Download Conditional Probability and Econometric Models

Conditional Probability and Econometric Models
Assoc. prof. Alexandru MANOLE PhD
„Artifex” University of Bucharest
Lecturer Andrei HREBENCIUC PhD
Academy of Economic Studies, Bucharest
Daniel DUMITRESCU PhD Student
Academy of Economic Studies, Bucharest
Abstract
The concept of conditional probability is fundamental because
econometrics regression models are probabilistic. Here, we consider the
conditional probabilities to a certain vector, which would be possible to
define more generally to the standard deviation σ.
Key words: conditional probability, econometrics, standard
deviation
JEL Classification: C60, C70
The concept of conditional probability is fundamental in econometrics
since the regression models are stochastic. Here, we take into consideration
conditional probabilities against a certain vector, which we might define more
generally as against the square mean deviation.
We place ourselves within the Hillbert space (L2) of integrating square
variables referring to a distribution of the probability of reference: if z is some real
variable belonging to this mass, then E( z2) < ∞. We should remember that L2 is a
standardized vectorial mass on R with a defined norm by || z || = [E(z2)]1/2. If z1
and z2 are two elements of this space, then we can write their product as E( z1 z2 );
these variables are considered as orthogonal in the sense of L2 if E( z1 z2 ) = 0.
zn within this space converges
towards an aleatory z if || zn − z ||→ 0 when n → +∞ . This notion of orthogonal
Moreover, we state that a series of numbers
and square mean convergence would farther allow us to use the notions of
orthogonal projection and the best approximations in terms of the smallest squares.
The usual requirement of rigorousness implies a careful distinction
between the equality of the aleatory variable and the doubtless equality.
Also, we can define the concept of conditional probability and enumerate
its main properties, mainly as regards he notion of the best approximation in the
spirit of the norm L2 and to appreciate the linear conditional probability as basis
out of which the linear regression derives.
Revista Română de Statistică Trim I/2013- Supliment
77
• The conditional probability
We shall insert the sample ( x1 ,..., xn ) generated depending on a series of
sampling probabilities Pθ . This interval acts as limits of the probability of
reference and we consider aleatory variables defined on this space. These variables
can be components xi of the series of studies or sub-vectors of I xi . The Hilbert
space of reference is that of the aleatory variables which depend on the sample and
are integrable as against P∞θ . Being rigorous, this interval depends on θ.
We shall mention some properties of the conditional probabilities which
we shall generalize to a vector ~
y defined in ℜ p . We shall consider the conditional
z by the vector of a dimension p, defined by the
probability of a vector ~
y given ~
relation:
All the aleatory vectors
~
y, ~
y (1) , ~
y ( 2 ) defined in ℜ p , as well as the
aleatory vectors ~
z,~
z (1) , ~
z ( 2) defined in ℜ q , have known properties (linearity,
positivity, non-equality and satisfy the rule of the three perpendiculars
The concept of conditional probability is a fundamental one because it
allow us to formalize the temporal dependence within the stochastic processes.
We may as well discuss about the conditional probability of an aleatory
matrix, which allows us to define the variation matrix – covariance of an aleatory
~
yi defined in ℜ p , conditioned on ~z , as well as in the following square matrix p
x p.
The definition of the conditional probability can be interpreted as the
orthogonality of ~
y − Eθ (~
y|~
z ) with all the functions ~z . Also, we can show a
theorem which characterizes the conditional probability in terms of orthogonal
projections, for instance in terms of the best approximation in the sense of L2
norm.
• The linear conditional probability
We have seen that the conditional probability E θ ( ~
y|~
z ) is the orthogonal
~
projection of the aleatory variable y in the sub-space of the square integrable
functions of ~
z in the sense of the L2 norm. We shall focus on the particular
z . We define the linear
situation when we consider the linear functions of ~
~
conditional probability of y by the orthogonal projection of ~
y on the sub-space
z , which we shall define through L*2( ~
z ). We know
of the linear functions of ~
2 ~
2 ~
L* ( z ) ⊂ L ( z ).
78
Revista Română de Statistică Trim. I/2013 - Supliment
The following two situations which will be submitted are showing two
simple cases in which we consider the projection of a scalar aleatory variable ~
y
z.
on a constant and on a scalar aleatory variable ~
• If ~
y is an aleatory variable, we wish to find out a constant „a” which
is the closest possible to ~
y in the sense of the L2 norm, „a” is the orthogonal
projection of ~
y defined on the sub-space of L established by the constant 1. Out
of this, it results:
and, further on,
z are aleatory variables, then the linear conditional probability
If ~
y and ~
z of the form ELθ ( ~
y|~
z ) = α~
z (where „a” is a scalar), is
of ~
y given ~
obtained by putting the orthogonality condition ELθ (( ~
y −~
z )~
z) = 0 .
•
z are both scalar aleatory variables and, in order to
When ~
y and ~
insert the linear regression coefficient, we deduce :
The linear regression coefficient of the pair ( ~
z ) is defined by the
y ,~
formula:
The coefficient ρ is always defined within the interval [—1. 1]. In
addition, we underline that | ρ | =1 if, and if only, ~
y is already a related function
z . Moreover, if ~
z are independent, then
of ~
y and ~
false).
•
p = 0 (its reciprocal is
z are two independent vectors, respectively defined on
If ~
y and ~
ℜ p and ℜ q , this means that in the case of the continuous distributions, we have the
function:
or
(in order to simplify the noting, both marginal densities are noted by
f m arg , the
systematic utilization of the arguments moves away any ambiguity). A first
consequence is that, for all the integrable square functions, h satisfies the relation:
Revista Română de Statistică Trim I/2013- Supliment
79
In this case, we do not express a different independence notion but one of
conditional independence. To mention only that ~
y (1) and ~
y () are independent on
~
z , given any two functions h1 and h2 defined on ℜ p .
References
Anghelache, C., Mitruţ, C. (coordonatori), Bugudui, E., Deatcu, C. (2009) –
„Econometrie: studii teoretice şi practice”, Editura Artifex, Bucureşti
Anghelache, C. (coord., 2012) – „Modele statistico – econometrice de analiză
economic – utilizarea modelelor în studiul economiei României”, Revista
Română de Statistică, Supliment Noiembrie 2012
Voineagu, V., Ţiţan, E. şi colectiv (2007) – “Teorie şi practică econometrică”,
Editura Meteor Press
80
Revista Română de Statistică Trim. I/2013 - Supliment