An Overview of ARMA-Like Models for Count and Binary Data

Abstract

A comprehensive overview of the literature on models for discrete valued time series is provided, with a special focus on count and binary data. ARMA-like models such as the BARMA, GARMA, M-GARMA, GLARMA and log-linear Poisson are illustrated in detail and critically compared. Methods for deriving the stochastic properties of specific models are delineated and likelihood-based inference is discussed. The review is concluded with two empirical applications. The first regards the analysis of the daily number of deaths from COVID-19 in Italy, under the assumption both of a Poisson and a negative binomial distribution for the data generating process. The second illustration analyses the binary series of signs of log-returns for the weekly closing prices of Johnson & Johnson with BARMA and Bernoulli GARMA and GLARMA models.

Appendices

Appendix

8.1.1 Technical Details

8.1.1.1 Markov Chain Specification

In order to derive strict stationarity and ergodicity conditions, the problem is reformulated in terms of Markov chain theory. Let us consider an observation driven model in the most general form:

$$\displaystyle \begin{aligned} Y_{t}\left|\right. \mathcal{F}_{t-1} \sim q(\cdot; \mu_{t}) {} \end{aligned} $$

(8.23)

$$\displaystyle \begin{aligned} \mu_{t}=c_{\delta}(Y_{0:t-1}) {} \end{aligned} $$

(8.24)

where we adopt the shorthand notation Y_t for the process and, as before, y_t its realization. The function q is simply the density function which comes from (8.1), whereas c_δ is some function which describes the form of the dependence from the observation. In general, Y_s:t = (Y_s, Y_s+1, …, Y_t) where s ≤ t. The symbol δ is the vector of parameters of the model. Of course, the initial values μ_0:p−1 are supposed to be known. The model in (8.24) can be rewritten as:

$$\displaystyle \begin{aligned} \mu_{t}=g_{\delta}(Y_{t-p:t-1},\mu_{t-p:t-1}). \end{aligned}$$

This way of writing the observation driven model [15] gives a Markov p-structure for μ_t and then implies that the vector μ_t−p:t−1 forms the state of a Markov chain indexed by t. In this case it is possible to prove stationarity and ergodicity of {Y_t} _{t ∈ℕ} by first showing these properties for the multivariate Markov chain {μ_t−p:t−1} _t≥p, then shifting the results back to the time series model {Y_t} _{t ∈ℕ}.

Some useful definitions for theorems based upon the theory of Markov chains asserted throughout the paper are introduced. Define a general Markov chain X = {X_t} _{t ∈ℕ} on a state space S with σ-algebra ℱ and define P^t(x, A) = P(X_t ∈ A |X₀ = x) for A ∈ℱ as the t-step transition probability starting from state X₀ = x.

Definition 8.1

A Markov chain X is φ-irreducible if there exists a non-trivial measure φ on ℱ such that, whenever φ(A) > 0, P^t(x, A) > 0 for some t = t(x, A), for all x ∈ S.

Also, the definition of aperiodicity as stated in [44] is needed. Define a period d(α) = gcd {t ≥ 1 : P^t(α, α) > 0}.

Definition 8.2

An irreducible Markov chain X is aperiodic if d(x) ≡ 1, x ∈ X.

Definition 8.3

A set A ∈ℱ is called a small set if there exists an m > 1, a non-trivial measure v on ℱ, and a λ > 0 such that for all x ∈ A and all C ∈ℱ, P^m(x, C) ≥ λv(C).

Let E_x(⋅) denote the expectation under the probability P_x(⋅) induced on the path space of the chain defined by Ω =∏ t=0∞X_t with respect to ℱ^∞ =∨ t=0∞ℬ(X_t) when the initial state X₀ = x; where ℬ(X_t) is the Borel σ-field on X_t.

