TUTORIAL IN PANEL DATA MODEL

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Available Online at www.ijpret.com 1

INTERNATIONAL JOURNAL OF PURE AND

APPLIED RESEARCH IN ENGINEERING AND

TECHNOLOGY

A PATH FOR HORIZING YOUR INNOVATIVE WORK

TUTORIAL IN PANEL DATA MODEL

AMEERA JABER MOHAISEN, SAJA YASEEN ABDULSAMAD

Mathematics Department College of Education for Pure Science AL-Basrah University-Iraq

Accepted Date: 09/04/2017; Published Date: 01/06/2017

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Abstract: - The contribution of this paper is to provide of theoretical results for panel data

model, We consider the random effect panel data model. Maximum likelihood method is employed to making inferences on the model, and we prove some properties about the parameters estimators.

Keywords: Panel Data Model, Maximum Likelihood Method, Best Linear Unbiased Estimator

(BLUE), Sufficiency, Efficiency, Likelihood Ratio Test.

Corresponding Author: MS. AMEERA JABER MOHAISEN Access Online On:

www.ijpret.com

How to Cite This Article:

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Available Online at www.ijpret.com 2 INTRODUCTION

The statistical data is important to study most of phenomena as economical, social, psychological phenomena. etc. The analysis of this data via the statistical methods gives the researcher or the decision maker more information about the studied phenomenon to make the suitable decision. The data availability needs to limit a mathematic model which represents them by the researcher and to put in consideration the type of the available data. One of the these data is panel which can be defined (that they are repeated measurements to the studied phenomena for N from the cross section and for T from the time series) which can be represented by one of the model (fixed effect model or random effect model).

One of the aims of science is to describe and predict events in the world in which we live. One way this is accomplished is by finding a formula or equation that relates quantities in the real world,[2].

In spite of the availability of highly innovative tools in statistics, the main tool of the applied statistician remains the linear model. The linear model involves the simplest and seemingly most restrictive statistical properties: independence, normality, constancy of variance, and linearity . However, the model and the statistical methods associated with it are surprisingly versatile and robust. More importantly, mastery of linear model is a prerequisite to work with advanced statistical tools because most advanced tools are generalizations of the linear model. The linear model is thus central to the training of any statistician, applied or theoretical, [6] .

In applied sciences, ones is often confronted with the collection of correlated data. This generic term embraces a multitude of data structures, such as multivariate observations, clustered data , repeated measurements , longitudinal data , and spatially correlated data .

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The problem we are interest in this paper is that of estimating the value of an (𝐾 + 1) −dimensional vector of parameters 𝜃 as well as the values of the variances parameters 𝜎𝜀2

and 𝜎₁2 for the panel data model. Two important problems in statistical inference are estimation and tests of hypotheses are to be the subject of many books for example references [4] and [5]. In the text context of this paper we are only interested in the maximum likelihood approach.

The contribution of this paper is to provide of theoretical results for panel data model, We consider the random effect panel data model . Maximum likelihood method is employed to making inferences on the model, and we prove some properties about the likelihood estimators and likelihood ratio test statistics are given here.

2. Panel Data Model

Consider the model:

Yit = μ + ∑ βjXjit

K

j=1

+ εit , i = 1, … , N, t = 1, … , T, (1)

Where, Yit the value of response variable for 𝑖𝑡ℎ unit at time t , Xjitthe explanatory variables ,

μ , β_j, j = 1, … , K are fixed parameters and ε_it is an error term with ε_it N(0, σ_ε2₎

~

iid _.

Now, if the parameter μ is specified as:

𝜇 = 𝛽0+ 𝑢𝑖 , (2)

Where, 𝑢_𝑖~𝑁(0,σ_u2_{) ,}_{then, the model (1) is}

Yit = β0+ ∑Kj=1βjxjit+ui+ εit . ( 3)

The model (3) is rewrite as follows:

Yit = β0 + ∑ βjxjit

K

j=1

+ 𝜔𝑖𝑡 , (4)

Where, 𝜔_𝑖𝑡 = u_i+ ε_it , 𝜔_𝑖𝑡~𝑁(0 , σ_ω2_{) ,}_σ ω 2₌_σ

ε 2₊_σ

u

2_{, thus by using matrix notation the model}

(4) is

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where, F = [e , X] , 𝑒 = [ 1,1, … ,1]𝑇 has length 𝑁𝑇 , Y =

