SlowGuess/NeurIPS_2023_OCR huggingface dataset:
DiD estimator returns  $\hat{\beta}$  in the above equation as the estimation of causal effect which is equal to:
$$
\hat {\beta} = \beta + \frac {\alpha_ {d} \left(\gamma - \alpha_ {z}\right) \mathbb {E} \left[ U ^ {2} \right]}{\mathbb {E} \left[ D ^ {2} \right]}. \tag {7}
$$
Thus,  $\hat{\beta}$  is an unbiased estimate of  $\beta$  only when  $\gamma = \alpha_{z}$ . In other words, latent variable  $U$  should have the same direct causal effect on  $Z$  and  $Y$ . This is akin to the so-called common trend assumption which says that the average natural drift (here, the effect of  $U$ ) is assumed to be the same across both the control and treatment groups. This result is consistent with the findings in [RSBP23], which studied a similar phenomenon in the DiD setting. In summary, whenever the common trend assumption is violated, the DiD estimator is biased.
4 Related work
In the past few years, there has been a growing interest in the literature to exploit proxy variables to de-bias the effect of latent confounders. A special type of such proxy variable is the so-called negative outcome which is a variable known not to be causally affected by the treatment [LTC10]. For instance, the variable  $Z$  in Figure 1 may be considered as negative outcome. In fact,  $[ \mathrm{SRC}^{+}16 ]$  interpreted DiD as a negative outcome control approach and proposed a method inspired by change-in-change [AI06] to identify the causal effect under the assumption that  $Y(0)$  and  $Z$  are monotonic increasing functions of latent confounders and some observed covariates.
[KP14] considered three settings in causal inference with proxy variables: 1- There exists only one proxy variable such as  $Z$  as a negative outcome. In this case, for discrete finite variables  $Z$  and  $U$ , they showed that the causal effect can be identified if  $\operatorname{Pr}(Z|U)$  is known from some external studies such as pilot studies. 2- Two proxy variables, for instance  $Z$  and  $W$  are considered where  $U$ ,  $Z$ , and  $W$  are all discrete finite variables and  $Z$  does not have a directed path to  $D$  or  $Y$ . It has been shown that the causal effect is identifiable under some assumptions on the conditional probabilities of  $\operatorname{Pr}(Y|D,U)$  and  $\operatorname{Pr}(Z,W|X)$ . In the setting, it is not necessary to know  $\operatorname{Pr}(Z|U)$  but two proxy variables are required to identify the causal effect. 3- In linear SCM, [KP14] showed that  $\beta$  (the average causal effect of  $D$  on  $Y$ ) can be recovered using two proxy variables. Later, [MGTT18] also considered a setting with two proxy variables  $Z$  and  $W$ . Unlike the second setting in [KP14], here,  $Z$  and  $W$  can be parents of  $D$  and  $Y$ , respectively. For the discrete finite variables, they showed that the causal effect can be identified if the matrix  $P(W|Z,D = d)$  is invertible. Moreover, they provided the counterpart of this condition for continuous variables. [SMNTT20] extended the identification result in [MGTT18], with a weaker set of assumptions. Still, they required two proxy variables to identify the causal effect. Based on the results in [MGTT18], [TYC+20] introduced a proximal causal inference in PO framework. Later, [CPS+20] provided an alternative proximal identification result to that of [MGTT18], again when two proxy variables were available. More recently, [SLZ+23] considered the PO framework under linearity assumptions for treatment and post-treatment phase. In this setting, the authors showed the identifiability of causal effect if in the pre-treatment phase, the latent confounder jointly with observed outcome follows a multivariate Gaussian distribution, and in the treatment phase, the exogenous noise of outcome variable is non-Gaussian.
