Testing for random coefficient autoregressive and stochastic unit root models

A Exact score vector computation method

Here, we briefly summarize the exact score computation method proposed in Nagakura (2020). Let y _t and α _t , for t = 1, …, n, be the v × 1 observation and m × 1 state vectors at time t, respectively. Nagakura (2020) considered the following general form of the linear Gaussian state space model:

where d _t , Z _t , S _t , c _t , T _t , R _t , and Q _t are the v × 1, v × m, v × v, m × 1, m × m, m × w, and w × w matrices, respectively (hereafter, we refer to these matrices as the system matrices); and ε _t and η _t are the v × 1 and w × 1 vectors of unobserved errors, respectively. The first and second equations in (A.1) are called the observation equation and state equation, respectively. Assume that each element of the system matrices is a function of an unknown h × 1 parameter vector θ = [ θ 1 , … , θ h ] ′ ∈ Θ ⊆ ℝ h , the functional form at time t is known before time t, and S _t and Q _t are positive semidefinite at any θ  ∈ Θ. Under the Gaussian assumption for ε _t and η _s , the joint density of Y n ≡ [ y n ′ , ⋯ y 1 ′ ] ′ is also Gaussian. Define Y t ≡ ( y t ′ , … , y 1 ′ ) ′ . Let p ( y t | Y u ) denote the conditional pdf of y _t conditional on Y _u . Then, L _n , the log-likelihood of Y _n , is defined as L n ≡ ∑ t = 1 n ℓ t , where ℓ t ≡ log p ( y t | Y t − 1 ) and p ( y 1 | Y 0 ) ≡ p ( y 1 ) . Under the assumption of Gaussian errors, the conditional distribution of y _t conditional on Y _t−1 is also Gaussian. Let a t | u ≡ E ( α t | Y u ) and P t | u ≡  var  ( α t | Y u ) . Then, ℓ _t is given by

where v t = y t − d t − Z t a t | t − 1 , and F t = Z t P t | t − 1 Z t ′ + S t . One can calculate a _t|t−1 and P _t|t−1 by running the Kalman filter, which is given as the following set of recursions for t = 1, …, n (e.g., see Harvey (1989)):

where L t ≡ T t + 1 − J t Z t and J t ≡ T t + 1 P t | t − 1 Z t ′ F t − 1 , with initial conditions a 1 | 0 ≡ a 0 and P 1 | 0 ≡ P 0 .

Here, we introduce some notations used below. Consider an m × k matrix A ≡ [ a i j ] , each of whose elements is a function of an h × 1 vector θ ≡ [ θ 1 , … , θ h ] ′ , θ ∈ Θ ⊆ ℝ h . Define G θ ( A ) ≡ ∂ [ vec  ( A ) ] ′ / ∂ θ , i.e.,  G θ ( A ) is an h × km matrix of the first-order partial derivatives of a _ij , i = 1, …, m, j = 1, …, k with respect to θ _r , r = 1, …, h.

Now, suppose that all of the elements of G θ ( d t ) , G θ ( Z t ) , G θ ( S t ) , G θ ( c s ) , G θ ( T s ) , G θ ( R s ) , G θ ( Q s ) , G θ ( a 0 ) ( = G θ ( a 1 | 0 ) ) , and G θ ( P 0 ) ( = G θ ( P 1 | 0 ) ) for t = 1, …, n and s = 2 , … , n + 1 exist and are finite at a point of θ . Nagakura (2020) derives a recursive formula for computing G θ ( ℓ t ) for t = 1, …, n, which is given by

and

where w t ≡ F t − 1 v t , a t | t ≡ a t | t − 1 + P t | t − 1 Z t ′ w t , and M t ≡ w t w t ′ − F t − 1 , N r = ( I r 2 + K r ) / 2 , and K _r is the unique r ² × r ² matrix given as K r = ∑ i = 1 r ∑ i = 1 r ( E i , j ⊗ E i , j ′ ) , where E i , j = e i ( r ) e j ( s ) ′ , and e i ( a ) is the a × 1 vector whose i-th element is 1 and all other elements are 0.^[6] Then the score vector evaluated at this point is given as s n ≡ G θ ( L n ) = ∑ t = 1 n G θ ( ℓ t ) .

This formula offers two advantages compared to extant methods. First, it can calculate all components of the score vector simultaneously, whereas most existing methods are designed to calculate only one component of the score vector at one pass of the formulas, except for Nagakura (2013). Second, the formula assumes neither time-invariant system matrices nor non-singularities of S _t and R t Q t R t ′ , unlike the existing methods.

When the system matrices are all time-invariant and the state vector α _t is stationary,^[7] a ₀ and P ₀ are often set to the stationary mean vector and covariance matrix of α _t , namely:

Then, Nagakura (2020) also shows that

Note that the formulas given in (A.5)–(A.7) need not assume time-invariant system matrices, whereas the formulas in (A.9) and (A.10) are valid only when the system matrices are time-invariant and initial conditions are set as in (A.8).

B ML estimation of a reparametrized RCA model

We reparametrize the model given in (1) as

Assume that y ₀ = 0. Under this parametrization, the log-likelihood function is given by

Solving the first-order conditions, ∂ L / ∂ d = 0 and ∂ L / ∂ σ ϵ 2 = 0 , with respect to d and σ ϵ 2 , we have

Substituting (B.2) into (B.1), and ignoring the terms irrelevant to the estimation, we have the concentrated log-likelihood function of κ:

from which we can easily obtain the ML estimate of κ, κ ˜ n , as the value that maximizes L n c ( κ ) . We can also calculate the ML estimates of d and σ ϵ 2 from d ˜ n = d n ( κ ˜ n ) and σ ˜ ϵ , n 2 = σ ϵ , n 2 ( κ ˜ n ) . Then, the ML estimate of σ ξ 2 is given as σ ˜ ξ , n 2 = κ ˜ n σ ˜ ϵ , n 2 .