Linearity tests under the null hypothesis of a random walk with drift

Abstract

This paper examines the linearity tests against smooth transition autoregressive models under the null hypothesis of a random walk with drift. The results show that the limiting distribution of the Wald-type statistic $W$ proposed by Teräsvirta (1994) and the$W_T$ statistic proposed by Harvey and Leybourne (2007) follow the standard ${\chi }^2$ distribution, whereas the robust $W_\lambda$ statistic proposed by Harvey et al. (2008) diverges at a rate of $T$. The finite sample simulations show $W$ is oversized, and the robust ${W_T^{*}}$ test proposed by Harvey and Leybourne (2007) has better power performance despite being slightly undersized. Therefore, the robust ${W_T^{*}}$ statistic is recommended to be used as a conservative test in practical applications.

Appendix

1.1 Proof of Theorem 1

(a) According to Eq. (2) and the assumption of Theorem 1, when the true DGP is a random walk with drift, $y_t =a_0 +y_{t-1} +\varepsilon _t$. This equation implies that $y_t =a_0 t+y_0 +\xi _t ,\xi _t =\sum {\varepsilon _t }$. Thus, we obtain

$$\begin{aligned} T^{-(i+1)}\sum {y_{t-1}^i } \buildrel p \over \longrightarrow \frac{a_0^i }{i+1},i=1,2,\cdots 8 \end{aligned}$$

(12)

Let $\Upsilon _1 ,\tilde{\Upsilon }_1$ be the scaling matrix,$\Upsilon _1 =diag\,(T^{1 / 2},T^{3 / 2},T^{5 /2},T^{7 /2},T^{9 / 2})$, $\tilde{\Upsilon }_1 =diag(T^{5 /2},T^{7 /2},T^{9 /2})$, and $\tilde{\Upsilon }_1 \mathbf{R}_1 =\mathbf{R}_1 \Upsilon _1$, ${\mathbf{R}}_1 =\left[ {\begin{array}{llllll} 0&{} 0 &{}1&{} 0 &{}0 \\ 0&{} 0&{} 0 &{}1 &{}0 \\ 0&{} 0&{} 0&{} 0&{} 1 \\ \end{array}} \right]$. Let X be the matrix of independent variables in Eq. (2), ${\mathbf {\beta }}$ the coefficient vector, $\mathbf{b}_T$ the OLS estimate of ${\mathbf {\beta }}$, and ${\mathbf {\varepsilon }}$ the error vector. Thus, we obtain

$$\begin{aligned} \left[ {\Upsilon ^{-1}_1 ({\mathbf{X}}^{\mathbf{'}}{\mathbf{X}})\Upsilon ^{-1}_1 } \right] \buildrel p \over \longrightarrow \left[ {\begin{array}{lllll} 1&{}\frac{a}{2}&{}\frac{a^2}{3}&{}\frac{a^3}{4}&{}\frac{a^4}{5} \\ \frac{a}{2}&{}\frac{a^2}{3}&{}\frac{a^3}{4}&{}\frac{a^4}{5}&{}\frac{a^5}{6} \\ \frac{a^2}{3}&{}\frac{a^3}{4}&{}\frac{a^4}{5}&{}\frac{a^5}{6}&{}\frac{a^6}{7} \\ \frac{a^3}{4}&{}\frac{a^4}{5}&{}\frac{a^5}{6}&{}\frac{a^6}{7}&{}\frac{a^7}{8} \\ \frac{a^4}{5}&{}\frac{a^5}{6}&{}\frac{a^6}{7}&{}\frac{a^7}{8}&{}\frac{a^8}{9} \\ \end{array}} \right] \equiv {\mathbf{Q}}_1 \end{aligned}$$

(13)

$$\begin{aligned}&\left\{ {\Upsilon ^{-1}_1 (\mathbf{X}^{'}{\mathbf {\varepsilon } })} \right\} =\nonumber \\&\quad \left[ {T^{\!-\!1/2}\sum {\varepsilon _t } T^{\!-\!3/2}\sum {y_{t\!-\!1} } \varepsilon _t T^{\!-\!5/2}\sum {y_{t\!-\!1}^2 } \varepsilon _t T^{\!-\!7/2}\sum {y_{t-1}^3 } \varepsilon _t T^{-9/2}\sum {y_{t-1}^4 } \varepsilon _t } \right] ^{\prime }\nonumber \\&\equiv \mathbf{h}_1 \end{aligned}$$

(14)

