<p>I would argue the answer to </p>

<blockquote>
  <p>Are the use of expectations augmented models useful for forecasting or is it just a way to explain forecast error?</p>
</blockquote>

<p>is neither.  There is nothing intrinsic to expectations that makes forecasting better unless a squared loss function makes sense under the circumstances.  In that case, it isn't "useful," so much as background information.  For some things such as stocks, it isn't possible to define a problem in terms of squared loss because the integrals diverge.</p>

<p>It also isn't a way to explain forecast error.  It may be the case that in a particular problem model errors naturally converge to quadratic loss.  In that case, it doesn't explain the errors, it just describes them.  If a quadratic loss function was inappropriate, as would be the case of stocks which naturally follow an all-or-nothing loss function, it would make understanding the errors more difficult.</p>

<p>Expectations are misused in economics as it implies all problems are quadratic loss problems.  All problems relevant to a particular research line may be, but that isn't mandatory and doesn't work for all distributions.</p>

<p>Expectations are one way to remove the stochastic component when they are defined for the density involved.  This allows differentiation because stochastic functions are not smooth even though they are continuous.  The use of expectations was an attempt to create a smooth curve over a random process.  If the first moment is defined, which it isn't for things like stocks, then you can construct a smooth path that is a function of time.</p>

<p>For stocks, you have to define the parameters as a function of time and minimize over the all-or-nothing loss function over a neighborhood of $\theta_t\pm\epsilon_t$, where $\theta_t$ is a parameter and $\epsilon_t$ is small where the loss is 1 outside the neighborhood and 0 in the neighborhood.  I am presenting this in a paper on generalized anticipation operators.</p>

CC BY-SA 3.0
2018-02-13T11:36:13.987Z
2018-02-13T11:36:13.987Z
Dave Harris
<p>If you prefer, think of the rational expectations framework as a benchmark -- much in the same way that perfect competition is a benchmark.  It's not about whether the benchmark is wrong, so much as what the benchmark allows us to conclude when it is wrong.</p>

<p>The canonical example for rational expectations is the Barro-Gordon model of central banking (or more generally, the Kydland-Prescott model of policy-making), which essentially couches monetary policy in the framework of an infinitely repeated game.</p>

<p>The idea there was to consider the question of whether it is better to have a central bank that sets monetary policy "by discretion" or by adhering to a stated rule.  In this case, the idealized assumption being made is that agents are rational and possess full information about the past behavior of the central bank (as well as the true value of pertinent measures like inflation &amp; unemployment).</p>

<p>In the model, the bank's objective function implies a tradeoff between unemployment and inflation via the Phillips Curve.  Both inflation and unemployment are costly, but setting both to the equilibrium value is not feasible.</p>

<p>In a one-shot game, the bank can improve the value of its objective function by "inflation surprises" -- basically, by injecting money into the economy without announcing it, so that in the short-run there is an increase to employment but no impact on inflation.  But since realized inflation is a function of inflation expectations (because if people expect their money to be worth less tomorrow, they will spend more of it today, thus leading to increased consumption demand which drives up prices and makes the expectation self-fulfilling), in the iterated game setting rational economic agents will come to expect the central bank to generate "inflation surprises".  What is interesting about the Barro-Gordon model is that this leads to an "inflation bias" -- realized inflation is always higher than the optimal value when the bank's actual policy targets don't match with the expectations they set out through statements, etc.</p>

<p>So rational expectations predicts that central banking done according to a rule is more efficient, because it keeps policy inflation in line with expectations and avoids this bias.  In the literature one hears about "credible banks" and this is essentially what this is about.</p>

<p>Getting back to your question, is this used to "explain" forecast error?  In a sense, yes:  if the model predicts a certain value of inflation under rational expectations, and that value of inflation isn't realized, then one either questions the rational expectation model itself or (more likely) that one or more of the assumptions behind it is violated.  Usually, this is the assumption of complete information, although one should also consider the possibility of additional shocks not accounted for explicitly in the model.</p>

CC BY-SA 3.0
2018-02-13T19:23:47.587Z
Why did you delete this?
EconJohn
CC BY-SA 3.0
2018-02-14T21:03:15.880Z
Short answer:  I'm new and have no idea what I'm doing.  The post came out a bit longer than I wanted, and I didn't have time to edit it down.

