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Not all economic variables have sufficient information available
to estimate
This has to do with
the type of underlying process that determines
and the
relative level of the necessary information for the forecast. In
order to shed light upon this conceptual problem, hypotheses have
been developed in the macro-economic theory, which do not require
the use of mathematical expectations
. The hypothesis on adaptive
expectations assumes for instance
that the forecast at time
of
,
, is
generated by the following mechanism:
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The process (10.7) is also referred to as the
integrated autoregressive moving average process (ARIMA) of order
(0,1,1). The family of ARIMA models will be discussed in more
detail in Chapter 11. In general
exogenous factors, for example, supply shocks, could also be
involved in determining the equilibrium prices. In this case
adaptive expectations would be suboptimal. If is the
stochastic exogenous factor and
is a family of
results which are determined from the observations
available at time
, then the optimal process
is
in general a function of
and of
. Special cases do exist in which adaptive
expectations coincide with rational expectations, for example, in
a linear supply/demand system with
as an exogenous shock
that follows a random walk. If
is instead an AR(1) process,
then forecasts with adaptive expectations have a larger mean
squared error than forecasts with rational expectations. If the
factor
is common knowledge, i.e., available to the
public, then rational expectations in this example would mean that
the price would be optimally forecasted by using this information.
However, when the factor is not observable for everyone, in
principle the uninformed agent could learn from the prices
offered by the informed agent. This means that through observation
of prices they could obtain information on the status of
,
above and beyond what is in their private information set
.
Here it is assumed that the information function of prices is
correctly interpreted.
In order to illustrate what role the price plays in forming expectations, imagine purchasing a bottle of wine. In the store there are three bottles to choose with the prices EUR 300, EUR 30 and EUR 3. Since the bottle for EUR 300 exceeds the budget, only two bottles for EUR 3 and EUR 30 are considered. Now assume that someone who is not a wine expert could not evaluate the quality of the wine from instructions written on the label. Since one is pressed by time, collecting information from other people is time consuming. What remains is the information included in the price. Assume further that one has learned through previous shopping experiences that the more expensive wine tends to be better than the cheaper wine. Thus one constructs a function of the price with respect to the quality, i.e., how good the wine is. One would choose the wine for EUR 30 if the better quality and more expensive wine was valued more in the utility function than the price advantage of the cheaper wine. The buyer behaved rationally, since he optimized his decision (here maximizing his utility function) with the help of the available information and the price function, assuming that the function was right.
In addition let's take a look at another example of an experimental market which is taken from the literature. We have a security that is traded in two periods P1 and P2 and in each period it pays various dividends according to the type of investor. The trading system is an auction in which at the time of an offer both bid and ask prices are verbally given. There are three types of investors and from each type there are three investors, i.e., a total of nine investors can trade the security, among other instruments. Each investor has an initial capital of 10000 Franks (1 `Frank' = 0.002 USD) and two securities. The initial capital of 10000 Franks must be paid back at the end of the second period. Every profit which results from trading the security may be kept. When the investor is in possession of the security at the end of P1 or P2, he will receive the dividend with respect to what type of investor he is. Table 10.2 displays information on the dividend payments.
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Every investor knows only his own dividend payment, no one else. The question is, whether and if so how fast the investors `learn' about the pricing structure, i.e., gain information on the value of the security to the other investors. There are two underlying hypotheses:
In repeated experiments it was discovered that the participants actually tended from naive behavior to rational behavior, although the transition did not occur immediately after the first experiment, it was gradual and took about 8 repetitions. Other experiments were run, including a forward and futures market in which in the first period P1 the price of the security in P2 could already be determined. Here it was shown that through the immediate transparency of the security's value in future periods the transition to rational expectations equilibrium was much quicker.
The observed market price is created through the interaction of
various supplies and demands an aggregation of the individual
heterogenous information sets. Assume that the price at time
is a function of the state of the economy, the price
function
,
. We define in the
following an equilibrium with rational expectations.
Definition 10.16 assumes in particular that every agent includes the information function of the prices correctly in his objective function.
The concept of efficient markets is closely related to the
concept of rational expectations. According to the original and
general definition, a market is efficient if at every point in
time all relevant information is reflected in the price. This
means, for example, that new information is immediately
incorporated into the price. In the following we define efficient
markets with respect to an information set .
Typically three cases are identified as weak, semi-strong and strong efficiency.
An equivalent definition says that under efficient markets no abnormal returns can be achieved. In order to test it one must first determine what a `normal' return is, i.e., one must define an econometric model. Efficient markets can then be tested only with respect to this model. If this combined hypothesis is rejected, it could be that markets are inefficient or that the econometric model is inadequate.
The following is a brief summary of the typical econometric models
that have been proposed for financial data. For each of the most
interesting financial instruments, stocks, exchange rates,
interest rates and options, a corresponding theory will be
presented, which are considered to be classic theory in the
respective areas. In later chapters we will refer back to these
theories when we discuss empirically motivated expansions.