Theorem 8.7 (Drift Conditions)

Suppose that X = {X_t}_{t ∈ℕ}is φ-irreducible on S. Let A ⊂ S be small, and suppose that there exist b ∈ (0, ∞), 𝜖 > 0, and a function V : S → [0, ∞) such that for all x ∈ S,

$$\displaystyle \begin{aligned} \mathrm{E}_{x}\left[ V(X_{1})\right] \leq V(x) - \varepsilon + b\boldsymbol{1}_{\left\lbrace x\in A\right\rbrace }, {} \end{aligned} $$

(8.25)

then X is positive Harris recurrent.

The function V  is called the Lyapunov function or energy function.

Positive Harris recurrent chains possess a unique stationary probability distribution π. Moreover, if X₀ is distributed according to π, then the chain X is a stationary process. If the chain is also aperiodic, then X is ergodic, in which case if the chain is initialized according to some other distribution, then the distribution of X_t will converge to π as t →∞.

A stronger form of ergodicity, called geometric ergodicity, arises if (8.25) is replaced by the condition

$$\displaystyle \begin{aligned} \mathrm{E}_{x}\left[ V(X_{1})\right] \leq \beta V(x) + b\boldsymbol{1}_{\left\lbrace x\in A\right\rbrace } {} \end{aligned} $$

(8.26)

for some β ∈ (0, 1) and some V : S → [1, ∞). Indeed, (8.26) implies (8.25). Eventually, stationarity and ergodicity for the GARMA model would be accomplished if at least one of the sufficient condition (8.25), (8.26) above is fulfilled.

Unfortunately, a problem can occur when the distribution in (8.23) is not continuous (that is, Bernoulli, Poisson,…). In fact, in these cases the Markov chain {μ_t−p:t−1} _n≥p may not be φ-irreducible. This occurs whenever Y_t can only take a countable set of values and the state space μ_t−p:t−1 is ℝ^p. Then, given a particular initial vector μ_0:p−1, the set of possible values for μ_t is countable. Definition 8.1 is not satisfied. For this reason, additional theoretical tools are required:

• Perturbation approach

• Feller conditions.

8.1.1.2 Perturbation Approach

First, define the perturbed form of an observation driven time series model:

$$\displaystyle \begin{aligned} Y_{t}^{(m)}\left|\right. Y^{(m)}_{0:t-1} \sim q(\cdot; \mu^{(m)}_{t}) {} \end{aligned} $$

(8.27)

$$\displaystyle \begin{aligned} \mu^{(m)}_{t}=g_{\delta, t}(Y^{(m)}_{0:t-1}, m Z_{0:t-1}), {} \end{aligned} $$

(8.28)

where Z_t ∼ ϕ are independent, identically distributed random perturbations having density function ϕ, m > 0 is a scale factor associated with the perturbation, and g_δ,t(⋅, mZ_0:t−1) is a continuous function of Z_0:t−1 such that g_δ,t(y, 0) = g_δ,t(y) for any y. The value μ0(m) is a fixed constant that is taken to be independent of m, so that μ0(m) = μ₀. The perturbed model is constructed to be φ-irreducible, so that one can apply usual drift conditions to prove its stationarity.

Then, it can be proved that the likelihood of the parameter vector δ calculated using (8.28) converges uniformly to the likelihood calculated using the unperturbed model as m → 0. More precisely, the joint density of the observations Y = Y 0:t(m) and first t perturbations Z = Z_0:t−1, conditional on the parameter vector δ, the perturbation scale m, and the initial value μ₀, is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} & f(Y,Z\left| \right. \delta, m, \mu_{0})&\displaystyle = f(Z\left| \right. \delta, m, \mu_{0}) \times f(Y\left| \right. Z, \delta, m, \mu_{0}) \\ & &\displaystyle = \left[ \prod_{k=0}^{t-1} \phi(Z_{k}) \right]\prod_{k=0}^{t}f\left( Y^{(m)}_{k}; \mu_{k}(m Z)\right) \end{array} \end{aligned} $$

where μ_k(mZ) is the value of μk(m) induced by the perturbation vector mZ through (8.28), with μ₀(mZ) = μ₀. The likelihood function for the parameter vector δ implied by the perturbed model is the marginal density of Y  integrating over Z, i.e.,

$$\displaystyle \begin{aligned} \mathcal{L}_{m}(\delta)=f(Y\left| \right. \delta, m, \mu_{0})=\int f(Y,Z\left| \right. \delta, m, \mu_{0})dZ. \end{aligned}$$

