Question

Estimate the demand curve using regression analysis. Write down the equational form. Interpret the coefficients, statistical significance and R2. What are the limitations of your specification (omitted variables, correlation vs. causality)?

Quantity	Price
84	59
80	65
85	54
83	61
81	64
84	58
87	48
78	68
82	63
76	70
79	65
75	80

Answer 1

1) Multiple regression, even given a perfect fit along some line in Rn, doesn't imply direction. It's similar to why correlation isn't causation: simplistically, correlation between A and B implies that either A causes B, B causes A, or C causes both. Multiple regression shows similar relationships (i.e., directionless and based upon the spread of data points from the regression "line").

2) Causal analyses are somewhat controversial in whether they do, in fact, show causal relationships. For one thing, there is no agreed upon definition of causality (even were we talking only about efficient causality). For example, a common model is counterfactual (granted that x and y occurred, x causes y iff had x not occurred, y wouldn't have occurred). This seems to fail in quantum physics. Another is very much mathematical in that a cause y is treated as a function of x (this runs into problems with circular causality, Rosen's [M,R] systems, etc.). In statistics, inferential models and methods can, theoretically, allow one to infer direction. Bayesian inference in particular is a fairly straightforward generalization of the epistemological basis of counterfactual causality to a quantitative framework.

3) There are a variety of causal analyses in the social sciences and not all of them correspond to or subscribe to the same model of causality. Stable association, for example, is a common basis for causal analyses but can be implemented via numerous statistical methods.

4) Some statistical methods are more or less designed from the ground up for this purpose (structural equation modeling is fairly prominent here). SEM and similar methods rely much on path analysis and graph-theoretic models. Simplistically, instead of showing relationships among variables as in multiple regression, causal models frequently rely on proposed/possible factors hypothesized to be causes (confirmatory analyses) or they use cluster, factor, path, etc., analyses to "uncover" hidden/latent relationships among variables by mathematically classifying the data points and projecting them onto a new lower-dimensional space. Whether exploratory or confirmatory, the main idea is to explore not just the relationship among variables but underlying relationships that explain the data.

5) Finally, when we're really lucky, someone uses predictive models. These are much easier to evaluate in terms of their validity: if your model says x is supposed to happen and it doesn't, your models is wrong (of course, "all models are wrong but some are useful"; however, this really means that there models will always be off at least a little not that they are "wrong").