I've undeleted it though, on the assumption that you're asking because you found it useful.  :P
Dan
CC BY-SA 3.0
2018-02-13T04:39:26.617Z
2018-02-14T21:04:54.823Z
2018-02-14T21:04:54.823Z
Dan
<p>I've been reading up on the topic of Rational expectations  econometrics, and have really been wondering if it is a theory useful for prediction or just used to "fit" a model.</p>

<p>Your standard rational expectations model (otherwise referred to as efficent markets model):
$$y_t=\tilde{y}+\sum_{i=1}^n\beta_i(X_{t-i}-X^e_{t-i})+\epsilon$$</p>

<ul>
<li>$y_t=$ Economic variable of interest at time $t$ like unemployment or real output.</li>
<li>$\tilde{y}_t=$Natural rate or equilibrium of economic variable at time $t$</li>
<li>$\beta_i=$ coefficents.</li>
<li>$X_{t-i}=$ an aggreagte demand variable like money growth, Inflation or Nominal GDP growth.</li>
<li>$X^e_{t-i}=$ Anticipated aggregate demand conditional on knowing all  information at point $t-i$</li>
<li>$\epsilon_t=$ an error term</li>
</ul>

<p>Where the property of "rational expectations" is defined as the true expected value of $X_t$ is equal to the markets expected value of $X_t$. Mathematically written as:
$$\mathbb{E}(X_{t}|\phi_{t-1})=\mathbb{E}_m(X_t|\phi_{t-1})$$
$$\mathbb{E}(X_{t}-\mathbb{E}_m(X_t)|\phi_{t-1})=0$$</p>

<p>If rational expectations hold, we should observe when expectations are taken:</p>

<p>$$y_t=\tilde{y}_t$$</p>

<p>that is $y_t$ should always be in equilibrium/ be forecasted based on its equilibrium values/growth process (providing rationale for the use of Univariate models in economics).</p>

<p><strong>My Question:</strong> Are the use of expectations augmented models useful for forecasting or is it just a way to explain forecast error?</p>

CC BY-SA 3.0
2018-02-13T01:26:24.527Z
I don't understand how it helps in fitting. The rational expectation hypothesis is used to kill off the degrees of freedom one has in modeling expectations.
Michael Greinecker
CC BY-SA 3.0
2018-02-13T01:42:25.920Z
@MichaelGreinecker can you elaborate/ provide a source.
EconJohn
CC BY-SA 3.0
2018-02-13T01:47:42.430Z
There is the [original paper](https://www.fep.up.pt/docentes/pcosme/S-E-1/Eco-29-3-315.pdf) by Muth. But to make my point briefly: You can add variables for the expectation of the beliefs of every agent and get a lot of new parameters in your model, or you set them all equal to their equilibrium values. That might not be warranted, but it certainly leads to a model with fewer parameters to be fitted.
Michael Greinecker
CC BY-SA 3.0
2018-02-23T19:43:02.613Z
I wouldn't call the model given here "your standard rational expectation model." For reference, I've discussed some of this here: https://economics.stackexchange.com/a/20556/59
jmbejara
<macroeconomics><econometrics>
Rational expectations econometrics: a theory or an excuse?
EconJohn
# Rational expectations econometrics: a theory or an excuse?

(Asked by: username_0 on 2018-02-13)

I've been reading up on the topic of Rational expectations econometrics, and have really been wondering if it is a theory useful for prediction or just used to "fit" a model.

Your standard rational expectations model (otherwise referred to as efficent markets model):
$$y\_t=\tilde{y}+\sum\_{i=1}^n\beta\_i(X\_{t-i}-X^e\_{t-i})+\epsilon$$

* $y\_t=$ Economic variable of interest at time $t$ like unemployment or real output.
* $\tilde{y}\_t=$Natural rate or equilibrium of economic variable at time $t$
* $\beta\_i=$ coefficents.
* $X\_{t-i}=$ an aggreagte demand variable like money growth, Inflation or Nominal GDP growth.
* $X^e\_{t-i}=$ Anticipated aggregate demand conditional on knowing all information at point $t-i$
* $\epsilon\_t=$ an error term

Where the property of "rational expectations" is defined as the true expected value of $X\_t$ is equal to the markets expected value of $X\_t$. Mathematically written as:
$$\mathbb{E}(X\_{t}|\phi\_{t-1})=\mathbb{E}\_m(X\_t|\phi\_{t-1})$$
$$\mathbb{E}(X\_{t}-\mathbb{E}\_m(X\_t)|\phi\_{t-1})=0$$

If rational expectations hold, we should observe when expectations are taken:

$$y\_t=\tilde{y}\_t$$

that is $y\_t$ should always be in equilibrium/ be forecasted based on its equilibrium values/growth process (providing rationale for the use of Univariate models in economics).

**My Question:** Are the use of expectations augmented models useful for forecasting or is it just a way to explain forecast error?

