# Causal identification and penalized splines

I just got a rejection from an economics journal. Among the reasons cited for rejection were:

the benefits of using the semi-parametric method are not clearly brought out compared to alternative simpler techniques with clean identification of causal relationships

It is certainly possible that I could have done a better job of motivating the methodology to a bunch of economists who generally stick to OLS. But have I violated "clean identification"? Please judge for yourself and let me know what you think:

My main estimating equation is $$y_{it} = \alpha_i + \beta_1 T_{it} + f\left(\begin{array}{l}Z_{it}\\ Z_{it} \times T_{it} \\ Z_{it}\times T_{it} \times X_t\end{array} \right) + \beta_2X_t + \epsilon_{it}$$ $Z$ is continuous, $X$ and $T$ are binary. I can justifiably assume that $$E[\epsilon|\alpha,T] = 0$$ Which is to say that the coefficient on $T$ is unbiased conditional on individual-level dummy variables ("fixed effects" in econometrics-speak). When I include continuous variable $Z$, I am simply looking at heterogeneity in estimated treatment effects over gradients of $Z$. So the average causal effect of the treatment $T$ is a an average of $\hat\beta_1 + \hat f_{Z\times T}$ for the various levels of $Z$ that I observe.

The model is etimated by penalized quadratic splines (e.g.: Ruppert et al. 2003). Specifically: $$y = \beta_0 +X'\beta + \displaystyle\sum_{1}^p (Z^{p})'\gamma + \displaystyle\sum_{j=1}^{\#vars} \displaystyle\sum_{k=1}^{\# knots_j}\delta_{jk}\left(\left(Z_j - \kappa_{jk} \right)^p \times \left(Z_j > \kappa_{jk} \right)\right) + \epsilon$$

This is solved by $$\left[\begin{array}{c} \hat\beta\\ \hat\gamma \\ \hat \delta \\ \end{array}\right] = (C'C + \lambda^{2p}D)^{-1}C'y$$

where $C$ includes the parametric terms and the knot terms, and where the ridge penalty $\lambda$ only applies to the knot terms, and is chosen to minimize AIC. (I can't do full justice to the methodology -- see Ruppert et al, or Simon Wood's textbook on GAM's).

Of course, I use these semiparametrics because I don't want to impose unfounded functional forms onto my data. Doing so would quite naturally bias my estimates as much as imposing a logarithmic fit onto a sinusoidal function would bias my estimates. But is there something inherent in penalized splines as I've described them that would inherently render the following statement untrue?

$$E[\hat\beta_1] = \beta_1 \text{ iff } E[\epsilon|\alpha,T] = 0$$

• I'm not qualified to answer your final question (although it seems suspicious), but perhaps to address the Journals concerns you should also include a OLS model in your paper and show that it performs badly by some metric? – GeorgeWilson Jun 4 '13 at 4:56
• You did not violate "clean identification." There's nothing inherent that makes semi-parametric model less capable of achieving clean identification. Indeed, your model encompasses a linear model. – user28544 Jul 28 '13 at 5:22
• @generic_user did you ever receive a resolution to this? If so, can you answer your question? If not, could you provide a definition of clean identification? I have some perspectives on publishing spline-adjusted analyses that may or may not be pertinent to this case. – AdamO Jul 20 '17 at 20:23
• Late to the party, but I think you're worrying about the wrong thing here. The refs are saying they don't like that you added complexity without proving that it's useful. An example showing a failure mode of their go-to simple methods would help motivate the extra complexity you're introducing. It should be possible to engineer (or even better find a real world example) of where splines are needed to properly identify a causal relationship. – Paul Nov 13 '17 at 22:57
• If this was published as some point can you please mention the name of the paper? It seems like an interesting application. – usεr11852 May 6 '18 at 21:00