All h $_{1}$ elements follow a Gaussian distribution, i.e.,

$$\begin{aligned}&T^{-1/2}\sum {\varepsilon _t } \Rightarrow N\left( 0,\sigma ^2\right) ,T^{-3/2}\sum {y_{t-1} } \varepsilon _t \Rightarrow N\left( 0,\frac{a_0^2 }{3}\sigma ^2\right) ,T^{-5/2}\sum {y_{t-1}^2 } \varepsilon _t\\&\Rightarrow N\left( 0,\frac{a_0^4 }{5}\sigma ^2\right) \text{, } \\&T^{-7 /2}\sum {y_{t-1}^3 } \varepsilon _t \Rightarrow N\left( 0,\frac{a_0^4 }{7}\sigma ^2\right) ,T^{-9/2}\sum {y_{t-1}^4 } \varepsilon _t\\&\Rightarrow N\left( 0,\frac{a_0^8 }{9}\sigma ^2\right) \\ \end{aligned}$$

Here, only the proof of $T^{-3/2}\sum {y_{t-1} } \varepsilon _t \Rightarrow N\left( 0,\frac{a_0^2 }{3}\sigma ^2\right)$ is given, and others can be readily obtained.

$$\begin{aligned}&T^{-3/2}\sum {y_{t-1} } \varepsilon _t =T^{-3 /2}\sum \Big [a_0 (t-1) +\xi _t +y_0\Big ]\varepsilon _t\nonumber \\&\buildrel p \over \longrightarrow a_0 T^{-1/2}\sum {{(t}/T} )\varepsilon _t \end{aligned}$$

(15)

Clearly, $t /T\varepsilon _t$ is a martingale difference sequence with a variance of $\sigma _t^2 =E(t/ T\varepsilon _t )^2={t^2} /T^2\sigma ^2$. Some conditions are satisfied, i.e., $\frac{1}{T}\sum {\sigma _t^2 } =\frac{1}{T}\sum {{t^2} /T^2\sigma ^2} \rightarrow \frac{1}{3}\sigma ^2$, $\frac{1}{T}\sum {(t/T\varepsilon _t )^2} \buildrel p \over \longrightarrow \frac{1}{3}\sigma ^2$, and $E(t/T\varepsilon _t )^r<\infty$ for some $r>$2 and all t. Thus, according to White (1984, corollary 5.25), we obtain $T^{-3 /2}\sum {y_{t-1} } \varepsilon _t \Rightarrow N(0,\frac{a^2}{3}\sigma ^2)$.

Consider the joint distribution of the h $_{1 }$elements. Any linear combination of these five elements takes the following form:

$$\begin{aligned}&T^{-1/2}\sum {\left( \lambda _1 +\lambda _2 T^{-1}y_{t-1} +\lambda _3 T^{-2}y_{t-1}^2 +\lambda _4 T^{-3}y_{t-1}^3 +\lambda _5 T^{-4}y_{t-1}^4\right) } \varepsilon _t \nonumber \\&\buildrel p \over \longrightarrow T^{-1/2}\sum {\left( \lambda _1 +\lambda _2 \frac{at}{T}+\lambda _3 \frac{a^2t^2}{T^2}+\lambda _4 \frac{a^3t^3}{T^3}+\lambda _5 \frac{a^4t^4}{T^4}\right) } \varepsilon _t \end{aligned}$$

(16)

Moreover,$\left( \lambda _1 +\lambda _2 \frac{at}{T}+\lambda _3 \frac{a^2t^2}{T^2}+\lambda _4 \frac{a^3t^3}{T^3}+\lambda _5 \frac{a^4t^4}{T^4}\right) \varepsilon _t$ is a martingale difference sequence with a positive variance given by