In linear SCMs, to the best of our knowledge, the methods that can identify the causal effect with only one proxy variable in Figure 2 are based on solving an OICA problem. In particular, [SGKZ20] considered linear SCM with non-Gaussian exogenous noises in the presence of latent variables. They showed that under some structural conditions, the causal effects among observed variables can be identified and the causal graph in Figure 2 satisfies such structural conditions.  $\mathrm{[YGN^{+}22]}$  extended the results in [SGKZ20]. They showed that the causal structure (direction) can always be identified, and the causal effect can be identified up to equivalence classes depending on the graphical
conditions (the causal graph in Figure 2 is still uniquely identifiable). However, both proposed methods [SGKZ20, YGN+22] require solving an OICA, and existing algorithms for solving such a problem might get stuck in bad local optima. Recently, [AHZ21] provided two graphical conditions for the same setting in [SGKZ20] which are necessary for the identification of the causal structure. These conditions are closely related to the sparsity of the causal graphs. For the causal graph in Figure 2, the method proposed in [AHZ21] for estimating the causal effect is the same as the one in [SGKZ20] and thus has the same drawback. Concurrent to our submission,  $\mathrm{[CHC^{+}23]}$  considered the problem of causal discovery for the linear non-Gaussian models. Under the specific assumptions on the true causal graph, they proposed an algorithm for learning the causal structure as well as the causal coefficients using high-order cumulants.
In PO framework, the setting of having just a pre-treatment phase and a post-treatment phase can be generalized to the case with multiple time slots in the panel data model  $\mathrm{[ABD^{+}21]}$ . In this paper, we mainly focus on the setting with two groups and two time slots but one can also study the extensions of the current work for other settings in the panel data model described in the following. Consider two  $N\times T$  matrices  $\mathbf{Y}$  and  $\mathbf{D}$  where  $N$  is the number of individuals in the population and  $T$  is the number of time slots. Assume that only the outcome for some individuals and time slots is observable. In particular:  $Y_{it} = (1 - D_{it})Y_{it}(0) + D_{it}Y_{it}(1)$ , where the realized outcome for individual  $i$  at time slot  $t$  is denoted by  $Y_{it}(D_{it})$ . DiD method has been proposed for the case  $T = 2$ , i.e., two time slots (pre-treatment and post-treatment phases). In the literature, other cases have been also studied for various assumptions on matrix  $\mathbf{Y}$ . For instance, in unconfounded case [RR83, IR], the number of individuals is much larger than the number of time slots ( $N\gg T$ ), and the treatment is provided only at the last time slot. Another setting is that of synthetic control [AG03, ADH10, ADH15, DI16] where  $T\gg N$ . In this setting, there is a single treated individual (suppose individual  $N$ ) and the goal is to estimate its missing potential outcomes for any  $t\in [T_0,T]$  after administering the treatment at time  $T_0$ . The last setting considers  $N\approx T$  and a two-way-fixed-effect (TWFE) regression model has been proposed to estimate the causal effect (see for a survey on TWFE in [DCd20]). It is noteworthy that TWFE estimator is equivalent to DiD estimator for two groups and two time slots.
In this section, we first evaluate our algorithm on synthetic data and compare it to DiD estimator and as well as the related work in [KP14] which estimates the causal effect in linear SCMs with two proxy variables. Further, we apply our algorithm to a real dataset provided by [CK93]. The implementation of the algorithm and additional experimental results are provided in https://github.com/ykivva/Cross-Moments-Method.
5.1 Synthetic data
We generated samples according to the SCM in (3) and with all the exogenous noises distributed according to exponential distribution. Note that the distribution of  $\epsilon_{u}$ , i.e., the exponential distribution satisfies Assumptions 2 and 3. Therefore  $\beta$  is identifiable according to the Corollary 1.
Given the observational data, we estimated the value of  $\beta$  from the following four approaches:
1. Cross-Moment algorithm (proposed in this work).
2. DiD estimator of (6).
3. A simple linear regression model based on the following equation:  $\hat{Y} = \alpha Z + \hat{\beta} D$
4. Causal effect estimate for linear SCM with two proxy variables (proposed in [KP14]). In the experiments, we call this estimate "two-proxy" method.
It is noteworthy that we also evaluated the method in [SGKZ20] which uses OICA as a subroutine. Unfortunately, the performance was too poor to be included.
