Panel conditional and multinomial logit with time-varying parameters

Likelihood function for static PCLE

The joint probability for (Y_i=Λ)∣(δ_i, X_i) is

(A.1) P ( Y i = Λ | δ i ,   X i ) = ∏ t P ( y i t = λ t | δ i ,   X i ) = ∏ t P ( y i t = λ t | δ i ,   x i t ) = ∏ t { 1 1 + e x p ( x ′ i t β t + δ i ) } 1 − λ t { e x p ( x ′ i t β t + δ i ) 1 + e x p ( x ′ i t β t + δ i ) } λ t = ∏ t e x p { λ t ( x ′ i t β t + δ i ) } 1 + e x p ( x ′ i t β t + δ i )   = e x p ( δ i ∑ t λ t ) ⋅ e x p ( ∑ t λ t x ′ i t β t ) ∏ t { 1 + e x p ( x ′ i t β t + δ i ) } .  (A.1)

Write P(Y=Λ∣δ, X) just as P(Λ∣δ, X) and replace Λ with Y to get the joint likelihood:

(A.2) P ( Y | δ ,   X ) = e x p ( δ ∑ t y t ) ⋅ e x p ( ∑ t y t x ′ t β t ) ∏ t { 1 + e x p ( x ′ t β t + δ ) } .  (A.2)

The probability of the sum of the random responses taking the sample value Σ_ty_t is

(A.3) P ( ∑ t y t | δ ,   X ) = ∑ λ ¯ = y ¯ P ( λ 1 ,..., λ T | δ ,   X ) .  (A.3)

Substituting (A.1) into (A.3) and then using Σ_tλ_t=Σ_ty_t, (A.3) becomes

(A.4) P ( ∑ t y t | δ ,   X ) = ∑ λ ¯ = y ¯ e x p ( δ ∑ t λ t ) e x p ( ∑ t λ t x ′ t β t ) ∏ t { 1 + e x p ( x ′ t β t + δ ) } = e x p ( δ ∑ t y t ) ∑ λ ¯ = y ¯ e x p ( ∑ t λ t x ′ t β t ) ∏ t { 1 + e x p ( x ′ t β t + δ ) } .  (A.4)

Divide P(Y∣δ, X) in (A.2) by P(Σ_ty_t∣δ, X) in (A.4) to obtain P(Y∣Σ_ty_t, δ, X)=P(Y/Σ_ty_tX) in (2.2). The division removes two common terms, exp(δΣ_ty_t) and ∏{1+exp(x′tβt+δ)}, and as the ratio is free of δ, Σ_ty_t is a sufficient statistic for δ given X.

Recursive algorithm for static PCLE

We show first how to obtain the “pre-normalization” denominator ∑λ¯=y¯iexp(∑tλtx′itβt), and then how to normalize this; T can be allowed to vary across i (i.e., T_i) for unbalanced panels or cross-section data with a group-structure. The key point of the algorithm is that

G ( T ,   ∑ t y i t ) ≡ ∑ λ ¯ = y ¯ i e x p ( ∑ t λ t x ′ i t β t )

satisfies the recursive formula

G ( T ,   ∑ t y i t ) = G ( T − 1,   ∑ t y i t ) + G ( T − 1,   ∑ t y i t − 1 ) ⋅ e x p ( x ′ i T β T )

where  G ( T ,   ∑ t y i t ) ≡ 0  when  T < ∑ t y i t  or  ∑ t y i t < 0   and   G ( T ,   0 ) ≡ 1.

Using this formula, G(T, Σ_ty_it) can be computed for each i and for each value of β. As was stated in the main text, this algorithm is a modified version of Krailo and Pike (1984).

The logic for the formula is simple. Suppose there are two ones in four periods and consider allocating the two ones over four periods. Then the formula becomes

G ( 4,   2 ) = G ( 3,   2 ) + G ( 3,   1 ) ⋅ e x p ( x ′ i 4 β 4 ) :

‘two ones over four’ = ‘two ones over the first three, and zero in the last’ + ‘one over the first three, and one in the last’ ;

e x p ( x ′ i 4 β 4 ) is for “one in the last.” The following special cases with T=2,3 will be helpful.