[ Y11, … , Y1T, Y21, … Y2T, … , YN1, … , YNT ]T has length NT , X = [X1, X2, … , XN ]T is a 𝑁𝑇 × 𝐾

design matrix of fixed effects , 𝜃 = [β₀, β₁, … , β_K]T_{has length}_{𝐾 + 1}_{, and}

ω = [ ω11, … , ω1T, ω21, … ω2T, … , ωN1, … , ωNT ]T has length NT . From model ( 5 ) , we have

Y~ N ( F θ , Ψ ) , where Ψ = E (ωωT₎_{= I}

N  (σε2 Іt+ σu 2eeT)

= σ_ε2_(І

NІt) + σu2(ІN  eeT),

replace І_Tby ( E_T+ J_T ) and eeT_by_{T Ј}

T , where ЈT = 1

𝑇𝑒𝑒

𝑇_and_E

T = ІT− ЈT, then

Ψ = σ_ε2_{[ І}

N ( ET+ ЈT ) ] + σu2 ( ІN T ЈT )

= σε2 (ІN ET ) + σε2 (ІN ЈT ) + Tσu2 ( ІN ЈT ) ,

by collecting terms with the same matrices, we get

Ψ = σε2 (ІN ET ) + (σε2+ Tσu2) (ІN ЈT ) = σε2 Q + σ12 P, where , σ12 = (σε2+ Tσu2) and

Ψ−1= Q

σε2+

P

σ12 , |Ψ| = product of its characteristic roots, [1] → |Ψ| = (σε

2₎N(T−1)_{( σ} 1 2₎N_.

Theorem1 : Let Y is N_NT( Fθ , Ψ ) , then the likelihood estimators of parameters θ , σ_ε2 , σ_u2 are

θ̂ = ( FT_Ψ−1_{F )}−1_{( F}T_Ψ−1_{Y )}_,_σ_̂

ε

2 ₌ 1

N (T−1 ) (Y − F θ̂)

T

Q ( (Y − Fθ̂) and

σ

̂_u2 = 1

N (Y − F θ̂)

T

P (Y − Fθ̂) − 1

Tσ̂ε 2_.

Proof :

Since Y~ NNT( Fθ , Ψ ) , then , the density function of Y is

𝑓( Y; θ, Ψ ) = (2π)−NT2 |Ψ|

−1

2 exp {−1

2 ( Y − Fθ)

T_Ψ−1_{(Y − Fθ)}_{}, then , the likelihood function}

is the joint density of the 𝑌′𝑠 that is

L(Y ; θ, Ψ ) = (2𝜋)−𝑁𝑇2 |Ψ|

−1

2 exp {−1

2 ( Y − Fθ)

T _Ψ−1_{(Y − Fθ)}_},

→ L(Y; θ, σ_ε2 , σ₁2) = (2π)−NT2 ( σ_ε2 )

−N(T−1)

2 (σ₁2) −N

2 exp{ −1

2 ( Y − Fθ)

T _Ψ−1_{(Y − Fθ)}_},

∴ L(Y; θ, σ_ε2_,_σ

1

2_{) = (2π}₎−NT₂ ₍_σ ε

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(σ₁2)−N2 exp{ −1

2 (Y – Fθ )

T_[Q

σε2+

P

σ12 ] (Y– Fθ )}, (6) Then,

ln L =− NT

2 ln(2π) −

N(T−1)

2 ln( σε

2_{) –}N

2ln ( σ1

2₎_–1

2 ( Y – Fθ )

T_[Q σε2+

P

σ12 ] ( Y – Fθ ).

Since (Y- F θ )TΨ−1 ( Y – F θ) = YTΨ−1Y − 2 θTFTΨ−1 Y + θTFTΨ−1Fθ , then

∂ ln L

∂θ =

−1

2 [ −2 F

T_Ψ−1_{Y + 2 F}T_Ψ−1_{Fθ ] = 0 → F}T_Ψ−1_{Fθ = F}T_Ψ−1_Y

∴ 𝜃̂ = ( FT Ψ−1 F )−1FT Ψ−1 Y . (7)

Now, derivative the ln L with respect to σ_ε2_and_σ 1

2_{, we obtain}

∂ ln L ∂σε2 =

−N(T−1) 2σε2 +

(Y− Fθ)TQ ( Y –Fθ)

2σε4 →

−N𝜎̂ε2 (T−1)+(Y –F𝜃̂ )TQ ( Y− F𝜃̂ )

