TA4 最近の更新履歴 Econometrics Ⅰ 2016 TA session

TA session# 4

Jun Sakamoto

May 17,2016

Contents

1

Properties of normal distribution 1

2

Reproducing property of normal distribution 3

3

Multivariate normal distribution 4

1 Properties of normal distribution

Normal distribution is a very common probability distribution. Density function is written as below.

f(x) = √¹ 2πσ^e

−^(x−µ)2_2σ2

If a random variable forrows normal distribution, we write

X ∼ N (µ, σ²). Property 1.1

xis normal distributed with mean µ and variance σ² . y is a variable as bellow.

y= ax + b

aand b are non-stochastic. Then, y follows normal distribution as below.

Y ∼ N (µ, σ²) Proof

We show moment generating function of y.

Figure 1: density function Figure 2: distribution function

M_y(t) = E(e^ty)

=E(e^t(ax+b))

=e^btE(e^atx)

=e^btM_x(at)

=exp[(aµ + b)t +¹₂(a²σ²)t²]

Q.E.D Standard normal distribution

If Y is nomal distributed with mean 0 and variance 1, then Y is called as standard normal distribution. X follows normal distribution can be transformed into Y follows stanndard normal distribution as follows.

Y = ^X−µ_σ

Moment generating function of standard normal distribution is written as below.

e^t22 Log normal distribution

Log normal distribution is very important in financial economics. For example the stock price is assumed to follow a lognormal distribution by many cases. Also log normal distribution is used by Black Scholes equation. If logX follows normal distribution, then X follows log normal distribution.

Proposition 1.2

Density function of log normal distribution is written as bellow

Figure 3: density function Figure 4: distribution function

√ 1

2πσx^exp[−

(logx−µ)² 2σ^{2 ]}

Proof

Let Y = logX and f (x) is a density function of X.Y follows normal distribution.

f(x) =_dx^dP(X < x) = _dy^d P(e^Y < x) =_dx^dP(Y < logx) =_dx^dy_dy^dP(Y < logx) = ¹_xf(y) =

√ 1

2πσx^exp[−

(logx−µ)² 2σ^{2 ]}

Q.E.D

2 Reproducing property of normal distribution

If random variable X and Y are independent, then covariance of X and Y is 0. Thus E[XY ] = E[X]E[Y ]. Proposition 2.1

If X and Y are independent, then moment generating function of X + Y is MX(t)MY(t). Proof

MX+Y(t) = E(e^{t(X+Y )})

=E(e^tXe^tY)

=E(e^tY)E(e^tX)

=M_X(t)M_Y(t)

Q.E.D Proposition 2.2

Let X ∼ N (µX, σ_X²) and Y ∼ N (µY, σ_Y²). Also X and Y are independent. Then X + Y follows

X+ Y ∼ N (µX+ µY^{, σ}_X² + σ²_Y). Proof

By independence of X and Y ,

MX+Y(t) = MX(t)MY(t)

=exp[µ_Xt+^σ2X₂^t2]exp[µ_Yt+^σ2Y₂^t2]

=exp[(µX+ µY)t + ^(σX2+σ₂^2Y)t2]

Q.E.D Proposition 2.3

If X = (X1, X2, ..., X_n) and Xi are mutualy independent and identicaly distributed, then (X1+ X2+ ... + Xn) ∼ N (µ,^σ_n²)

Proof

X1

n ^{∼ N (} Xi

n ^, σ² n²⁾

X1

n ⁺ X2

n ^{∼ N (} 2µ

n^, 2σ²

n^{2 )} So we can get Pn

i=1^Xn^{i ∼ N ((} nµ

n ^{= µ), (} nσ²

n^{2 =} σ²

n⁾⁾

Q.E.D

3 Multivariate normal distribution

LetX = (X1, X2, ..., Xn) and Xi are standard normal distributed. A joint density function can be written by the product of density function. So,

f(x1, x2, ..., x_n) = f (x1)f (x2)...f (xn)

=^Qn_i=1 ¹

(2π)^12e

−^x22ⁱ

=^Qn_i=1 ¹

(2π)^12e

−¹2^xx′

We assume X = (X1, X2, ..., X_n)^′ are mutually independent and follows standard normal distribution, and random variable Y = (Y₁, Y₂, ..., Y_n)^′ is given by bellow.

