Editing Causality (section)

==== Statistics and economics ====
{{See also|Causal graph}}

[[Statistics]] and [[economics]] usually employ pre-existing data or experimental data to infer causality by regression methods. The body of statistical techniques involves substantial use of [[regression analysis]]. Typically a linear relationship such as

:<math>y_i = a_0 + a_1x_{1,i} + a_2x_{2,i} + \dots + a_kx_{k,i} + e_i</math>

is postulated, in which <math>y_i</math> is the ''i''th observation of the dependent variable (hypothesized to be the caused variable), <math>x_{j,i}</math> for ''j''=1,...,''k'' is the ''i''th observation on the ''j''th independent variable (hypothesized to be a causative variable), and <math>e_i</math> is the error term for the ''i''th observation (containing the combined effects of all other causative variables, which must be uncorrelated with the included independent variables). If there is reason to believe that none of the <math>x_j</math>s is caused by ''y'', then estimates of the coefficients <math>a_j</math> are obtained. If the null hypothesis that <math>a_j=0</math> is rejected, then the alternative hypothesis that <math>a_{j} \ne 0 </math> and equivalently that <math>x_j</math> causes ''y'' cannot be rejected. On the other hand, if the null hypothesis that <math>a_j=0</math> cannot be rejected, then equivalently the hypothesis of no causal effect of <math>x_j</math> on ''y'' cannot be rejected. Here the notion of causality is one of contributory causality as discussed [[#Necessary and sufficient causes|above]]: If the true value <math>a_j \ne 0</math>, then a change in <math>x_j</math> will result in a change in ''y'' ''unless'' some other causative variable(s), either included in the regression or implicit in the error term, change in such a way as to exactly offset its effect; thus a change in <math>x_j</math> is ''not sufficient'' to change ''y''. Likewise, a change in <math>x_j</math> is ''not necessary'' to change ''y'', because a change in ''y'' could be caused by something implicit in the error term (or by some other causative explanatory variable included in the model).

The above way of testing for causality requires belief that there is no reverse causation, in which ''y'' would cause <math>x_j</math>. This belief can be established in one of several ways. First, the variable <math>x_j</math> may be a non-economic variable: for example, if rainfall amount <math>x_j</math> is hypothesized to affect the futures price ''y'' of some agricultural commodity, it is impossible that in fact the futures price affects rainfall amount (provided that [[cloud seeding]] is never attempted). Second, the [[instrumental variables]] technique may be employed to remove any reverse causation by introducing a role for other variables (instruments) that are known to be unaffected by the dependent variable. Third, the principle that effects cannot precede causes can be invoked, by including on the right side of the regression only variables that precede in time the dependent variable; this principle is invoked, for example, in testing for [[Granger causality]] and in its multivariate analog, [[vector autoregression]], both of which control for lagged values of the dependent variable while testing for causal effects of lagged independent variables.

Regression analysis controls for other relevant variables by including them as regressors (explanatory variables). This helps to avoid false inferences of causality due to the presence of a third, underlying, variable that influences both the potentially causative variable and the potentially caused variable: its effect on the potentially caused variable is captured by directly including it in the regression, so that effect will not be picked up as an indirect effect through the potentially causative variable of interest. Given the above procedures, coincidental (as opposed to causal) correlation can be probabilistically rejected if data samples are large and if regression results pass [[Cross-validation (statistics)|cross-validation]] tests showing that the correlations hold even for data that were not used in the regression. Asserting with certitude that a common-cause is absent and the regression represents the true causal structure is ''in principle'' impossible.<ref>{{Cite journal|last=Henschen|first=Tobias|date=2018|title=The in-principle inconclusiveness of causal evidence in macroeconomics|journal=European Journal for the Philosophy of Science|volume=8|issue=3|pages=709–733|doi=10.1007/s13194-018-0207-7|s2cid=158264284}}</ref>

The problem of omitted variable bias, however, has to be balanced against the risk of inserting [[Collider (statistics)|Causal colliders]], in which the addition of a new variable <math>x_{j+1}</math> induces a correlation between <math>x_j</math> and <math>y</math> via [[Berkson's paradox]].<ref name="Pearl" />

Apart from constructing statistical models of observational and experimental data, economists use axiomatic (mathematical) models to infer and represent causal mechanisms. Highly abstract theoretical models that isolate and idealize one mechanism dominate microeconomics. In macroeconomics, economists use broad mathematical models that are calibrated on historical data. A subgroup of calibrated models, [[dynamic stochastic general equilibrium]] (DSGE) models are employed to represent (in a simplified way) the whole economy and simulate changes in fiscal and monetary policy.<ref>{{Cite journal|last=Maziarz Mariusz|first=Mróz Robert|date=2020|title=A rejoinder to Henschen: the issue of VAR and DSGE models|journal=Journal of Economic Methodology|volume=27|issue=3|pages=266–268|doi=10.1080/1350178X.2020.1731102|s2cid=212838652}}</ref>