2𝜎̂ε4 = 0

→(Y – F𝜃̂)T_{Q (Y – F𝜃̂)}₌_{N(T − 1)𝜎̂}

ε2

∴ σ̂ε2= 1

N (T−1 ) (Y – Fθ̂ )

T_{Q ( Y – Fθ̂), (8)}

and, ∂ ln L

∂σ12

=_2σ−N 1

2 – (Y – Fθ)

T _{P ( Y – Fθ )}

2σ14 = 0 →N 𝜎̂1

2₌_{(Y – F𝜃̂)}T_{P ( Y – F𝜃̂) .}

∴ σ̂₁2= 1

N (Y – Fθ̂)

T_{P ( Y – Fθ̂), (9)}

→( σ̂_ε2₊_Tσ_̂ u 2 _{) =}1

N( Y – Fθ̂)

T _{P ( Y – Fθ̂)}

∴ σ̂_u2 = 1

NT (Y – Fθ̂ )

T_{P ( Y – Fθ̂)}₋1

Tσ̂ε

2_∎

Theorem2: Let Y is N_NT( Fθ , Ψ ) , then the maximum likelihood estimator of parameter θ is the best linear unbiased estimator (BLUE).

Proof:

Since θ̂ =( FT_Ψ−1_F)−1_FT_Ψ−1_Y_→_E(_θ̂_{) = E[}_{( F}T_Ψ−1_F)−1_FT_Ψ−1_{Y ]}

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𝑉𝑎𝑟(θ̂) = 𝑉𝑎𝑟 [ ( FTΨ−1F)−1FTΨ−1 Y ]

= ( FT_Ψ−1_F)−1_FT_Ψ−1_{{𝑉ar (Y)} Ψ}−1_{F ( F}T_Ψ−1_F)−1

= ( FT_Ψ−1_F)−1₍_FT_Ψ−1_{F )}_{( F}T_Ψ−1_F)−1₌_{( F}T_Ψ−1_F)−1_.

Now, le t 𝜃̂•₌_DY_{is another unbiased estimator for θ, where}

D =(FT_Ψ−1_F)−1_FT_Ψ−1_{+ G ,}_where_G_is_{(𝐾 + 1) × 𝑁𝑇}_matrix,

Since E(𝜃̂•) = θ → E(DY) = E { [ ( FTΨ−1F)−1FTΨ−1+ G ] Y }

= [ ( FT_Ψ−1_F)−1_FT_Ψ−1_{+ G ] E ( Y)}₌_{[ ( F}T_Ψ−1_F)−1_FT_Ψ−1_{+ G ] Fθ}

= ( FT_Ψ−1_F)−1_FT_Ψ−1_{Fθ + G Fθ}₌_{I . θ + 0 = θ}_{, that is}_{G F = 0}

𝑉𝑎𝑟(𝜃̂•_{) = 𝑉𝑎𝑟 (DY)}_{= D 𝑉𝑎𝑟 𝑌 D}T

= [ ( FTΨ−1F)−1FTΨ−1+ G ] Ψ[ ( FTΨ−1F)−1FTΨ−1+ G ]T

= [ ( FTΨ−1F)−1 ( FTΨ−1F) ( FTΨ−1F)−1+ G Ψ GT = ( FTΨ−1F)−1+ G Ψ GT

= 𝑉𝑎𝑟( θ̂) + G Ψ GT , thus 𝑉𝑎𝑟 (θ̂) < 𝑉𝑎𝑟(𝜃̂•) ∎

Theorem3: suppose that Y is NNT( Fθ , Ψ ) ,where F is 𝑁𝑇 × (𝐾 + 1)of rank(𝐾 + 1) < 𝑁𝑇 and

𝜃 = [β₀, β₁, … , β_K]T_{.Then the maximum likelihood estimator of θ is an efficient statistic for θ .}

Proof:

Since Y~ N_NT( Fθ , Ψ ), then , the density function of Y is

𝑓( Y; θ, Ψ ) = (2π)−NT2 |Ψ|

−1

2 exp {−1

2 ( Y − Fθ)

T_Ψ−1_{(Y − Fθ)}_{}, and the likelihood function is}

L(Y ; θ, Ψ ) =(2π)−NT2 |Ψ|

−1

2 exp {−1

2 ( Y − Fθ)

T_Ψ−1_{(Y − Fθ)}_},

then,

ln L =− NT

2 ln (2π) −

N(T−1)

