

b
b(n,q) is notation for a binomial distribution with parameters n and q, where n is the number of draws and q is the probability that each is a one; the value of X~b(n,q) is a count of the number of ones drawn.

b Sequential equilibrium
Kind of refinement of Perfect Bayesian Equilibrium that puts sharper requirements on the beliefs
which cannot be formed by Bayes' rule, but which are hold after moves off the equilibrium path.
These beliefs have to be formed in a 'continuous' way from the information available in the
extensive form of the game.
Further refinements of Perfect Bayesian equilibrium restrict the players' beliefs about moves off
the equilibrium path to the set of those types only for which the observed offequilibrium move
could have been worthwhile at all.

B1
B^{1} denotes the Borel sigmaalgebra of the real line. It will contain every open interval by definition, which implies that it contains every closed interval and every countable union of open, halfopen, and closed intervals. What won't it contain? In practice, only obscure sets. Here's an example: Define the equivalence class ~ on the real line such that x~y (read: x is in the same equivalence class as y) if xy is a rational number. Now consider the set of all numbers in [0,1] such that none of them are in the same equivalence class. How many members of that set are there? Well, it's not a countable number. This set is not in B^{1}.

balance of payments
A country's balance of payments is the quantity of its own currency flowing out of of the country (for purchases, for example, but also for gifts and intrafirm transfers) minus the amount flowing in.
[Ed: this next part is partly speculation; feel free to correct it.] For some purposes this term refers to a stock value and for others a flow value. It is well defined over a period in the sense that it has changed from time A to time B.

balanced growth
A macro model exhibits balanced growth if consumption, investment, and capital grow at at a constant rate while hours of work per time period stays constant.

Banach space
Any complete normed vector space is a Banach space.

bandwidth
In kernel estimation, a scalar argument to the kernel function that determines what range of the nearby data points will be heavily weighted in making an estimate. The choice of bandwidth represents a tradeoff between bias (which is intrinsic to a kernel estimator, and which increases with bandwidth), and variance of the estimates from the data (which decreases with bandwidth). Crossvalidation is one way to choose the bandwidth as a function of the data. Has a variety of similar definitions in spectral analysis. Generally, a bandwidth is some way of defining the range of frequencies that will be included by the estimation process. In some estimations it is an argument to the estimation process.

bank note
In periods of free banking, such as most states in the U.S. from 18391863, banks could issue their own money, called bank notes. A bank note was a risky, perpetual debt claim on a bank which paid no interest, and could be redeemed on demand at the original bank, usually in gold. There was a risk that the bank would not be able or willing to redeem it.

Barriers to Entry
The obstacles to producers of entering a market. These can be manifested as:
 requiring a specialized type of license (such as having to be a certified electrician to perform electrical work);
 a governmentimposed monopoly (such as specifying a single provider of power and electricity);
 the costs of a business license;
 patents;
 having to compete with other wellentrenched firms already in the market (having to compete with economies of scale);
 other things that may prevent a new firm from entering a market.

barter economy
An economy that does not have a medium of exchange, or money, and where trade occurs instead by exchanging useful goods for useful goods.

base point pricing
The practice of firms setting prices as if their transportation costs to all locations were the same, even if all the vendors are distant from one another and have substantially different costs of transportation to each location. One might interpret this as a form of monitored collusion between the vendor firms.

Baserate fallacy
In making probabilistic inferences perceivers ought to take account of general, broadly based
information about population characteristics, and more specifically the prior probability of an
event occuring. The tendency to under use, sometimes even ignore, such information is called
the base rate fallacy.
Some authors
(Kahneman & Tversky, 1973)
explain this phenomenon with respect to the representativeness heuristic.
Gigerenzer and Hoffrage (1995)
argue that the baserate fallacy is due to the presentation of
the information in probability format and that natural sampling reduces the baserate fallacy.

basin of attraction
the region of states, in a dynamical system, around a particular stable steady state, that lead to trajectories going to the stable steady state. (E.g. the region inside the event horizon around a black hole.)

basis point
Onehundredth of a percentage point. Used in the context of interest rates.

basket
A known set of fixed quantites of known goods, needed for defining a price index.