For each sample size, we sampled parameters  $\alpha_{z},\alpha_{d},\beta ,\gamma$  randomly and then generated the samples of  $Z,D,Y$  accordingly to (3) (More details regarding the data generation mechanism can be found in Appendix B). We ran an experiment 10 times and reported the the average relative error for each value of sample size: err  $= \mathbb{E}\left[\left|\frac{\beta - \hat{\beta}}{\beta}\right|\right]$
Figure 4: The performance measure err against the number of samples. Colored regions show the standard deviation of the err.
Figure 4a, depicts the performances of Cross-Moment algorithm, DiD estimator, and the simple linear regression when we have access to only one proxy variable. The colored region around each curve shows the empirical standard deviation of  $|(\beta - \hat{\beta}) / \beta|$ . Cross-Moment algorithm outperforms the other two methods significantly. In fact, DiD estimator is biased if  $\alpha_z \neq \gamma$  which occurs with measure one as  $\alpha_z$  and  $\gamma$  are generated randomly. Moreover, DiD estimate is no better than simple linear regression if  $\gamma$  is not close to  $\alpha_z$ . In the literature, it has been noted that the parallel trend assumption (in linear SCM, this assumption is equivalent to the condition  $\alpha_z = \gamma$ ) is violated if the scale of the proxy variable  $Z$  and outcome variable  $Y$  are different which can be the case in many practical applications [L+11].
We compared Cross-Moment with the two-proxy method in [KP14] when we have access to two proxy variables. In particular, we assumed that there is an additional proxy variable  $W$  such that  $W := \alpha_w U + \epsilon_w$ . For Cross-Moment algorithm, we considered three versions: I - "Cross-Moment: Z", which estimates the causal effect by using only proxy variable  $Z$  (we denote this estimate by  $\beta_Z$ ), II - "Cross-Moment: W", which estimates  $\beta$  from only proxy variable  $W$  (which we denote the estimate by  $\beta_W$ ), III - "Cross-Moment: W-Z", which estimates  $\beta$  from aggregating the estimates of the methods I and II. In particular, "Cross-Moment: W-Z" uses bootstrapping method (Monte Carlo algorithm for case resampling [ET94]) to estimate the variances of estimates  $\beta_Z$  and  $\beta_W$ , denoted by  $\sigma_{\beta_Z}^2$  and  $\sigma_{\beta_W}^2$ , respectively. Subsequently,  $\beta$  is estimated by combining two estimates  $\beta_Z$  and  $\beta_W$  with an inverse-variance weighting scheme [SHK11] where we give a higher weight to the estimate with the lower variance:  $\frac{\sigma_{\beta_Z}^2}{\sigma_{\beta_Z}^2 + \sigma_{\beta_W}^2} \beta_W + \frac{\sigma_{\beta_W}^2}{\sigma_{\beta_Z}^2 + \sigma_{\beta_W}^2} \beta_Z$ . When  $\mathrm{Var}(\epsilon_w) / \mathrm{Var}(W)$  and  $\mathrm{Var}(\epsilon_z) / \mathrm{Var}(Z)$  are small, the causal effect can be estimated with a low estimation error from either  $Z$  or  $W$  as they contain low noise versions of the latent confounder  $U$ . In our experiments, we considered the case where one of the proxy variables (herein,  $W$ ) is too noisy but not the other one. Specifically, we choose  $\mathrm{Var}(\epsilon_w) / \mathrm{Var}(\epsilon_u) = 10$  and  $\mathrm{Var}(\epsilon_z) / \mathrm{Var}(\epsilon_u) = 0.1$ . Figure 4b illustrates the performances of the three aforementioned variants of Cross-Moment algorithm and the two-proxy method in [KP14]. "Cross-Moment: Z" has the best performance since it uses  $Z$  with less noise as the proxy variable. Moreover, "Cross-Moment: W-Z" has a comparable performance by combining the estimates of  $\beta_Z$  and  $\beta_W$ . The two-proxy estimate does not exhibit robustness and has a large average relative error for various values of sample size.