2 ln( σε

2_{) –}N

2ln ( σ1

2₎_–1

2 ( Y – Fθ )

T_[Q σε2+

P

σ12 ] ( Y – Fθ ),

→ ∂ ln L

∂θ =

−1 2 [-2F

T_Ψ−1_{Y + ( F}T_Ψ−1_{F ) θ] →}∂2ln L

∂θ2 = − ( F

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Then , the Rao – Cramer lower bounded is

C. R. L = −E [ ∂2ln L ∂θ2 ]

−1

= ( FT Ψ−1F )−1= 𝑉𝑎𝑟( 𝜃̂ ) ∎

Theorem4: Let Y is NNT( Fθ , Ψ ) , then the maximum likelihood estimators of parameters θ

, σ_ε2_,_and_σ 1

2_{are jointly sufficient for}_θ_and_σ ε

2_{, and σ}

1 2_.

Proof:

Since Y~ N_NT( Fθ , Ψ ), then , the density function of Y is

𝑓( Y; θ, Ψ ) = (2π)

−NT

2 |Ψ|

−1

2 exp {−1

2 ( Y − Fθ)

T _Ψ−1_{(Y − Fθ)}_{} , we add and subtract}_Fθ̂_to

obtain

𝑓( Y; θ, Ψ ) = (2π)−NT2 |Ψ|

−1

2 exp [- 1

2[(Y − Fθ̂ + Fθ̂ − Fθ)

T _Ѱ−1_{(Y − Fθ̂ + Fθ̂ − Fθ)]}_,

= (2π)−NT2 |Ψ| −1

2 𝑒𝑥𝑝{−1

2 [(Y − Fθ̂) + F(θ̂ − θ)]

T _Ѱ−1_{[(Y − Fθ̂) + (θ̂ − θ)]}.}

→ 𝑓(Y; θ, Ψ) =(2π)−NT2 |Ψ|

−1

2𝑒𝑥𝑝[−1

2 {(Y − Fθ̂)

T_Ѱ−1_{(Y − F𝜃̂) + (θ̂ − θ)}T_FT_Ѱ−1_F_.

(θ̂ − θ)}].

𝑓( Y; θ, Ψ) = (2π)−NT2 |Ψ|

−1

2 exp[−1

2{(Y − Fθ̂)

T_[𝑄 σε2 +

𝑃

σ12 ](Y − F𝜃̂) + (θ̂ − θ)

T_FT_Ψ−1_F

(θ̂ − θ)}].

𝑓( Y; θ, Ψ) = (2π)−NT2 |Ψ|

−1

2 exp [−1

2{(Y − Fθ̂)

T 𝑄

σε2 (Y − F𝜃̂) + (Y − Fθ̂)

T 𝑃

σ12 (Y − F𝜃̂) +

(θ̂ − θ)T_FT_Ψ−1_{F (θ̂ − θ)}]}

𝑓( Y; θ, Ψ) = (2π)−NT2 |Ψ|

−1

2 exp [−1

2{(

𝑁(𝑇−1)𝜎̂𝜀2

σε2 +

𝑁𝜎̂12

σ12 ) + (θ̂ − θ)

T_FT_Ψ−1_{F (θ̂ − θ)}]}_.

We can now write the density as:

𝑓( Y ; θ, Ψ ) = g (θ̂, σ̂_ε2_{, σ}_̂

12, θ , σε2 , σ12) h (Y), where h (Y) =1, therefore by the Neyman

factorization theorem θ̂and 𝜎̂_𝜀2_{, 𝜎̂}

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Available Online at www.ijpret.com 8 Theorem5: Let Y is N_NT( Fθ , Ψ ) , the likelihood ratio test for H₀: θ = 0 is

( YTY)

−NT₂

[ (Y – Fθ̂ )T_{( Y – Fθ}_{̂) ]}−NT₂

~ F ( NT , NT − K − 1 )

Proof:

Since the likelihood estimators for θ,σ_ε2_{and σ}

12 are

θ̂ = ( FT Ψ−1 F )−1FT Ψ−1 Y , 𝜎̂𝜀2= 1

N (T−1 )(Y – Fθ̂)

T_{Q ( Y – Fθ̂) and}

𝜎̂₁2=_N1 (Y – F𝜃̂)T_{P ( Y – F𝜃̂),}_{and, the maximum likelihood function is}

L = (2π)−NT2 (σ_ε2 )

−N(T−1) 2 (σ₁2 )

−N

2 exp{ −1

2 (Y − Fθ)