Bayes theorem
This theorem deals with the impact of new information on the revision of probability estimates, and provides a normative model to assess how well people use empirical information to update the probability that a hypothesis is true.
P(HO) = P(H) x P(OH) / [ P(H) x P(OH) + P(nonH) x P(OnonH) ]
Bayes's theorem tells us that the probability that a hypothesis is true given that we
have made some observation (called the "posterior odds") P(HO) is a function of:
P(H) = The probability you would have assigned to the hypothesis before you made the observation, called the "prior probability" of the hypothesis.
P(OH) = The probability the observation would occur if the hypothesis were true.
P(nonH) = The prior probability the hypothesisis not true, 1P(H).
P(OnonH) = The probability the event would have occured even if the hypothesis were not true.
For example, when the baserates of women having breast cancer and having no breast
cancer are known to be 1% and 99%, respectively, and the hit rate is given as
P(positive mammography/ breast cancer) = 80 %, applying the Bayes theorem leads to
a normative prediction as low as P(breast cancer/ positive mammography) = 7.8%.
That means that the probability that a woman who has a positive mammography actually
has breast cancer is less than 8%. Studies show
(e.g. Gigerenzer & Hoffrage, 1995)
that subjective estimates clearly exceed the normative prediction and are often very
close to the hit rate (80% in the example).

Bayesian analysis
"In Bayesian analysis all quantities, including the parameters, are random variables. Thus, a model is said to be identified in probability if the posterior distribution for [the parameter to be estimated] is proper."

BayesNash equilibrium
In normal form games of incomplete information, the players have no possibility to
update their prior beliefs about their opponents
payoffrelevant characteristics, called their types. All that a player knows, except from
the game itself (and the priors), is his own type, and the fact that the other players do not know
his own type as well.
As their best responses, however, depend on the players' actual types, a player must see himself
through his opponents' eyes and plan an optimal reaction against the possible strategies of his
opponents for each potential type of his own. Thus, a strategy in a Bayesian game of
incomplete information must map each possible type of each player into a plan of actions.
Then, since the other players' types are unknown, each player forms a best response against the
expected strategy of each opponent, where he averages over the (wellspecified) reactions
of all possible types of an opponent, using his prior probability measure on the type space.
Such a profile of typedependent strategies which are unilaterally unimprovable in expectations
over the competing types' strategies forms a Bayes Nash equilibrium. Basically, a Bayes Nash
equilibrium is thus a Nash equilibrium 'at the interim stage' where each player selects a best
response against the average best responses of the competing players.

Behavioral economics
In neoclassical economic theory, it is assumed that decision makers, given their knowledge of
utilities, alternatives, and outcomes, can compute which
alternative will yield the greatest subjective (expected) utility.
The term bounded rationality is used to designate models of
rational choice that take into account the cognitive limitations of both knowledge
and cognitive capacity. Bounded rationality is a central theme in behavioral
economics. It is concerned with the ways in which the actual decisionmaking
process influences the decisions that are eventually reached. To this end, behavioral
economics departs from one or more of the neoclassical assumptions underlying the
theory of rational behavior. The two most important questions
that can be posed are:
•Are the assumptions of utility or profit maximization good approximations of real behavior?
•Do individuals maximize subjective expected utility?
Simon (1987b) provides an overview of the literature
on these issues.
Research in behavioral economics has adopted specific methodological approaches that complement
traditional statistical and econometric tests of economic models. For
example, experiments are commonly used in behavioral economics, and
survey data are also becoming more important in the process of learning about individuals'
actual decisionmaking processes.

Behavioral finance
Despite strong evidence that securities markets are highly efficient, there
have been scores of studies that have documented longterm historical phenomena
in securities markets that contradict the efficient market hypothesis and cannot
be captured plausibly in models based on perfect investor rationality. Such
phenomena are often referred to as stock market anomalies.

