by In order to make causal inferences from simple regression, it is now conventional to assume something like the setting in equation (1) … The equation makes very strong invariance assumptions, which cannot be tested from data on X and Y. (1) Y = a + bx + δ What happens without invariance? The answer will be obvious. If intervention changes the intercept a, the slope b, or the mean of the error distribution, the impact of the intervention becomes difficult to determine. If the variance of the error term is changed, the usual confidence intervals lose their meaning. How would any of this be possible? Suppose, for instance, that — unbeknownst to the statistician — X and Y are both the effects of a common cause operating through linear statistical laws like (1). Suppose errors are independent and normal, while Nature randomizes the common cause to have a normal distribution. The scatter diagram will look lovely, a regression line is easily fitted, and the straightforward causal interpretation will be wrong. Since econometrics doesn’t content itself with only making ‘optimal predictions’ but also aspires to explain things in terms of causes and effects, econometricians need loads of assumptions. And invariance is not the only limiting assumption that has to be made. Equally important are the ‘atomistic’ assumptions of additivity and linearity.
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Lars Pålsson Syll considers the following as important: Statistics & Econometrics
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In order to make causal inferences from simple regression, it is now conventional to assume something like the setting in equation (1) … The equation makes very strong invariance assumptions, which cannot be tested from data on X and Y.
(1) Y = a + bx + δ
What happens without invariance? The answer will be obvious. If intervention changes the intercept a, the slope b, or the mean of the error distribution, the impact of the intervention becomes difficult to determine. If the variance of the error term is changed, the usual confidence intervals lose their meaning.
How would any of this be possible? Suppose, for instance, that — unbeknownst to the statistician — X and Y are both the effects of a common cause operating through linear statistical laws like (1). Suppose errors are independent and normal, while Nature randomizes the common cause to have a normal distribution. The scatter diagram will look lovely, a regression line is easily fitted, and the straightforward causal interpretation will be wrong.
Since econometrics doesn’t content itself with only making ‘optimal predictions’ but also aspires to explain things in terms of causes and effects, econometricians need loads of assumptions. And invariance is not the only limiting assumption that has to be made. Equally important are the ‘atomistic’ assumptions of additivity and linearity.
These assumptions are of paramount importance and ought to be much more argued for — on both epistemological and ontological grounds — if at all being used.
Limiting model assumptions in economic science always have to be closely examined since if we are going to be able to show that the mechanisms or causes that we isolate and handle in our models are stable in the sense that they do not change when we ‘export’ them to our ‘target systems,’ we have to be able to show that they do not only hold under ceteris paribus conditions and a fortiori only are of limited value to our understanding, explanations or predictions of real economic systems.
The kind of fundamental assumption about the character of material laws, on which scientists appear commonly to act, seems to me to be much less simple than the bare principle of uniformity. They appear to assume something much more like what mathematicians call the principle of the superposition of small effects, or, as I prefer to call it, in this connection, the atomic character of natural law.
The system of the material universe must consist, if this kind of assumption is warranted, of bodies which we may term (without any implication as to their size being conveyed thereby) legal atoms, such that each of them exercises its own separate, independent, and invariable effect, a change of the total state being compounded of a number of separate changes each of which is solely due to a separate portion of the preceding state. We do not have an invariable relation between particular bodies, but nevertheless each has on the others its own separate and invariable effect, which does not change with changing circumstances, although, of course, the total effect may be changed to almost any extent if all the other accompanying causes are different. Each atom can, according to this theory, be treated as a separate cause and does not enter into different organic combinations in each of which it is regulated by different laws …
The scientist wishes, in fact, to assume that the occurrence of a phenomenon which has appeared as part of a more complex phenomenon, may be some reason for expecting it to be associated on another occasion with part of the same complex. Yet if different wholes were subject to laws qua wholes and not simply on account of and in proportion to the differences of their parts, knowledge of a part could not lead, it would seem, even to presumptive or probable knowledge as to its association with other parts. Given, on the other hand, a number of legally atomic units and the laws connecting them, it would be possible to deduce their effects pro tanto without an exhaustive knowledge of all the coexisting circumstances.
Econometrics may be an informative tool for research. But if its practitioners do not investigate and make an effort of providing a justification for the credibility of the assumptions on which they erect their building, it will not fulfill its tasks. There is a gap between its aspirations and its accomplishments, and without more supportive evidence to substantiate its claims, critics like Keynes — and yours truly — will continue to consider its ultimate argument as a mixture of rather unhelpful metaphors and metaphysics.
The marginal return on its ever higher technical sophistication in no way makes up for the lack of serious under-labouring of its deeper philosophical and methodological foundations that already Keynes complained about. Firmly stuck in an empiricist tradition, econometrics is only concerned with the measurable aspects of reality, and a rigorous application of econometric methods in economics really presupposes that the phenomena of our real world economies are ruled by stable causal relations.
Unfortunately, real world social systems are usually not governed by stable causal mechanisms or capacities. The kinds of ‘laws’ and relations that econometrics has established, are laws and relations about entities in models that presuppose causal mechanisms being invariant, atomistic and additive. But — when causal mechanisms operate in the real world they only do it in ever-changing and unstable combinations where the whole is more than a mechanical sum of parts. If economic regularities obtain they do it as a rule only because we engineered them for that purpose. Outside man-made ‘nomological machines’ they are rare, or even non-existant.