T₍Q σε2+

P

σ₁2 ) (Y − Fθ) } , then, the likelihood

function under 𝐻1, (maxH1L ( θ , σε

2_{, σ} 12 ) ) is

𝐿₁ = (2π)−NT2 [ 1

N (T−1 )(Y – Fθ̂)

T_{Q ( Y – Fθ̂) ]}−N(T−1)₂ _[1

N (Y – Fθ̂)

T_{P ( Y – Fθ̂) ]}−N₂

exp {−1 2 [

(Y –Fθ̂)TQ ( Y –Fθ̂)

1

N (T−1 ) (Y –Fθ̂)TQ ( Y –Fθ̂)

+ 1(Y –Fθ̂)TP ( Y –Fθ̂) N (Y –Fθ̂)T P ( Y –Fθ̂)

]}

𝐿₁ = (2π)−NT2 [ 1

N (T−1 ) (Y – Fθ̂)

T_{Q ( Y – Fθ̂) ]}−N(T−1)₂ _[ 1

N (Y – Fθ̂)

T_{P ( Y – Fθ̂) ]}−N₂ _e−NT

2.

And the likelihood function under 𝐻0, maxH0L ( θ , σε

2_{, σ} 1

2_{) = Max L( 0 , σ}

ε 2_{, σ}

12 )

→ 𝐿₀( 0 , σ_ε2_{, σ}

1

2_{) = (2π)}−NT₂ _(σ

ε

2₎−N(T−1)₂ _(σ 12 )

−N

2 exp [ −1

2 Y

T₍Q

σ_ε2+

P

σ₁2 ) Y ] →

ln𝐿0( 0 , σε2 , σ12 ) = −

NT

2 ln(2π) −

N(T − 1)

2 ln σε

2 ₋N

2ln σ1

2₋1

2 Y

T₍Q

σε2

+ P

σ₁2 ) Y,

→∂ ln L

∂σε2 =

−N(T−1) 2σε2 +

YTQ Y

2σε4 = 0 →

−N (T−1)σε2+YTQ Y

2σε4 = 0 → Y

T_{Q Y}₌_{N(T − 1)𝜎̂}

ε2

∴ 𝜎̂_𝜀2₌ 1

N (T−1 ) Y

T_{Q Y}_,

and ∂ ln L

∂σ12

=_2σ−N 1

2 + Y_2σTP Y 1

4 = 0 →

−N σ12+ YTP Y

2σ₁4 = 0 →N 𝜎̂1

2₌_YT_{P Y → 𝜎̂}

12= 1

NY

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Then, the likelihood function under 𝐻₀ is

max_H₀ L ( θ , σ_ε2 , σ₁2 ) = Max L( 0 , σ_ε2 , σ₁2 )

= (2π)−NT2 [ 1

N (T−1 )Y

T_{Q Y ]}−N(T−1)₂ _[1

NY

T_{P Y]}

−N 2

= (2π)−NT2 [ 1

N (T−1 )Y

T_{Q Y ]}−N(T−1)₂ _[1

NY

T_{P Y]}

−N 2

e−NT2.

Then , the likelihood ratio is

λ =MaxH0

MaxH1 =

(2π)

−NT

2 _[ 1

N (T−1 ) Y

T_{Q Y ]}−N(T−1)₂ _[1 NY

T_{P Y]} −N

2 _e−NT₂

(2π)

−NT

2 [_{N (T−1 )}1 (Y –F θ̂)T_{Q ( Y –F 𝜃}_{̂) ]}−N(T−1)₂ _[1

N (Y – Fθ̂)TP ( Y – Fθ̂) ] −N

2 e−

NT 2

= ( YTY) −NT₂

[ (Y – Fθ̂)T_{( Y – Fθ}_{̂) ]}−NT2

=

𝜔_{̂ 0}𝑇 𝜔̂0 NT 𝜔_{̂ 1}𝑇 𝜔̂ 1 NT−k−1

~ F ( NT , NT − K − 1 ). (10)

Where, 𝜔̂₁𝑇 𝜔̂₁ = (Y – Fθ̂ )T ( Y – Fθ̂) is 𝜒2(NT − K − 1) and 𝜔̂₀𝑇 𝜔̂₀ = ( YT Y ) is

𝜒2_{(NT). ∎}

Theorem6: Let Y is NNT( Fθ , Ψ ) , the likelihood ratio test for H0 : Aθ = 0, where A is 𝑞 ×

