360 The Basics of financial economeTrics
The general problem that we address is the process of gaining informa-
tion on the true population parameter such as, for example, the mean of
some portfolio returns. Since we do not actually know the true value of
θ, we merely are aware of the fact that it has to be in Θ. For example, the
normal distribution has the parameter θ = (μ, σ^2 ) where the first component,
the mean, denoted by μ, can technically be any real number between minus
and plus infinity. The second component, the variance, denoted by σ^2 , is any
positive real number.
Sample Let Y be some random variable with a probability distribution that
is characterized by parameter θ. To obtain the information about this popu-
lation parameter, we draw a sample from the population of Y. A sample
is the total of n drawings X 1 , X 2 ,... , Xn from the entire population. Note
that until the drawings from the population have been made, the Xi are still
random. The actually observed values (i.e., realizations) of the n drawings
are denoted by x 1 , x 2 ,... , xn. Whenever no ambiguity will arise, we denote
the vectors (X 1 , X 2 ,... , Xn) and (x 1 , x 2 ,... , xn) by the short hand notation
X and x, respectively.
To facilitate the reasoning behind this, let us consider the value of the
Dow Jones Industrial Average (DJIA) as some random variable. To obtain a
sample of the DJIA, we will “draw” two values. More specifically, we plan
to observe its closing value on two days in the future, say June 12, 2009,
and January 8, 2010. Prior to these two dates, say on January 2, 2009, we
are still uncertain as to value of the DJIA on June 12, 2009, and January 8,
- So, the value on each of these two future dates is random. Then, on
June 12, 2009, we observe that the DJIA’s closing value is 8,799.26, while
on January 8, 2010, it is 10,618.19. Now, after January 8, 2010, these two
values are realizations of the DJIA and not random any more.
Let us return to the theory. Once we have realizations of the sample, any
further decision will then be based solely on the sample. However, we have
to bear in mind that a sample provides only incomplete information since it
will be impractical or impossible to analyze the entire population. This pro-
cess of deriving a conclusion concerning information about a population’s
parameters from a sample is referred to as statistical inference or, simply,
inference.
Formally, we denote the set of all possible sample values for samples of
given length n (which is also called the sample size) by X.
Sampling Techniques There are two types of sampling methods: with
replacement and without replacement. Sampling with replacement is pre-
ferred because this corresponds to independent draws such that the Xi are
independent and identically distributed (i.i.d.).