Y_{= a + BX}

Where a is non-stochastic n × 1 vector and B is a non-stochastic regular n × n matrix. The mean and variance of Y are

E[Y] = a, V[Y] = BB^′

Define

µ≡ a, Σ ≡ BB^′

Also a density function of Multivariate normal distribution is written as below. f(X) = ¹

(2π)¹²_|Σ|¹2

exp[−¹₂(x − µ)^′Σ−1(x − µ)] Where µ is a mean vector and Σ is a variance-covariance matrix.

µ=









 µ1

µ2

... µ_n









 , Σ =

















V(x1) Cov(x1, x2) · · · Cov(x1, xn) Cov(x2, x1) V(x2) ^...

... ^{. .. ..}.

Cov(xn^{, x}1) · · · V(xn)

















Let u^′ = t^′B. Moment generating function of multivariate normal distribution function become MY(t) = E[e^t′Y] = E[e^t′(BX+µ)]

=exp[(t’µ)]E[exp[t^′BX]] = exp[(t^′µ)]E[exp[u^′X]]

=exp[(t’µ)]Πⁿ_i=1E[exp[uixi]] = exp[t^′µ]Πⁿ_i=1mxi(ui)

=exp[(t’µ)]Πⁿ_i=1exp[¹₂u²_i] = exp[t^′µ+¹₂u^′u]

=exp[t’µ + t^′BB^′t] = exp[t^′µ+ t^′Σt] Proposition 3.1

If X1, X2, ..., Xn follow multivariate normal distribution, then random variables are mutually independent if and only if the Cov[xi, xj] = 0.

Proof

If Cov[xi, xj] = 0, then Σ is diagonal matrix. So |Σ| = σ²₁, σ₂², ..., σ²_n. Thus density function is written as bellow.

Qn i=1 ¹

(2πσ²_i)^12exp[− (xi−µi)²

2σ²_i ^].

This shapes like a product of a probability density function of one-dimensional normal distribution.

Q.E.D Example

Let Y = a + BX a=^2

1



, B =^{2 3} 4 5

 .

X is multivariate standard normal distrbution vector, Then

Y1= 2X1+ 3X2+ 2, Y2= 4X1+ 5X2+ 1 and E[Y ] =^2

1



, V [Y ] =^{13 23} 23 45

 .

Figure 5: Cov[x1, x2] = 0 Figure 6: Cov[x1, x2] = 0.6

Proposition 3.2

If error term of OLS follows normal distribution, then OLSE follows normal distribution. Proof

A regression model

y= Xβ + u, u ∼ N (0, σ²I_n) The moment generating function of u is given by:

φu(tu) = E[exp(t^′_uu)] = exp(^σ₂²t^′_utu) From ˆβ = β + (X^′X)⁻¹X^′uthus the moment generating function of is:

φ_βˆ= E[exp(t^′_ˆ

β^βˆ)]

=E[exp(t’_βˆβ+ t^′_ˆ

β^(X

′_X)−1_X′_u)]

=exp(t’_βˆβ)E[exp(t^′_ˆ

β^(X

′_X)−1_X′_u)]

=exp(t’_βˆβ)φ_u[X(X^′X)⁻¹t_βˆ]

=exp(t’_βˆβ+¹₂t^′_ˆ

β^[σ

2_{(X′X)−1]t} βˆ⁾

which corresponds to the moment generating function of ˆβ. Thus βˆ∼ N (β, σ²(X^′X)⁻¹)

Q.E.D

Appendix

Correlation absence, but the case which isn’t independent.Such case often exists as bellow. Example

X and Y are lottery as bellow.

(X, Y ) = ((100, 0), (0, 100), (−100, 0), (0, −100)) A probability above-mentioned is 25% respectively. Thus

E[X] = 0, E[Y ] = 0, and E[XY ] = 0. So E[XY ] = E[X]E[Y ], Then X and Y are decorrelation.

On the other hand, joint probability and product of probability are

P(X = 100, Y = 0) = 25% 6= P (X = 100)P (Y = 100) = 12.5%.

So X and Y are non-independent. It’s being just called correlation absence that there are no straight relations. So independence is a concept stronger than correlation absence.