(𝐾 + 1) matrix of constants is

[

𝜔̂0𝑇𝜔̂0+𝜃̂𝑇𝐴𝑇 [𝐴 (𝐹𝑇𝐹) −1

𝐴𝑇_]−1_𝐴𝜃_̂

𝑞 𝜔̂₁𝑇𝜔̂1

𝑁𝑇−𝐾−1 _]

−𝑁𝑇

2

~𝐹(𝑞, 𝑁𝑇 − 𝐾 − 1)

Proof:

The likelihood function is

L = (2π)

−NT 2 (σ_ε2 )

−N(T−1) 2 (σ₁2 )

−N

2 exp{ −1

2 (Y − Fθ)

T₍Q σε2+

P

σ12 ) (Y − Fθ) }, then for test

H₀: Aθ = 0. Let L•_{= ln L + λ}T_{A θ}

= −NT

2 ln(2π) −

N(T−1)

2 ln(σε

2_{) −}N

2 ln (σ1

2_{) −}1

2 (Y − Fθ)

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(Q σε2+

P

σ12 ) (Y − Fθ) + λ

T_{A θ}

→∂L•

∂θ = 2 F

T₍Q

σε2+

P

σ12 ) Y − 2 F

T₍Q

σε2+

P

σ12 ) Fθ + A

T_{λ = 0 , (11)}

→∂L∗

∂λ = A θ = 0 , (12)

→∂ ln 𝐿• ∂σε2 =

−N(T−1) 2σε2 +

(Y− Fθ̂)TQ ( Y –Fθ̂)

2σε4 = 0 , (13)

→∂ ln 𝐿• ∂σ12 =

−N 2σ12 –

(Y – Fθ̂)T P ( Y – Fθ̂)

2σ14 = 0 . (14)

Multiply ( 11 ) by A [ FT(Q σε2+

P σ12 ) F ]

−1_{to obtain}

2A [ FT(Q

σε2

+ P

σ₁2 ) F ]−1 FT(

Q σε2

+ P

σ₁2 ) Y − 2A [ FT(

Q σε2

+ P

σ₁2 ) F ]−1 [ FT(

Q σε2

+ P

σ₁2 ) F ]θ

+ A [ FT(Q σε2+

P σ12 ) F ]

−1_AT_{λ = 0}

→ 2A 𝜃̂ − 2Aθ + A [ FT₍Q

σε2+

P σ₁2 ) F ]

−1_AT_{λ = 0,}_since_{Aθ = 0}

→ A [ FT(Q

σε2+

P σ12 ) F ]

−1_AT_{λ = 2A𝜃̂}_{→ λ = −2 ( A [ F}T₍Q

σε2+

P σ12 ) F ]

−1_AT₎−1_A𝜃̂.

From (13) and (14) ,we get

𝜎̂_𝜀2₌ 1

N (T−1 )(Y – Fθ̂)

T_{Q ( Y – Fθ̂) , (15)}

𝜎̂₁2= 1

N(Y – Fθ̂)

T_{P( Y – Fθ̂) . (16)}

Then ,

λ = −2 ( A [ FT₍ Q

1

N (T−1 ) (Y –F𝜃̂)TQ ( Y –F𝜃̂)

+ 1 P

N (Y –F𝜃̂ )T P ( Y – F𝜃̂)

) F ]−1_AT₎

−1 A𝜃̂

→ λ = −2 (𝐴 [𝑁(𝑇−1)𝐹𝑇𝐹+𝑁𝐹𝑇𝐹

(𝑌−𝐹𝜃̂)𝑇(𝑌−𝐹𝜃̂) ] −1

𝐴𝑇) 𝐴𝜃̂ → λ = −2 (𝐴 [ 𝑁𝑇 𝐹𝑇𝐹

(𝑌−𝐹𝜃̂)𝑇(𝑌−𝐹𝜃̂) ] −1

(11)

Now, substitute λ in (11) to obtain

2 FT( 1 Q

N (T−1 ) (Y –F𝜃̂)TQ ( Y –F𝜃̂)

+ 1 P

N (Y –F𝜃̂)TP ( Y –F𝜃̂)

) Y −2 FT( 1 Q

N (T−1 ) (Y –F𝜃̂)TQ ( Y –F𝜃̂)

+

P

1

N (T−1 ) (Y –F𝜃̂)

