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Convergence in Probability

One of the handiest tools in regression is the asymptotic analysis of estimators as thenumber of observations becomes large. This is handy for the following reason. When you have

a nonlinear function of a random variable g(X), when you take an expectation E[g(X)], this is not

the same as g(E[X]). For example, for a mean centered X, E[X

2

] is the variance and this is notthe same as (E[X])2=(0)2=0. However, as we will soon show, plim[g(Xn)] is equal tog(plim[Xn]), so asymptotic properties can be passed through nonlinear operators. What is this

plim in the above?

Suppose that we have a sequence of random variables X1, X2,...,Xn,each withcumulative distribution functions F1(X), F2(X),...,Fn(X), We say that this sequence converges

in probability to a constant c if 0cobPrlim nn

!I"gp

for any small I>0. In terms of the

CDFs, this says that Fn gets very close to a staircase step function at the value c, as seen in the

figure below.

That is, the limiting random variable is very unlikely to be either below or above the value of c.

We use the notation plim[Xn]=c to denote the above 0cXobrlim nn

I

, and we say that

the probability limit of Xn is c.

If g(X) is a continuous function then values of X close to c produce values of g(X) that

are close to g(c), since that is what continuous means. If the plim[Xn]=c, then almost certainlyXn is near c, so almost certainly g(Xn) is near g(c). That is, plim[g(Xn)]=g(plim[Xn]). For

example, if g(X)=X2

and plim[ nx ]=Q, then the plim[ nx2]= (plim[ nx ])

2=Q2. It is not true, of

course, that E[ nx2

]=( E[ nx ])2

=Q2

. In fact, since var( nx )=E[( nx -Q)2

]=E[ nx2

]-Q2

, we know thatE[ nx

2]=Q2+var( nx )=Q

2+W2/n Q2.

c X

.0

F

c X

.0

Fn

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Chebychevs Inequality: 2k

1kXobr eW

Q .

Chebychevs inequality (sometimes written Chebysheff) says that only an insignificant portionof the probability falls beyond a given number of standard deviations of the mean, regardless ofthe probability distribution. For example, less than 25% of the probability can be more than 2

standard deviations of the mean; of course, for a normal distribution, we can be more specific less than 5% of the probability is more than 2 standard deviations from the mean.

roof: g

WQ

WQ

gg

gQQuQ|W

k

2k 222dx)x(f)x(dx)x(f)x(dx)x(f)x( because we have

left out the middle piece of the sum of positive numbers.

a. Look at the first integral on the right of the inequality. Notice that in this region of integration

x eQ-kW or x-Qe-kW, but x

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Application of plim in regression.

Suppose Y=XF+I and the OLS estimator is b=(XX)-1XY=F+(XX/n)-1XI/n. The plimb=F+(XX/n)

-1plim(XI/n). For the kth variable, plim(7XikIi/n)=0, since E[7XikIi/n]=0 and

var(7XikIi/n)=W2

7Xik2

/n2

. By Chebychevs inequality 2

2

ik

iikk

1

n

X

kn/Xobr e

7W

I7 . Set

7W

n

Xk

2

ikand we have

2

2ik

2

iikn

Xn/Xobr

7We

!I7 . Taking the limit with respect

to n, plim(7XikIi/n)=0. That is, the plim b =F and we say that the OLS estimator is consistent.