Let the likelihood function without the perturbations be denoted by ℒ, so that

$$\displaystyle \begin{aligned} \mathcal{L}(\delta)=\prod_{k=0}^{t}f\left( Y^{(m)}_{k}; \mu_{k}(0)\right). \end{aligned}$$

Theorem 8.8

Under regularity conditions 1 and 2 below, the likelihood function ℒ_mbased on the perturbed model (8.27)–(8.28) converges uniformly on any compact set K to the likelihood function ℒ based on the original model, i.e.,

for any fixed sequence of observations y_0:tand conditional on the initial value μ₀.

So if ℒ is continuous in δ and has a finite number of local maxima and a unique global maximum on K, the maximum-likelihood estimate of δ based on ℒ_m converges to that based on ℒ. The proof is in [42]. Regularity Conditions:

• 1. For any fixed y the function q(y;μ) is bounded and Lipschitz continuous in μ, uniformly in δ ∈ K.

• 2. For each t, μ_t(mZ) is Lipschitz in some bounded neighbourhood of zero, uniformly in δ ∈ K.

Regularity condition 1 holds, e.g., for q(y;μ) equal to a Poisson or binomial density with mean μ, or a negative binomial density with mean μ and precision parameter φ. μ_t(mZ) can easily be constructed to satisfy condition 2. One can choose to use the perturbed model (with fixed and sufficiently small perturbation scale m) instead of the original model, without significantly affecting finite-sample parameter estimates, in order to get the strong theoretical properties associated with stationarity and ergodicity.

Although, it has been shown that the perturbed and original models are closely related, and although one can use drift conditions to show the stationarity and ergodicity properties of the perturbed model, this approach does not yield stationarity and ergodicity properties for the original model. In fact, this approach addresses consistency of parameter estimation for the perturbed model when t →∞ for fixed m and then shows that as m → 0 the finite sample estimates (for a fixed number of observations t) of the perturbed model approach those of the original one. In order to show real proprieties of the original model one should consider both limits t →∞ together with m → 0 in which a substantial technical difficulty associated with interchanging the limits arises. For this reason, the Feller properties introduced in the next section are needed.

8.1.1.3 Feller Conditions

To deal with the lack of the φ-irreducibility condition, the Feller properties can be used instead.

Definition 8.4

A chain evolving on a complete separable metric space S is said to be “weak Feller” if P(x, ⋅) satisfies P(x, ⋅) ⇒ P(y, ⋅) as x → y, for any y ∈ S and where ⇒ indicates convergence in distribution.

In the absence of φ-irreducibility, the “weak Feller” condition can be combined with a drift condition (8.25) or (8.26) to show the existence of a stationary distribution [55]:

Theorem 8.9

Suppose that S is a locally compact complete separable metric space with ℱ the Borel σ-field on S, and the Markov chain {X_t}_{t ∈ℕ}with transition kernel P is weak Feller. Let A ∈ℱ be compact, and suppose that there exist b ∈ (0, ∞), ε > 0, and a function V : S → [0, ∞) such that for all x ∈ S, the drift condition (8.25) holds. Then there exists a stationary distribution for P.

Uniqueness of the stationary distribution can be established using the “asymptotic strong Feller” property, defined in [35]. Before doing it, further definitions are required:

Definition 8.5

Let S be a Polish (complete, separable, metrizable) space. A “totally separating system of metrics” {d_t} _{t ∈ℕ} for S is a set of metrics such that for any x, y ∈ S with x ≠ y, the value d_t(x, y) is non-decreasing in t and lim_t→∞d_t(x, y) = 1.

