Collider attributes in graph theory (wonkish)

7 Nov, 2019 at 11:49 | Posted in Statistics & Econometrics | 3 Comments

causal-inference-in-statisticsWhy would two independent variables suddenly become dependent when we condition on their common effect? To answer this question, we return again to the definition of conditioning as filtering by the value of the conditioning variable. When we condition on Z, we limit our comparisons to cases in which Z takes the same value. But remember that Z depends, for its value, on X and Y. So, when comparing cases where Z takes some value, any change in value of X must be compensated for by a change in the value of Y — otherwise, the value of Z would change as well.

The reasoning behind this attribute of colliders — that conditioning on a collision node produces a dependence between the node’s parents — can be difficult to grasp at first. In the most basic situation where Z = X + Y, and X and Y are independent variables, we have the follow- ing logic: If I tell you that X = 3, you learn nothing about the potential value of Y, because the two numbers are independent. On the other hand, if I start by telling you that Z = 10, then telling you that X = 3 immediately tells you that Y must be 7. Thus, X and Y are dependent, given that Z = 10.

People usually find this collider attribute rather perplexing. Why? My guess is the reason is most people wrongly think there can be no correlation without causation.

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  1. In math physics we say the two parents are not determined but they are now constrained. For example, in fluid dynamics, a constraint is introduced by the requirement that the fluid must remain continuous, i.e. voids cannot open up no matter how low the pressure falls. The resulting “continuity equation”, viz. divergence of velocity equals the time derivative of density, prevents a fluid from doing pretty much whatever it wants, no matter the initial data and boundary conditions.
    The analogy in economics would be stock/flow consistent modeling of financial networks in a monetary economy. Without the formalism of imposing constraints on the creation and destruction of money, and its conservation as it flows between network nodes, you find yourself hopelessly muddled in thinking about causality.
    Here:
    http://folk.ntnu.no/tronda/econ/andresen-phd-finished.pdf
    Is a perfect example of what I’m describing.

    • “A prerequisite for the analysis to follow is that the country in question
      issues its own currency (it is not in the eurozone or dollarised).”
      .
      Eurodollars are created outside the Fed’s control, so the analysis does not apply to the US dollar, which is the world’s hub currency.
      .
      Money is emergent. Surprisingly, the dollar gets stronger the more dollars there are.

      • Yes. Money is an emergent property of payment data networks. The value (utility) of money is itself a network externality that scales on the square of the number of nodes (users) in the network. However, the relative price of currencies used in assorted financial networks is likely set by constraints and linkages on that network of networks. For example, the global financial markets are generally shorting the dollar. Technically a position is short if the seller benefits by the shorted asset declining in price. In that sense, every dollar debtor on the planet is short the dollar, and we may now be seeing a dollar short squeeze of epic proportions. Macroeconomic theory does not address this. Keynes developed the idea of liquidity preference and noted how interest rates could influence macro behavior, but in his day the importance of currency price swings was much lower than it is today. This, imo, is the main weakness of Andresen’s thesis. I like his methodology though.


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