5.2 Minimum Wage and Employment Dataset
We evaluate our method on the real data which contains information about fast-food stores (Burger King, Roy Rogers, and Wendy's stores) in New Jersey and Pennsylvania in 1992, and some details about them such as minimum wage, product prices, open hours, etc [CK93]. The goal of study was to estimate the effect of the increment in minimum wage in New Jersey from  $4.25 to$ 5.05 per hour on the employment rate.
TWFE	Cross-Moment
With X	2.68	2.68
Without X	3.24	4.03
Table 1: Causal effect estimation of minimum wage on employment level in the real dataset in [CK93].
The data was collected by interviews in two waves, before and after the rise in the minimum wage. The information was gathered from 410 restaurants with similar average food prices, store hours, and employment levels. In this experiment, stores from Pennsylvania are treated as a control group and stores from New Jersey are considered as the treatment group. We define employment level  $Y$  as  $Y = Y_{f} + \frac{1}{2} Y_{p}$ , where  $Y_{p}$  is a number of employees working part-time and  $Y_{f}$  is a number of employees working full-time.
First, we reproduced the results in [CK93]. We considered an extended version of TWFE model [CK93]:
$$
\hat {Y} = \hat {\beta} _ {1} T + \mathbf {X} ^ {T} \hat {\alpha} + \hat {\beta} _ {2} D + \hat {\beta} D T,
$$
where  $\hat{Y}$  is the estimate of number of employees in the store,  $T$  is a binary variable that equals 0 prior to raising the minimum wage and equals to 1 after the raise.  $D$  is equal to 0 if the store is in Pennsylvania and equal to 1 if the store is in New Jersey.  $\mathbf{X}$  is a vector that contains additional information such as the opening hours, product prices, etc.  $\hat{\alpha}$  is also a vector of parameters corresponding to the vector  $\mathbf{X}$ . We dropped all the stores from the dataset that contain NaN values after which 246 restaurants were left. The estimate of  $\beta$  computed by TWFE is given in the first row of Table 1. According to [CK93], the estimate of  $\beta$  is equal to 2.76. The difference in estimation is due to the slight difference in the features of the vector  $\mathbf{X}$ , i.e., [CK93] added a few additional manually computed features to  $\mathbf{X}$ .
For the Cross-Moment algorithm, in order to incorporate the features  $\mathbf{X}$  in estimating  $\beta$ , we first regressed  $Y$  on  $\mathbf{X}$  and then used  $Y - \mathbf{X} \hat{\alpha}$  instead of  $Y$  as the outcome. The result of applying Cross-Moment algorithm to this newly defined outcome is given in the first row of Table 1 and is very close to the estimate by TWFE.
Finally, we assumed that the additional information  $\mathbf{X}$  gathered during the interview is not available. Then TWFE model for the employment level takes the following form
$$
\hat {Y} = \hat {\beta} _ {1} T + \hat {\beta} _ {2} D + \hat {\beta} D T.
$$
We used the previously pre-processed dataset but dropped the columns corresponding to  $\mathbf{X}$ . Subsequently, we applied TWFE and Cross-Moment method to estimate  $\beta$ . The respective estimates appear in the second row of Table 1, which stipulate the rise in the minimum wage had a positive effect on the employment level.
We considered the problem of estimating the causal effect of a treatment on an outcome in the linear SCM where we have access to a proxy variable for the latent confounder of the treatment and the outcome. This problem has been studied in both PO framework (such as DiD estimator) and SCM framework (such as the negative outcome control approach). We proposed a method that uses cross moments between the treatment, the outcome, and the proxy variable and recovers the true causal effect if the latent confounder is non-Gaussian. We also showed that the causal effect cannot be identified if the joint distribution over the observed variable are Gaussian. Unlike previous work which requires solving an OICA problem, our performs simple arithmetic operations on the cross moments. We evaluated our proposed method on synthetic and real datasets. Our experimental results show the proposed algorithm has remarkable performance for synthetic data and provides consistent results with previous studies on the real dataset we tested on.
This research was in part supported by the Swiss National Science Foundation under NCCR Automation, grant agreement 51NF40_180545 and Swiss SNF project 200021_204355/1.
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