To see that the recursive formula holds, suppose T=2. From the definition of G(·, ··),

G ( T − 1,   0 ) = G ( 1,   0 ) = 1  and  G ( T − 1,   1 ) = G ( 1,   1 ) = e x p ( x ′ i 1 β 1 ) ; G ( T ,   0 ) = G ( 2,   0 ) = 1 , G ( T ,   1 ) = G ( 2,   1 ) = e x p ( x ′ i 1 β 1 ) + e x p ( x ′ i 2 β 2 ) , G ( T ,   2 ) = G ( 2,   2 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 ) .

The recursive formula holds because (the left-hand side from the last display)

G ( 2,   0 ) = G ( 1,   0 ) + G ( 1,   − 1 ) e x p ( x ′ i 2 β 2 ) = 1 + 0 = 1 ; G ( 2,   1 ) = G ( 1,   1 ) + G ( 1,   0 ) e x p ( x ′ i 2 β 2 ) = e x p ( x ′ i 1 β 1 ) + e x p ( x ′ i 2 β 2 ) ; G ( 2,   2 ) = G ( 1,   2 ) + G ( 1,   1 ) e x p ( x ′ i 2 β 2 ) = 0 + e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 ) .

Now suppose T=3. Then, from the definition of G,

G ( T − 1,   0 ) = G ( 2,   0 ) = 1,   G ( T − 1,   1 ) = G ( 2,   1 ) = e x p ( x ′ i 1 β 1 ) + e x p ( x ′ i 2 β 2 ) , G ( T − 1,   2 ) = G ( 2,   2 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 ) , G ( T ,   0 ) = G ( 3,   0 ) = 1, G ( T ,   1 ) = G ( 3,   1 ) = e x p ( x ′ i 1 β 1 ) + e x p ( x ′ i 2 β 2 ) + e x p ( x ′ i 3 β 3 ) , G ( T ,   2 ) = G ( 3,   2 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 ) + e x p ( x ′ i 1 β 1 + x ′ i 3 β 3 ) + e x p ( x ′ i 2 β 2 + x ′ i 3 β 3 ) , G ( T ,   3 ) = G ( 3,   3 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 + x ′ i 3 β 3 ) .

The recursive formula holds because (the left-hand side from the last display)

G ( 3,   0 ) = G ( 2,   0 ) + G ( 2,   − 1 ) = 1 + 0 = 1 ; G ( 3,   1 ) = G ( 2,   1 ) + G ( 2,   0 ) e x p ( x ′ i 3 β 3 ) = e x p ( x ′ i 1 β 1 ) + e x p ( x ′ i 2 β 2 ) + e x p ( x ′ i 3 β 3 ) ; G ( 3,   2 ) = G ( 2,   2 ) + G ( 2,   1 ) e x p ( x ′ i 3 β 3 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 ) + { e x p ( x ′ i 1 β 1 ) + e x p ( x ′ i 2 β 2 ) } e x p ( x ′ i 3 β 3 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 ) + e x p ( x ′ i 1 β 1 + x ′ i 3 β 3 ) + e x p ( x ′ i 2 β 2 + x ′ i 3 β 3 ) ; G ( 3,   3 ) = G ( 2,   3 ) + G ( 2,   2 ) e x p ( x ′ i 3 β 3 ) = e x p ( x ′ i 1 β 1 + x ′ i 2 β 2 + x ′ i 3 β 3 ) .

With time-constant regressors c_i in x_it, do the normalization with x_i1β₁:

(A.5) set  x ′ i 1 β 1 = 0  and replace  x ′ i t β t  by  x ′ i t β t − x ′ i 1 β 1    ∀ t = 2,   … ,   T  (A.5)

where c_i appears as c′i(βtc−β1c) in x′itβt−x′i1β1. With this modification for the denominator of the likelihood function, the normalized version exp{∑t≥2yit(x′itβt−x′i1β1)} should be used accordingly for the numerator of the likelihood function.