Definition 8.6

A metric d on S implies the following distance between probability measures μ₁ and μ₂:

$$\displaystyle \begin{aligned} \left\| \mu_{1} - \mu_{2} \right\|{}_{d} = \sup_{\mathrm{Lip}_{d}\phi=1}\left( \int\phi(x)\mu_{1}(dx) - \int\phi(x)\mu_{2}(dx)\right) {} \end{aligned} $$

(8.29)

where

$$\displaystyle \begin{aligned} \mathrm{Lip}_{d}\phi= \sup_{x,y\in S:x\neq y} \frac{\left|\phi(x)-\phi(y)\right|}{d(x,y)} \end{aligned}$$

is the minimal Lipschitz constant for ϕ with respect to d.

Definition 8.7

A chain is “asymptotically strong Feller” if, for every fixed x ∈ S, there is a totally separating system of metric {d_t}  for S and a sequence t_n > 0 such that

$$\displaystyle \begin{aligned} \lim_{\delta \to \infty} \limsup_{t \to \infty} \sup_{y\in B(x,\delta)} \left\| P^{t_{n}}(x,\cdot) - P^{t_{n}}(y,\cdot) \right\|{}_{d_{t}} = 0 \end{aligned}$$

where B(x, δ) is the open ball of radius δ centred at x, as measured using some metric defining the topology of S.

Definition 8.8

A “reachable” point x ∈ S means that for all open sets A containing x, ∑ t=1∞P^t(y, A) > 0 for all y ∈ S.

Theorem 8.10

Suppose that S is a Polish space and the Markov chain {X_t}_{t ∈ℕ}with transition kernel P is asymptotically strong Feller. If there is a reachable point x ∈ S then P can have at most one stationary distribution.

This is an extension of [35]. The results of this section lay the foundation for showing the convergence and asymptotic properties of maximum likelihood estimators for the discrete-valued observation driven models.

8.1.1.4 Coupling Construction

Introduce a kernel H̄ from (X², X^⊗2) to (Y², Y^⊗2) satisfying the following conditions on the marginals: for all (x, x^′) ∈X² and A ∈Y,

$$\displaystyle \begin{aligned} \bar{H}((x,x^\prime);A \times \mathsf{Y})=H(x,A), \quad \bar{H}((x,x^\prime); \mathsf{Y} \times A)=H(x^\prime,A). {} \end{aligned} $$

(8.30)

Let C ∈Y^⊗2 such that H̄((x, x^′);C) ≠ 0 and the chain {Z_t = (X_t, Xt′, U_t)}_{t ∈ℤ} on the“extended” space (X² × 0, 1, X^⊗2 ⊗P(0, 1)) with transition kernel Q̄ implicitly defined as follows. Given Z_t = (x, x^′, u) ∈X² ×{0, 1}, draw (Y_t+1, Yt+1′) according to H̄((x, x^′);⋅) and set

$$\displaystyle \begin{aligned} X_{t+1}=f_{Y_{t+1}}(x), \quad X^{\prime}_{t+1}=f_{Y^{\prime}_{t+1}}(x^\prime), \end{aligned}$$

$$\displaystyle \begin{aligned} U_{t+1}={\mathbf{1}}_C(Y_{t+1},Y^{\prime}_{t+1}), \end{aligned}$$

$$\displaystyle \begin{aligned} Z_{t+1}=(X_{t+1},X^{\prime}_{t+1},U_{t+1}). \end{aligned}$$

The conditions on the marginals of H̄, given by (8.30) also imply conditions on the marginals of Q̄: for all A ∈X and z = (x, x^′, u) ∈X² ×{0, 1},

$$\displaystyle \begin{aligned} \bar{Q}(z; A \times \mathsf{X} \times \left\lbrace 0,1\right\rbrace)=Q(x,A), \quad \bar{Q}(z;\mathsf{X} \times A \times \left\lbrace 0,1\right\rbrace)=Q(x^\prime,A). {} \end{aligned} $$

(8.31)