Belief
In incomplete
information
games, in order to predict the optimal behavior of his opponent, a player has to form
expectations and assessments of his opponent's
type.
In a simultaneous game of incomplete information, each player's belief about
any other player's type is exogenously given, or it is inferred by Bayes'
rule from an intial draw by nature that determines the various types of the
players. In sequential games of incomplete information, the players' beliefs
about their opponents' types must be updated according to Bayes' rule during
the play of the game whenever this is possible by having observed another
player's move.

Bellman equation
Any value or flow value equation. For a discrete problem it can generally be of the form: v(k) = max over k' of { u(k,k') + b*v(k') } where: u() is the oneperiod return function (e.g., a utility function) and v() is the value function and k is the current state and k' is the state to be chosen and b is a scalar real parameter, the discount rate, generally slightly less than one.

Benefit Principle
The idea that the tax burden should be proportional to an individual's use of governmentsupplied goods and services.

Bertrand competition
A bidding war in which the bidders end up at a zeroprofit price. See Bertrand game.

Bertrand duopoly
The two firms producing in a market modeled as a Bertrand game.

Bertrand game
Model of a bidding war between firms each of which can offer to sell a certain good (say, widgets), but no other firms can. Each firm may choose a price to sell widgets at, and must then supply as many as are demanded. Consumers are assumed to buy the cheaper one, or to purchase half from each if the prices are the same. Best for the firms (both collectively and individually) is to cooperate, charge monopoly price, and split the profits. Each firm could seize the whole market by lowering price slightly, however, and the noncooperative Nash equilibrium outcome of a Bertrand game is that both charge a zeroprofit price.

Between subjects design
In a between subjects design the values of the dependent variable for one subject or group of
subjects (e.g., the experimental group) are compared with the values for another subject or
another group of subjects (e.g., the control group).

Beveridge curve
The graph of the inverse relation of unemployment to job vacancies.

BHHH
A numerical optimization method from Berndt, Hall, Hall, and Hausman (1974). Used in Gauss, for example. The following discussion of BHHH was posted to the newsgroup sci.econ.research by Paul L. Schumann, Ph.D., Professor of Management at Minnesota State University, Mankato (formerly Mankato State University). It is included here without any explicit permission whatsoever.
BHHH usually refers to the procedure explained in Berndt, E., Hall, B.,
Hall, R., & Hausman, J. (1974), 'Estimation and Inference in Nonlinear
Structural Models,' Annals of Economic and Social Measurement, 3/4: 653665.
BHHH provides a method of estimating the asymptotic covariance matrix of a
Maximum Likelihood Estimator. In particular, the covariance matrix for a MLE
depends on the second derivatives of the loglikelihood function. However,
the second derivatives tend to be complicated nonlinear functions. BHHH
estimates the asymptotic covariance matrix using first derivatives instead
of analytic second derivatives. Thus, BHHH is usually easier to compute than
other methods.
In addition to the original BHHH article referenced above, BHHH is also
discussed in Greene, W.H., Econometric Analysis, 3rd Edition, PrenticeHall,
1997. Greene's econometric software program, LIMDEP, uses BHHH for some of
the estimation routines.
Someone (perhaps BHHH themselves?) wrote a Fortran subroutine in the 1970's
to do BHHH. I do not have a copy of this subroutine at the present time. You
may want to check out Green's econometric software, LIMDEP, to see if it
will do what you require, rather than writing your own program to use an
existing BHHH subroutine. The Web address for LIMDEP is:
http://www.limdep.com/index.htm
Cheers,
Paul.

Paul L. Schumann, Ph.D., Professor of Management
Minnesota State University, Mankato (formerly Mankato State University)
Mankato, MN 56002
mailto:paul.schumann@mankato.msus.edu
http://krypton.mankato.msus.edu/~schumann/www/welcome.html

BHPS
British Household Panel Survey. A British government database going back to 1990. Web page: http://www.iser.essex.ac.uk/bhps/index.php

bias
the difference between the parameter and the expected value of the estimator of the parameter.

bidding function
In an auction analysis, a bidding function (often denoted b()) is a function whose value is the bid that a particular player should make. Often it is a function of the player's value, v, of the good being auctioned. Thus the common notation b(v).