T_{Q ( Y –F𝜃}_̂) ) Fθ − 2 ( A [

NT FTF (Y –F𝜃̂)T_{( Y –F𝜃}_̂)]

−1

AT )

−1

A𝜃̂ = 0

→ 2NT

(Y – F𝜃̂)T_{( Y – F𝜃̂)} [ F

T_{Y − F}T_{Fθ − A}T_{[ A (F}T_F)−1_AT_]−1_{A𝜃̂ = 0}

→ FTY − FTFθ − AT[ A (FT_F)−1_AT_]−1_{A𝜃̂ = 0}

𝜃̂₀ = (FT_F)−1_{( F}T_{Y − A}T_{[ A (F}T_F)−1_AT_]−1_{A𝜃̂). (17)}

Then ,

Y − F𝜃̂₀ = Y − F𝜃̂ − F (FTF )−1 AT[ A (FT_{F )}−1_AT_]−1_A𝜃̂,

u₀ = 𝜔̂ − F(FTF)−1AT[ A (FT_{F )}−1_AT_]−1_A𝜃̂_,

Where u₀ is estimated residual from the restricted model

u₀Tu0 = { 𝜔̂T− 𝜃̂𝑇AT[A (FTF)−1AT]−1A(FTF)−1FT } {𝜔̂ − F(FTF)−1𝐴𝑇[A (FTF)−1AT]−1A𝜃̂}

→ u₀Tu₀ = 𝜔̂T_𝜔_{̂ − 𝜔}_̂T_{F A(F}T_F)−1_{[A (F}T_F)−1_AT_]−1_{A𝜃̂ − 𝜃̂}𝑇_AT_{[A (F}T_F)−1_AT_]−1

A(FTF)−1FT𝜔̂ + AT𝜃̂𝑇 [A (FTF)−1 AT]−1_A(FT_F)−1_FT_{F A (F}T_F)−1

[A (FT_F)−1_AT_]−1_{A𝜃̂ → u} 0 T_u

0 = 𝜔̂T𝜔̂ + AT 𝜃̂𝑇[A (FTF)−1 AT]−1A𝜃̂.

We can note that 𝜔̂TF = (𝑌 − 𝐹𝜃̂)𝑇𝐹 = (𝑌𝑇− 𝜃̂𝑇𝐹𝑇)𝐹

= 𝑌𝑇𝐹 − (𝑌𝑇𝛹−1𝐹(𝐹𝑇𝛹−1𝐹)−1𝐹𝑇)𝐹

= 𝑌𝑇_{𝐹 − 𝑌}𝑇_{𝐹 = 0}_,

(12)

→ u₀Tu0 − 𝜔̂T𝜔̂ = AT𝜃̂𝑇 [A (FTF)−1 AT]−1A𝜃̂

→

u0Tu0 −𝜔̂𝑇𝜔̂ q 𝜔̂ 𝑇𝜔̂ NT−K−1

~ F( q , NT − K − 1).

The likelihood function under 𝐻₀ is

𝑀𝑎𝑥_𝐻₀𝐿 ( 𝜃 , 𝜎_𝜀2 , 𝜎₁2 ) = 𝐿 ( 𝜃̂₀ , 𝜎̂_𝜀(0)2 , 𝜎̂_𝜀(0)2 )

=(2π)

−NT

2 [ 1

N (T−1 ) (Y – F𝜃̂0)

T_{Q ( Y – F𝜃̂}

0) ]

−N(T−1)

2 [1

N (Y – F𝜃̂0)

T_{P ( Y – F𝜃̂}

0) ]

−N 2 e−

NT 2,

and the likelihood function under 𝐻₁ is

𝑀𝑎𝑥𝐻1 𝐿 ( 𝜃 , 𝜎𝜀

2_{, 𝜎}

12 ) = 𝐿 (𝜃̂ , 𝜎̂𝜀2, 𝜎̂12)

= (2π)

−NT

2 [ 1

N (T−1 )(Y – F𝜃̂)

T_{Q ( Y – F𝜃̂) ]}−N(T−1)₂ _[1

N (Y – F𝜃̂ )

T_{P ( Y – F𝜃̂) ]}−N₂ _e−NT₂

.