For z = (x, x^′, u) ∈X² ×{0, 1}, write

$$\displaystyle \begin{aligned} \alpha(x,x^\prime)=\bar{Q}(z;\mathsf{X}^2 \times \left\lbrace 1\right\rbrace )=\bar{H}((x,x^\prime);C)\neq0. {} \end{aligned} $$

(8.32)

The quantity α(x, x^′) is thus the probability of the event {U₁ = 1} conditionally on Z₀, taken on Z₀ = z. Denote by Q^♯ the kernel on (X², X^⊗2) defined by: for all z = (x, x^′, u) ∈X² ×{0, 1} and A ∈X^⊗2,

$$\displaystyle \begin{aligned} Q^\sharp((x,x^\prime);A)=\frac{\bar{Q}(z;A \times \left\lbrace 1\right\rbrace )}{\bar{Q}(z;\mathsf{X}^2 \times \left\lbrace 1\right\rbrace )} \end{aligned}$$

so that using (8.32),

$$\displaystyle \begin{aligned} \bar{Q}(z;A \times \left\lbrace 1\right\rbrace)=\alpha(x,x^\prime)\,Q^\sharp((x,x^\prime);A) . {} \end{aligned} $$

(8.33)

This shows that Q^♯((x, x^′);⋅) is the distribution of (X₁, X1′) conditionally on (X₀, X0′, U₁) = (x, x^′, 1).

8.1.1.5 Assumptions and Results of the Alternative Markov Chain Approach Without Irreducibility

In what follows, if (E, ℰ) is a measurable space, ξ a probability distribution on (E, ℰ), and R a Markov kernel on (E, ℰ), denote by PξR the probability induced on (E^ℕ, ℰ^⊗ℕ) by a Markov chain with transition kernel R and initial distribution ξ. Denote by EξR the associated expectation. Consider the following assumptions.

• (A1) The Markov kernel Q is weak Feller. Moreover, there exist a compact set C ∈X,(b, ε) ∈ℝ∗+ ×ℝ∗+ and a function V : X →ℝ⁺ such that

  $$\displaystyle \begin{aligned} QV\leq V-\varepsilon+b{\mathbf{1}}_C . \end{aligned}$$

• (A2) The Markov kernel Q has a reachable point.

• (A3) There exists a kernel Q̄ on (X² ×{0, 1}, X^⊗2 ⊗P({0, 1})), a kernel Q^♯ on (X², X^⊗2), and a measurable function α : X² → [1, ∞), and real numbers (D, ζ₁, ζ₂, ρ) ∈ (ℝ⁺)³ ×(0, 1) such that for all (x, x^′) ∈X²,

  $$\displaystyle \begin{aligned} 1-\alpha(x,x^\prime)\leq d(x,x^\prime)W(x,x^\prime) {} \end{aligned} $$

  (8.34)
  $$\displaystyle \begin{aligned} \mathrm{E}^{Q^\sharp}_{\delta_x\otimes\delta_{x^\prime}}[d(X_t,X^{\prime}_t)] \leq D\rho^td(x,x^\prime) {} \end{aligned} $$

  (8.35)
  $$\displaystyle \begin{aligned} \mathrm{E}^{Q^\sharp}_{\delta_x\otimes\delta_{x^\prime}}[d(X_t,X^{\prime}_t)W(X_t,X^{\prime}_t)] \leq D\rho^td^{\zeta_1}(x,x^\prime)W^{\zeta_2}(x,x^\prime) . {} \end{aligned} $$

  (8.36)

  Moreover, for all x ∈X, there exists γ_x > 0 such that

  $$\displaystyle \begin{aligned} \sup_{x^\prime\in B(x,\gamma_x)}W(x,x^\prime)<\infty \end{aligned}$$

Some practical conditions from checking (8.35) and (8.36) in (A3) can be denoted.

Lemma 8.1

Assume that either (i) or (ii) or (iii) (defined below) holds.