bill of exchange
From the late Middle Ages. A contract entitling an exporter to receive immediate payment in the local currency for goods that would be shipped elsewhere. Time would elapse between payment in one currency and repayment in another, so the interest rate would also be brought into the transaction.

billon
A mixture of silver and copper, from which small coins were made in medieval Europe. Larger coins were made of silver or gold.

bimetallism
A commodity money regime in which there is concurrent circulation of coins made from each of two metals and a fixed exchange rate between them. Historically the metals have almost always been gold and silver. Bimetallism was tried many times with varying success but since about 1873 the practice has been generally abandoned.

BJE
Bell Journal of Economics, the previous name of the RAND Journal of Economics or RJE.

Black Market
An illegal market. This may include markets for illegal goods and services (for example, illegal drugs or prostitution), or markets for otherwise legal goods that are sold illegally (for example, making cash payments for goods and services to avoid recordkeeping and therefore to to avoid paying taxes).
Because of the illegal nature of these markets, they are difficult to study and to obtain a precise measure of the size and extent of blackmarket activity in an economy.

BlackScholes equation
An equation for option securities prices on the basis of an assumed stochastic process for stock prices.
The BlackScholes algorithm can produce an estimate the value of a call on a stock, using as input:  an estimate of the riskfree interest rate now and in the near future  current price of the stock  exercise price of the option (strike price)  expiration date of the option  an estimate of the volatility of the stock's price Click here for a derivation of BlackScholes equation. From the BlackScholes equation one can derive the price of an option.

BLS
Abbrevation for the U.S. government's Bureau of Labor Statistics, in the Labor Department.

Bonferroni criterion
Suppose a certain treatment of a patient has no effect. If one runs a test of statistical significance on enough randomly selected subsets of the patient base, one would find some subsets in which statistically significant differences were apparently distinguished by the treatment. The Bonferroni criterion is a redefinition of the statistical signficance criterion for the testing of many subgroups: e.g. if there are five subgroups and one of them shows an effect of the treatment at the .01 significance level, the overall finding is significant at the .05 level. This is discussed in more detail (and probably more correctly) in Bland and Altman (1995) in the statistics notes of the British Medical Journal. Either of these links should go there: Llink 1. Link 2; search for Bonferroni.

bootstrapping
The activity of applying estimators to each of many subsamples of a data sample, in the hope that the distribution of the estimator applied to these subsamples is similar to the distribution of the estimator when applied to the distribution that generated the sample.
It is a method that gives a sense of the sampling variability of an estimator. "After the set of coefficients b0 is computed, M randomly drawn samples of T observations are drawn from the original data set with replacement. T may be less than or equal to n, the sample size. With each such sample the ... estimator is recomputed."  Greene, p 6589. The properties of this distribution of estimates of b0 can then be characterized, e.g. its variance. If the estimates are highly variable, the investigator knows not to think of the estimate of b0 as precise.
Bootstrapping could also be used to estimate by simulation, or empirically, the variance of an estimation procedure for which no algebraic expression for the variance exists.

Borel set
Any element of a Borel sigmaalgebra.

Borel sigmaalgebra
The Borel sigmaalgebra of a set S is the smallest sigmaalgebra of S that contains all of the open balls in S. Any element of a Borel sigmaalgebra is a Borel set.
Example: The set B^{1} is the Borel sigmaalgebra of the real line, and thus contains every open interval.
Example: Consider a filled circle in the unit square. It can be constructed by a countable number of nonoverlapping open rectangles (since a series of such rectangles can be defined that would cover every point in the circle but no point outside of it. Therefore it is in the smallest sigmaalgebra of open subsets of the unit square.

bounded rationality
Models of bounded rationality are defined in a recent book by Ariel Rubinstein as those in which some aspect of the process of choice is explicitly modeled.

