Rational Expectations

Dynamic stochastic general equilibrium modelling (abbreviated DSGE or sometimes SDGE or DGE) is a branch of applied general equilibrium theory that is influential in contemporary macroeconomics

As VARs are entirely econometric in nature, they may tell us what happens, but they cannot tell us why things happen in a particular way. The goal of the modern DSGE approach is to develop models that can explain macroeconomic dynamics as well as the VAR approach, but that are based upon the fundamental idea of optimising firms and households.

A key sense in which DSGE models differ from VARs is that while VARs just have backward-looking dynamics, DSGE models are backward-looking and forward-looking dynamics. The backward-looking dynamics stem, for instance, from identities linking today’s capital stock with last period’s capital stock and this period’s investment,

Kt = (1 − δ)Kt−1 + It .

 The forward-looking dynamics stem from optimising behaviour. What agents expect to happen tomorrow is very important for what they decide to do today. Modelling this idea requires an assumption about how people formulate expectations. This approach relies on the idea that people have so-called rational expectations.

Economic decisions have an intertemporal element to them. A key issue in macroeconomics is how people formulate expectations about the in the presence of uncertainty. Prior to the 1970s, this aspect of macro theory was largely ignored. Generally, it was assumed that agents used some simple extrapolative rule whereby the expected future value of a variable was close to some weighted average of its recent past values.

This approach was strongly criticised in the 1970s by economists such as Robert Lucas and Thomas Sargent. Lucas and Sargent instead promoted the use of an alternative approach which they called “rational expectations.”

 In economics, rational expectations usually means two things:

1.       They use publicly available information in an efficient manner. Thus, they do not make systematic mistakes when formulating expectations.

2.       They understand the structure of the model economy and base their expectations of variables on this knowledge

Rationality is a strong assumption, largely due to the complexity of the economy. Behavioural economists have now found lots of examples of deviations from rationality in people’s economic behaviour.

Rational expectations requires one to be explicit about the full limitations of people’s knowledge and exactly what kind of mistakes they make. And while rational expectations is a clear baseline, once one moves away from it there are lots of essentially ad hoc potential alternatives. In addition, rational expectations models need to be assessed on the basis of their ability to fit the data.

Take the following model

yt = xt + aEtyt+1

The equation just says that y today is determined by x and by a weighted tomorrow’s expected value of y. Rational expectations implies a very specific answer for what the expected value will be. Under rational expectations, the agents in the economy understand the equation and formulate their expectation in a way that is consistent with it.

Etyt+1 = Etxt+1 + aEtyt+2

This is known as the Law of Iterated Expectations: It is not rational for me to expect to have a different expectation next period for yt+2 than the one that I have today.

Substituting this into the previous equation, we get

yt = xt + aEtxt+1 + a 2Etyt+2

 Repeating this by substituting for Etyt+2, and then Etyt+3 and so on gives

yt = xt + aEtxt+1 + a 2Etxt+2 + .... + a N−1Etxt+N−1 + a N Etyt+N

Which can be written in more compact form as

yt = N X−1 k=0 a kEtxt+k + a N Etyt+N

Usually, it is assumed that lim N→∞ a N Etyt+N = 0, So the solution is

yt = X∞ k=0 a kEtxt+k

The model

yt = xt + aEtyt+1

can also be written as

yt = xt + ayt+1 + aet+1

where et+1 is a forecast error that cannot be predicted at date t. Moving the time subscripts back one period and re-arranging this becomes

yt = a −1 yt−1 − a −1 xt−1 − et

This backward-looking equation which can also be solved via repeated substitution to give

yt = − X∞ k=0 a −k et−k − X∞ k=1 a −k xt−k

The forward and backward solutions are both correct solutions to the first-order stochastic difference equation (as are all linear combinations of them). Which solution we choose to work with depends on the value of the parameter a.

 If |a| > 1, then the weights on future values of xt in the forward solution will explode. In this case, it is most likely that the forward solution will not converge to a finite sum. Even if it does, the idea that today’s value of yt depends more on values of xt far in the distant future than it does on today’s values is not one that we would be comfortable with. In this case, practical applications should focus on the backwards solution.

However,­ if |a| < 1 then the weights in the backwards solution are explosive and the forward solution is the one to focus on. Also, this solution is determinate. Knowing the path of xt will tell you the path of yt.