$$\begin{aligned} \sigma _t^2 =&E\left( \lambda _1 +\lambda _2 \frac{at}{T}+\lambda _3 \frac{a^2t^2}{T^2}+\lambda _4 \frac{a^3t^3}{T^3}+\lambda _5 \frac{a^4t^4}{T^4}\right) ^2\varepsilon ^2_t\nonumber \\&\!\!\!\!=\sigma ^2\left( \lambda ^2_1 +\lambda ^2_2 \frac{a^2t^2}{T^2}+\lambda ^2_3 \frac{a^4t^4}{T^4}+\lambda ^2_4 \frac{a^6t^6}{T^6}+\lambda ^2_5 \frac{a^8t^8}{T^8}\right. \nonumber \\&\left. \,+2\lambda _1 \lambda _2 \frac{ta}{T}+2\lambda _1 \lambda _3 \frac{a^2t^2}{T^2}+2\lambda _1 \lambda _4 \frac{a^3t^3}{T^3}+2\lambda _1 \lambda _5 \frac{a^4t^4}{T^4}\right. \nonumber \\&\left. \,+2\lambda _2 \lambda _3 \frac{a^3t^3}{T^3}+2\lambda _2 \lambda _4 \frac{a^4t^4}{T^4}+2\lambda _2 \lambda _5 \frac{a^5t^5}{T^5}+2\lambda _3 \lambda _4 \frac{a^5t^5}{T^5}\right. \nonumber \\&\left. +2\lambda _3 \lambda _5 \frac{a^6t^6}{T^6}+2\lambda _4 \lambda _5 \frac{a^7t^7}{T^7}\right) \end{aligned}$$

(17)

and $\frac{1}{T}\sum {\sigma _t^2 } \rightarrow \sigma ^2 \mathbf{{\lambda }^{\prime }Q}_1 {\mathbf {\lambda }}$. Furthermore,

$$\begin{aligned} \frac{1}{T}\sum {\left( \lambda _1 +\lambda _2 \frac{at}{T}+\lambda _3 \frac{a^2t^2}{T^2}+\lambda _4 \frac{a^3t^3}{T^3}+\lambda _5 \frac{a^4t^4}{T^4}\right) ^2\varepsilon _t^2 } \buildrel p \over \longrightarrow \sigma ^2 \mathbf{{\lambda }^{\prime }Q}_1 {\mathbf {\lambda }} \end{aligned}$$

(18)

where ${\mathbf {\lambda }}=\left[ {\lambda _1 \lambda _2 \lambda _3 \lambda _4 \lambda _5 } \right] ^\prime$. Thus, any linear combination of the five h $_{1}$ elements is asymptotically Gaussian, which implies a joint Gaussian distribution of h $_{1 }$according to the Cramer-Wold theorem. Thus, ${\mathbf{h}}_1 \Rightarrow N( {{\mathbf{0}},\sigma ^2{\mathbf{Q}}_1 })$ and

$$\begin{aligned} \Upsilon _1 (\mathbf{b}_T -{\mathbf {\beta } })=\left[ {\Upsilon _1^{-1} (\mathbf{{X}'X})\Upsilon _1^{-1} } \right] ^{-1}\Upsilon _1^{-1} \mathbf{{X}'\varepsilon }\Rightarrow N\left( {{\mathbf{0}},\sigma ^2 \mathbf{Q}_1^{-1} }\right) \end{aligned}$$

(19)

The limiting distribution of statistic $W$ can be derived by

$$\begin{aligned} W&=({\mathbf{b}}_T -{{\mathbf {\beta } }})^{{'}}{\mathbf{R}}^{{'}}_1 \left\{ {{\mathbf{R}}_1 s_T ^2({\mathbf{X}}^{\mathbf{{'}}}{\mathbf{X}})^{-1}{\mathbf{R}}^{'}_1 } \right\} ^{-1}{\mathbf{R}}_1 \text{( }{\mathbf{b}}_T -{{\mathbf {\beta } }}) \nonumber \\&=({\mathbf{b}}_T -{{\mathbf {\beta } }})^{'}{\mathbf{R}}^{'}_1 \tilde{\Upsilon }_1 \left\{ {\tilde{\Upsilon }_1 {\mathbf{R}}_1 s_T ^2({\mathbf{X}}^{\mathbf{{'}}}{\mathbf{X}})^{-1}{\mathbf{R}}^{'}_1 \tilde{\Upsilon }_1 } \right\} ^{-1}\tilde{\Upsilon }_1 {\mathbf{R}}_1 \text{( }{\mathbf{b}}_T -{{\mathbf {\beta } }}) \nonumber \\&=\left[ {{\mathbf{R}}_1 \left[ {\Upsilon _1^{-1} ({\mathbf{{X}{'}X}})\Upsilon _1^{-1} } \right] ^{-1}\Upsilon _1^{-1} {\mathbf{{X}{'}\varepsilon }}} \right] ^\prime \left\{ {s_T^2 {\mathbf{R}}_1 \left[ {\Upsilon _1^{-1} ({\mathbf{{X}{'}X}})\Upsilon _1^{-1} } \right] ^{-1}{\mathbf{{R}{'}}}_1 } \right\} ^{-1}\nonumber \\&\quad {\mathbf{R}}_1 \left[ {\Upsilon _1^{-1} ({\mathbf{{X}{'}X}})\Upsilon _1^{-1} } \right] ^{-1}\Upsilon _1^{-1} {\mathbf{{X}{'}\varepsilon }} \buildrel p \over \longrightarrow {\mathbf{z}}^\prime \left[ {\sigma ^2{\mathbf{R}}_1 {\mathbf{Q}}_1^{-1} {\mathbf{{R}{'}}}_1 } \right] ^{-1}{\mathbf{z}} \end{aligned}$$