• (i) There exist (ρ, β) ∈ (0, 1) × R such that for all (x, x^′) ∈X²

  $$\displaystyle \begin{aligned} d(X_1,X^\prime_1)\leq\rho d(x,x^\prime),\quad \mathrm{P}^{Q^\sharp}_{\delta_x\otimes\delta_{x^\prime}}-a.s. {} \end{aligned} $$

  (8.37)
  $$\displaystyle \begin{aligned} Q^\sharp W\leq W+\beta {} \end{aligned} $$

  (8.38)

• (ii) Equation (8.35) holds and W is bounded.

• (iii) Equation (8.35) holds and there exist 0 < α < α^′and β ∈ℝ⁺such that for all (x, x^′) ∈X²

  $$\displaystyle \begin{aligned} d(x,x^\prime)\leq W^\alpha(x,x^\prime) \end{aligned}$$
  $$\displaystyle \begin{aligned} Q^\sharp W^{1+\alpha^\prime}\leq W^{1+\alpha^\prime}+\beta \end{aligned}$$

Then, (8.35) and (8.36) hold.

Assumption (A1) implies, by Tweedie [55], that the Markov kernel Q admits at least one stationary distribution. Assumptions (A2)–(A3) are then used to show that this stationary distribution is unique.

Note that assumptions (A1)–(A2) are the same as those of Theorems 8.9 and 8.10 of Appendix Section “Feller Conditions” and they can be proved for each observation driven model as has been done for the GARMA model in Sect. 8.5.1; assumption (A3) weakens the Lipschitz condition (8.18) by introducing a function W in (8.34). This allows to treat models which do not satisfy the Lipschitz condition (8.18); for example the log-linear Poisson autoregression of [28], see Sect. 8.5.2.

Theorem 8.11

Assume that (A1) –(A3) hold. Then, the Markov kernel Q in (8.19) admits a unique invariant probability measure.

Proposition 8.1

Assume that the Markov kernel Q admits a unique invariant probability measure. Then, there exists a strict-sense stationary ergodic process on ℤ, {Y_t}_{t ∈ℤ}, the solution to the recursion (8.19).

These results can be found in [23].

Main Proofs

8.1.1 Proof of Theorem 8.2

Following [42], Theorem 8.9 is applied to the chain {g(μ_t)}_{t ∈ℕ} to show that it has a stationary distribution; this implies the same result for the chain {μ_t} _{t ∈ℕ}. The state space S = ℝ of {g(μ_t)}_{t ∈ℕ} is a locally compact complete separable metric space with Borel σ-field. A drift condition for {g(μ_t)}_{t ∈ℕ} is given under the conditions of Theorem 8.1, for the compact set A = [−M, M] (the drift condition holds when the perturbation m = 0). All that remains is to show that the chain {g(μ_t)}_{t ∈ℕ} is weak Feller. Let X_t = g(μ_t). For X₀ = x, the GARMA model can be rewritten as

$$\displaystyle \begin{aligned} X_1(x)=\gamma+\phi(g(Y_0^*(g^{-1}(x)))-\gamma)+\theta(g(Y_0^*(g^{-1}(x)))-x). \end{aligned}$$

Since g⁻¹ is continuous, Y₀(g⁻¹(x)) ⇒ Y₀(g⁻¹(x^′)) as x → x^′. Since the ^∗ that maps Y₀ to the domain of g is continuous, it follows that Y 0∗(g⁻¹(x)) ⇒ Y 0∗(g⁻¹(x^′)) as x → x^′. Since g is continuous, then g(Y 0∗(g⁻¹(x))) ⇒ g(Y 0∗(g⁻¹(x^′))). So X₁(x) ⇒ X₁(x^′) as x → x^′, showing the weak Feller property. This ends the proof.

8.1.2 Proof of Theorem 8.4

The proof is based on the results of Appendix Sections “Coupling Construction”–“Assumptions and Results of the Alternative Markov Chain Approach Without Irreducibility”. The conditions (A1)–(A2) for the log-linear Poisson autoregression are proved as in Sect. 8.5.1 for the GARMA model. We report the proof of (A3).

Lemma 8.2

If |a + b|∨|a|∨|b| < 1, then (A3) holds.

