Maximum Likelihood Inference And Prediction For Panel Data

Panel data is also named time series and cross section data or pool data, which is the planar data gained from time and section space at the same time. Observing from cross section, it is the observation value structured by several individual units at a certain moment; while, from longitudinal section, it is a time sequence.The model that studies and analyzes panel data is called panel data model, and its variable value has the dualism of time sequence and cross section. The generic model can only deal with cross section or time sequence, and is not able to analyze and compare with each other synchronously; however, comparing with the generic model, there are two advantages of panel data model. One is that it considers the commonness of cross section data existing; the other is that it can analyze the individual special effect of cross section factor in the model. As a result, the panel data model is becoming the hotspot for the scholars to study and also the important embranchment of econometrics.Regarding to the bidirectional error stochastic effect model, we suppose the stochastic part has normal school, and use maximum likelihood principle to estimate the parameter. This is the first important part we will discuss about.The bidirectional error stochastic effect model is as below, and is the observation to individual i at time point t partly as the interpretative variable;αis the scalar quantity;β= (β₁ ,...,β_k)'is the coefficient vector; X_it = ( X_1it , X_2it ,..., X_kit)' is the arrange vector of k as the interpretative variable;μ_i is the proper effect of individual i,λ_t is the proper effect of timedimensionality, u_it is the random fluctuation.We supposeμ_i IID (0,σ_μ²),λ_t IID (0,σ_λ²),υ_it IID (0,σ_ν²) andμ_i,λ_t andν_it are all independent mutually.The model is written as matrix form,thereinto, Regarding to the parameter estimate of the bidirectional error stochastic effect model, we can use the broad sense GLS to estimate underσ_μ² andσ_ν² known, and the broad sense under unknown. Further more, ifμ_i IID (0,σ_μ²),λ_t ^IID (0,σ_λ²),υ_it^IID N(0,σ_ν²),we can use maximum likelihood estimate law.Firstly, we can get the logarithm equation, Accordinglyδ,σ_ν²,σ_μ² andσ_λ² can be gotten from the simultaneous equations below:It is known from (1)If the expression ofσ_νMLE²,σ_μMLE²,σ_λMLE² is known, then can be worked out. However, Amemiya pointed out that even if u could be observed, the equation group above was still very supernal nonlinear and hard to be worked out. So he had suggested a kind of iterative method. The result was below, I was the identity matrix and J was whole1 matrix.At the same time, we can get the congruence and approximate normal according to the estimate.Theorem 1σ_ν²,σ_μ² andσ_λ² goes through (1) and then gets . They are the coincidence estimates of maximum likelihood estimate toσν2,σ_μ²,σ_λ²andδ.Theorem 2σ_ν²,σ_μ² andσ_λ² has the approximate normal below Fuller and Battese also suggested different iterative methods. It is easy to calculate used ordinary decomposition methodλ₁ =σ_ν²,λ₂ = Tσ_μ² +σ_ν²,λ₃ = Nσ_λ² +σ_ν²,λ₄ = Tσ_μ² + Nσ_λ² +σ_ν² are partly the scale latent roots of (N-1)(T-1) scale,(N-1) scale,(T-1)scale and 1 scale tosymmetry idempotent of Q_i . They are plus to the identity matrix. The analyse is below logarithm likelihood equation can be written as below. c is the constant. If d = y-Xβ, and then u = d- l_NTα. We can supposeβ,φ₂² andφ₃² are any value, (2) only hasαandσ_ν² as the unknown parameters. We can get their estimate used maximum likelihood estimate. It is as below and u = d-l_NTα(the estimated value ofαinstead ofα).Simultaneously, so,the Simplification of (2) is On the base, if the value ofφ₂²andφ₃² is certain, Baltagi and Li got the Maximum likelihood estimate ofβ. Similarly, if the if the value of and is certain, we can also seek the estimate of used maximum likelihood estimate mathod. The process is as On the certain supposed condition, Baltagi and Li worked out the only plus root equation(5), Ifφ₃² is certain, making Q₁ = Q₁ +φ₃²Q₃, well then (4) will change to Ifφ₃² is sure,0<φ₃²<1 can go through (6) and (7) and iterative gradually betweenβandφ₂² , and then go back to (3)to get all the estimate of all parameters. On the certain supposed condition, the iterative process has good character. Proposition 2.1 supposedβ_i andβ_j replacedφ₂²⁽ⁱ⁾ andφ₂^2(j) got the result from(4)or(6) ,thereintoφ2 2 ( i)≥0,φ₂^2(j)≥0,then ifβi =βjand onlyφ2 2 ( i)=φ₂^2(j). Proposition 2.2 supposed T>1, then anyβ⁽ⁱ⁾ goes through (5) and getsφ₂²⁽ⁱ⁺¹⁾ , to satisfy 0<φ₂²⁽ⁱ⁺¹⁾<∞, And the more important one is the proposition that is called remarkable property. Proposition 2.3 if 0≤φ₂²⁽ⁱ⁾<φ₂^2(j),则0<φ₂²⁽ⁱ⁺¹⁾<φ₂^2(j+1). Number i individual's BLUP at the moment T+S is another important part in the paper. In order to seek BLUP of y_i.T+S, which is the same to seek constant vector c, to make p = c 'y satisfy toσ_p² = E (p-y_i.T+S)( p-y_i.T+S)' be the least with the limitation of E ( p-y_i.T+S) = 0. We can use Lagrange multiplier method to get the result and w=E(u_i,T+s,u).And then BLUP can be predigested to the formula below.If there is constant parameter in the original model, the formula can be further predigested as below.