Thus, the likelihood ratio is

𝜆 =𝑀𝑎𝑥𝐻0

𝑀𝑎𝑥_𝐻1 =

(2𝜋)

−𝑁𝑇

2 _[ 1

𝑁 (𝑇−1 ) (𝑌 –𝐹 𝜃̂0)

𝑇_{𝑄 ( 𝑌 –𝐹𝜃}_̂ 0) ]

−𝑁(𝑇−1)

2 _[1

𝑁 (𝑌 – 𝐹𝜃̂0)

𝑇_{𝑃 ( 𝑌 –𝐹𝜃}_̂ 0) ]

−𝑁 2_𝑒−

𝑁𝑇 2

(2𝜋)

−𝑁𝑇

2 _[ 1

𝑁 (𝑇−1 ) (𝑌 –𝐹 𝜃ˆ)𝑇𝑄 ( 𝑌 –𝐹 𝜃ˆ) ] −𝑁(𝑇−1)

2 _[1

𝑁 (𝑌 – 𝐹𝜃ˆ )𝑇𝑃 ( 𝑌 – 𝐹𝜃ˆ) ] −𝑁

2 _𝑒− 𝑁𝑇

2

= [ (𝑌−𝐹𝜃̂0)𝑇( 𝑌 –𝐹𝜃̂0)]− 𝑁𝑇

2

[ (𝑌 –𝐹𝜃̂)𝑇_{( 𝑌 – 𝐹𝜃}_̂)]−𝑁𝑇2

= [

𝜔̂ 𝑇𝜔̂ +𝜃̂𝑇𝐴𝑇 [𝐴 (𝐹𝑇𝐹)−1 𝐴𝑇] −1 𝐴𝜃̂ 𝑞 𝜔̂ 𝑇𝜔̂ 𝑁𝑇−𝐾−1 ] −𝑁𝑇₂

∎

3. CONCLUSION

The conclusions which are obtained throughout this paper are given as follows:

1. The maximum likelihood estimators of parameters θ , σε2 , σu2 of panel data model are

θ̂ = ( FT Ψ−1 F )−1 ( F Ψ−1 Y ) , σ̂_ε2 = 1

N (T−1 ) (Y − F θ̂)

T

Q ( (Y − Fθ̂) and

σ ̂u2 =

1

N (Y − F θ̂)

T

P (Y − Fθ̂) − 1

(13)

2. The maximum likelihood estimator of parameter θ of panel data model is the best linear unbiased estimator (BLUE).

3. The maximum likelihood estimator of parameter θ of panel data model is an efficient statistic for θ.

4. The maximum likelihood estimators of parameters θ, σ_ε2 , and σ₁2 of panel data model are jointly sufficient for θ and σε2 , and σ12.

5. The likelihood ratio test for H0 : θ = 0 in the panel data model is

( YTY)−

NT 2

[ (Y – Fθ̂ )T_{( Y – Fθ}_{̂) ]}−NT2

~ F ( NT , NT − K − 1 ).

6. The likelihood ratio test for H₀: Aθ = 0 ,where A is 𝑞 × (𝐾 + 1) matrix of constants in the panel data model is

[

𝜔̂̂ 𝑇𝜔̂̂ +𝜃̂𝑇𝐴𝑇 [𝐴 (𝐹𝑇𝐹)−1 𝐴𝑇] −1

𝐴𝜃̂ 𝑞

𝜔̂̂ 𝑇𝜔̂̂ 𝑁𝑇−𝐾−1

] −𝑁𝑇₂

~𝐹(𝑞, 𝑁𝑇 − 𝐾 − 1).

4. REFERENCES

1. Baltagi, badi,” Econometric Analysis of panel data “, John Wily & Sons Inc. third edition,(2005).

2. Graybill, F.A. “Theory and Application of the Linear Model “North Scituate, MA; Duxbury Press,(1976).

3. Husio, Cheng, “Analysis of panel data “, Second edition, Cambridge university Press, (2003). 4. Hogg, Robert V., McKean, Joseph W.& Craig, Allen T., “ Introduction to Mathematical Statistics “, seven edition ,Pearson Education , Inc., (2013).

5. Mood, Alexander M., Graybill, Franklin A. & Boes , Duane C., “ Introduction to the theory of statistics “, third edition. McGraw- Hill, Inc., (1974).

6. Rencher , Alvin C. and Schaalje, G. Bruce, “ Linear Models in Statistics “ ,second edition, Wiley-Interscience, A John Wiley & Sons ,Inc., Publication,(2008).

7. Searal ,S.R.,” Linear Model .New York: Wiley, (1971).

8. Seber , G .A .F. and A.J. Lee, “ Linear Regression Analysis ( 2nd_{ed.). Hoboken, NJ:} Wiley,(2003).

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