Bounded rationality
Rational behavior, in economics, means that individuals maximize some target
function under the constraints they face (e.g., their utility function) in
pursuit of their selfinterest. This is reflected in the theory of
(subjective) expected utility
(Savage, 1954).
The term bounded rationality is used to designate rational choice that
takes into account the cognitive limitations of both knowledge and cognitive
capacity. Bounded rationality is a central theme in
behavioral economics. It
is concerned with the ways in which the actual decisionmaking process
influences decisions. Theories of bounded rationality relax one or more
assumptions of standard expected utility theory.

BoxCox transformation
The BoxCox transformation, below, can be applied to a regressor, a combination of regressors, and/or to the dependent variable in a regression. The objective of doing so is usually to make the residuals of the regression more homoskedastic and closer to a normal distribution: {  y(l) = ((y^l)  1) / l  for l not equal to zero  y(l)=log(y)  l=0   Box and Cox (1964) developed the transformation.
Estimation of any BoxCox parameters is by maximum likelihood.
Box and Cox (1964) offered an example in which the data had the form of survival times but the underlying biological structure was of hazard rates, and the transformation identified this.

BoxJenkins
A "methodology for identifying, estimating, and forecasting" ARMA models. (Enders, 1996, p 23). The reference in the name is to Box and Jenkins, 1976.

BoxPierce statistic
Defined on a time series sample for each natural number k by the sum of the squares of the first k sample autocorrelations. The k^{th} sample autocorrelation is denoted r: BP(k)=S_{s=1}^{k} [r_{s}^{2}] Used to tell if a time series is nonstationary. Below is Gauss code with a procedure that calculates the BoxPierce statistic for a set of residuals.
/* A series of residuals eps_hat[] is generated from a regression, e.g.: */
eps_hat = y  X*betaols;
/* Then the BoxPierce statistic for each k can be calculated this way: */
print 'BoxPierce statistic for k=1 is' BP(eps_hat,1);
print 'BoxPierce statistic for k=2 is' BP(eps_hat,2);
print 'BoxPierce statistic for k=3 is' BP(eps_hat,3);
proc BP(series, k);
local beep, rho;
beep = 0;
do until k < 1;
rho = autocor(series, k);
beep = beep + rho * rho;
k = k  1;
endo;
beep = beep * rows(series); /* BP = T* (the sum) */
retp(beep);
endp;
/* This functions calculates autocorrelation estimates for lag k */
proc autocor(series, k);
local rowz,y,x,rho;
rowz = rows(series);
y = series[k+1:rowz];
x = series[1:rowzk]; rho = inv(x'x)*x'y; /* compute autocorrelation by OLS */
retp(rho);
endp;

BPEA
An abbreviation for the Brookings Papers on Economic Activity.

bPerfect Bayesian Nash equilibrium
Parallel to the extension of Nash equilibrium to subgame perfect equilibrium in games of
complete information, the concept of Bayesian Nash equilibrium loses much of its bite in
extensive form games and is accordingly refined to 'Perfect Bayesian' equilibrium.
In a sequential game, it is often the threats about certain reactions 'off the equilibrium
path' that force the players' actions to be best responses to one another 'onto
the equilibrium path'.
In sequential games with incomplete information, where the players hold beliefs
about their opponents' types and optimize given their beliefs, a player then effectively
'threatens by the beliefs' he holds about his opponents' types after moves that deviate
from the equilibrium path. Different beliefs about other players' types after deviations
typically yield different reactions, some of which force the players back on the (candidate)
equilibrium path, some of which lead them even farer away. In the first case, the plans
of actions are confirmed by the beliefs about them, and the crucial selfconfirming
property of equilibrium beliefs and equilibrium strategies is met.
The concept of Perfect Bayesian equilibrium makes precise this selfconfirming 'interaction'
of beliefs about types selecting certain actions and their 'actual' strategies. First, it
requires that players forms a complete system of beliefs about the opponents' types
at each decision node that can be reached. Next, this system of beliefs is updated
according to Bayes' rule whenever possible (in particular, 'along the equilibrium
path'), and finally, given each player's system of beliefs, the strategies from best responses to
one another in the sense of ordinary Bayesian Nash equilibrium.
A Bayesian equilibrium thus is a profile of complete strategies and a profile of
complete beliefs such that (i) given the beliefs, the strategies are unilaterally
unimprovable at each potential decision node that might be reached, and such that
(ii) the beliefs are consistent with the actual evolution of play as prescribed by the
equilibrium strategies.

