametric assumption is invoked, and OLS to the CRF is consistent to E w o w ( W ) τ 1 ( W ) for any LDV, as long as Y¹ − Y⁰ makes sense. Extending this finding to more general cases, such as multi-periods, non-binary D and endogenous Q, remains to be explored.

The fact that OLS estimates an OW-average of heterogeneous effects is well known (e.g. Angrist 1998; Angrist and Pischke 2009) under the “saturated model” assumption – i.e., p _W is equal to the linear projection of Q on W – which is, however, not necessary in the above OLS. See Choi and Lee (2023) for details on OW and its advantages. OW has been gaining popularity in statistics, epidemiology, and medical science; see, e.g. Li et al. (2018), Mao et al. (2019), Li (2019), Zhou et al. (2020), Thomas et al. (2020), Aminian et al. (2021), Cheng et al. (2022), Anderson et al. (2023), Wei et al. (2023), Matsouaka and Zhou (2024), and Xu et al. (2025).

3.1 Proportional Effect Identification with RR

To simplify notation when covariates W are allowed for, define

note μ_Q0(w) = E(Y₂|W₂ = w, Q) and μ_Q1(w) = E(Y₃|W₃ = w, Q). Analogously to (1.6), define RR or ‘proportional-effect +1 given W = w’:

The identification condition for RR(w) is “unity multiplicative trends”:

keep in mind that S is independent of the other random variables. In ID _RR , E Y 3 0 | w , Q = 1 is the counterfactual, because only Y 3 1 is realized for Q = 1 at t = 3.

ID _RR is analogous to the usual linear-model DD parallel trend condition:

which dictates that the counterfactual E Y 3 0 | w , Q = 1 be constructed as E Y 2 0 | w , Q = 1 + E Y 3 0 | w , Q = 0 − E Y 2 0 | w , Q = 0 . In the same vein, ID _RR dictates that the counterfactual be constructed multiplicatively as the baseline untreated outcome of the Q = 1 group times the ratio of the control group means at t = 3 to t = 2:

The main point is that RR(w) − 1 is the ‘proportional effect on the treated at the post-treatment period’, as the first and last terms in the following show:

Instead of the difference effect E Y 3 1 − Y 3 0 | w , Q = 1 , examining the proportional effect can be beneficial (Yadlowsky et al. 2021). E.g., suppose E Y 3 0 | w , Q = 1 = G ( w ) for a function G(⋅) and the proportional effect is a constant β ̆ d . Then the difference effect E Y 3 1 − Y 3 0 | w , Q = 1 = β ̆ d G ( w ) introduces effect heterogeneity unnecessarily, compared with the simple β ̆ d . Proportional effects for exponential models have been advocated in many studies: Lee and Kobayashi (2001), Dukes and Vansteelandt (2018) and Ciani and Fisher (2019), among others.

If the dimension of W is low (or if W is discrete), RR(w) can be estimated nonparametrically, but typically the dimension of W is high in practice, and thus we explore a simpler semiparametric exponential regression next – semiparametric because only E(Y|W, Q, S) is specified, not the full distribution of Y|(W, Q, S).

3.2 Poisson Quasi-MLE (QMLE)

In view of (1.5), suppose that a panel data exponential model holds for Y i t d :

If periods 1, …, T were available, then W₁, …, W _T would appear in the first term. Then ID _RR holds with (3.3):

For the corresponding RCS model, the observed Y in RCS is: due to D = QS,

Take E(⋅|W₂, W₃, Q, S) on this Y and invoke (3.3): recalling β _τ ≡ β₃ − β₂ in (1.1),

the second equality can be verified by substituting Q, S = 0, 1 into both sides. If β _d is near zero, then β _d is an approximate proportional effect. RCS data and the model (3.5) can be used to estimate β _d along with the other parameters.

The heterogeneous effect can be easily allowed by replacing β _d with β d ( W t ) = β d 0 + β d w ′ W t . Then we have β d ( W ) = β d 0 + β d w ′ W in RCS, and (3.5) becomes

It is often believed that if a treatment effect is random but a constant effect is assumed, then its mean effect would be estimated. However, this is false as the following explains.

The staggered DD literature revealed that, if a constant effect is assumed when the true effect is heterogeneous, then the popular two-way fixed effect estimator is consistent to a weighted average of heterogeneous effects with some weights negative. This paints a rather pessimistic picture, but positive findings also exist: if OLS is applied to a constant-effect model when the effect is actually heterogeneous in W, then the OLS is consistent to an OW average of heterogeneous effects; see Lee and Han (2024) and references therein. This kind of positive scenario was already noted when Kim and Lee (2025) was reviewed. It would take another full paper to investigate the analogous question for proportional effect in general settings, which we thus eschew.

