## The shaky mathematical basis of DSGE models

28 March, 2016 at 14:14 | Posted in Economics | 1 CommentIn most aspects of their lives humans must plan forwards. They take decisions today that affect their future in complex interactions with the decisions of others. When taking such decisions, the available information is only ever a subset of the universe of past and present information, as no individual or group of individuals can be aware of all the relevant information. Hence, views or expectations about the future, relevant for their decisions, use a partial information set, formally expressed as a conditional expectation given the available information.

Moreover, all such views are predicated on there being no un-anticipated future changes in the environment pertinent to the decision. This is formally captured in the concept of ‘stationarity’. Without stationarity, good outcomes based on conditional expectations could not be achieved consistently. Fortunately, there are periods of stability when insights into the way that past events unfolded can assist in planning for the future.

The world, however, is far from completely stationary. Unanticipated events occur, and they cannot be dealt with using standard data-transformation techniques such as differencing, or by taking linear combinations, or ratios. In particular, ‘extrinsic unpredictability’ – unpredicted shifts of the distributions of economic variables at unanticipated times – is common. As we shall illustrate, extrinsic unpredictability has dramatic consequences for the standard macroeconomic forecasting models used by governments around the world – models known as ‘dynamic stochastic general equilibrium’ models – or DSGE models …

Many of the theoretical equations in DSGE models take a form in which a variable today, say incomes (denoted as yt) depends inter alia on its ‘expected future value’… For example, yt may be the log-difference between a de-trended level and its steady-state value. Implicitly, such a formulation assumes some form of stationarity is achieved by de-trending.

Unfortunately, in most economies, the underlying distributions can shift unexpectedly. This vitiates any assumption of stationarity. The consequences for DSGEs are profound. As we explain below, the mathematical basis of a DSGE model fails when distributions shift … This would be like a fire station automatically burning down at every outbreak of a fire. Economic agents are affected by, and notice such shifts. They consequently change their plans, and perhaps the way they form their expectations. When they do so, they violate the key assumptions on which DSGEs are built.

A great article, not only showing on what shaky mathematical basis DSGE models are built, but also confirming much of Keynes’s critique of econometrics, underlining that to understand real world ”non-routine” decisions and unforeseeable changes in behaviour, stationary probability distributions are of no avail. In a world full of genuine uncertainty — where real historical time rules the roost — the probabilities that ruled the past are not those that will rule the future.

Advocates of DSGE modeling want to have deductively automated answers to fundamental causal questions. But to apply “thin” methods we have to have “thick” background knowledge of what’s going on in the real world, and not in idealized models. Conclusions can only be as certain as their premises — and that also applies to the quest for causality and forecasting predictability in DSGE models.

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They remind me of Hiawatha’s experiments.

Hiawatha Designs an Experiment

Hiawatha, mighty hunter,

He could shoot ten arrows upward,

Shoot them with such strength and swiftness

That the last had left the bow-string

Ere the first to earth descended.

This was commonly regarded

As a feat of skill and cunning.

Several sarcastic spirits

Pointed out to him, however,

That it might be much more useful

If he sometimes hit the target.

“Why not shoot a little straighter

And employ a smaller sample?”

Hiawatha, who at college

Majored in applied statistics,

Consequently felt entitled

To instruct his fellow man

In any subject whatsoever,

Waxed exceedingly indignant,

Talked about the law of errors,

Talked about truncated normals,

Talked of loss of information,

Talked about his lack of bias,

Pointed out that (in the long run)

Independent observations,

Even though they missed the target,

Had an average point of impact

Very near the spot he aimed at,

With the possible exception

of a set of measure zero.

“This,” they said, “was rather doubtful;

Anyway it didn’t matter.

What resulted in the long run:

Either he must hit the target

Much more often than at present,

Or himself would have to pay for

All the arrows he had wasted.”

Hiawatha, in a temper,

Quoted parts of R. A. Fisher,

Quoted Yates and quoted Finney,

Quoted reams of Oscar Kempthorne,

Quoted Anderson and Bancroft

(practically in extenso)

Trying to impress upon them

That what actually mattered

Was to estimate the error.

Several of them admitted:

“Such a thing might have its uses;

Still,” they said, “he would do better

If he shot a little straighter.”

Hiawatha, to convince them,

Organized a shooting contest.

Laid out in the proper manner

Of designs experimental

Recommended in the textbooks,

Mainly used for tasting tea

(but sometimes used in other cases)

Used factorial arrangements

And the theory of Galois,

Got a nicely balanced layout

And successfully confounded

Second order interactions.

All the other tribal marksmen,

Ignorant benighted creatures

Of experimental setups,

Used their time of preparation

Putting in a lot of practice

Merely shooting at the target.

Thus it happened in the contest

That their scores were most impressive

With one solitary exception.

This, I hate to have to say it,

Was the score of Hiawatha,

Who as usual shot his arrows,

Shot them with great strength and swiftness,

Managing to be unbiased,

Not however with a salvo

Managing to hit the target.

“There!” they said to Hiawatha,

“That is what we all expected.”

Hiawatha, nothing daunted,

Called for pen and called for paper.

But analysis of variance

Finally produced the figures

Showing beyond all peradventure,

Everybody else was biased.

And the variance components

Did not differ from each other’s,

Or from Hiawatha’s.

(This last point it might be mentioned,

Would have been much more convincing

If he hadn’t been compelled to

Estimate his own components

From experimental plots on

Which the values all were missing.)

Still they couldn’t understand it,

So they couldn’t raise objections.

(Which is what so often happens

with analysis of variance.)

All the same his fellow tribesmen,

Ignorant benighted heathens,

Took away his bow and arrows,

Said that though my Hiawatha

Was a brilliant statistician,

He was useless as a bowman.

As for variance components

Several of the more outspoken

Make primeval observations

Hurtful of the finer feelings

Even of the statistician.

In a corner of the forest

Sits alone my Hiawatha

Permanently cogitating

On the normal law of errors.

Wondering in idle moments

If perhaps increased precision

Might perhaps be sometimes better

Even at the cost of bias,

If one could thereby now and then

Register upon a target.

Maurice G. Kendall

The American Statistician 13 (1959) 23-24

Comment by Greg Orme— 5 April, 2016 #