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# The Accuracy of Long-Run Forecasts

## Can Predicting The Long Run Be More Accurate Than Predicting Tomorrow?

Forecasting of future events is used a great deal in economics; from financial predictions such as future levels of the stock market or a future value of an exchange rate, to demographic ones such as the future population density of an area. Long-run models are difficult to validate empirically, since their results do not come about until years in the future. So naturally, we should take any long-run predictions we hear with a dose of skepticism.

That skepticism can go too far, however. You often hear those skeptical of long-run predictions make a comparison between short-run predictions and long-run ones. This is particularly true of those skeptical of climate change models - "If we can't predict with any reliability what the weather will be like tomorrow, how can we possibly know what it will be like 50 years from now?!?"

They fallacy in this argument is that nobody is predicting what the weather will be like on, say, August 12th, 2057. That would be far more difficult than predicting what the weather will be like tomorrow. Rather the long-run forecast predicts what weather will be like, on average in, say, the year 2057. And all else being equal, averages are far easier to easier to predict that individual events.

To give you an example of what I mean, consider a coin that is weighted to come up "heads" slightly more often than it comes up "tails". Suppose that the probability of heads is 55% and the probability of heads is 45%.

We cannot predict with any real accuracy what will happen when we flip the coin. Predicting heads is the obvious choice, but we will still be wrong 45%. Not a very accurate prediction.

Now what if we flipped the coin 3 times to see what came up (overall) more often, heads or tails. Turns out that by using the binomial theorem that 57.5% of the time we will see more heads than tails. That's an improvement on our one-shot 55% odds, but still not very accurate.

What if we tried this experiment 11 times? 51 times? What would we see then? At the bottom of this article is the probability that we will see a greater number of heads, given a set number of coin flips. Note that when we flip the coin 1001 times, that we will see more heads than tails 99.9% of the time. This makes sense - since the coin is weighted towards heads, we would expect that in the long-run, we would end up seeing more heads than tails. That's exactly what happens.

Note that this does not mean that on the 1001th flip that we can accurately predict if we'll have a head or a tail - the probability of a head is still 55%. Instead, we see that on the average of 1001 flips, we see more heads than tails. This is typically what we mean when we make long-run forecasts. We are not forecasting what the stock market will be at on July 12, 2034. Rather we are forecasting an average level for a much longer period of time - say the 500 or so business days in 2034 and 2035.

There are many reasons to be skeptical of long-run models, particularly ones dealing with complex systems such as an economy or a climate. The idea that because we cannot predict with a great deal of accuracy what will happen tomorrow, so we cannot possibly predict 10 years from now, is not a particularly compelling reason for skepticism.

## Probability We Will See More Heads Than Tails

 Number of Flips Probability of Greater Number of Heads 1 .550 3 .575 11 .633 21 .679 51 .764 101 .844 501 .988 1001 .999