The simplest estimator for (3.5) is Poisson QMLE, whose maximand is:

as in Poisson MLE. The first order condition at b = β is ∑ i Y i − exp X i ′ β X i = 0 for β ≡ β 2 , β τ , β q , β d , β w ′ ′ , and the second order derivative − ∑ i X i X i ′ ⁡ exp X i ′ b is n.d. for all b. Hence, just under E(Y|X) = exp(X′β), Poisson QMLE is consistent for β. For heterogeneous effects, we may use β d ( W ) = β d 0 + β d w ′ W in Poisson QMLE.

The difference between Poisson QMLE and Poisson MLE is that the first-order condition is taken just as a moment condition to estimate β in the former. Hence, the asymptotic variance of Poisson QMLE is estimated with a “sandwich-form” estimator. Since Poisson QMLE requires only E(Y|X) = exp(X′β), not the full distribution of Y|X, it is advocated by Santos Silva and Tenreyro (2006), which has been cited about 9,300 times in Google Scholar as of this writing.

3.3 Remarks

Here we make remarks on the applicability of the above RR identification and Poisson QMLE to count and zero-censored outcomes. The semiparametric exponential model (3.5) is appropriate for these, as it requires only the lower bound 0 ≤ Y.

4.1 Proportional Odds Effect Identification with ROR

For binary Y, define the ‘odds conditional on (W = w, Q = q, S = s)’ for RCS as

Also define ‘Ratio in Odds Ratios (ROR) conditional on W = w’:

The identification condition to be invoked for ROR is

where R 11 Y 3 0 ; w is a counterfactual. Just as in (3.1), ID _ROR dictates that the counterfactual R 11 Y 3 0 ; w be constructed multiplicatively:

Doing analogously to (3.2), ROR(Y; w) − 1 is the ‘proportional odds effect on the treated at the post-treatment period t = 3’:

Although odds ratio is extensively used in the medical and epidemiological literature, still it is not necessarily easy to interpret the proportional odds effect. For this, suppose Y = 1 is a rare event in the sense

e.g. Y = 1 is a rare cancer occurrence such that P Y 3 d = 0 | w , Q = 1 ≃ 1 for all w and d = 0, 1. Under (4.3), R O R ( Y ; w ) = R 11 Y 3 1 ; w / R 11 Y 3 0 ; w in (4.2) becomes

Hence, the proportional odds effect (4.2) becomes the RR proportional effect (3.2):

ROR(Y; w) can be estimated nonparametrically by substituting sample analogs into the components of ROR(Y; w). However, as was the case for DD and RR(w), this is not what practitioners would do. Instead, we apply logistic regression next.

4.2 Logit for Binary Outcome

Analogously to (3.3), suppose that a panel data logistic model holds for Y i t d :

The logistic panel data model for Y i t 0 renders, with β _τ ≡ β₃ − β₂,

Hence, ID _ROR holds analogously to (3.4), because (4.5) yields

For the corresponding RCS, Y = ( 1 − S ) Y 2 0 + S ( 1 − Q ) Y 3 0 + D Y 3 1 holds. Analogously to (3.5) and (3.6), take E(⋅|W₂, W₃, Q, S) on Y and invoke (4.4) to get

Also, R 11 Y 3 1 ; w = exp β 2 + β τ + β q + β d + w ′ β w and R 11 Y 3 0 ; w in (4.5) yield

Estimate β _d by the MLE with (4.6) to use exp(β _d ) − 1 as the proportional odds effect, which is also the RR proportional effect when Y = 1 is a rare event as in (4.3).

Suppose β _qτ tQ with β _qτ ≠ 0 appears as an extra regressor in (4.4). Then the parallel trends do not hold for the latent Y*. The appearance of β _qτ tQ also ruins ID _ROR for binary Y because ID _ROR becomes (3.7), just as β _qτ tQ ruins ID _RR in (3.7). As in (3.7), using tQ is an easy way to test or allow for non-parallel trends in Y*. Overall, the comments made for (3.7) hold more or less the same for ROR as well.

To allow for heterogeneous effect, suppose now that the slope of d in (4.4) is β _d (W _t ); e.g. β d ( W t ) = β d 0 + β d w ′ W t . Then (4.6) and ROR(Y; w) − 1 become, respectively,