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Archive for February, 2015

Happy belated Valentine’s Day! It’s time to think about love. Math love.

While Scarlet O’Hara was brazenly optimistic in proclaiming tomorrow a new day, the reality is that she was just being naive – no day exists in a vacuum.  It’s unlikely that Rhett would feel differently about Miss Scarlett a day after walking out on their marriage. It’s also probably likely that the mood you were in when you went to bed will affect how you start your day eight hours later, especially if you were in a really sour one. But as the day goes on, you are likely to be affected by little bumps in the road, little surprises throughout the day that you weren’t expecting (either a welcome or negative).

This is an especially tough problem for statisticians and macroeconomists to figure out, for all macroeconomic indicators, such as unemployment, inflation or GDP.  Today’s unemployment rate is a function of yesterday’s.  Yesterday’s rate is affected by the day before that, and so on. It becomes one of those picture in a picture in a picture headaches that you’ve probably seen before.

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I’ve recently been thinking about this issue, but in the context of love. What happens if you meet someone amazing and wonderful, and he or she is perfect, but the beginning of the relationship itself is rather tumultuous? Can you make things peaceful despite a crazy start?  Can something that starts crazy, not implode? I think that maybe it’s going to be fine.  Actually, I know it’s going to be more than fine.

Let’s say this is the function of how you feel about someone in month t. But this is also a function of how you felt about them for the last 12 consecutive months. In each month you also introduce the new shock – a major life event, money problems, babies, etc.

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So t=1.  It’s the beginning, there are no past values to tell you how to feel about someone. The craziness of the beginning manifests itself in the error term. But the next month, you know that your relationship is also a function of the first month, which includes with it, the first period shock. This shows up again and again, month after month, influencing the relationship every month for the first year. Furthermore, you know that one year from how, your relationship is a function of the state it was in for each of the twelve months prior, which are all affected by that one first period shock. It’s one big chain reaction.

The stability depends on whether the entire relationship is covariance stationary. It’s a bit of math that boils down to whether you can solve for the zeros in an equation where you are trying to solve for L values. If it is, then over time, over enough time, even a big shock in the beginning will be absorbed by the inherent stability of the system and become smoothed over time.  If it isn’t then, the system isn’t stable, and any small disruptions, will cause the relationship to spiral wildly out of control.

The covariance stationary single-variable relationship will be fine, as long as you can ride out the initial crash landing.  Anything that isn’t, is already doomed. So in a way, there’s really no point in wondering whether a relationship can survive an initial big shock, or what you can do – it’s already been pre-determined. For Miss Scarlett, there’s nothing she can do to bring Rhett back. For the rest of us, it may be a relief to know that as time goes on, a system that is inherently resilient, will be able to weather the occasional big storm.

On an interesting sidenote, I’m not the only one who’s been thinking about applying econometric models to relationships – there’s been work on determining whether this can be used to estimate, in a two variable case, whether two countries will stabilise a chilly relationship or spiral into nuclear war.  In reality, using vector autoregressions to figure out whether or not nuclear escalation is likely, even when looking at ex post examples, is actually pretty tricky. Read about it here.

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