Likelihood for dynamic PCLE with no regressors

With T=3 and β_t=0, the likelihood function for (y₁, y₂, y₃)∣(y₀, δ) is

(A.6) P ( y 1 ,   y 2 ,   y 3 | y 0 ,   δ ) = e x p { ( α 1 y 0 y 1 + α 2 y 1 y 2 + α 3 y 2 y 3 ) + δ ( y 1 + y 2 + y 3 ) } { 1 + e x p ( α 1 y 0 + δ ) } { 1 + e x p ( α 2 y 1 + δ ) } { 1 + e x p ( α 3 y 2 + δ ) } .  (A.6)

Given (y₀, δ), consider (y₁=0, y₂=1, y₃) and (y₁=1, y₂=0, y₃):

(A.7) P ( y 1 = 0,   y 2 = 1, y 3   | y 0 , δ ) = e x p { α 3 y 3 + δ ( 1 + y 3 ) } { 1 + e x p ( α 1 y 0 + δ ) } { 1 + e x p ( δ ) } { 1 + e x p ( α 3 + δ ) } ; P ( y 1 = 1, y 2 = 0, y 3   | y 0 , δ ) = e x p { α 1 y 0 + δ ( 1 + y 3 ) } { 1 + e x p ( α 1 y 0 + δ ) } { 1 + e x p ( α 2 + δ ) } { 1 + e x p ( δ ) } .  (A.7)

The two denominators are the same under α₂=α₃, and thus only the numerators matter in the ratio of the first probability to the sum of the two probabilities in (A.7):

(A.8) P ( y 1 = 0,   y 2 = 1,   y 3 | y 0 ,   δ ) P ( y 1 = 1,   y 2 = 0,   y 3 | y 0 ,   δ ) + P ( y 1 = 0,   y 2 = 1,   y 3 | y 0 ,   δ ) = e x p { α 3 y 3 + δ ( 1 + y 3 ) } e x p { α 1 y 0 + δ ( 1 + y 3 ) } + e x p { α 3 y 3 + δ ( 1 + y 3 ) } = e x p ( α 3 y 3 − α 1 y 0 ) 1 + e x p ( α 3 y 3 − α 1 y 0 ) .  (A.8)

(A.8) can be written also as

P ( y 1 = 0,   y 2 = 1,   y 1 + y 2 = 1,   y 3 | y 0 ,   δ ) P ( y 1 + y 2 = 1,   y 3 | y 0 ,   δ ) = P ( y 1 = 0,   y 2 = 1 | y 1 + y 2 = 1,   y 0 ,   y 3 ,   δ ) .

Hence

P ( y 1 = 0,   y 2 = 1 | y 1 + y 2 = 1,   y 0 ,   y 3 ,   δ ) = e x p ( α 3 y 3 − α 1 y 0 ) 1 + e x p ( α 3 y 3 − α 1 y 0 ) .

The conditional log-likelihood function for (α₁, α₃) under α₂=α₃ with four waves is thus (Lc3); 1[y_i1+y_i2=1] is to use only the informative observations (y_i1≠y_i2) as in (2.4)

Likelihood for dynamic PCLE with the same last two-period regressors

Analogous to (A.7) is the likelihood for (y₁, y₂, y₃)∣(y₀, δ, X) with ytw′tζt in:

P ( y 1 = 0,   y 2 = 1,   y 3 | y 0 ,   δ ,   X ) = e x p { α 3 y 3 + y 3 w ′ 3 ζ 3 + ( x ′ 2 β 2 + y 3 x ′ 3 β 3 ) + δ ( 1 + y 3 ) } { 1 + e x p ( α 1 y 0 + y 0 w ′ 1 ζ 1 + x ′ 1 β 1 + δ ) } { 1 + e x p ( x ′ 2 β 2 + δ ) } { 1 + e x p ( α 3 + w ′ 3 ζ 3 + x ′ 3 β 3 + δ ) } ; P ( y 1 = 1,   y 2 = 0,   y 3 | y 0 ,   δ ,   X ) = e x p { α 1 y 0 + y 0 w ′ 1 ζ 1 + ( x ′ 1 β 1 + y 3 x ′ 3 β 3 ) + δ ( 1 + y 3 ) } { 1 + e x p ( α 1 y 0 + y 0 w ′ 1 ζ 1 + x ′ 1 β 1 + δ ) } { 1 + e x p ( α 2 + w ′ 2 ζ 2 + x ′ 2 β 2 + δ ) } { 1 + e x p ( x ′ 3 β 3 + δ ) } .

The two denominators are equal if α₂=α₃, β₂=β₃, ζ₂=ζ₃ and x₂=x₃, under which we can proceed analogously to the preceding no-regressor case.

