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I suppose that the main scope of an econometric models should be predictive or causal inference. Following this perspective it was shown that underspecified model can perform better than the correct specified (see here: When will a less true model predict better than a truer model?). Note that in this vein the so called “true model” is to be considered like a structural/causal model.

Now, It is known that models like ARMA are used for pure prediction and do not have a role in causal inference (see here: https://en.wikipedia.org/wiki/Autoregressive%E2%80%93moving-average_model#Estimating_coefficients); Indeed sometimes authors speak about of “free of theory models”. Moreover we can argue that endogeneity in forecasting do not have a role, and the precise value of parameters is not important; Indeed only the minimization of measure like $MSE$ is (see here: Endogeneity in forecasting). Indeed, regardless of the specification, parameters of predictive regression always maintain a some correlational meaning but this is usually of little interest.

However the tool “true model” is traditionally used in ARMA framework too. Then, considering what said above, we can show that, for example, if the $AR(2)$ is the true model we can achieve a better predictive performance from an estimated $AR(1)$ regression (underspecified) than an $AR(2)$ (rightly specified).

Now, what role the "true" $AR(2)$ model have? More precisely, what meaning/legitimacy have his two parameters? This question can be immediately generalized to any ARMA model.

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  • $\begingroup$ Related (although not related to Pearl's language): stats.stackexchange.com/questions/192809/… $\endgroup$ Commented Mar 29, 2019 at 15:34
  • $\begingroup$ Indeed had the ancients tried to difference a velocity time series obtained from smooth slowed down slope, we wouldn't need to wait for Newton to discover the extremely simple looking 2nd classical dynamics (causal) law. $\endgroup$
    – cinch
    Commented Dec 25, 2022 at 5:15
  • $\begingroup$ It isn't clear to me what "something more" refers to. $\endgroup$
    – Galen
    Commented Sep 15, 2023 at 14:27
  • $\begingroup$ Refers to the “true parameters”, them meaning seems neither correlational or causal. The question is about their meaning/legitimacy. $\endgroup$
    – markowitz
    Commented Sep 15, 2023 at 14:39
  • $\begingroup$ In other words we can use the concept of structural parameters (or structural quantities more in general) as something like a third way, no pure correlational quantities but not causal? If yes what substantive meaning such parameters/quantities encode. I suspect that such concept is a chimera and even for this structural or causal parameters/quantities must be considered only as synonym. $\endgroup$
    – markowitz
    Commented Sep 15, 2023 at 14:40

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Currently I think that in the framework of ARMA the "true model", and related parameters, do not have a relevant meaning. It do not have counterparts in the real world, neither correlational or causal, then lose its legitimacy. We can, maybe should be, looking for ARMA specification (regression) without refer to something like the "true ARMA structure". In practical point of view we have to looking for the ARMA(p,q) specification, choice of $p$ and $q$, that is the best for forecasting. Best in MSE (or other metric) sense.

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