Brent method
An algorithm for choosing the step lengths when numerically calculating maximum likelihood estimates.

Bretton Woods system
The international monetary framework of fixed exchange rates after World War II. Drawn up by the U.S. and Britain in 1944. Keynes was one of the architects. The system ended on August 15, 1971, when President Richard Nixon ended trading of gold at the fixed price of $35/ounce. At that point for the first time in history, formal links between the major world currencies and real commodities were severed.

BreuschPagan statistic
A diagnostic test of a regression. It is a statistic for testing whether dependent variable y is heteroskedastic as a function of regressors X. If it is, that suggests use of GLS or SUR estimation in place of OLS. The test statistic is always nonnegative. Large values of test statistic reject the hypothesis that y is homoskedastic in X. The meaning of 'large' varies with the number of variables in X.
Quoting almost directly from the Stata manual: The Breusch and Pagan (1980) chisquared statistic  a Lagrange multiplier statistic  is given by
l = T * [S_{m=1}^{m=M} [S_{n=1}^{n=m1} [r_{mn}^{2} ]]
where r_{mn}^{2} is the estimated correlation between the residuals of the M equations and T is the number of observations. It has a chisquared distribution with M(M1)/2 degrees of freedom.

bubbles
A substantial movement in market price away from a price determined by fundamental value. In practice, "bubble" always refers to a situation where the market price is higher than the conjectured fundamentally supported price. The idea of a fundamental value requires some model or outside knowledge of what the security (or other good) is worth.
Bubbles are often described as speculative and it is conjectured that bubbles could be risky ventures for speculators who earn a fair rate of return on them. [ed: I believe these are "rational" bubbles.] There exist statistical models of a bubbles. For example, stochastic collapsing bubbles are cited to Blanchard and Watson (1982)  in this form, "the bubble continues with a certain conditional probability and collapses otherwise."

budget
A budget is a description of a financial plan. It is a list of estimates of revenues to and expenditures by an agent for a stated period of time. Normally a budget describes a period in the future not the past.

Budget Constraint
A budget constraint is the maximum amount an individual can consume, given current income and prices. For an example, suppose one's income is $100 and there are two goods in the economy; The price of Good A is $2 and the price of Good B is $4. Points on the budget constraint include:
50 units of Good A and 0 units of Good B
48 units of Good A and 1 unit of Good B
46 units of Good A and 2 units of Good B
...
2 units of Good A and 24 units of Good B
0 units of Good A and 25 units of Good B
The budget constraint can be represented graphically, in a table, or in words.

budget line
A consumer's budget line characterizes on a graph the maximum amounts of goods that the consumer can afford. In a two good case, we can think of quantities of good X on the horizontal axis and quantities of good Y on the vertical axis. The term is often used when there are many goods, and without reference to any actual graph.

budget set
The set of bundles of goods an agent can afford. This set is a function of the prices of goods and the agents endownment.
Assuming the agent cannot have a negative quantity of any good, the budget set can be characterized this way. Let e be a vector representing the quantities of the agent's endowment of each possible good, and p be a vector of prices for those goods. Let B(p,e) be the budget set. Let x be an element of R_{+}^{L}; that is, the space of nonnegative reals of dimension L, the number of possible goods. Then: B(p,e) = {x: px <= pe}

bureaucracy
A form of organization in which officeholders have defined positions and (usually) titles. Formal rules specify the duties of the officeholders. Personalistic distinctions are usually discouraged by the rules.

Burr distribution
Has density function (pdf): f(x) = ckx^{c1}(1+x^{c})^{k+1} for constants c>0, k>0, and for x>0. Has distribution function (cdf): F(x) = 1  (1+x^{c})^{k}.

business
business

business cycle frequency
Three to five years. Called the business cycle frequency by Burns and Mitchell (1946), and this became standard language.

BVAR
Bayesian VAR (Vector Autoregression)