In most cases, it is assumed that |a| < 1. In this case, the assumption that

lim N→∞ a N Etyt+N = 0

 Amounts to a statement that yt can’t grow too fast. If this doesn’t hold, there may be a bubble.

Llet yt = y ∗ t + bt. The solution must satisfy

y ∗ t + bt = xt + aEty ∗ t+1 + aEtbt+1

By construction, one can show that y ∗ t = xt + aEty ∗ t+1. This means that s the additional component satisfies

bt = aEtbt+1

Because |a| < 1, this means b is always expected to get bigger in absolute value, going to infinity in expectation. For example, if a= ½, then the value of b is expected to double in the next period. This is a bubble. Note that the term bubbles is usually associated with irrational behaviour by investors. But, in this model, the agents have rational expectations. This is a rational bubble. There may be restrictions in the real economy that stop b growing forever.

The above model is a very useful way of forecasting values of yt. Without some assumptions about how xt evolves over time, it cannot be used to give precise predictions about the dynamics of yt. One reason there is a strong linkage between DSGE modelling and VARs is that this question is usually addressed by assuming that the exogenous “driving variables” such as xt are generated by backward-looking time series models like VARs.

While this example is obviously a relatively simple one, it illustrates the general principal for getting predictions from DSGE models:

1.       Obtain structural equations involving expectations of future driving variables, (in this case the Etxt+k terms).

2.       Make assumptions about the time series process for the driving variables (in this case xt)

3.       Solve for a reduced-form solution than can be simulated on the computer along with the driving variables.

Finally, note that the reduced-form of this model also has a VAR-like representation, which can be shown as follows:

yt = 1 1 − aρ (ρxt−1 + t) = ρyt−1 + 1 1 − aρ t

So both the xt and yt series have purely backward-looking representations. Even this simple model helps to explain how theoretical models tend to predict that the data can be described well using a VAR.

In a famous 1976 paper, Robert Lucas pointed out that the assumption of rational expectations implied that the coefficients in reduced-form models would change if expectations about the future changed. Lucas stressed that this could make reduced-form econometric models based on historical data useless for policy analysis. This problem is now known as the Lucas critique of econometric models.

Suppose the government is thinking about a temporary one-period income tax cut. Consider yt to be after-tax labour income, so it would be temporarily boosted by the tax cut. They ask their economic advisers for an estimate of the effect on consumption of the tax cut. The advisers run a regression of consumption on after-tax income.

 If, in the past, consumers had generally expected income growth of g, then these regressions will produce a coefficient of approximately γ /1−β(1+g) on income. So, the advisers conclude that for each €1 of income produced by the tax cut, there will be an increase in consumption of € γ/ 1−β(1+g)

However, if the households have rational expectations, then then each €1 of tax cut will produce only €γ of extra consumption. Suppose β = 0.95 and g = 0.02. In this case, the advisor concludes that each unit of tax cuts is worth extra 32γ (= γ/ 1−β(1+g) ) in consumption. In reality, the tax cut will produce only γ units of extra consumption.

Today’s DSGE models feature policy equations that describe how monetary policy is set via rules relating interest rates to inflation and unemployment; how fiscal variables depends on other macro variables; what the exchange rate regime is. These models all feature rational expectations, so changes to these policy rules will be expected to alter the reduced-form VAR-like structures associated with these economies. This is an important “selling point” for modern DSGE models. These models can explain why VARs fit the data well, but they can be considered superior tools for policy analysis. They explain how reduced-form VAR-like equations are generated by the processes underlying policy and other driving variables. However, while VAR models do not allow reduced-form correlations change over time, a fully specified DSGE model can explain such patterns as the result of structural changes in policy rules.

Variables that are characterized by yt = X∞ k=0 a kEtxt+k are jump variables. They only depends on what’s happening today and what’s expected to happen tomorrow. If expectations about the future change, they will jump. Nothing that happened in the past will restrict their movement. This may be an ok characterization of financial variables like stock prices but it’s harder to argue with it as a description of variables in the real economy like employment, consumption or investment. Many models in macroeconomics feature variables which depend on both the expected future values and their past values. They are characterized by second-order difference equations of the form yt = ayt−1 + bEtyt+1 + xt