Proof

Define Q̄ as the transition kernel Markov chain {Z_t} _{t ∈ℤ} with Z_t = (X_t, Xt′, U_t) in the following way. Given Z_t = (x, x^′, u), if x ≤ x^′, draw independently Y_t+1 ∼P(e^x) and V_t+1 ∼P(e^x′− e^x) and set Yt+1′ = Y_t+1 + V_t+1. Otherwise, draw independently Yt+1′∼P(e^x′) and V_t+1 ∼P(e^x − e^x′) and set Y_t+1 = Yt+1′ + V_t+1.

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle X_{t+1}=d+a\,x+b\ln(Y_{t+1}+1),\\ & &\displaystyle X_{t+1}^\prime=d+a\,x^\prime+b\ln(Y_{t+1}^\prime+1),\\ & &\displaystyle U_{t+1}={\mathbf{1}}_{Y_{t+1}=Y_{t+1}^\prime}={\mathbf{1}}_{V_{t+1}=0},\\ & &\displaystyle Z_{t+1}=(X_{t+1},X_{t+1}^\prime,U_{t+1}) \end{array} \end{aligned} $$

where Q̄ satisfies the marginal condition (8.31). Moreover, define for all x^♯ = (x, x^′) ∈X²,Q^♯(x^♯, ⋅) as the law of (X₁, X1′) where

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle X_1=d+a\,x+b\ln(Y+1),\quad Y\sim\mathcal{P}(e^{x\wedge x^\prime}), {}\\ & &\displaystyle X_1^\prime=d+a\,x^\prime+b\ln(Y+1), \end{array} \end{aligned} $$

(8.39)

and set for all x^♯ = (x, x^′) ∈ℝ²,

$$\displaystyle \begin{aligned} \alpha(x^\sharp)=\left\lbrace \exp-e^{x\vee x^\prime}+e^{x\wedge x^\prime}\right\rbrace . \end{aligned}$$

Then, Q̄ and Q^♯ satisfy (8.33). Using twice 1 − e^−u ≤ u,it follows that

$$\displaystyle \begin{aligned} \begin{array}{rcl} 1-\alpha(x^\sharp)& =&\displaystyle 1-\left\lbrace \exp-e^{x\vee x^\prime}+e^{x\wedge x^\prime}\right\rbrace\leq e^{x\vee x^\prime}-e^{x\wedge x^\prime} \\ & &\displaystyle e^{x\vee x^\prime}(1-e^{-|x-x^\prime|})\leq W(x,x^\prime)|x-x^\prime| \end{array} \end{aligned} $$

with W(x, x^′) = e^|x|∨|x′| so that (8.34) holds true. To check (8.35) and (8.36), Lemma 8.1 is applied, by checking option (i). Note first that

$$\displaystyle \begin{aligned} \mathrm{P}^{Q^\sharp}_{\delta_x\otimes\delta_{x^\prime}}\left\lbrace |X_1-X_1^\prime|=|a||x-x^\prime|\right\rbrace =1 , \end{aligned} $$

(8.40)

so that (8.37) is satisfied. To check (8.38), it can be shown that

$$\displaystyle \begin{aligned} \lim_{|x|\vee|x^\prime|\to\infty} \frac{Q^\sharp W(x,x^\prime)}{W(x,x^\prime)}=0 {} \end{aligned} $$

(8.41)

and for all M > 0,

$$\displaystyle \begin{aligned} \sup_{|x|\vee|x^\prime|\leq M} Q^\sharp W(x,x^\prime)<\infty {} \end{aligned} $$

(8.42)

Without loss of generality, assume x ≤ x^′. Using (8.39) provides

$$\displaystyle \begin{aligned} Q^\sharp W(x,x^\prime)=\mathrm{E}\left( e^{|X_1|\vee|X_1^\prime|}\right) \leq \mathrm{E}\left( e^{|X_1|}\right)+\mathrm{E}\left( e^{|X_1^\prime|}\right) . {} \end{aligned} $$

(8.43)

