A Unit Root Test with Markov Switching Deterministic Components: A Special Emphasis on Nonlinear Optimization Algorithms

Abstract

In this study, we investigate the performance of different optimization algorithms in estimating the Markov switching (MS) deterministic components of the traditional ADF test. For this purpose, we consider Broyden, Fletcher, Goldfarb, and Shanno (BFGS), Berndt, Hall, Hall, Hausman (BHHH), Simplex, Genetic, and Expectation-Maximization (EM) algorithms. The simulation studies show that the Simplex method has significant advantages over the other commonly used hill-climbing methods and EM. It gives unbiased estimates of the MS deterministic components of the ADF unit root test and delivers good size and power properties. When Hamilton’s (Econometrica 57:357–384, 1989) MS model is re-evaluated in conjunction with the alternative algorithms, we furthermore show that Simplex converges to the global optima in stationary MS models with remarkably high precision and even when convergence criterion is raised, or initial values are altered. These advantages of the Simplex routine in MS models allow us to contribute to the current literature. First, we produce the exact critical values of the generalized ADF unit root test with MS breaks in trends. Second, we derive the asymptotic distribution of this test and provide its invariance feature.

Appendix A

The asymtotic distribution of the generalized ADF test with MS breaks is not given in the previous literature. However, following Hamilton (1994, Chp 17) and using Proposition 2 of Cavaliere (2003) instead of Proposition 17.1 of Hamilton (1994, Chp 17, p. 486), one can easly derive the asymptotic distribution of the newly proposed tests. The distribution is similiar to the distribution given in equation 17.4.55 of Hamilton (1994, p. 500). The density functions of the proposed tests are given in Fig. 4:

Fig. 4

Density function of $\tau_{M\mu }$ and $\tau_{M,\tau }$

Following Cavaliere (2003) first we assume the following:

Assumption 1

The Markov Chain $\left\{ {s_{t} } \right\}_{0}^{\infty }$ is irreducible, aperiodic with one-step transition probability matrix denoted by $P: = (p_{ij} ),p_{ij} : = \Pr ob\left\{ {\left. {s_{t + 1} = j} \right|s_{t} = i} \right\},i, = 1,...,K$. The initial probabilities are $\Pr ob\left\{ {s_{0} = j} \right\} = :\pi_{j}$ and $p_{j}^{(t)} : = \Pr ob\left\{ {s_{t} = j} \right\} = :\pi_{j}$ for all j, t.

The above given assumption is followed by the following definition and proposition.

Definition 1

(Markov-switching Trend). Let $\mu : = \left( {\mu^{1} ,...,\mu^{K} } \right){\prime}$ be such that $\mu^{i} \ne \mu^{j}$ for at least a pair of indices $(i,j)$, $i \ne j$, and let $\mu_{t} : = \mu{\prime} s_{t}$ be a Markov chain satisfying Assumption 1. A Markov-switching trend is a stochastic process $\left\{ {\tau_{t} } \right\}_{0}^{\infty }$ that satisfies $\tau_{t} = \tau_{t - 1} + \mu_{t}$, with initial condition $\tau_{0} = 0$.

Proposition 1

Let $\left\{ {\tau_{t} } \right\}$ be a Markov-switching trend and let $\overline{\mu } = \pi{\prime} \mu$ and $\lambda_{\mu }^{2} : = \mu{\prime} \Lambda_{s} \mu$, $\Lambda_{s} : = \Gamma_{s} (0) + \sum\nolimits_{n = 1}^{ + \infty } {\left( {\Gamma_{s} (n) + \Gamma_{s} (n){\prime} } \right)}$. Furthermore, assume that $\lambda_{\mu }^{2} > 0$. Then, as $T \uparrow \infty$ $T^{ - 1/2} \left( {\tau \left[ {sT} \right] - \overline{\mu }\left[ {sT} \right]} \right)\mathop{\longrightarrow}\limits^{w}\lambda_{\mu } B\left( s \right)$, where $B\left( . \right)$ is a standard Brownian motion on $C\left[ {0,1} \right]$

Proposition 1¹³, which is proposition 2 of Cavaliere (2003), shows that a MS trend, if properly normalized, converges weakly to a standard Brownian motion. Hence $\Delta \tau_{t} = \mu_{t}$ is Renyi mixing or I,0) in a broad sense. The only requirement for invariance principle to hold is that $\lambda_{\mu }^{2} > 0$,i.e., long-run variance is strictly positive).

Theorem 1

Let the assumptions of given for Theorem 3 of Cavaliere (2003) hold. Moreover, assume that $\left\{ {\Delta Z_{t} } \right\}$ is strictly stationary with finite fourth moments and that has $\left\{ {\Delta X_{t} } \right\}$ invertible ARMA representation. If $p \to \infty$ as $T \to \infty$ and $k = o(T^{\gamma } )$, $0 < \gamma \le 1/2$, then, $T \uparrow \infty$,

$$ \tau_{M,i} \mathop{\longrightarrow}\limits^{w}\frac{{B_{F} (1)^{2} - B_{F} (0)^{2} - 1}}{{2\left( {\int {B_{F}^{2} (s)ds} } \right)^{1/2} }}, \, i = \mu ,\tau $$

(A.1)

$$ \tau_{M,i} \mathop{\longrightarrow}\limits^{w}\frac{{\int {B_{F} } dW(r)}}{{2\left( {\int {B_{F}^{2} (s)ds} } \right)^{1/2} }}, \, i = \mu ,\tau $$

(A.2)

where $B_{F} (.)$ is a demeaned and detrended Brownian motion. $W(r)$ is Brownian motion.

$$ \tau_{M,i} \mathop{\longrightarrow}\limits^{w}\frac{{B_{F} (1)^{2} - B_{F} (0)^{2} - 1}}{{2\left( {\int {B_{F}^{2} (s)ds} } \right)^{1/2} }} = \frac{{(1/2)\left( {B_{F} (1)^{2} - B_{F} (0)^{2} - 1} \right)}}{{\left( {\int {B_{F}^{2} (s)ds} } \right)^{1/2} }} $$

Proof of this theorem directly follows from Cavaliere (2003). As stated in Cavaliere (2003); this theorem, which is based on a recent work by Chang and Park (2002), shows that the presence of a Markov linear trend does not change the asymptotic distribution of the ADF, $t$-test provided that $p$ is chosen properly. However, the assumptions given in Cavaliere (2003) for Theorem 3 need to be strengthened to obtain the convergence given in Eq. (A.1).¹⁴