The ratio of the first probability to the sum of the two probabilities given α₂=α₃, β₂=β₃, ζ₂=ζ₃ and x₂=x₃ is

P ( y 1 = 0,   y 2 = 1,   y 3 | y 0 ,   δ ,   X ,   x 2 = x 3 ) P ( y 1 = 1,   y 2 = 0,   y 3 | y 0 ,   δ ,   X ,   x 2 = x 3 ) + P ( y 1 = 0,   y 2 = 1,   y 3 | y 0 ,   δ ,   X ,   x 2 = x 3 ) = P ( y 1 = 0,   y 2 = 1 | y 0 ,   y 3 ,   y 1 + y 2 = 1,   δ ,   X ,   x 2 = x 3 ) .

With A≡exp{α3y3+y3w′3ζ3+x′2β2+y3x′3β3+δ(1+y3)}, this equals

A e x p { α 1 y 0 + y 0 w ′ 1 ζ 1 + x ′ 1 β 1 + y 3 x ′ 3 β 3 + δ ( 1 + y 3 ) } + A   = e x p ( α 3 y 3 − α 1 y 0 + y 3 w ′ 3 ζ 3 − y 0 w ′ 1 ζ 1 + x ′ 2 β 2 − x ′ 1 β 1 ) 1 + e x p ( α 3 y 3 − α 1 y 0 + y 3 w ′ 3 ζ 3 − y 0 w ′ 1 ζ 1 + x ′ 2 β 2 − x ′ 1 β 1 ) .

Here, the parameters are (α₁, α₃, ζ₁, ζ₃, β₁, β₂) for the regressors (–y₀, y₃, –y₀w₁, y₃w₃, –x₁,x₂). This leads to (3.5), which becomes (Lhk3) when ζ_t=0 ∀t.

Likelihood for dynamic PCLE conditional on y_T of Bartolucci and Nigro (2010)

Under (3.6),

P ( Y | y 0 ,   δ ,   X ) = ∏ t P ( y t | y t − 1 ,   δ ,   X ) = ∏ t e x p { α t y t − 1 y t + y t − 1 y t w ′ t ζ t + y t x ′ t β t + δ y t + y t e t * ( δ ,   X ) } 1 + e x p { α t y t − 1 + y t − 1 w ′ t ζ t + x ′ t β t + δ + e t * ( δ ,   X ) } = ∏ t e x p { α t y t − 1 y t + y t − 1 y t w ′ t ζ t + y t x ′ t β t + δ y t } ⋅ ∏ t [ e x p { e t * ( δ ,   X ) } ] y t ∏ t [ 1 + e x p { α t y t − 1 + y t − 1 w ′ t ζ t + x ′ t β t + δ + e t * ( δ ,   X ) } ] .

∏ t [ e x p { e t * ( δ ,   x ) } ] y t in the numerator equals

[ 1 + e x p { α 2 + w ′ 2 ζ 2 + x ′ 2 β 2 + δ + e 2 * ( δ ,   X ) } 1 + e x p { x 2 ' β 2 + δ + e 2 * ( δ , X ) } ] y 1 ⋯ [ 1 + e x p { α T + w ′ T ζ T + x ′ T β T + δ + e T * ( δ ,   X ) } 1 + e x p { x ′ T β T + δ + e T * ( δ ,   X ) } ] y T − 1 ⋅ e x p { e T * ( δ ,   X ) } ] y T .

The denominator of P(Y∣y₀, δ, X) can be written as

∏ t [ 1 + e x p { α t + w ′ t ζ t + x ′ t β t + δ + e t * ( δ ,   X ) } ] y t − 1 ⋅ [ 1 + e x p { x ′ t β t + δ + e t * ( δ ,   X ) } ] 1 − y t − 1 = ∏ t [ 1 + e x p { α t + w ′ t ζ t + x ′ t β t + δ + e t * ( δ ,   X ) } 1 + e x p { x ′ t β t + δ + e t * ( δ ,   X ) } ] y t − 1 [ 1 + e x p { x ′ t β t + δ + e t * ( δ ,   X ) } ] .