First, consider the second term of the right-hand side of (8.43),

$$\displaystyle \begin{aligned} \mathrm{E}\left( e^{|X_1^\prime|}\right)\leq e^{|d|}\mathrm{E}(e^{|ax^\prime+b\ln(1+Y)}). {} \end{aligned} $$

(8.44)

Noting that if u and v have different signs or if v = 0, then |u + v|≤|u|∨|v|. Otherwise, |u + v| = (u + v)1_v>0 ∨ (−u − v)1_v<0. This implies that

$$\displaystyle \begin{aligned} e^{|u+v|}\leq e^{|u|}+e^{|v|}+e^{u+v}{\mathbf{1}}_{v>0}+e^{-u-v}{\mathbf{1}}_{v<0} . \end{aligned}$$

and plugging this into (8.44),

$$\displaystyle \begin{aligned} \mathrm{E}(e^{|X_1^\prime})&\leq e^{|d|}\left( e^{|a||x^\prime|}+\mathrm{E}[(1+Y)^{|b|}]+e^{ax^\prime}\mathrm{E}[(1+Y)^b]{\mathbf{1}}_{b>0}\right.\\&\quad \left.+e^{-ax^\prime}\mathrm{E}[(1+Y)^{-b}]{\mathbf{1}}_{b<0}\right) . \end{aligned} $$

Note that for all γ ∈ [0, 1],

$$\displaystyle \begin{aligned} \mathrm{E}[(1+Y)^\gamma]\leq[\mathrm{E}(1+Y)]^\gamma=(1+e^x)^\gamma\leq1+e^{\gamma x}\leq1+e^{\gamma x^\prime} . \end{aligned}$$

Moreover, since |b|∈ [0, 1], b1_b>0 ∈ [0, 1] and − b1_b<0 ∈ [0, 1]. Therefore,

$$\displaystyle \begin{aligned} \begin{array}{rcl} & \mathrm{E}(e^{|X_1^\prime})&\displaystyle \leq e^{|d|}\left( e^{|a||x^\prime|}+1+e^{|b||x|}+e^{ax^\prime}(1+e^{bx^\prime}){\mathbf{1}}_{b>0}+e^{-ax^\prime}(1+e^{-bx^\prime}){\mathbf{1}}_{b<0}\right) \\ & &\displaystyle \leq e^{|d|}\left( e^{|a||x^\prime|}+1+e^{|b||x|}+e^{|a||x^\prime|}+e^{|a+b||x^\prime|}\right) \\ & &\displaystyle \leq e^{|d|}\left( 1+4e^{\gamma(|x|\vee|x^\prime|)}\right)\,, \end{array} \end{aligned} $$

where γ = |a|∨|b|∨|a + b| < 1. The first therm of the right hand side of (8.43) is treated as the second term by setting x^′ = x. So

$$\displaystyle \begin{aligned} \mathrm{E}(e^{|X_1|})\leq e^{|d|}\left( 1+4e^{\gamma(|x|\vee|x^\prime|)}\right)\,, \end{aligned}$$

so that using (8.43),

$$\displaystyle \begin{aligned} Q^\sharp W(x,x^\prime)\leq 2 e^{|d|}\left( 1+4e^{\gamma(|x|\vee|x^\prime|)}\right)\,. \end{aligned}$$

Since γ ∈ (0, 1) and W(x, x^′) = e^|x|∨|x′|, and (8.43) clearly implies (8.41) and (8.42), this proves (A3) and together with (A1)–(A2) provides stationarity conditions for the process {Y_t}  of Theorem 8.4.

Computational Aspects

The replication code for the application in Sect. 8.7 is available at https://github.com/mirkoarmillotta/covid_code. First, a function for the log-likelihood and the gradient of the log-linear Poisson autoregression is provided. The code for the other models works in a similar way and it is available upon request. Then, a function to perform the QMLE is presented. Finally, we give the code for the COVID-19 example and the relative plots. The code to perform the PIT is due to [17] and it is available in the reference therein.