**Tags:** macroeconomics, econometrics

> **Comments:**
> **username_1**: I don't understand how it helps in fitting. The rational expectation hypothesis is used to kill off the degrees of freedom one has in modeling expectations.  
> **username_0**: @MichaelGreinecker can you elaborate/ provide a source.  
> **username_1**: There is the [original paper](https://www.fep.up.pt/docentes/pcosme/S-E-1/Eco-29-3-315.pdf) by Muth. But to make my point briefly: You can add variables for the expectation of the beliefs of every agent and get a lot of new parameters in your model, or you set them all equal to their equilibrium values. That might not be warranted, but it certainly leads to a model with fewer parameters to be fitted.  
> **username_2**: I wouldn't call the model given here "your standard rational expectation model." For reference, I've discussed some of this here: https://economics.stackexchange.com/a/20556/59  

---

## 2 answers

### Answer by username_3 on 2018-02-13. Score: 1

I would argue the answer to

> Are the use of expectations augmented models useful for forecasting or is it just a way to explain forecast error?

is neither. There is nothing intrinsic to expectations that makes forecasting better unless a squared loss function makes sense under the circumstances. In that case, it isn't "useful," so much as background information. For some things such as stocks, it isn't possible to define a problem in terms of squared loss because the integrals diverge.

It also isn't a way to explain forecast error. It may be the case that in a particular problem model errors naturally converge to quadratic loss. In that case, it doesn't explain the errors, it just describes them. If a quadratic loss function was inappropriate, as would be the case of stocks which naturally follow an all-or-nothing loss function, it would make understanding the errors more difficult.

Expectations are misused in economics as it implies all problems are quadratic loss problems. All problems relevant to a particular research line may be, but that isn't mandatory and doesn't work for all distributions.

Expectations are one way to remove the stochastic component when they are defined for the density involved. This allows differentiation because stochastic functions are not smooth even though they are continuous. The use of expectations was an attempt to create a smooth curve over a random process. If the first moment is defined, which it isn't for things like stocks, then you can construct a smooth path that is a function of time.

For stocks, you have to define the parameters as a function of time and minimize over the all-or-nothing loss function over a neighborhood of $\theta\_t\pm\epsilon\_t$, where $\theta\_t$ is a parameter and $\epsilon\_t$ is small where the loss is 1 outside the neighborhood and 0 in the neighborhood. I am presenting this in a paper on generalized anticipation operators.


---

### Answer by username_4 on 2018-02-13. Score: 1

If you prefer, think of the rational expectations framework as a benchmark -- much in the same way that perfect competition is a benchmark. It's not about whether the benchmark is wrong, so much as what the benchmark allows us to conclude when it is wrong.

The canonical example for rational expectations is the Barro-Gordon model of central banking (or more generally, the Kydland-Prescott model of policy-making), which essentially couches monetary policy in the framework of an infinitely repeated game.

The idea there was to consider the question of whether it is better to have a central bank that sets monetary policy "by discretion" or by adhering to a stated rule. In this case, the idealized assumption being made is that agents are rational and possess full information about the past behavior of the central bank (as well as the true value of pertinent measures like inflation & unemployment).

In the model, the bank's objective function implies a tradeoff between unemployment and inflation via the Phillips Curve. Both inflation and unemployment are costly, but setting both to the equilibrium value is not feasible.

In a one-shot game, the bank can improve the value of its objective function by "inflation surprises" -- basically, by injecting money into the economy without announcing it, so that in the short-run there is an increase to employment but no impact on inflation. But since realized inflation is a function of inflation expectations (because if people expect their money to be worth less tomorrow, they will spend more of it today, thus leading to increased consumption demand which drives up prices and makes the expectation self-fulfilling), in the iterated game setting rational economic agents will come to expect the central bank to generate "inflation surprises". What is interesting about the Barro-Gordon model is that this leads to an "inflation bias" -- realized inflation is always higher than the optimal value when the bank's actual policy targets don't match with the expectations they set out through statements, etc.

So rational expectations predicts that central banking done according to a rule is more efficient, because it keeps policy inflation in line with expectations and avoids this bias. In the literature one hears about "credible banks" and this is essentially what this is about.

Getting back to your question, is this used to "explain" forecast error? In a sense, yes: if the model predicts a certain value of inflation under rational expectations, and that value of inflation isn't realized, then one either questions the rational expectation model itself or (more likely) that one or more of the assumptions behind it is violated. Usually, this is the assumption of complete information, although one should also consider the possibility of additional shocks not accounted for explicitly in the model.

> **Comments:**
> **username_0**: Why did you delete this?  
> **username_4**: Short answer:  I'm new and have no idea what I'm doing.  The post came out a bit longer than I wanted, and I didn't have time to edit it down.  I've undeleted it though, on the assumption that you're asking because you found it useful.  :P  

---