Substituting these two displays gives

P ( Y | y 0 ,   δ ,   X ) = μ ( y 0 ,   y T ,   δ ,   X ) ⋅ ∏ t e x p { α t y t − 1 y t + y t − 1 y t w ′ t ζ t + y t x ′ t β t + δ y t }

where μ(y₀, y_T, δ, X) equals

[ 1 + e x p { α 1 + w ′ 1 ζ 1 + x ′ 1 β 1 + δ + e 1 * ( δ ,   X ) } 1 + e x p { x ′ 1 β 1 + δ + e 1 * ( δ ,   X ) } ] − y 0 [ e x p { e T * ( δ ,   X ) } ] y T ∏ t [ 1 + e x p { x ′ t β t + δ + e t * ( δ , X ) } ] .

From this P(Y∣y₀, δ, X), P(Σ_ty_t, y_t∣y₀, δ, X) becomes

μ ( y 0 ,   y T ,   δ ,   X ) ⋅ ∑ λ ¯ 1, T − 1 = y ¯ e x p { ∑ t α t λ t − 1 λ t + ∑ t λ t t − 1 λ t w ′ t ζ t + ∑ t λ t x ′ t β t + δ ∑ t λ t } .

Dividing P(Y∣y₀, δ, X) by P(Σ_ty_t, y_T∣y₀, δ, X) removes μ(y₀, y_T, δ, X) and δΣ_ty_t to result in (3.8) after the normalization with exp(∑tyt⋅x′1β1)=exp(∑tλt⋅x′1β1) and exp(yTx′TβT)=exp(λTx′TβT). When T=3, recalling (Lhk3) to use y₁y₂=λ₁λ₂=0 for the two useful sequences (y₀, 0, 1, y₃) and (y₀, 1, 0, y₃), (3.8) becomes (3.9).

Likelihood for PML with three alternatives and two waves

The likelihood function for Y=(y_a1, y_b1, y_c1, y_a2, y_b2, y_c)′ is (note Σ_jy_jt=1 ∀t)

(A.9) P ( Y | W ,   δ a ,   δ b ,   δ c ) = Π j ( e x p ( w ′ j 1 γ j 1 + δ j − δ a ) ∑ j e x p ( w ′ j 1 γ j 1 + δ j − δ a ) ) y j 1 ( e x p ( w ′ j 2 γ j 2 + δ j − δ a ) ∑ j e x p ( w ′ j 2 γ j 2 + δ j − δ a ) ) y j 2 = e x p { ∑ j y j 1 ( w ′ j 1 γ j 1 + δ j − δ a ) + ∑ j y j 2 ( w ′ j 2 γ j 2 + δ j − δ a ) } ∑ j e x p ( w ′ j 1 γ j 1 + δ j − δ a ) ⋅ ∑ j e x p ( w ′ j 2 γ j 2 + δ j − δ a ) = e x p { ∑ j y j 1 w ′ j 1 γ j 1 + ∑ j y j 2 w ′ j 2 γ j 2 + ∑ j ( y j 1 + y j 2 ) ( δ j − δ a ) } ∑ j e x p ( w ′ j 1 γ j 1 + δ j − δ a ) ⋅ ∑ j e x p ( w ′ j 2 γ j 2 + δ j − δ a ) ;  (A.9)

y _j1+y_j2, j=a, b, c, are candidates to condition on to remove δ_j–δ_a, j=a, b, c. Observe

(A.10) P ( y j 1 + y j 2 ,   j = a ,   b ,   c | W ,   δ a ,   δ b ,   δ c ) = ∑ λ ¯ j = y ¯ j ∀ j e x p { ∑ j λ j 1 w ′ j 1 γ j 1 + ∑ j λ j 2 w ′ j 2 γ j 2 + ∑ j ( λ j 1 + λ j 2 ) ( δ j − δ a ) } ∑ j e x p ( w ′ j 1 γ j 1 + δ j − δ a ) ⋅ ∑ j e x p ( w ′ j 2 γ j 2 + δ j − δ a ) = e x p { ∑ j ( y j 1 + y j 2 ) ( δ j − δ a ) } ∑ λ ¯ j = y ¯ j ∀ j e x p ( ∑ j λ j 1 w ′ j 1 γ j 1 + ∑ j λ j 2 w ′ j 2 γ j 2 ) ∑ j e x p ( w ′ j 1 γ j 1 + δ j − δ a ) ⋅ ∑ j e x p ( w ′ j 2 γ j 2 + δ j − δ a ) .  (A.10)

Dividing (A.9) by (A.10) renders (4.4).