(20)

where $s_T ^2$ is the sample estimate of $\sigma ^2$, and ${\mathbf{R}}_1 \Upsilon _1 ({\mathbf{b}}_T\! -\!{{\mathbf {\beta }}})\equiv {\mathbf{z}}\Rightarrow N({\mathbf{0}},\sigma ^2{\mathbf{R}}_1 {\mathbf{Q}}_1^{-1} {\mathbf{{R}^{\prime }}}_1 )$. Therefore,$W\Rightarrow \chi ^2(3)$.

(b) According to the proof of (a), the limiting distribution of the Wald statistic $W_{0}$ can be readily derived as $\chi ^2(2)$.

Under the assumption of Theorem 1, the unrestricted model and restricted model of Equation (4) are misspecified because neither nests the true DGP. This result in the population value $\gamma _2 ,\gamma _3$ is non-zero, and thus, the OLS estimator $\hat{\gamma }_2 ,\hat{\gamma }_3$ converges in probability to the non-zero constants. As a result, ${RSS_1^r } /{RSS_1^u }=O_p (1)$, but it does not converge in probability to 1. Therefore,

$$\begin{aligned} W_1&= T( {O_p (1)-1}) \\&= O_p (T) \\ \end{aligned}$$

The robust Wald statistic is $W_\lambda =(1-\lambda )W_0 +\lambda W_1$, where $\lambda (U,S)=\exp ( {-g( {\frac{U}{S}})^2})$, $U$ and $S$ denote the standard ADF unit root statistic and the nonparametric stationarity statistic proposed by Harris et al. (2003), respectively, and $g$ is a finite positive constant derived through simulation.

According to the Lemma 1 developed by Harvey et al. (2008),

$$\begin{aligned} \lambda (U,S)&= \exp (-g(\frac{U}{S})^2) \\&= \frac{1}{1+g(\frac{U}{S})^2+\frac{g^2}{2}(\frac{U}{S})^4+\frac{g^3}{6}(\frac{U}{S})^6+\cdots } \end{aligned}$$

Under the null hypothesis of $H_{0,1} :\gamma _2 =\gamma _3 =0$, $y_{t}$ is a linear unit root, and then$\left| U \right| =O_p (1)$ because it converges to the Dickey-Fuller limit distribution; under the alternative $H_{1,1} :\gamma _2 \ne \text{ and }/\text{ or } \gamma _3 \ne 0$, $y_{t}$ is nonlinear and a unit root, and then $\left| U \right| =O_p (1)$ according to Harvey et al. (2008). Similarly, according to Harris et al. (2003), under both $H_{0,1}$ and $H_{1,1}$, $\left| S \right|$diverge to $+\infty$, and thus, under both $H_{0,1}$ and$H_{1,1}^{, }\lambda \buildrel p \over \longrightarrow 1$ and $W_\lambda =O_p (T)$.

(c) Harvey and Leybourne (2007) construct an auxiliary regression in terms of the observed $y_{t}$

$$\begin{aligned} y_t ={\beta }'_0 \!+{\beta }'_1 y_{t-1} +\!{\beta }'_2 y_{t-1}^2 \!+{\beta }'_3 y_{t-1}^3 +{\beta }'_4 \Delta y_{t-1} +{\beta }'_5 (\Delta y_{t-1})^2+{\beta }'_6 (\Delta y_{t-1})^3+\varepsilon _t^*\nonumber \\ \end{aligned}$$

(21)

In terms of 21, the null hypothesis of linearity test can be stated as

$$\begin{aligned} H_0 :{\beta }'_2 ={\beta }'_3 ={\beta }'_5 ={\beta }'_6 =0 \end{aligned}$$

They define a Wald statistic $W_{T}$ under the null of $H_0$ . Under the assumption of Theorem 1, the limiting distribution of $W_T$ can be derived directly as $\chi ^2(4)$ using the proof of (a) and standard asymptotic results of Wald tests.